Optimization Advances in Electric Power Systems [1 ed.] 9781614704805, 9781606926130

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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Optimization Advances in Electric Power Systems, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Optimization Advances in Electric Power Systems, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

OPTIMIZATION ADVANCES IN ELECTRIC POWER SYSTEMS

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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Optimization Advances in Electric Power Systems, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

OPTIMIZATION ADVANCES IN ELECTRIC POWER SYSTEMS

EDGARDO D. CASTRONUOVO

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EDITOR

Nova Science Publishers, Inc. New York

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All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Optimization advances in electric power systems / Edgardo D. Castronuovo, editor. p. cm. Includes bibliographical references and index. HISBN H%RRN 1. Electric power systems. 2. Mathematical optimization. I. Castronuovo, Edgardo D. (Edgardo Daniel) TK1005.O625 2009 621.319'1--dc22 2008043951

Published by Nova Science Publishers, Inc.

New York

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CONTENTS Preface

vii

About the Editor

xi

Chapter 1

A Mathematical Programming Approach to State Estimation Eduardo Caro, Antonio J. Conejo and Roberto Mínguez

Chapter 2

Trust Region Optimization Methods via Givens Rotations Applied to Power System State Estimation Antônio J. Simões Costa, Roberto S. Salgado and Paulo Hass

27

Chapter 3

The Impact of Deregulation on Mathematical Models Using Optimization Techniques to Aid System Planning and Operations Narayan S. Rau

53

Chapter 4

Metaheuristic-Based Optimization Methods for Transmission Expansion Planning Considering Unreliability Costs Armando M. Leite da Silva, Cleber E. Sacramento, Luiz A. da Fonseca Manso, Leandro S. Rezende, Leonidas C. de Resende and Warlley S. Sale

59

Chapter 5

A Voltage Control Optimization for Distribution Networks with DG and MicroGrids João A. Peças Lopes and André Madureira

87

Chapter 6

Tools for the Effective Integration of Large Amounts of Wind Energy in the System Jorge Martínez-Crespo, Jorge L. Angarita, Edgardo D. Castronuovo, Hortensia Amaris and Julio Usaola García

113

Chapter 7

Application of Cost Functions for Large Scale Integration of Wind Power using a Multi-Scheme Ensemble Prediction Technique Markus Pahlow, Corinna Möhrlen and Jess U. Jørgensen

151

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1

vi

Contents

Chapter 8

Security Optimization of Bulk Power Systems in the Market Environment Alberto Berizzi, Cristian Bovo, Maurizio Delfanti and Marco Merlo

181

Chapter 9

Optimal Placement in Power System Gabriel Olguin and Tuan A. Le

225

Chapter 10

Non-Linear Mathematical Programming Applied to Electric Power Systems Stability Carlos F. Moyano and Edgardo D. Castronuovo

253

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Index

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283

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PREFACE The increasing search for the efficiency, the computational continuous improvement and the development of new effective mathematical methods are three impelling forces for the utilization of optimization in electric power systems. Nowadays, it is unlikely to find an electric company that does not use optimization methods. This kind of processes is utilized in both planning and operation calculations for the generation, transmission and distribution areas of power systems. Electrical engineers face these new operational methods, in some cases without the adequate preparation. This book aims to include some of the present and foreseen applications of the optimization in electric power systems, explained by main experts in the field. Furthermore, this book may serve as state-of-the-art for undergraduate and graduate students worldwide. Optimization is the systemized search for the best action. Probably, the first non-linear formulations for optimization applications in electric power systems were presented in the sixties, included in the works of Carpentier (CARPENTIER, J.; 1962. Contribuition à l'étude du dispatching économique, Bulletin de la Société Française des Electriciens, ser.8, vol.3, pp. 431-447) and Dommel and Tinney (DOMMEL, H.W.; TINNEY, W.; 1968. Optimal Power Flow Solutions, IEEE Transactions on Power Apparatus and Systems, Piscataway, NJ, USA , PAS 87, n. 10, Oct.). However, the difficulties associated with formulate and solve real problems in the industry delayed the diffusion of the optimization in electric companies. In the last two decades, the maturity of mathematical programming (in particular, the development of Interior Point methods) and the amazing increase in the computational capacities allowed the solution of real large-size power system problems. In addition, an emphasis in the market environment gives prominence to the pursuit for improved results in the electric industry. It is a fact that the implementation of optimization tools in the electric industry leads to the rationalization of use of resources, decreasing the operational costs. However, the representation of a real electric problem through optimization techniques is frequently not a simple task. Optimization requires a mathematical representation of the physical problem and objectives. Due to the complexity of the electric systems, approximated models of the reality may be used. These models condition the performance of the applications and the accuracy of the results. As a greater part of the optimization tools in the electric industry is customized, the electrical engineers must know what and how can be made by using optimization. In the proposed book, the up-to-date solutions are shown, explained by principal researchers and

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viii

Edgardo D. Castronuovo

developers. Techniques, know-how and examples of application are included, showing some advantages of the optimization in the operation and planning of electric power systems. This book is intended for a wide audience, including researchers and practitioners. Electrical engineers in the industry can use it to understand implemented optimization tools, to know practical solutions applied in other places and, possibly, to evaluate their available operational tools. As educational tool, it can be used in undergraduate courses of Optimization, Electric Power Systems, Economy of the Energy, Technology of the Energy, Electric Power Markets and others. Likewise, this book can integrate the basis of the graduate course of Optimization in Electric Power Systems. In Chapter 1, Eduardo Caro, Antonio J. Conejo and Roberto Mínguez analyse an optimization method to solve the state estimation problem. The traditional solution method for the state estimation problem is compared with the mathematical programming approach, using illustrative examples. The optimization method presents significant advantages, as result of the treatment of inequality constraints, post-solution sensitivity analyses and decomposition for several independently operated regions. In Chapter 2, Antônio J. Simões Costa, Roberto S. Salgado and Paulo Hass consider other optimization approach for the power system state estimation problem, using Trust Region estimators. Two innovative alternatives are analysed in the chapter, showing improved convergence characteristics when contrasted with the traditional formulation. Test systems of small and large size are used to reveal the benefits of the optimization method. With an extended performance in the industry, Narayan S. Rau offers in Chapter 3 a specialist vision of the application of optimization in the new deregulated environment. Starting from an incisive visualization of the actual situation, Dr. Rau proposes challenges and questions that must be solved in the next future for the players of the electric power system. Reactive power influence in the locational marginal prices, proper market signals, the cost of CO2 emissions and others issues are stood out by the author. Armando M. Leite da Silva, Cleber E. Sacramento, Luiz A. da Fonseca Manso, Leandro S. Rezende, Leonidas C. de Resende and Warlley S. Sale consider in Chapter 4 the application of three metaheuristic-based methodologies (Evolution Strategies, Tabu Search and Ant Colony algorithms) to the transmission expansion planning problem. This problem aims to determinate the reinforcements required by a power system, to adequately operate in the future. This is a complex and CPU-time consuming problem, if the minimum cost for the reinforcements and satisfactory security conditions are requested. The three methods are fully explained, with illustrative examples and applications to a real sub-transmission network. In chapter 5, João A. Peças Lopes and André Madureira study the feasibility and the technical advantages of exploiting reactive power generation capability of distributed generation and microgeneration in the operation of distribution power systems. For this objective, a hierarchical voltage with three levels is considered. The results, obtained in two large-scale distribution systems, show the efficiency of the application. Worldwide, and especially in Europe and USA, wind power has largely incremented in the last decades in electric power systems. Wind power has advantages with regard to other energy sources, as reduction of CO2 emissions, local availability, use of an unexploited resource, and others. However, large penetration of wind power in the power system can produce some difficulties in the operation, due to the particular characteristics of the primary energy and the special features of the wind farms. In chapter 6, Jorge Martínez Crespo, Hortensia Amarís, Jorge L. Angarita, Julio Usaola García and I summarize different practical

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Preface

ix

tools for the effective integration of large amounts of wind energy in power systems. Procedures to bid the energy in the electric market, cooperation among wind farms and hydro plants or water pump stations, delegated dispatches of renewable producers and voltage stability enhancement in grids with wind farms are considered in this chapter. The text includes complete optimization formulations, application examples and results obtained from test and real networks. In Chapter 7, Markus Pahlow, Corinna Möhrlen and Jess U. Jørgensen analyse the use of ensemble prediction systems for wind power forecasting. The economic value of wind power is related to the predictability of the resource. The effective participation of the wind power producers in the market depends on the capacity to obtain a good prevision of the wind power production. In the text, the effect of the wind power forecast in different optimization scenarios is presented. The results are obtained from real electric systems. In Chapter 8, Alberto Berizzi, Cristian Bovo, Maurizio Delfanti and Marco Merlo study the problem of the optimal reactive power flow. Different objective functions are analysed, including minimization of real power losses, minimization of reactive power produced, maximization of the distance to the voltage collapse and others. Multiobjective approaches are also considered by using Pareto sets. The features of all the methods and the procedures presented are shown by numerical tests and examples in a real large-scale system. Gabriel Olguin and Tuan A. Le deal in Chapter 9 with the optimal placement of equipments. Two examples are considered in this area of power system planning: optimal location of voltage sags monitors and optimal position of flexible alternating current transmission systems (FACTS) devices. Voltage sags (short duration reductions in rms voltage) must be monitored in power systems. FACTS can be very profitable to manage transmission congestions. However, the determination of the best arrangement of these power systems equipments can be complex and computationally demanding. In the text, optimization alternatives for this calculation are analysed, showing results in test and real systems. Finally, in Chapter 10 Carlos F. Moyano and I review applications of non linear optimization in power system stability. In the last decade, there has been an increasing interest of research to deal with stability problems using optimization algorithms. Two main directions can be recognized: the utilization of static analysis to estimate stability margins and, most recently, the inclusion of transient stability constraints into the optimization problems. Formulations and results for different approaches in these two research lines are presented in the text, including solution algorithms. The 10 chapters of the book aim to take in consideration the wide spectrum of applications of the optimization in the modern power systems. I am grateful to the authors for adapting to the pressing editorial schedule and the requirements of the book. This work can not be performed without the support and love of my dear wife Marcela and the tenderness of my daughters Fabiana and Lorena. Thank you.

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Edgardo D. Castronuovo The Editor

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ABOUT THE EDITOR

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Edgardo D. Castronuovo was born at General Belgrano, Buenos Aires, Argentina, and lived at Salta, Argentina, where he completed technical instruction in electrotechniques at ENET 2. He received the Electrical Engineer degree from National University of La Plata, Argentina, and both M.Sc. and Ph.D. degrees from Federal University of Santa Catarina, Brazil. He performed a Post-Doctorate at the Institute for Systems and Computer Engineering of Porto (INESC-Porto), Portugal. He worked at the Power System areas of the federal Research Centre for Electrical Studies (CEPEL), Brazil, and INESC-Porto, Portugal. Nowadays, Dr. Castronuovo is a professor of the Electrical Engineering Dept., Carlos III de Madrid University, Spain. He is author and reviewer of papers in the most prestigious electric journals. He is Senior Member of the Institute of Electrical and Electronic Engineers (IEEE) and is associated to the Power & Energy Society of this Institute. His interests are on optimization methods applied to power systems problems, operations planning and deregulation of the electric energy systems. Edgardo D. Castronuovo is married and has two daughters.

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In: Optimization Advances in Electric Power Systems ISBN: 978-1-60692-613-0 c 2008 Nova Science Publishers, Inc. Editor: Edgardo D. Castronuovo, pp. 1-25

Chapter 1

A M ATHEMATICAL P ROGRAMMING A PPROACH TO S TATE E STIMATION Eduardo Caro, Antonio J. Conejo and Roberto M´ınguez University of Castilla-La Mancha, Ciudad Real, Spain

Abstract

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This chapter revisits the state estimation problem in power systems and proposes a direct mathematical programming solution approach. Additionally, a sensitivity analysis assessing the relevance of measurements and parameters on the solution is provided. Finally, a decomposition technique is presented to solve in a decentralized fashion the multi-region state estimation problem. All procedures are illustrated by simple but insightful examples.

1.

Introduction

This chapter revisits the state estimation problem in power systems and proposes a direct mathematical programming solution approach. Traditionally, power system state estimation is tackled by solving the system of nonlinear equations corresponding with the first order necessary optimality conditions of the estimation problem. Alternatively to the traditional approach, we propose to solve directly the estimation problem, which has a number or advantages such as (i) including inequality constraints representing physical hard limits, (ii) taking advantage of currently available state-of-the-art nonlinear programming solvers, (iii) treating sparsity in an efficient and implicit manner, etc. A sensitivity analysis assessing the relevance of both measurements and parameters (resistances, susceptances, etc.) on the solution obtained is straightforwardly derived from a state estimation solution obtained via mathematical programming. We derive and provide compact sensitivity formulae useful to further characterize a state estimation solution. A state estimation model considering a large-scale, multi-region power system (e.g, the power system of the US or the EU) can be efficiently solved in a decentralized fashion by mathematical programming decomposition techniques. We explore a Lagrangian decomposition technique specifically adapted to the state estimation problem.

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Eduardo Caro, Antonio J. Conejo and Roberto M´ınguez

All procedures considered are illustrated by simple but insightful examples. For the interested reader, pioneering state estimation works include [1]-[9], and a detailed description of state estimation in power system is provided in [10], which include an appropriate literature review. The rest of this chapter is organized as follows. Section 2 formulates the state estimation problem in an electric energy system, Section 3 addresses the observability issue, Section 4 describes the classical solution approach, Section 5 provides the mathematical programming approach that is advocated in this chapter, Sections 6 and 7 consider the problem of detecting and identifying bad measurements, respectively, Section 8 presents a sensitivity analysis for the state estimation problem, Section 9 provides a decomposition technique that allows solving the state estimation problem in a decentralized manner, and, finally, Section 10 concludes the chapter providing some relevant conclusions. An appendix provides formulae pertaining to the sensitivity analysis of Section 8.

2.

Formulation

The state estimation problem has the form: minimize J(x) = x

m X i=1

wi (hi (x) − zi )2

(1)

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subject to f (x) = 0 ,

(2)

g(x) ≤ 0 ,

(3)

where x is the n × 1 state-variable vector, J(x) the objective function (weighted quadratic error), h(x) a m×1 functional vector expressing the measurements as a function of the state variables, w a m × 1 weighting factor vector, z the m × 1 measurement vector, f (x) the p × 1 equality-constraint vector mainly enforcing conditions at transit buses (no generation and no demand), and g(x) the q × 1 inequality-constraint vector enforcing physical limits of the system. Among the components of vector h(x), note that active and reactive and power injections at bus i are computed as

Pi = vi

X j∈Ξ

Qi = vi

X j∈Ξ

vj (Gij cos(θi − θj ) + Bij sin(θi − θj )) ,

(4)

vj (Gij sin(θi − θj ) − Bij cos(θi − θj )) ,

(5)

where Pi and Qi are the active and reactive power injection at bus i, respectively, vi and θi are the voltage magnitude and angle at bus i, respectively, Gij and Bij are the real and imaginary part of the bus admitance matrix, respectively, and Ξ is the set of all buses. Gii and Bii can be computed as indicated in [10].

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A Mathematical Programming Approach to State Estimation

3

Figure 1. One-line diagram and measurement configuration of a 4-bus power system.

The active and reactive power flow ij are computed as Pij

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Qij

= vi vj (Gij cos(θi − θj ) + Bij sin(θi − θj )) − Gij vi2 ,

= vi vj (Gij sin(θi − θj ) − Bij cos(θi − θj )) +

vi2 (Bij

(6) −

bSij /2)

,

(7)

where Pij and Qij are the active and reactive power flow from bus i to bus j, respectively, and bSij the shunt susceptance of line ij. As it is customary, we consider that measurements are independent Gaussian-distributed random variables. Alternative assumptions are analyzed in [11]. Example 1: Traditional formulation The 4-bus power system depicted in Figure 1 is considered throughout this chapter for illustrative purposes. The network data is provided in Table 1. This example includes a generating bus, two demand buses and a transit bus. Considering bus 4 as the reference bus, the state-variable vector x has the form: x = (θ1 , θ2 , θ3 , v1 , v2 , v3 , v4 )T . Ten measurements are considered as shown in Figure 1: two voltage measurements at bus 1 and 2; two active/reactive power injection measurements at bus 1 and 3; three active power flow measurements at lines 1–4, 3–2 and 3–4; and one reactive power flow measurement at line 3–4. The standard deviation of any measurement other than voltage is 0.02, while the standard deviation of voltage measurements is 0.01. The considered measurement configuration provides a redundancy ratio of (10 + 2)/7 = 1.71. Note that the measurement

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Eduardo Caro, Antonio J. Conejo and Roberto M´ınguez Table 1. Line Parameters Bus i 1 1 2 3

Bus j 2 4 3 4

Bij (p.u.) 10 10 10 10

set includes ten actual measurements plus two exact measurements corresponding to active and reactive power injections at the transit bus. The state estimation problem for this example has the form: minimizex J(x) = w1 (v1 (v2 B12 sin(θ1 − θ2 ) + v4 B14 sin(θ1 − 0)) − z1 )2 + w2 (v3 (v2 B32 sin(θ3 − θ2 ) + v4 B34 sin(θ3 − 0)) − z2 )2

+ w3 (v1 v4 B14 sin(θ1 − 0) − z3 )2

+ w4 (v3 v2 B32 sin(θ3 − θ2 ) − z4 )2

+ w5 (v3 v4 B34 sin(θ3 − 0) − z5 )2 + w6 (v1 − z6 )2

+ w7 (v2 − z7 )2

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+ w8 (v1 (−v1 B11 − v2 B12 cos(θ1 − θ2 ) − v4 B14 cos(θ1 − 0)) − z8 )2

+ w9 (v3 (−v3 B33 − v2 B32 cos(θ3 − θ2 ) − v4 B34 cos(θ3 − 0)) − z9 )2
2 + w10 −v3 v4 B34 cos(θ3 − 0) + v32 B34 − z10 ,

subject to two equality constraints for the transit bus, and seven inequality constraints enforcing physical limits: v2 (v1 B21 sin(θ2 − θ1 ) + v3 B23 sin(θ2 − θ3 )) = 0 , v2 (−v1 B21 cos(θ2 − θ1 ) − v2 B22 − v3 B23 cos(θ2 − θ3 )) = 0 , P1min ≤ v1 (v2 B12 sin(θ1 − θ2 ) + v4 B14 sin(θ1 − 0)) ≤ P1max ,

Qmin ≤ v1 (−v1 B11 − v2 B12 cos(θ1 − θ2 ) − v4 B14 cos(θ1 − 0)) ≤ Qmax , 1 1 −π ≤ θi ≤ π, i = 1, . . . , 3 .

3.

Observability

Functional vector h(x) provides the structural relationships among measurements and state variables. These relations remain unchanged if h(x) is linearized and can be studied Optimization Advances in Electric Power Systems, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

A Mathematical Programming Approach to State Estimation

5

through the m × n Jacobian matrix [12], H = ∇x h(x) .

(8)

The general solution of the system of linear equations Hy = dz, where y = dx, is: y = yP + N ρ ,

(9)

where N is a n × k null-space matrix and ρ a k × 1 parameter vector. Equation (9) provides a unique solution if and only if null-space matrix N is nil. Thus, problem (1)-(3) is well posed if and only if N is nil. Row-wise, (9) is:

yi = yiP +

k X

Nij ρj ,

(10)

j=1

thus, variable yi (or state variable xi ) is observable if and only if Nij = 0, ∀j; otherwise it is not.

Example 2: Observability example

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Considering the measurement set in Figure 1, the functional vector h(x) and the 10 × 7 Jacobian matrix H(x) are:  v1 (v2 B12 sin(θ1 − θ2 ) + v4 B14 sin(θ1 − 0))   v3 (v2 B32 sin(θ3 − θ2 ) + v4 B34 sin(θ3 − 0))     v v B sin(θ − 0) 1 4 14 1     v3 v2 B32 sin(θ3 − θ2 )     v3 v4 B34 sin(θ3 − 0)  , h(x) =   v1     v 2   v (−v B − v B cos(θ − θ ) − v B cos(θ − 0)) 1 11 2 12 1 2 4 14 1  1  v (−v B − v B cos(θ − θ ) − v B cos(θ − 0)) 3 3 33 2 32 3 2 4 34 3 −v3 v4 B34 cos(θ3 − 0) + v32 B34 

∂h1 (x)  ∂x1  .. H(x) =  .   ∂h (x) 10 ∂x1 

··· ..

.

···

 ∂h1 (x) ∂x7   .. . .  ∂h10 (x)  ∂x7

In order to study the observability, the Jacobian is evaluated at flat voltage level Optimization Advances in Electric Power Systems, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Eduardo Caro, Antonio J. Conejo and Roberto M´ınguez

(vi = 1, ∀i; θi = 0, ∀i):

H(x) =

P1 P3 P14 P32 P34 v1 v2 Q1 Q3 Q34

               

θ 1 θ 2 θ 3 v1 v 2 v3 v4  20 −10 0 0 0 0 0 0 −10 20 0 0 0 0   10 0 0 0 0 0 0   0 −10 10 0 0 0 0   0 0 0  0 0 10 0  . 0 0 0 1 0 0 0   0 0 0 0 1 0 0   0 0 0 20 −10 0 −10   0 0 0 0 −10 20 −10  0 0 0 0 0 10 −10

This Jacobian can be divided into two blocks, linking the P – θ and the Q – v variables, H P and H Q , respectively. These submatrices and their null spaces, N P and N Q , respectively, are: θ1 θ2 θ3  P1 20 −10 0  P3   0 −10 20    T  0 0  H P (x) = P14  10  , NP = 0 0 0 ; P32  0 −10 10  P34 0 0 10

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

v1 v2 v3 v1 1 0 0 v2  0 1 0  H Q (x) = Q1   20 −10 0 Q3  0 −10 20 Q34 0 0 10 

v4  0 0     T 0 0 0 0 −10  , N = . Q  −10  −10

Since all columns in both null spaces include only zeros, all state variables are observable. If we consider neither the reactive power flow measurement Q34 nor the reactive power injection measurement Q1 , the H P and H Q matrices and their null spaces are: θ1 θ2 θ3  P1 20 −10 0  P3   0 −10 20    T 0 0  H P (x) = P14   , NP = 0 0 0 ;  10 P32  0 −10 10  P34 0 0 10 

v1 v1 1  0 H Q (x) = v2 Q3 0 

v2 v3 v4  0 0 0   1 0 0  , N TQ = 0 0 1 2 . −10 20 −10

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In this case, the N TQ matrix includes non-zero elements. Therefore, the system is not observable. Voltage magnitudes of buses 1 and 2 are observable, since the first and second columns of N TQ include only zeroes. However, voltage magnitudes of buses 3 and 4 are not observable. Voltage angles are observable throughout the system.

4.

Classical Solution

Traditionally, problem (1)-(3) is simplified by ignoring inequality constraints (3) and then solving the system of nonlinear equations constituted by the first order optimality conditions of (1)-(2), i.e., m X i=1

p h i X ∇x wi (hi (x) − zi )2 + λi ∇x fi (x) = 0 ,

(11)

i=1

fi (x) = 0 i = 1, . . . , p ,

(12)

or H T W [z − h(x)] + F T λ = 0 ,

(13)

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f (x) = 0 ,

(14)

where W is a m × m diagonal matrix of the measurement weights wi , F = ∇x f (x) is the p×n equality constraint Jacobian and λ is the p×1 Lagrangian multiplier vector associated with equality constraints (2). The nonlinear system of equations (13)-(14) can be solved by Newton through the iteration below 

HT W H F T F 0



∆x(ν+1) −λ(ν+1)



=



H T W ∆z (ν) −f (x(ν) )



,

(15)

where ∆z (ν) = z − h(x(ν) ). The linear system of equations (15) is iteratively solved until ∆x is sufficiently small. Further details can be found in [10]. It should be noted that the solution approach based on (15) was developed at the time that no efficient mathematical programming solvers (in terms of accuracy, required computing time and sparsity treatment) were available. However, such solvers are nowadays available. Example 3: Classical solution example The system in Figure 1 is considered in this example. In order to solve the nonlinear system (13)-(14) by the Newton method, matrices H and F and vectors ∆z and f (x)

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should be computed at each iteration. The diagonal terms of the measurement weights matrix W correspond with the inverse of the variance of the measurement errors. Expressions for these matrices are:   ∂h1 (x) ∂h1 (x) ···  ∂x1 ∂x7    . .. .  , .. .. H(x) =  .   ∂h (x) ∂h10 (x)  10 ··· ∂x1 ∂x7   ∂f1 (x) ∂f1 (x) ···  ∂x7  1 F (x) =  ∂f∂x(x) ∂f2 (x)  , 2 ··· ∂x1 ∂x7 W

= Diag(2500, 2500, 2500, 2500, 2500, 10000, 10000, 2500, 2500, 2500) .

The minimum and maximum active/reactive power injections in puMW/puMVar are: = 3.5 and Qmin = −3.5. Measurements are generated P1max = 3, P1min = 0.8, Qmax 1 1 by solving the power flow with a high accuracy and then adding zero-mean independent Gaussian-distributed errors to the exact values of the measurements. These measurements and the exact values are provided in Tables 2-4. In these tables, superscript “true” indicates true value, superscript “m” measurement, and a “∧” estimated value.

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Table 2. Voltages: measured, true and estimated values Bus # 1 2 3 4

vim (p.u.) 1.0934 1.0252 – –

vitrue (p.u.) 1.0965 1.0220 0.9531 0.9174

vˆi (p.u.) 1.0960 1.0222 0.9540 0.9175

θitrue (rad) 0.1819 0.1122 0.0320 0.0000

θˆi (rad) 0.1820 0.1128 0.0334 0.0000

Table 3. Power injections: measured, true and estimated values Bus # 1 3

Pitrue Pˆi (MW p.u.) 2.6093 2.6000 2.5940 -0.4634 -0.5000 -0.4819 Pim

ˆi Qtrue Q i (MVAr p.u.) 2.9597 2.9743 2.9578 -0.2606 -0.2800 -0.2674 Qm i

The initial solution considered for state variables is the flat voltage level, and a convergence tolerance of 10−5 on the largest ∆xi is considered. The convergence, illustrated in Table 5, is obtained after four iterations.

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Table 4. Power flows: measured, true and estimated values Line # 1- 4 3- 2 3- 4

Pˆij Pijtrue (MW p.u.) 1.8120 1.8198 1.8200 -0.7615 -0.7802 -0.7740 0.2650 0.2802 0.2921 Pijm

ˆ ij Q Qtrue ij (MVAr p.u.) – – – – – – 0.3448 0.3447 0.3524 Qm ij

Table 5. State-variable updates and convergency summary

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State variable θ1 θ2 θ3 v1 v2 v3 v4 J(ˆ x)

5.

0 0.0000 0.0000 0.0000 1.0000 1.0000 1.0000 1.0000 49850.5529

Iterations 1 2 0.1824 0.1821 0.1056 0.1130 0.0287 0.0336 1.1003 1.0959 1.0183 1.0221 0.9363 0.9538 0.8895 0.9174 562.7272 4.3639

3 0.1820 0.1128 0.0334 1.0960 1.0222 0.9540 0.9175 4.2674

4 0.1820 0.1128 0.0334 1.0960 1.0222 0.9540 0.9175 4.2674

Mathematical Programming Solution

Problem (1)-(3) can be directly solved using mathematical programming techniques through a nonlinear solver. We advocate this approach because currently available mathematical programming solvers treat efficiently sparsity, are robust and computationally efficient, and provide highly accurate results. In addition, these solvers allow incorporating easily inequality constraints representing physical limits [13]. For instance, problem (1)-(3) can be solved using solver CONOPT [14] or MINOS [15] under the General Algebraic Modeling System (GAMS) [16], which is a high-level modeling system for mathematical programming. It consists of a language compiler and a set of integrated high-performance solvers. GAMS is tailored for complex, large scale modeling applications, and makes it possible to build large maintainable models that can be adapted quickly to new situations. Modeling systems similar to GAMS are AMPL [17] and AIMMS [18].

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Example 4: Mathematical programming problem In order to solve the example in Figure 1 through mathematical programming techniques, the solver CONOPT under GAMS is used [14]. The solution is provided in Taˆ is J(ˆ bles 2-4. The objective function evaluated at the estimated state x x) = 4.2674. Needless to say, the solutions provided by the classical approach and the mathematical programming one are the same.

6.

Bad Measurement Detection

If measurements are Gaussian-distributed unbiased and independent, and if the weighting factors wi correspond with the inverse of the measurement variances (wi = 1/σi2 ), the distribution of J(x) is a χ2 with m + r − n degrees of freedom [19], where r is twice the number of transit buses (exact measurements). Thus, we can write
Prob J(ˆ x) ≤ χ2 (1 − α, m + r − n) = 1 − α , (16) where 1 − α is the confidence level. Therefore, for a given α (e.g., 0.01), the value χ2 (1 − α, m + r − n) can be computed and the χ2 test applied, i.e., if J(ˆ x) < χ2 (1 − α, m + r − n) there is no bad measurement at the 1 − α confidence level, otherwise there is. Further details can be found in [10].

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Example 5: Bad measurement detection example The voltage measurement at bus 1 (Figure 1) is considered to be 1.05 instead of 1.0933, i.e., a bad measurement is introduced. Solving the estimation problem, the optimal objective function value is J(ˆ x) = 16.6008. The test threshold at 0.99 confidence level (α = 0.01) with 10 + 2 − 7 = 5 degrees of freedom is χ2 (0.99, 5) = 15.0863. Therefore, since χ2 (0.99, 5) < 16.6008, we conclude that bad data plague the measurement set with a 0.99 confidence level. For the initial case and since J(ˆ x) = 4.2674 < χ2 (0.99, 5) no bad data affect the measurement set with at 0.99 confidence level.

7.

Identification of Erroneous Measurements

If the bad measurement test detects bad measurements, these measurements should be identified. This is accomplished below. Consider the nonlinear measurement model: e = z − h(xtrue ) ,

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(17)

A Mathematical Programming Approach to State Estimation

11

where xtrue is the unknown true state vector and e is the measurement error vector. Note that E[e] = E[z − h(xtrue )] = E[z] − E[h(xtrue )] (18) = E[z] − h(xtrue ) = 0 , which is a typical assumption in state estimation. Considering the differential measurement equation: de = dz − Hdx ,

(19)

and if measurements are Gaussian-distributed with zero mean and independent, the least squares estimator dˆ x can be obtained by minimizing the weighted sum of square deviations of the differential errors: minimize (dz − Hdx)T W (dz − Hdx) . dx

(20)

This minimization problem leads to the system of equations: (H T W H)dx = H T W dz ,

(21)

known as the system of normal equations [19]. Assuming that the gain matrix, G = HT W H ,

(22)

has an inverse, the least squares estimates dˆ x can be written explicitly as

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dˆ x = G−1 H T W dz ,

(23)

from which we conclude that dˆ x is a linear function of dz. The residual and the differential residual are defined as r = z − h (ˆ x) ,

(24)

dr = dz − Hdˆ x,

(25)

dr = dz − Hdˆ x = dz − P dz = (I m − P )dz ,

(26)

thus where P = HG−1 H T W is known as hat or projection matrix. Integrating (26), the linear transformation from z to r at the optimum is: r = [I m − P ] z + k = Sz + k ,

(27)

where k is a constant vector, and S is known as the residual sensitivity matrix. If measurements z are Gaussian-distributed and independent with zero mean and covariance matrix R (R = W −1 ) and h(ˆ x) is an unbiased estimator of h(xtrue ), i.e.,

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E[h(ˆ x)] = h(xtrue ), the residual vector r provided by the linear transformation (27) is Gaussian-distributed with parameters given by: E[r] = E[z − h(ˆ x)] = E[z] − E[h(ˆ x)]

= E[z] − h(xtrue ) = E[e] = 0 ,

(28)

T

Ω = E[(r − E[r]) (r − E[r]) ]

= E[(Sz +k −SE[z]−k)(Sz+k−SE[z]−k)T]

= E[S (z − E[z]) (z − E[z])T S T ] ,

(29)

and due to (17) and (28): Ω = E[SeeT S T ] = SE[eeT ]S T = SRS T = SR .

(30)

Note that the sensitivity matrix S is a idempotent matrix [20]. The normalized residual (N (0, 12 )) of measurement i is: 1 riN = √ (zi − hi (ˆ x)) . Ωii

(31)

The largest residual identifies a bad measurement with a 1 − α confidence level (e.g., 0.99) if riN > Φ−1 (1 − α/2). Further details can be found in [10].

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Example 6: Bad measurement identification example The considered covariance matrix R is:

R = 10−4 Diag(4, 4, 4, 4, 4, 1, 1, 4, 4, 4) . Matrices P and Ω are computed using expressions (22) and (30), respectively. The residual (from (24)) and the normalized residual of measurements (from (31)) are also calculated and provided in Table 6. The threshold for detection at a 0.99 confidence level (α = 0.01) is −1 Φ (1 − 0.01/2) = 2.5758. The normalized residual with highest absolute value in Table 6 (column 3) corresponds to v1 and, because its value is larger than 2.5758, it is identified as a bad measurement and, therefore, it is removed from z. After removing this bad measurement, observability is checked again. In this case, the system remains observable, and the state estimation is carried out again. After estimating the state, the objective function J(ˆ x) = 4.1388 < χ2 (0.99, 4), and therefore no additional bad measurement is identified.

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Table 6. Residuals and normalized residuals Measurement P1 P3 P14 P32 P34 v1 v2 Q1 Q3 Q34

8.

riN 2.2757 1.5836 -0.5386 0.8996 -1.9219 -3.6530 3.5700 2.6631 1.0033 -1.0586

ri -0.0154 -0.0183 0.0078 -0.0127 0.0275 0.0258 -0.0245 -0.0032 -0.0018 0.0041

Sensitivity

Once problem (1)-(3) is identified as observable, solved, and verified that no bad measurements alter the solution, a sensitivity analysis can be carried out (see [21]). For this purpose, problem (1)-(3) is recast as minimize J(x, a, z) (32) x

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subject to f (x, a, z) = 0

:

λ,

(33)

g(x, a, z) ≤ 0

:

µ,

(34)

where a is a parameter vector, and the measurement vector z is explicity written for clarity. The optimal solution of problem (32)-(34) is (ˆ x, λ∗ , µ∗ , J ∗ ), where λ and µ are the dual variable vectors related to equality and inequality constraints, respectively. The Karush-Kuhn-Tucker (KKT) first order optimality conditions for problem (32)-(34) are: ∇x J(ˆ x, a, z) + λ∗T ∇x f (ˆ x, a, z) + µ∗T ∇x g(ˆ x, a, z) = 0 ,

(35)

f (ˆ x, a, z) = 0 ,

(36)

g(ˆ x, a, z) ≤ 0 ,

(37)

∗

µ

µ∗i gi (ˆ x, a, z)

≥ 0,

= 0 , ∀i .

(38) (39)

Once an optimal solution of the estimation problem is known, binding inequality constraints are considered equality constraints and non-binding ones are disregarded. Therefore, vector f (x, a, z) includes p equality constraints and qΩ active inequality constraints,

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where Ω is the set of active inequality constraints. Thus, system (35)-(38) becomes: ∇x J(ˆ x, a, z) + λ∗T ∇x f (ˆ x, a, z) = 0 , f (ˆ x, a, z) = 0 .

(40) (41)

ˆ , a, z, λ∗ , J ∗ in such a way that the KKT To obtain sensitivity equations, we perturb x conditions still hold [22]. To this end, we differentiate the objective function J(x, a, z) and the optimality conditions (40)-(41), obtaining the following linear system of equations:   dx   J | J a | J z | 0 | −1    da   x    T dz  (42)  J xx | J xa | J xz | F x | 0    =0,   dλ  Fx | Fa | Fz | 0 | 0 dJ

ˆ , λ∗ , J ∗ , where the vectors and submatrices in (42) are evaluated at the optimal solution, x and provided in the Appendix. To compute sensitivities with respect to the components of the vectors a and z, system (42) can be written as U [ dx dλ dJ ]T

= S a da + S z dz ,

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where the matrices U , S a , and S z are   J x | 0 | −1   U =  J xx | F Tx | 0  , Fx | 0 | 0 S Ta = − [ J a J xa F a ] , S Tz = − [ J z J xz F z ] .

(43)

(44) (45) (46)

Therefore h

dx dλ dJ

iT

= U −1 S a da + U −1 S z dz .

(47)

System (47) can be solved in a decoupled manner applying the principle of superposition by replacing da and dz by the np -dimensional and m-dimensional identity matrices, respectively. Considering the matrices: " # " # # " J xx | F Tx J xz J xa Hx = and H z = , , Ha = Fx | 0 Fa Fz where H x is symmetrical, the sensitivity matrices containing all derivatives with respect to parameters and measurements, respectively, are:   ∂J ∂x ∂x ∂λ T = −H −1 (48) x H a ; ∂a = J a + J x ∂a ; ∂a ∂a

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A Mathematical Programming Approach to State Estimation 15  T ∂x ∂λ ∂J ∂x = −H −1 Hz ; = Jz + Jx . (49) x ∂z ∂z ∂z ∂z Matrix H x is generally invertible as power flow equations (33) and binding physical limits (34) are generally regular, non-degenerated constraints [23]. Moreover, matrix H x is highly sparse as a result of the sparsity of its building blocks and it can be easily factorized using sparse-oriented LU algorithms. Example 7: Sensitivity analysis example First, matrices J xx , J xa , J xz , F x , F a and F z are evaluated. Then, matrices H x , H a and H z are evaluated. Finally, (48) and (49) are used to obtain the sensitivities. The most relevant sensitivities for the example in Figure 1 are presented below. 1. Sensitivities of the voltage estimates with respect to the voltage measurements are (dimensionless matrix): v1m vˆ1 0.46591 " # ∂ˆ vi vˆ  0.49405 = 2 m vˆ3  0.51395 ∂vj vˆ4 0.52464 

v2m  0.49405 0.52821  . 0.55358  0.56107

Note that all sensitivities above are fairly similar, which is a consequence of the small size of the system considered.

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2. Sensitivities of the voltage estimate in each bus with respect to reactive power injection and reactive power flow measurements are (units are puV/puMVar):

"

∂ˆ vi ∂Qm j

Qm Qm Qm 1 3 34  v ˆ 0.00782 −0.02823 0.01989 1 # ∂ˆ vi vˆ  −0.01669 −0.01131 0.03107  . = 2 m vˆ3  −0.04085 0.00625 0.04184  ∂Q34 vˆ4 −0.05423 −0.04245 0.00208 

The sensitivities of the voltage magnitudes at nodes 1 and 3 are positive with respect to the reactive power injections at nodes 1 and 3, respectively. Other sensitivities of voltage magnitude with respect to reactive power injections are negative. On the other hand, all sensitivities of voltage magnitude with respect to the reactive power flow 3-4 are positive. 3. Sensitivities of the angular differences between the bus voltages of each line with respect to active power injections are (units are rad/puMW): P1m P3m   θˆ1 − θˆ2 0.0163 −0.0180   θˆ − θˆ  0.0419 0.0093  ∂ θˆi − θˆj . = ˆ1 ˆ4  m θ2 − θ3  0.0184 −0.0206  ∂Pk θˆ3 − θˆ4 0.0072 0.0478

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Eduardo Caro, Antonio J. Conejo and Roberto M´ınguez The largest sentivities correspond to angles differences 1-4 and 3-4 with respect to active power injection measurements at buses 1 and 3, respectively. 4. Sensitivities of the angular differences between the bus voltages of each line with respect to active power flows are (units are rad/puMW): m m m P14 P32 P34   θˆ1 − θˆ2 0.0071 −0.0087 −0.0092 # " ∂ θˆi − θˆj θˆ − θˆ  0.0336 0.0176 −0.0083  . = ˆ1 ˆ4  m θ2 − θ3  0.0079 −0.0101 −0.0105  ∂Pk,l θˆ3 − θˆ4 0.0186 0.0364 0.0114

The largest sensitivities correspond to angles differences 1-4, 3-4 and 3-4 with respect to active power flows 1-4, 3-2 and 3-4, respectively.

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9.

Decomposition

To solve the state estimation problem in a power system comprising several independently operated regions, mathematical programming decomposition techniques are useful to attain the global solution for the whole system while preserving the independent operation of each region. For the sake of simplicity and without loss of generality, we consider a power system comprising just two areas. Additionally, we assume that the constraints coupling the two areas are just equality constraints (which is generally the case in state estimation). The two-area state estimation problem is then formulated as minimize Ja (xa ) + Jb (xb ) + Jc (xS,a , xS,b ) xa , xb

(50)

subject to f a (xa ) = 0 ,

(51)

g a (xa ) ≤ 0 ,

(52)

f ab (xS,a , xS,b ) = 0

:

λab ,

(53)

f b (xb ) = 0 ,

(54)

g b (xb ) ≤ 0 ,

(55)

f ba (xS,b , xS,a ) = 0

:

λba ,

(56)

where a and b identify the two areas, and xS,a and xS,b are vectors containing subsets of the variables in vectors xa and xb , respectively. The remaining notation of (50)-(56) mimics that of (1)-(3). The objective function term Jc (xS,a , xS,b ) includes variables pertaining to both areas, a and b. Constraints f ab (xS,a , xS,b ) couple together the two area but are mostly

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related to area a, while constraints f ba (xS,b , xS,a ) also couple the two areas but are mostly related to area b. We consider the following decomposed subproblems for areas a and b, respectively: (ν)

(ν)

(ν)

minimize Ja (xa ) + Jc (xS,a , xS,b ) + (λba )T f ba (xS,b , xS,a ) xa

(57)

subject to f a (xa ) = 0 ,

(58)

g a (xa ) ≤ 0 ,

(59)

(ν) f ab (xS,a , xS,b )

= 0

:

λab ,

(60)

and (ν)

(ν)

(ν)

minimize Jb (xb ) + Jc (xS,a , xS,b ) + (λab )T f ab (xS,a , xS,b ) xb

(61)

subject to f b (xb ) = 0 ,

(62)

g b (xb ) ≤ 0 ,

(63)

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(ν) f ba (xS,b , xS,a )

= 0

:

λba .

(64)

If multiplier vectors λab and λba are equal to their respective optimal values, it is simple to show that the first order necessary optimality conditions of problems (57)-(60) and (61)-(64) are identical to the first order necessary optimality conditions of problem (50)-(56) (see [24] and [25]). Thus, solving (57)-(60) and (61)-(64) is equivalent to solve (50)-(56). The decentralized solution algorithm works as follows. Problem (57)-(60) is solved for (ν) (ν) (ν) (ν) trial values of λba and xS,b , which result in updated values for λab and xS,a , which in turn are used to solve again problem (61)-(64). The iteration continues until convergence in xa and xb (and in λab and λba ) is attained, which is guaranteed in most practical situations [25]. It should be noted that instead of solving each subproblem until optimality at each iteration, it is sufficient to perform a descent step for each subproblem per iteration, which results in high computational efficiency. The communication burden between problems (57)-(60) and (61)-(64) is small due to the generally reduced dimensions of xS,a , xS,b , λab and λba . Additionally, areas need to interchange power measurement values at border lines and buses. Example 8: Decomposition example We consider the 8-bus electric energy system depicted in Figure 2. The network data is provided in Table 7. This example includes two generating buses, four demand buses and two transit buses. Considering bus 4a as the reference bus, the state-variable vectors have

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Figure 2. One-line diagram and measurement configuration of a 2-region 8-bus system.

the form: xa = (θ1a , θ2a , θ3a , v1a , v2a , v3a , v4a )T , xb = (θ1b , θ2b , θ3b , θ4b , v1b , v2b , v3b , v4b )T , xS,a = (θ1a , θ2a , v1a , v2a )T , xS,b = (θ1b , θ2b , v1b , v2b )T .

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The measurement system, shown in Figure 2, includes sixteen measurements: four voltage measurements, two active/reactive power injection measurements, six active power flow measurements and four reactive power flow measurements. The measurement standard deviations are the same of those used in Example 1. The measurement redundancy ratio provided by this measurement configuration is (16 + 4)/15 = 1.33. The decomposed two-area estimation problem is formulated below. Table 7. Line Parameters Bus i 1a 2a 2a 3a 1b 2b 2b 3b 1a

Bus j 2a 3a 4a 4a 2b 3b 4b 4b 1b

Bij (p.u.) 10 10 10 10 10 10 10 10 2

The objective function for area A is:

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(ν)

(ν)

(ν)

Ja (xa ) + Jc (xa , xS,b ) + (λba )T f ba (xS,b , xS,a ) = w1a (v4a (v2a B4a,2a sin(0 − θ2a ) + v3a B4a,3a sin(0 − θ3a )) − z1a )2 + w3a (v2a v3a B2a,3a sin(θ2a − θ3a ) − z3a )2 + w4a (v2a v4a B2a,4a sin(θ2a − 0) − z4a )2 + w5a (v3a − z5a )2 + w6a (v4a − z6a )2 + w7a (v4a (−v2a B4a,2a cos(0 − θ2a ) − v3a B4a,3a cos(0 − θ3a ) −v4a B4a,4a ) − z7a )2
2 2 B + w9a −v2a v3a B2a,3a cos(θ2a − θ3a ) + v2a 2a,3a − z9a 2  (ν) (ν) + w2a v1a v1b B1a,1b sin(θ1a − θ1b ) − z2a 2  (ν) (ν) 2 B − z + w8a −v1a v1b B1a,1b cos(θ1a − θ1b ) + v1a 8a 1a,1b  2 (ν) (ν) + w2b v1b v1a B1b,1a sin(θ1b − θ1a ) − z2b  2   (ν) (ν) (ν) 2 + w8b −v1b v1a B1b,1a cos(θ1b − θ1a ) + v1b B1b,1a − z8b (ν)

(ν)

(ν)

(ν)

+ λba,1 fba,1 (xS,a , xS,b ) + λba,2 fba,2 (xS,a , xS,b ) . Optimization Advances in Electric Power Systems, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

20

Eduardo Caro, Antonio J. Conejo and Roberto M´ınguez Equality and inequality constraints for area A are: h (ν) (ν) (ν) fab,1 (xS,a , xS,b ) = 0 = v1a v1b B1a,1b sin(θ1a − θ1b ) i +v2a B1a,2a sin(θ1a − θ2a ) : λab,1 , h (ν) (ν) (ν) fab,2 (xS,a , xS,b ) = 0 = v1a −v1b B1a,1b cos(θ1a − θ1b )

−v1a B1a,1a − v2a B1a,2a cos(θ1a − θ2a )

i

: λab,2 ,

min max P4a ≤ v4a (v2a B4a,2a sin(0 − θ2a ) + v3a B4a,3a sin(0 − θ3a )) ≤ P4a ,

Qmin ≤ v4a (−v2a B4a,2a cos(0 − θ2a ) − v3a B4a,3a cos(0 − θ3a ) − v4a B4a,4a ) ≤ Qmax 4a 4a , −π ≤ θi

≤

π, i = 1a, . . . , 3a .

The optimization variable vector for area A is: xa = (θ1a , θ2a , θ3a , v1a , v2a , v3a , v4a )T .

The objective function for area B is:

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(ν)

(ν)

(ν)

Jb (xb ) + Jc (xS,a , xb ) + (λab )T f ab (xS,a , xS,b ) = w1b (v4b (v2b B4b,2b sin(θ4b − θ2b ) + v3b B4b,3b sin(θ4b − θ3b )) − z1b )2 + w3b (v2b v3b B2b,3b sin(θ2b − θ3b ) − z3b )2 + w4b (v2b v4b B2b,4b sin(θ2b − θ4b ) − z4b )2 + w5b (v3b − z5b )2 + w6b (v4b − z6b )2 + w7b (v4b (−v2b B4b,2b cos(θ4b − θ2b ) − v3b B4b,3b cos(θ4b − θ3b ) −v4b B4b,4b ) − z7b )2
2 2 B + w9b −v2b v3b B2b,3b cos(θ2b − θ3b ) + v2b 2b,3b − z9b 2  (ν) (ν) + w2a v1a v1b B1a,1b sin(θ1a − θ1b ) − z2a  2   (ν) (ν) (ν) 2 + w8a −v1a v1b B1a,1b cos(θ1a − θ1b ) + v1a B1a,1b − z8a  2 (ν) (ν) + w2b v1b v1a B1b,1a sin(θ1b − θ1a ) − z2b  2 (ν) (ν) 2 B + w8b −v1b v1a B1b,1a cos(θ1b − θ1a ) + v1b 1b,1a − z8b (ν)

(ν)

(ν)

(ν)

+ λab,1 fab,1 (xS,b , xS,a ) + λab,2 fab,2 (xS,b , xS,a ) . Optimization Advances in Electric Power Systems, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

A Mathematical Programming Approach to State Estimation Equality and inequality constraints for area B are: h (ν) (ν) (ν) fba,1 (xS,b , xS,a ) = 0 = v1b v1a B1b,1a sin(θ1b − θ1a ) i +v2b B1b,2b sin(θ1b − θ2b ) : λba,1 , h (ν) (ν) (ν) fba,2 (xS,b , xS,a ) = 0 = v1b −v1a B1b,1a cos(θ1b − θ1a )

−v1b B1b,1b − v2b B1b,2b cos(θ1b − θ2b )

i

21

: λba,2 ,

min max P4b ≤ v4b (v2b B4b,2b sin(θ4b − θ2b ) + v3b B4b,3b sin(θ4b − θ3b )) ≤ P4b ,

Qmin ≤ v4b (−v2b B4b,2b cos(θ4b − θ2b ) − v3b B4b,3b cos(θ4b − θ3b ) − v4b B4b,4b ) ≤ Qmax 4b 3b , −π ≤ θi ≤ π, i = 1b, . . . , 4b .

The optimization variable vector for area B is: xb = (θ1b , θ2b , θ3b , θ4b , v1b , v2b , v3b , v4b )T .

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Table 8. Voltage and angle evolution State variable v1a v2a v3a v4a θ1a θ2a θ3a v1b v2b v3b v4b θ1b θ2b θ3b θ4b

0 1.0000 1.0000 0.9791 1.0782 0.0000 0.0000 0.0000 1.0000 1.0000 1.0204 1.0764 0.0000 0.0000 0.0000 0.0000

1 1.0097 1.0121 0.9774 1.0695 -0.0607 -0.0727 -0.1093 1.0204 1.0225 0.9989 1.0782 -0.0668 -0.0680 -0.1071 0.0016

Iteration 2 ... 1.0190 1.0188 0.9836 1.0754 . . . -0.0720 -0.0730 -0.1079 1.0262 1.0277 1.0039 1.0830 ... -0.0867 -0.0896 -0.1283 -0.0206

8 1.0225 1.0213 0.9860 1.0777 -0.0769 -0.0732 -0.1074 1.0285 1.0297 1.0058 1.0848 -0.0955 -0.0992 -0.1377 -0.0305

9 1.0225 1.0213 0.9860 1.0777 -0.0769 -0.0732 -0.1074 1.0285 1.0297 1.0058 1.0848 -0.0955 -0.0992 -0.1377 -0.0305

Measurements are generated by solving with high accuracy the corresponding power flow and then adding zero-mean independent Gaussian-distributed errors to the exact values Optimization Advances in Electric Power Systems, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

22

Eduardo Caro, Antonio J. Conejo and Roberto M´ınguez

of the measurements. These measurements and the exact values are provided in Tables 1012. The same minimum and maximum active/reactive power injections as in Example 3 are considered for buses 4a and 4b. The example in Figure 2 is solved using the proposed decomposition method and a convergence tolerance of 10−5 . Tables 8 and 9 provide information on the convergence procedure while Tables 10-12 provide final results. Solving this problem using a centralized algorithm results in the same solution. Table 9. Multiplier evolution Multiplier λab,1 λab,2 λba,1 λba,2

0 0.0000 0.0000 0.0000 0.0000

1 -265.9133 53.7406 -0.0476 39.7697

Iterations 2 ... -82.8998 7.4591 ... -0.0604 20.6668

8 -0.1307 -13.2624 -0.0488 13.2169

9 -0.0755 -13.1867 -0.0488 13.2238

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Table 10. Voltage results. Measured, true and estimated values Bus # 1a 2a 3a 4a 1b 2b 3b 4b

vim (p.u.) – – 0.9791 1.0782 – – 1.0204 1.0764

vitrue (p.u.) 1.0219 1.0186 0.9822 1.0749 1.0388 1.0423 1.0158 1.0947

vˆi (p.u.) 1.0225 1.0213 0.9860 1.0777 1.0285 1.0297 1.0058 1.0848

θitrue (rad) -0.0783 -0.0735 -0.1088 0.0000 -0.1018 -0.1064 -0.1458 -0.0392

θˆi (rad) -0.0769 -0.0732 -0.1074 0.0000 -0.0955 -0.0992 -0.1377 -0.0305

Table 11. Power injection results. Measured, true and estimated values Bus # 4a 4b

Pitrue Pˆi (MW p.u.) 1.9354 1.9500 1.9440 1.9306 1.9500 1.9353 Pim

ˆi Qtrue Q i (MVar p.u.) 1.6859 1.6938 1.6858 1.5458 1.5272 1.5443 Qm i

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Table 12. Power flow results. Measured, true and estimated values From To 1a-1b 2a-3a 2a-4a 1b-1a 2b-3b 2b-4b

10.

Pijm 0.0348 0.3536 -0.8217 -0.0431 0.4035 -0.7764

Pijtrue (MW p.u.) 0.0500 0.3536 -0.8036 -0.0500 0.4167 -0.7667

Pˆij

Qm ij

0.0391 0.3444 -0.8049 -0.0391 0.3988 -0.7673

0.0037 0.3630 – 0.0097 0.2571 –

ˆ ij Q Qtrue ij (MVAr p.u.) -0.0339 -0.0119 0.3774 0.3664 – – 0.0357 0.0127 0.2846 0.2534 – –

Conclusions

This chapter revisits the state estimation problem in an electric energy system and advocates a direct solution approach based on mathematical programming. This approach allows carrying out a sensitivity analysis that enriches the solution derived. Moreover, mathematical programming decomposition techniques can be directly applied to obtain the solution in a decentralized manner, which is desirable in multi-regional systems.

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A. Vectors and Matrices for Sensitivity Analysis Auxiliary matrices and vectors needed for sensitivity analysis are provided below. Dimensions are indicated in parenthesis. J x (1×n) = [∇x J(ˆ x, a, z)]T ,

(65)

J a (1×np ) = [∇a J(ˆ x, a, z)]T ,

(66)

T

x, a, z)] J z (1×m) = [∇z J(ˆ

,

(67)

J xx (n×n) = ∇xx J(ˆ x, a, z) p+q XΩ

λ∗k ∇xx fk (ˆ x, a, z) ,

p+q XΩ

λ∗k ∇xa fk (ˆ x, a, z) ,

(69)

J xz (n×p) = ∇xz J(ˆ x, a, z) p+q XΩ λ∗k ∇xz fk (ˆ + x, a, z) ,

(70)

+

k=1

(68)

J xa (n×p) = ∇xa J(ˆ x, a, z) +

k=1

k=1

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Eduardo Caro, Antonio J. Conejo and Roberto M´ınguez F x ((p+qΩ )×n) = [∇x f (ˆ x, a, z)]T ,

(71)

F a ((p+qΩ )×np ) = [∇a f (ˆ x, a, z)]T ,

(72)

F z ((p+qΩ )×m) = [∇z f (ˆ x, a, z)]T .

(73)

References [1] F. C. Schweppe and J. Wildes. Power system static state estimation. Part I: Exact model. IEEE Transactions on Power Apparatus and Systems, 89(1):120–125, January 1970. [2] F. C. Schweppe and D.B. Rom. Power system static state estimation. Part II: Approximate model. IEEE Transactions on Power Apparatus and Systems, 89(1):125–130, January 1970. [3] F. C. Schweppe. Power system static state estimation. Part III: Implementation. IEEE Transactions on Power Apparatus and Systems, 89(1):130–135, January 1970. [4] R. Larson, W. Tinney, L. Hadju, and D. Piercy. State estimation in power systems. part II: Implementations and applications. IEEE Transactions on Power Apparatus and Systems, 89(3):353–362, March 1970. [5] A. Garc´ıa, A. Monticelli, and P. Abreu. Fast decoupled state estimation and bad data processing. IEEE Transactions on Power Apparatus and Systems, 98(5):1645–1652, September/Octuber 1979.

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[6] A. Monticelli and A. Garc´ıa. Fast decoupled state estimators. IEEE Transactions on Power Systems, 5(2):556–564, May 1990. [7] J. J. Allemong, L. Radu, and A. M. Sasson. A fast and reliable state estimation algorithm for AEP’s new control center. IEEE Transactions on Power Apparatus and Systems, 101(4):933–944, April 1982. [8] L. Holten, A. Gjelsvik, S. Aam, F. Wu, and W. H. E. Liu. Comparison of different methods for state estimation. IEEE Transactions on Power Systems, 3(4):1798–1806, November 1988. [9] A. Monticelli. Electric power system state estimation. Proceedings of the IEEE, 88(2):262–282, 2000. [10] A. Abur and A. G. Exp´osito. Electric Power System State Estimation. Theory and Implementations. Marcel Dekker, New York, 2004. [11] R. M´ınguez, A. J. Conejo, and A. S. Hadi. Non-Gaussian state estimation in power systems. In B. C. Arnold, N. Balakrishnan, J. M. Sarabia, and R. M´ınguez, editors, Advances in Mathematical and Statistical Modeling, Statistics for Industry and Technology (SIT). Birkhauser Boston, 141–156, 2008.

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A Mathematical Programming Approach to State Estimation

25

[12] E. Castillo, A. J. Conejo, R. E. Pruneda, and C. Solares. Observability analysis in state estimation: A unified approach. IEEE Transactions on Power Systems, 21(2):877– 886, May 2006. [13] A. J. Conejo, S. de la Torre, and M. Ca˜nas. An optimization approach to multi-area state estimation. IEEE Transactions on Power Systems, 22(1):213–221, 2007. [14] A. Drud, CONOPT, in GAMS-The Solver Manuals. GAMS Development Corporation, Washington, 2007. Available: http://www.gams.com/solvers/conopt.pdf. [15] B. A. Murtagh, M. A. Saunders, W. Murray, P. E. Gill, R. Raman, and E. Kalvelagen MINOS: A solver for large-scale nonlinear optimization problems, in GAMS-The Solver Manuals, 2007. Available: http://www.gams.com/solvers/minos.pdf. [16] A. Brooke, D. Kendrick, A. Meeraus, R. Raman, and R. E. Rosenthal. GAMS: a Users Guide. Washington, DC: GAMS Development Corporation, 2008. Available: http://www.gams.com/. [17] R. Fourer, D.M. Gay, and B.W. Kernighan. AMPL: A Modeling Language for Mathematical Programming, 2nd Ed.. Pacific Grove, CA: Brooks/Cole–Thomson Learning, 2003. Available: http://www.ampl.com/. [18] Bisschop, J., Roelofs, M. AIMMS - The User’s Guide, 2007. Paragon Decision Technology B.V., Haarlem. Available: http://www.aimms.com/. [19] S. Chatterjee, A. S. Hadi, and B. Price. Regression Analysis by Example. John Wiley & Sons, New York, 1990.

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[20] G. B. Golub, and Van Loan, C. F. Matrix Computations. The Johns Hopkins University Press, Third Edition USA, 1996. [21] R. M´ınguez and A. J. Conejo. State estimation sensitivity analysis. IEEE Transactions on Power Systems, 22(3):1080–1091, August 2007. [22] E. Castillo, A. J. Conejo, C. Castillo, R. M´ınguez, and D. Ortigosa. A perturbation approach to sensitivity analysis in nonlinear programming. Journal of Optimization Theory and Applications, 128(1):49–74, January 2006. [23] D. G. Luenberger. Linear and Nonlinear Programming. Addison-Wesley, Reading, Massachusetts, Second edition, 2003. [24] A. J. Conejo, E. Castillo, R. M´ınguez, and R. Garc´ıa-Bertrand. Decomposition techniques in mathematical programming. Engineering and science applications. Springer, Heidelberg, New York, 2006. [25] A. J. Conejo, F. J. Nogales, and F. J. Prieto. A decomposition procedure based on approximate Newton directions. Mathematical Programming, Series A, 93(3):495– 515, 2002.

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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Optimization Advances in Electric Power Systems, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

In: Optimization Advances in Electric Power Systems ISBN: 978-1-60692-613-0 c 2008 Nova Science Publishers, Inc. Editor: Edgardo D. Castronuovo, pp. 27-51

Chapter 2

T RUST R EGION O PTIMIZATION M ETHODS VIA G IVENS ROTATIONS A PPLIED TO P OWER S YSTEM S TATE E STIMATION A.J. Sim˜oes Costa, R.S. Salgado and P. Haas Power Systems Group, Department of Electrical Engineering, Universidade Federal de Santa Catarina, Florian´opolis, SC, Brazil

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1.

Introduction

The restructuring of the power industry worldwide has brought about significant changes affecting the environment under which several power systems applications are executed. Transmission networks covered nowadays by Independent System Operators (ISO), for instance, tend to be much larger, since they often include several control areas. In addition, the diversified ownership of transmission facilities makes the system operation dependent on real-time data to be provided by others to the system operator. As a result, power system functions in charge of ensuring operational reliability currently face much more stringent conditions to successfully fulfill their tasks. Power System State Estimation (PSSE) is one of the security related functions most affected by the above mentioned changes. Concurrently, in restructured environments PSSE importance has increased as the basic monitoring tool available to the ISO. This combination of factors leads to the need of developing more robust state estimators, capable of improved convergence characteristics in the presence of unfavorable conditions (larger networks, poor measurement redundancy, occurrence of bad data/topology errors, etc). Under such severe conditions, even the superior numerical properties of orthogonal techniques [1],[2] may fail to yield convergence. Fortunately, recent advances in the Numerical Optimization area have paved the way to the development of algorithmically more robust power system state estimators. The purpose of this chapter is to describe recent research efforts towards the application of Trust Region (TR) methods to power system state estimation. These methods pertain to a class of recently developed tools devised to enhance the convergence capability of iterative solutions for optimization problems in which the objective function is approximated in each iteration by a quadratic model. Since this is precisely the case of PSSE solved by

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28

A.J. Sim˜oes Costa, R.S. Salgado and P. Haas

the Gauss-Newton method, such robust algorithms are well suited for the problem at hand. TR methods basically consist in implementing careful schemes to control the optimization step as a function of how well the local quadratic model represents the actual objective function [3],[4]. Both the direction and the step length of the next optimization step are simultaneously chosen while solving the minimization problem based on the approximate model. This type of step control imparts ability to PSSE algorithms to cope with occasional power system modelling errors. The TR approach for PSSE problems has been originally proposed by Pajic and Clements [5],[6]. In their implementation of a TR-based state estimator, the authors make use of an orthogonal factorization techniques to avoid building the normal equation gain matrix, thus reducing the chance of numerical ill-conditioning. In addition, to obtain the intermediate solutions required in the minor loop of the TR algorithm, the authors resort to a scheme in which only the submatrix of the Jacobian matrix related to the adjustable parameter λ needs to be reprocessed. This version of orthogonal TR-based state estimators is referred to in this chapter as λ − refactorization method.

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On the other hand, it has been recently shown that the Levenberg-Marquardt equation solved as a basic step of the TR approach can be interpreted as the solution of a linearized least-squares problem taking into account a priori information on state variables [7]. This approach builds on previous results presented in [8], according to which a priori state information can be processed at virtually no computational cost by orthogonal estimators based on the 3-multiplier version of Givens rotations (G3M). Therefore, the state estimator presented in [7] is an attempt to combine the numerical robustness and versatility of G3M rotations with the algorithmic robustness of the TR approach. Such a TR-based state estimator is referred to as A Priori State Information (APSI) method. In this chapter, both orthogonal implementations of TR-based state estimators, namely, the λ-refactorization method and the APSI method, are described in detail. The respective analytical bases, computational requirements and numerical performances are discussed and compared. Attention is given to strategies to properly adjust the TR radius during the PSSE iterative process. Simulation results obtained with test systems of different sizes are used to illustrate and compare the performance of both approaches. This chapter is organized as follows. Section 2. reviews the power system state estimation background. In addition to the conventional normal equation approach, attention is also given to the consideration of a priori information on the state variables, since this constitutes the basis for the APSI strategy. The solution of the PSSE problem through Givens rotations is also reviewed, including the processing of a priori information through orthogonal state estimators. Trust Region methods are introduced in Section 3., which includes detailed descriptions of both the generic TR algorithm and the strategy employed to determine the λ parameter so as to enforce the TR equality constraint. Section 4. is devoted to the main issue dealt with in this chapter, that is, orthogonal implementations of TR methods. The λ-refactorization method is presented in Subsection 4.1., while Subsection 4.2. introduces the APSI method. In addition, a qualitative appraisal of both strategies is presented in Subsection 4.3.. Section 5. reports the results of several numerical simulations involving both methods conducted on four distinct power networks. Finally, the concluding remarks on the performance of both orthogonal TR state estimators are presented in Section 6..

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2. 2.1.

29

State Estimation Background Measurement Model and Weighted Least Squares Estimator

Consider a N bus power network for which m measurements are taken. Assuming a nonlinear model for the electrical network, the relationships between measured quantities and state variables can be expressed as: z = h(x) + ε

(1)

where z is the m × 1 measurement vector, x is the n × 1 vector of state variables to be estimated, h(x) is the m × 1 vector of nonlinear functions relating measurements to states, and ε is the m × 1 measurement error vector, whose m × m covariance matrix is assumed diagonal and denoted by R. Power system state estimation is formulated as the following weighted least squares problem: 1 x)]t R−1 [z − h(ˆ x)] (2) M in J(ˆ x) = [z − h(ˆ 2 where x ˆ represents the vector of state estimates, and the function to be minimized is the weighted sum of the squared residuals.

2.2.

Solution through Gauss-Newton Method

The solution of the minimization problem represented by equation (2) through the GaussNewton method requires at each step of the iterative process the solution of the linear system [9], [10]  t −1  H R H ∆x = Ht R−1 ∆z (3)

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where H is the Jacobian matrix of h(x) computed at a given point xk and ∆z = z − h(xk ).

Solving Eq. (3) provides the vector ∆x of increments to the states, so that the solution of the nonlinear problem is obtained by an iterative procedure given by xk+1 = xk + ∆x, which goes on until ∆x becomes smaller than a pre-specified tolerance. It is important to notice that the coefficient matrix on the left-hand side of Eq. (3) (often referred to as gain matrix) is in fact an approximation for the Hessian matrix ∇2 J(ˆ x) near the solution, and the right-hand side is the negative of the gradient of J(ˆ x), both computed at point xk [9], [10]. Furthermore, in practical applications the gain matrix is often kept constant after a given iteration of the iterative process, thereby adding further approximations to the Hessian matrix. Although under usual conditions the Gauss-Newton method is able to solve many practical problems, it exhibits one significant drawback: the condition number of the gain matrix 1 is the square of the condition number of the matrix (R 2 H) [1]. For this reason, the solution obtained through the normal equation may be much less accurate than those provided by

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A.J. Sim˜oes Costa, R.S. Salgado and P. Haas

algorithms that avoid the explicit calculation of the gain matrix. If the Jacobian matrix is ill-conditioned, the factorization of the gain matrix may fail due to the amplified effects of round-off errors. Additionally, the speed of convergence of the Gauss-Newton method close to the optimal solution depends on how much the leading term represented by the gain matrix prevails in the composition of the (exact) Hessian matrix.

2.3.

A Priori State Information in Least-Squares Problems

The conventional least-squares problem expressed by equation (2) can be extended in order to account for the availability of a priori information on the state variables, here referred to as APSI. To take such information into account, the objective function of problem (2) is augmented by the term [8] 1 (ˆ x−x ¯)t P−1 (ˆ x−x ¯) (4) 2 where x ¯ is a n × 1 vector formed by the a priori values for the state variables, whose n × n covariance matrix is P. In practice, matrix P is usually assumed as diagonal, its i-th ¯i . It can be easily shown diagonal entry being the variance σ ¯i2 of the a priori information x that the optimality conditions for the augmented problem lead to the following equation:  t −1  H R H + P−1 ∆x = HT R−1 ∆z + P−1 ∆x

(5)


∆ ¯ − xk . Therefore, in the presence of a priori information, Eq. (5) replaces where ∆x = x Eq. (3) in the process of solving the weighted least squares problem.

2.4.

Solution through Givens Rotations

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2.4.1. A Macro View of PSSE via Givens Rotations To overcome the numerical difficulties associated to the normal equation approach, the use of decomposition methods based on orthogonal transformations has been proposed. Methods based on the application of Givens rotations [1], [2] have proved to be particularly successful to provide robust solutions, since they prevent further degradation of the problem’s numerical conditioning. In this chapter, we focus attention on the solution of the weighted least-squares problem (2) by the three-multiplier version of Givens rotations (G3M) [11]. To outline the procedure, consider that successive orthogonal transformations are applied to matrix H and vector 1 ∆z (both previously scaled by matrix R− 2 ) in order to obtain an upper triangular linear system of equations. If Q represents the matrix that stores the individual rotations, we have [1],[12]:     1  U c H ∆z = (6) Q R− 2 0 e

where U is an upper n × n triangular matrix and c and e are n × 1 and (m − n) × 1 vectors, respectively. The fast versions of Givens rotations are based on the decomposition of matrix U as [11],[1]: 1 ¯ U = D2 U (7)

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¯ is a unit upper triangular matrix. Vector ∆z is conwhere D is a diagonal matrix and U sidered as an extra column of H, so that vector c is also scaled in the transformation. The resulting scaled vector is denoted by ¯ c. The artifice of scaling U as above has a number of computational benefits, such as the elimination of square-root computations during the 1 factorization given by Eq. (6) [11]. In practice, D 2 is not required, and only D needs to be computed. Following the transformation step given by Eq. (6), vector ∆x is obtained by simply solving the upper triangular system ¯ ∆x = ¯ U c

(8)

by back-substitution. The weighted sum of squared residuals is determined from e, as a by-product of the estimation process.

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2.4.2. The Elementary G3M Givens Rotations - Interpretation of Scaling Factors To sequentially implement the scaling of matrix U as defined in Eq. (7), each new row of matrix H to be processed (augmented with the corresponding entry of vector ∆z) is √ also assumed as scaled by a factor w. Therefore, prior to a particular rotation between a generic row η of H and the i − th row u of the triangular matrix U, we have: √ √ √ u = [0 . . . 0 d ... d uk . . . du ] √ √ √ n+1 (9) η = [0 . . . 0 w ηi . . . w ηk . . . w ηn+1 ] √ where d is the current scaling factor of row u of U. The next elementary rotation aims at zeroing out the i − th entry of η, producing the following result: √ √ √ d′ . . . √ d′ u′k . . . √ d′ u′n+1 ] u′ = [0 . . . 0 (10) ′ η ′ = [0 . . . 0 0 . . . w′ ηk . . . w′ ηn+1 ] The operations which implement each elementary operation are detailed next. √ √ In order to zero out the i-th entry of η both rows u and η are scaled by d and w before the rotation, as shown in Eqs. (9). The result of the rotation is represented in Eqs. (10). The equations which define the relationships between the transformed and the original entries of u and η are given by [11]: d′ w′ c¯ s¯ ′

¯k ηk = ηk − ηi u ′ ¯k + s¯ηk u ¯k = c¯u

d + wηi2 d w/d′ d/d′ wηi /d′

= = = = 

,

k = i + 1...n + 1

(11)

(12)

where c¯ and s¯ are the parameters which define each elementary rotation. The 3-multiplier variant of Givens rotations (G3M) owns its name from the number of multiplications required in the transformations shown in Eqs. (12). Weighting factors d and w must be initialized, as further discussed in the sequel.

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After the initialization of the weighting factors, the rows of H are processed sequentially, each of them undergoing the number of elementary rotations necessary to annihilate all its nonzero entries. The rows of U, as well as matrix D, are updated during this process, through the above equations. A very attractive feature of the G3M rotations is that the scaling mechanism required for their implementation allows the solution of weighted least squares problems for free, that is, at no extra computational cost [11]. In PSSE problems, the value initially assigned to the row scaling factor w is the weight attributed to the corresponding measurement. Hence, if row η corresponds to, say, measurement zj , then w = 1/σj2 [1]. Having established the role of the measurement scaling fator w, an interpretation of the scaling factors d of the U rows is now in order. Analogy with w suggests that the initial value of d can also be seen as a weight, but in this case assigned to the states (notice that there are as many d’s as state variables) before any measurement is processed. In other words, di is the weighting factor for the a priori information possibly available on state variable xi . In addition, the value for di must be in agreement with expression (4), which establishes how a priori information is taken into account in the estimation process. This leads to the conclusion that

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di = 1/¯ σi2 ,

i = 1, . . . , n

(13)

where σ ¯i2 is the variance of the a priori information on state variable i. The practice in conventional applications of the G3M rotations to PSSE (which neglect a priori state information) has been to initialize di = 0 and u ¯ii = 1.0 for every U row, which amounts to assuming that U is initially a null triangular matrix [1]. This is consistent with the discussion in Subsection 2.3., since such an initialization actually means that nothing is known in advance about the states, so that their a priori variances are considered infinite. Therefore, we conclude that prior information on the states can be easily considered in the G3M rotations framework by simply initializing the extra element u ¯n+1 in Eq. (9) as x ¯i , and di as given by Eq. (13). Since the same variables are already present in the conventional formulation, there is no extra computational cost for taking the a priori state information into account. It should be also emphasized that, in terms of the weighted leastsquares method, the solution so obtained is theoretically equivalent to solving Eq. (5) by conventional, non-orthogonal techniques. To conclude this section, it should be stressed that even orthogonal techniques mail fail to yield convergence when severe modeling errors are present. Convergence in such extreme cases is still desirable even when it leads to impractical solutions, since from those one may proceed to identify and correct the existing modeling errors which caused the problem in the first place.

3.

Trust Region Theoretical Basis

3.1.

Rationale of Trust Region Approach

The convergence of the Gauss-Newton method can be made more robust by using trust regions [3]. Trust-region methods generate steps based on a quadratic model of the objective Optimization Advances in Electric Power Systems, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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function. A region around the current solution is defined, within which the model is supposed to be an adequate representation of the objective function. Then a step is selected to minimize this approximate model in the trust region. In case a step is not acceptable, the size of the region is reduced and the computation of a new step is carried out. Changing the radius of the trust region affects not only the size, but also the direction of the next step. In order to properly adjust the size of the trust region, a merit function is defined as a scalarvalued function of the optimization variables to indicate whether a new candidate iterate is better or worse than the current available iterate, in the sense of the progress made toward the optimal solution. In the PSSE case, the sum of weighted squared residuals qualifies as a suitable component of the merit function to be used for trust region definition. To illustrate the rationale behind the TR approach, consider a simple problem involving only two state variables. Let J(xk ) and m(xk ) be the values of the summation of the weighted squared residuals for the actual objective function and the corresponding approximated quadratic model, respectively, evaluated at the k-th iteration. In the situation depicted in Fig. 1, the contours of J describe a curved valley, the current point xk being at one end of the valley and the actual optimal solution x∗ lying at the other. A search based on the quadratic local model given by m(xk ), whose elliptical contours are shown in grey in the figure, would lead to a point far away from the actual minimizer, allowing only a small reduction in J. On the other hand, the step calculated by the trust region method is confined to the dotted circle around xk , so that the optimization step generates point xk+1 . This points yields a more significant reduction in J and also remains much closer to the path leading to the actual minimizer. Denoting the gradient of objective function by g(xk ) and the approximation to the Hessian matrix computed at the current point by G(xk ), the general trust-region problem is represented by: 1 M in gt (xk )∆x + ∆xt G(xk )∆x 2 subject to k∆xk ≤ δ

(14)

where the index to be minimized is the local quadratic model of the objective function and δ is the trust region radius. In the Gauss-Newton case, we use the Gauss approximation for the Hessian matrix of the objective function of Eq. (14) [9]: G(xk ) = Ht R−1 H

(15)

g(xk ) = −Ht R−1 ∆z.

(16)

whereas the gradient is given by

As qualitatively described above, the Trust Region method solves the subproblem (14) to obtain a point xk+1 . Then the value of the true objective function is calculated at xk+1 and compared to the value predicted by the quadratic model, in order to verify if the point located in the trust region represents an effective progress towards the optimal solution. According to the magnitude of the progress achieved, the trust region may need to be redimensioned. In the next subsection, a metric is defined to quantify the amount of the progress provided by a candidate step.

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Figure 1. Step based on Trust Region.

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3.2.

Basic Trust Region Algorithm

The size of the trust region is critical to the effectiveness of each step. If the region is too small, larger steps that would lead the current point much closer to the optimal solution can not be taken. If too large, the quadratic model representing the objective function can be inaccurate, such that the reduction of the trust region is needed. In practice, the size of the region is determined according to the evolution of the iterative process. If the model is sufficiently accurate, the size of the trust region is steadily increased to allow bigger steps. Otherwise, the quadratic model is inadequate, so that the size of the trust region must be reduced. In order to establish an algorithm to control the trust region radius, define the reduction ratio evaluated at the k-th iteration (denoted ρk ) as ρk =

J(xk ) − J(xk+1 ) m(xk ) − m(xk+1 )

(17)

Note that ρk can be seen as the ratio between the actual reduction and the predicted reduction of the objective function. Whenever the inequality constraint of Problem (14) is not binding, the solution is given by the familiar normal equation, as it would be expected. On the other hand, when the step becomes constrained by the current value of the trust region radius δ, it can be shown that the solution will be given by   G(xk ) + λI ∆x(λ) = −g(xk )

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with k∆x(λ)k = δ and λ is the (scalar) Lagrange multiplier corresponding to the TR constraint. By using Eqs. (15) and (16), the above equation can be rewritten for the GaussNewton approximation as
Ht R−1 H + λI ∆x = Ht R−1 ∆z (18)

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Equation (18) exhibits the same form as the basic Levenberg-Marquardt method to solve nonlinear least-squares problems [13],[14] (although the original Levenberg-Marquardt (LM) approach actually preceded the trust region concepts and adjusts λ through a different rationale [3],[13],[14]). The basic trust region algorithm can be summarized as follows [3]. ¯ and η ∈ [0, 1 ) : Given: δ¯ > 0, δ 0 ∈ (0, δ), 4 for k = 0, 1, 2, . . . Solve Equation (i.e., Eq. (18) with λ = 0);

thek Normal k

if ∆x > δ : Define ∆x(λ) = −(G + λI)−1 g, Compute: λ > 0 such that k∆x(λ)k = δ; ∆xk = ∆x(λ); end Compute the reduction ratio ρk ; if ρk < 14 :

δ k+1 = 14 ∆xk ; (shrink TR) else

if ρk > 34 and ∆xk = δ k : ¯ (expand TR) δ k+1 = min{2δ k , δ}, else δ k+1 = δ k ; (maintain TR size) end (if ) end if ρk > η : xk+1 = xk + ∆xk , (update solution) else xk+1 = xk ; (stay at same point & try again) end end Note that δ¯ represents the maximum limit of the step length. The radius δ k is increased

only if it indicates that enough progress is being achieved with the local model and ∆xk has reached the limit of the trust region. If the step stays strictly inside the region, one concludes that the current value of δ k is not interferring with the progress of the algorithm, so that its value is left unchanged for the next iteration. The crucial step in the above algorithm is the computation of a λ > 0 which ensures that k∆x(λ)k ≤ δ. An efficient technique to reach this goal is based on the Mor´e and

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A.J. Sim˜oes Costa, R.S. Salgado and P. Haas

Sorensen algorithm [4]. This strategy provides nearly exact solutions at a relatively low computational effort. The theoretical basis of this algorithm is described next.

3.3.

Ensuring Step Feasibility via λ > 0

In order to obtain a feasible step when the constraint of Problem (14) is binding, define  −1 ∆x(λ) = − G(xk ) + λI g(xk )

(19)

A sufficiently large value of λ > 0 is sought such that matrix G(xk ) + λI is positive definite and k∆x(λ)k = δ (20) At first sight, the solution of ∆

φ1 (λ) = k∆x(λ)k − δ = 0,

(21)

could be obtained by Newton’s method. However, it turns out (see Eq. (39) of Appendix A.) that φ1 is a strongly nonlinear function of λ, reducing therefore the efficiency of Newton’s method. Better results can be achieved by solving the equation φ2 (λ) =

1 1 − =0 δ k∆x(λ)k

(22)

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whose solution is clearly the same as that of equation (21). Furthermore, it can be more easily obtained, since φ2 is approximately linear close to solution λ (see Appendix A.). In this case, Newton’method provides φ2 (λ) ≈ φ2 (λk ) + φ′2 (λk ) δλ and thus ∆λ = −

φ2 (λk ) φ′2 (λk )

(23)

such that λ can be updated by λk+1 = λk + ∆λ = λk −

φ2 (λk ) φ′2 (λk )

(24)

If we assume that G3M rotations are used to decompose matrix G(xk ) of Eq. (18), then it can be shown [4] that Eq. (24) becomes λ ∆

1

k+1

k

=λ +

k∆xk2 2

kqk k

×



k∆xk − δ δ



∆

(25)

˜ and q ˜ = U−t ∆x, with D and U as defined in Eqs. (6) and (7). Details where qk = D− 2 q of the formal derivation of expression (25) are given in Appendix A.. Optimization Advances in Electric Power Systems, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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4.

37

Trust Region Methods through Givens Rotations

This section deals with the implementation of TR state estimators through orthogonal techniques based on G3M rotations. Two distinct strategies are considered, namely, the λrefactorization method [5], [6], and the APSI method presented in [7]. Both strategies are described in detail in the following subsections.

4.1.

The λ−Refactorization Method

The λ−refactorization method starts by recognizing that the coefficient matrix of the Levenberg-Marquardt equation, Eq. (18), can be rewritten as t

−1

HR

H + λI =



Ht

I





R λ−1 I

−1 

H I



(26)

or, if we define ˜ , H



H I



,

˜ , R



R λ−1 I



(27)

then

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˜ tR ˜ −1 H ˜ Ht R−1 H + λI = H

(28)

Therefore, the effect of the extra term λI added to the gain matrix is equivalent to augmenting the Jacobian matrix with extra rows (those of the identity matrix), whose weighting factors are all equal to λ−1 . Similarly to what has been previously done for the least squares method (see Eq. (6)), we can also perform an orthogonal factorization on such augmented matrix, that is, " #     − 12 ˜ R H 1 U − ˜ ˜ ˜ ˜ Q R 2H = Q (29) = 1 0 λ2 I ˜ is an orthogonal matrix and U ˜ is an n × n upper triangular matrix. where Q Although the above developments are based the use of the 4-multiplier Givens rotations, the same arguments of Subsection 2.4. can be employed to conclude that the factorization is also implementable through the 3-multiplier version. Eq. (29) suggests that the factorization can be seen as composed of two steps. In the first one, the rows of the original Jacobian matrix (scaled, as usual, by the inverse of the measurements standard deviations) are processed by the G3M algorithm in the same way as already discussed in connection to Eq. (6). The second step consists in applying additional 1 rotations to process the rows of the identity matrix scaled by λ 2 . Furthermore, the above two steps do not need to be simultaneously performed. In fact, it is computationally advantageous to split the factorization into the two above mentioned steps. This is so because, in each major iteration of the state estimator, several minor iterations on λ, as given by Eq. (25), are usually required in order to satisfy the trust region constraint. Since in the course of the minor iterations matrix H remains constant and only λ varies, the first factorization step is performed just once, while the second step is carried out

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as many times as λ is updated. Accordingly, if U is the upper triangular matrix produced ˜ is obtained as by the factorization of H in the first step, then U     ˜ U U = Q1 (30) 1 0 λ2 I Since U is already available, Q1 comprises only the additional Givens rotations necessary to re-triangularize the augmented matrix on the left-hand side of Eq. (30), by processing 1 the entries of the diagonal matrix λ 2 I. The final stage of the λ-refactorization method is the computation of ∆x to be used in Eq. (25) in order to generate a λ which enforces the TR constraint. From the above discussion and Eq. (6), ∆x is obtained by solving ¯ t U∆x(λ) ¯ U = Ut c

4.2.

(31)

The APSI Method

This alternative way to solve the Levenberg-Marquardt equation using Givens rotations relies on the ability of G3M-based state estimators to efficiently process a priori state information. The connection between the L-M equation and the processing of a priori information through least-squares estimators is established in the sequel. The comparison between Eq. (5), that provides estimates in the presence of a priori state information and the Levenberg-Marquardt equation (18) reveals relevant similarities among them. In fact, one can readily see that the latter equation can be obtained from the former by making P−1 = λI (32)

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and ∆x = 0.

(33)

This result indicates that a state estimator capable of efficiently processing a priori data can be used to solve the L-M equation (18). As shown in Subsection 2.4., the G3M estimator exhibits those desired properties. We thus conclude that an orthogonal solution for the L-M equation (18) can be obtained through a G3M state estimator. This is tantamount to saying that the repeated solutions of the L-M equation required by the TR algorithm for candidate λ′ s are obtained by taking advantage of the G3M rotations capability to process a priori information. For that purpose, we make use of Eqs. (13) and (32) to simply initialize the scaling factors for the rows of the triangular matrix U as di = λ,

i = 1, . . . , n

(34)

In addition, the a priori state information must be defined as given by Eq. (33). The execution of the G3M state estimation as outlined in subsection 2.4. under such an inialization scheme provides the desired solution for the L-M equation. As a final note on the APSI strategy, we remark that the characteristics of the method does not allow splitting the coefficient matrix factorization into two stages according to its rows’ dependence on λ, as provided by the λ-refactorization approach.

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4.3.

39

Qualitative Appraisal

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Subsections 4.1. and 4.2. describe two distinct strategies to combine the trust region concept and the orthogonal implementations of power system state estimators. In both cases, the TR-based methodology requires the dynamic definition of the trust region during the iterative process which, at the expense of some additional computational effort, yields significant enhancements to the state estimator convergence properties. The main differences between these strategies refer to the way of applying Givens rotations during the iterative process. In the λ−refactorization method, matrix H does not change during the inner λ−iteration loop, and then the first step given by Eq. (6) is performed only once at each major iteration. The second step, described by Eq. (30), consists of applying the QR factorization to an augmented matrix. This is performed by rotating the rows of a diagonal matrix with the rows of the triangular matrix U computed in the first step. The fact that U is previously available and the diagonal nature of the matrix whose rows are to be rotated significantly reduce the computational effort required by the QR factorization in the second step. 1 On the other hand, the APSI works solely on matrix R− 2 H, so that no augmented matrix is required. Also, the factorization takes full advantage of the G3M rotations properties to process the a priori information. However, the very conception of the APSI approach does not allow the matrix factorization decoupling provided by the λ-refactorization method, so that the rotations must be applied each time from the very beginning, as updated values of λ become available. Consequently, although the processing of a priori information is expected to enhance the numerical robustness of the APSI-method, an increase in computational effort with system size can also be expected, as a result of the need to re-apply all rotations each time to the scaled Jacobian matrix.

5.

Simulation Results

In order to illustrate the application of the TR-based strategies described in the previous sections, computational codes written in Fortran have been developed to implement the following estimators: • The conventional Gauss-Newton estimator solved by Givens rotations, referred to in this Section as GN-orthogonal estimator; • The λ-refactorization trust region-based state estimator described in Section 4.1.; • The APSI trust region-based state estimator presented in Section 4.2.. All estimators are initialized at the flat voltage profile. Aiming at reducing the computational effort required by QR factorization, ordering schemes have been also considered in the implementation of the orthogonal estimators. Numerical results have been obtained by performing state estimation with the above computer programs for four test systems: a 6-bus network and the 30-bus, 118-bus and 300-bus IEEE test systems. The branch and bus data and one-line diagrams of the IEEE

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networks can be found in [15], while data for the 6-bus test system are presented in Appendix B.. Data sets for PSSE purposes were composed of 15, 73, 403 and 911 measurements, respectively, yielding corresponding measurement redundancies of 1.36, 1.23, 1.71 and 1.52. The measurements sets comprise line flow, bus power injection and bus voltage magnitude measurements, and have been generated by adding random errors with normal distribution to power flow results. In order to assess the performance of the TR-based state estimators in terms of the PSSE iterative process convergence and computational effort, significant modelling errors comprising bad analog data, network topology errors, or both, have been inserted into the data sets. In addition, in some cases the power system operating point has been made more stringent by considering heavier loading conditions. It is well known that the combination of severe bad data and system heavy loading produce challenging conditions to the convergence of the PSSE iterative process. The convergence criterion adopted for all cases is based on the norm of the difference between two consecutive estimates, that is, the maximum increment on the state estimates. A tolerance of 1 × 10−3 pu has been adopted. In addition, the evolution of gradient norms and cost function values through the iterations are also used to evaluate the performance of the three state estimators. For the sake of improving visualization, semilogarithmic scales are used in some plots.

5.1.

6-bus System

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Figure 2 shows the 6-bus system and the corresponding measurement set, composed of active and reactive power injections at buses 1, 4, 5 and 6, active and reactive power flows on transmission lines 1-3, 1-6 and 2-5, and voltage magnitude at bus 1.

Figure 2. 6-bus test system and metering scheme. The first test aims at comparing the accuracy of results produced by the proposed TR approach with respect to those obtained with the orthogonal G-N estimator. For this reason, no modelling errors or severe operating conditions are considered. Figures 3 and 4 show the sums of weighted squared residuals and the norms of gradient vectors and state increment vectors provided by both estimators. As one can notice from the plots, all estimators converge in three iterations, tracking similar paths to the optimal solution. Since there are no

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large residuals, the full Gauss-Newton steps are taken at each iteration by both estimators. In the case of the trust region estimators, it can be concluded that the trust region radius is effectively adjusted by the algorithm of Section 3.2. to prevent undue control actions which could have affected the convergence rate. −4

Objective function value

20

x 10

Gauss−Newton Trust Region

15

10

5

0

Gadient Norm

2 0.6

0.4

0.2

0

1

2 Iterations

3

Figure 3. 6 Bus System - gradient vector and objective function: absence of bad data.

0.2 0.18 0.16 Gauss−Newton 0.14

Trust Region

Step norm

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0.12 0.1 0.08 0.06 0.04 0.02 0

1

1.2

1.4

1.6

1.8

2 Iterations

2.2

2.4

2.6

2.8

3

Figure 4. 6 Bus System - vector of increments in the state variables: absence of bad data. The next test consists of introducing two analog bad data affecting both the active and reactive power injection measurements taken at bus 1. In addition, a topology error of the exclusion type is simulated involving branch 1-4 of the network. Figure 5 shows the evolution of the gradient norm and the sum of weighted squared residuals during the iterative process for both estimators, under the influence of the modelling errors. It can be observed that the orthogonal G-N iterations eventually blow up, as a consequence of extremely large residuals. Figure 6 shows that the same occurs with the

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A.J. Sim˜oes Costa, R.S. Salgado and P. Haas Log of objective function value

−3.5

Gauss−Newton Trust Region

−4

−4.5

−5 Log of gradient norm

−2

2

3

4

5

6

7

8

2

3

4

5

6

7 8 Iterations

9

10

11

12

13

9

10

11

12

13

−4 −6 −8 −10 1

14

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Figure 5. 6 Bus System - norm of the gradient vector and value of the objective function: measurement and topology errors. norm of the state increment vector, although the instability manifests itself somewhat earlier than with other indices. On the other hand, the trust region estimators not only prevent the divergence of the iterative process, but also promote an steadily decrease of objective function values throughout the iterations. Recall that the sum of the weighted squared residuals composes the merit figure based on which the trust region is calculated. Thus, if the trust region is properly defined, such an index must decrease continuously. As for the norm of the gradient vector, despite some occasional oscillations it eventually becomes smaller than the specified tolerance. Recall that this condition must be effectively satisfied at convergence. Note also that the plot goes on up to the 14-th iteration only for visualization purposes. The norm of the state increment vector decreases until the convergence criterion is satisfied, as indicated in Figure 6.

5.2.

30-bus System

The test case based on the IEEE 30-bus system considers a severe operating point generated by uniformly increasing the active load at all buses. In addition, a single analog bad data and a topology error are also introduced into the data set available to the state estimator. The combination of such factors produce hard conditions for a state estimator to converge. The tests conducted with this network are mainly aimed at comparing the trajectories generated by both TR-based estimators toward a possible converged solution. The results are presented in Table 1 and indicate that both orthogonal TR-based estimators follow the same path to convergence. This could be expected, since the only difference in the definition of the TR region is the way the QR factorization is applied. Additionally, the algorithm used to compute λ is the same for both methods. In Table 1, we refer to major step as the stage in which the increments in the state variables are computed, and minor steps as those steps necessary to define the TR radius. A zero value attributed to λ in a minor step means that a full Gauss-Newton step is taken,

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0.6

0.5 Gauss−Newton Trust Region Step norm

0.4

0.3

0.2

0.1

0

2

4

6

8 Iterations

10

12

14

Figure 6. 6 Bus System - norm of the vector of increments in the state variables: measurement and topology errors. that is, there is no need to define the TR-radius to restrain the step. The table shows that that is the case in major iterations 1, 7, 10 and 13. On the other hand, a non-zero value for λ indicates that the step has been limited by the TR radius. For instance, at the second major step three minor iterations are needed to determine a λ equal to 7.43 ×103 in order to confine the step within the TR region.

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Table 1. Iterative process: 30-bus system APSI and λ-refactorization methods Main Step Number of Minor Steps λ-value 1 1 0 2 3 7.43 ×103 3 2 4.71 ×103 4 1 1.82 ×103 5 1 7.74 ×102 6 1 2.60 ×102 7 1 0 8 2 7.35 ×103 9 1 3.60 ×103 10 1 0 11 2 6.16 ×102 12 1 3.21 ×102 13 1 0 14 2 6.35 ×102

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The second quantitative aspect considered in the comparison of the orthogonal TRbased methods has been the evolution of objective function values as well as of the increments in the state variables during the iterations. Figure 7 shows the norm of the state increment vector and the sum of weighted squared residuals (here referred to as step norm and objective function value) provided by the three estimators for the 30-bus system. We again observe that, due to the large residuals resulting from the presence of modelling errors, the GN-orthogonal estimator fails to converge. On the other hand, both TR-based estimators take fourteen iterations to converge to the same values, following similar paths to the optimal solution. Again, the convergence can be attributed to the effective control of the trust region radius. The sum of the weighted squared residuals steadily decreases, whereas the step norm, despite some occasional oscillations, becomes eventually smaller than the specified tolerance, thereby satisfying the convergence criterion. 2 1

Gauss−Newton

Step Norm

TR − ASPI TR − λ−refact.

4 3 2 1

2.0

5

Objective function value

x10 1.5

TR − ASPI TR − λ−refact.

1.0

0.5

0

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Gauss−Newton

1

2

3

4

5

6

7 8 Iterations

9

10

11

12

13

14

Figure 7. 30 Bus System - norm of the state increment vector and objective function.

5.3.

118-bus and 300-bus Systems

As in the previous cases, stringent conditions for convergence of a state estimator have been also simulated for the IEEE 118-bus and 300-bus networks. In the former case, this is achieved through the introduction of a topology error affecting an important transmission corridor, whereas the load of the 300-bus system has been increased by an unusually large amount. Figures 8 and 9 show the evolution along the iterations of the major step norm and objective function value for the two test systems. The analysis of those plots conducts to conclusions similar to those of the previous cases. Once more, the orthogonal G-N estimator fails to converge in both cases, as one can infer from the plots of the objective function and norm of the state increment vector. The figures also show that the TR-based estimators are able to overcome the problems caused either by bad data or unusually heavy loading, leading the iterative process to convergence in 6 iterations (118-bus) and 4 iterations (300bus) through the application of the APSI-method, and 9 iterations (118-bus) and 3 iterations

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(300-bus) when the λ-refactorization method is applied. 0.8 Gauss−Newton TR − ASPI

Step Norm

0.6

TR − λ−refact. 0.4

0.2

4.0 5

Objective function value

x10

Gauss−Newton

3.5

TR − ASPI TR − λ−refact.

3.0

2.5

2.0

1

2

3

4

5

6

7

8

Iterations

Figure 8. 118 Bus System - norm of the state increment vector and objective function.

0.5 Gauss−Newton TR − ASPI TR − λ−Refat.

Step Norm

0.4 0.3 0.2 0.1

Objective Function Value

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15

4

x10

Gauss−Newton TR − ASPI TR − λ−refat.

10

5

0

1

2

3

4

5 6 Iterations

7

8

9

10

Figure 9. 300 Bus System - norm of the state increment vector and objective function.

5.4.

Comparison of Numerical Performance

To carry out a preliminary evaluation of the computational effort demanded by the orthogonal TR-based state estimators, the CPU times required to execute both the λ-refactorization and the APSI methods as applied to the three IEEE test systems have been determined and compared. As anticipated by the qualitative analysis presented in Subsection 4.3., the results for the 30-bus system confirms that the APSI strategy exhibits a better performance for networks of small size. However, as both network size and measurement redundancy increase, the λ-refactorization method tends to outperform the APSI approach, as perfor-

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A.J. Sim˜oes Costa, R.S. Salgado and P. Haas

mance indices based on computational efforts obtained for the 118-bus and 300-bus systems clearly indicate.

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6.

Conclusions

Recent research efforts in the Numerical Optimization field have led to the development of the so-called Trust Region algorithms, which exhibit superior convergence properties in the presence of approximate, and sometimes even inaccurate, quadratic models for the objective function. Trust Region methods are applicable to distinct classes of optimization problems, including those formulated according to the least-squares framework, such as Power System State Estimation. The main advantage of applying Trust Region-based methods to PSSE is to enhance the ability of state estimators to provide converged solutions under unfavorable modelling conditions. Such conditions may arise as a consequence of poor measurement redundancy, or the occurrence of bad analog data and/or topology errors. In this chapter, the basic principles of the Trust Region approach and its application to PSSE have been reviewed in detail. Emphasis is given to orthogonal implementations of Trust Regions state estimators. Two distinct strategies have been described, namely, the λ-refactorization method and the a priori state information method. Although both algorithms are implemented through the use of Givens rotations, they are conceptually distinct concerning the way each of them deals with the parameter that enforces the Trust Region constraint. A number of test cases involving four electrical networks, including three IEEE test systems, are employed to compare the performances of the TR-based estimators, both against each other and against the performance of a conventional Gauss-Newton estimator. The first conclusion drawn from that comparative analysis indicates that, in the absence of modelling errors or severe operating conditions, the trust region radius is successfully controlled so as not to interfere with the evolution of the iterative process. As a consequence, the TRbased estimators exhibit the same trajectory towards the solution as the conventional state estimator, which successfully converges for those cases. On the other hand, in the presence of severe modelling errors the effective definition of the trust region size achieved with the use of the TR-based estimators prevents the divergence of the iterative process. The same conclusion is not applicable to the performance of the conventional Gauss-Newton estimator, which fails to converge under the same conditions. Some case studies also show that, although both orthogonal TR-based state estimators have successfully converged to the same solution in all test cases, the trajectories of the corresponding iterations may be somewhat distinct. Such differences may be due to the numerical behavior of the λ-refactorization and APSI approaches, since the orthogonal factorizations are applied in distinct forms according to each method. In terms of computational behavior, the superior performance of the APSI approach for small networks with low redundancy level has not been confirmed when it is applied to large test systems monitored through more redundant metering schemes. In those cases, the λ-refactorization technique provides the same state estimation results at significantly less computational costs.

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Trust Region Optimization Methods via Givens Rotations...

47

Appendix A. Definition of the Trust Region This appendix presents the formal derivation of the Eq. (25) of Section 3.2., used to evaluate the step with respect to the trust region radius. In order to solve equation k∆x(λ)k = δ (35)

firstly recall that, since matrix G(xk ) of equation   G(xk ) + λI ∆x(λ) = −g(xk )

(36)

is symmetric, it can be written as

G(xk ) = TΛTt ,

(37)

where Λ = diag(λ1 , λ2 , . . . , λn ) and T is an orthogonal matrix. Additionally, let λ1 be the smallest eigenvalue of G(xk ) (which has real eigenvalues as a consequence of its symmetry). From Eqs. (36) and (37) , it can be proved that ∆x(λ) is given by [4], [3] ∆x(λ) =

n X ttj g(xk ) j=1

λj + λ

tj

(38)

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where tj is the j-th column of T; and taking into account the orthogonality of the columns of matrix T, k∆x(λ)k2 = ∆x(λ)t ∆x(λ) 2  n ttj g(xk ) X = (λj + λ)2 j=1

(39)

From Eq. (39), it can be shown that, if λ > −λ1 , then there are only one

solution of k k −1

Eq. (35) [3]. Particularly, if G(x ) is positive definite but G(x ) g(ˆ x) > δ, there is a strictly positive value of λ that satisfies Eq. (35). In this case, the value of λ is sought in the interval (0, ∞). If matrix G(xk ) is indefinite, and supposing that ttj g(xk ) 6= 0, the solution is in the interval (−λ1 , ∞).

Computation of φ′2 The determination of the roots of equation φ2 (λ) =

1 1 − =0 δ k∆x(λ)k

(40)

by Newton’s method, requires the calculation of increments ∆λ = −

φ2 (λk ) φ′2 (λk )

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(41)

48

A.J. Sim˜oes Costa, R.S. Salgado and P. Haas

at each iteration, with the first derivative φ′2 with respect to λ given by   d 1 ′ φ2 (λ) = − dλ k∆xk where, for the sake of simplicity, the argument λ of ∆x is omitted. The right hand side of Eq. (42) can be expressed as   − 1 1 d  d 2 2 k∆xk = dλ k∆xk dλ − 3 d 1 2 = − k∆xk2 k∆xk2 2 dλ

(42)

(43)

From Eq. (39),

where ∆

α(λ) =

d k∆xk2 = −2α(λ) dλ

(44)

 2 n ttj g(xk ) X

(45)

j=1

(λj + λ)3

=

∆xt ∆x (λj + λ)

Recalling that the terms (λj + λ) are the diagonal of matrix (Λ + λI), Eq. (45) can be re-written as α(λ) = ∆xt × M × (Λ + λI)−1 × N × ∆x (46)

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where M and N are matrices n × n, and for consistency of Eqs. (45) and (46), M = T and N = Tt . On the other hand, since T(Λ + λI)−1 Tt = (G(xk ) + λI)−1 then α(λ) = ∆xt (G(xk ) + λI)−1 ∆x

(47)

Assuming that matrix (G(xk ) + λI) is available in the factorized form; that is, (G(xk ) + λI) = Ut DU where U is a upper triangular matrix with unit diagonal and D is a diagonal matrix; Eq. (47) can be expressed as  t   α(λ) = ∆xt U−1 D−1 U−t ∆x = U−t ∆x D−1 U−t ∆x and defining

∆

˜ = U−t ∆x q then ˜ = kqk2 ˜ t D−1 q α(λ) = q ∆

1

˜. where q = D− 2 q Optimization Advances in Electric Power Systems, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(48)

Trust Region Optimization Methods via Givens Rotations...

49

Finally, the combination of Eqs. (43), (44) and (48) results   1 d = k∆xk−3 kqk2 dλ k∆x(λ)k and therefore φ′2 (λ) = − k∆xk−3 kqk2

(49)

Computation of ∆λ Defining ∆xk = ∆x(λk ) and qk = q(λk ), from Eqs. (40), (41) and (49), !



∆xk −1

∆xk − δ φ2 (λk ) ∆λ = − ′ k = −3 2 × δ φ2 (λ ) k∆xk k kqk k or simply ∆λ =



∆xk 2 2

kqk k

×

!

∆xk − δ δ

(50)

Thus, the strategy to update λ can be summarized as: λ

k+1

k

=λ +



∆xk 2 2

kqk k

×

!

∆xk − δ δ

(51)

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The steps of a computational algorithm to implement the method of Mor´e and Sorensen [4] are presented as follows. Given:

δ > 0, λ0 ≥ 0, itmax; matrices R and H, vector ∆z.

for k = 1 : itmax - Fatorize (G(xk ) + λk I) = Ut DU; -

Solve (Ut DU) ∆xk = −g(xk );

-

˜ k = ∆xk ; Solve lower triangular system (Uk )t q

-

2 P Calculate ∆xk = ni=1 (∆xk )2 ;

-

2 P  k 2 k  qi ) /dii ; Calculate qk = ni=1 (˜

Update λ :

λk+1 end

=

λk

+



∆xk 2 2

kqk k



∆xk − δ × δ

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B.

A.J. Sim˜oes Costa, R.S. Salgado and P. Haas

Data for the 6-Bus Test System

Tables 2 and 3 show the transmission line data and the results of a converged power flow for the 6-bus test system.

Table 2. Transmission Lines Data - 6 bus system Line

Buses

1 2 3 4 5 6 7 8 9 10 11

1-3 3-6 4-5 3-5 5-6 1-4 1-2 2-4 2-5 1-6 1-5

R (%) 5.00 2.00 20.00 12.00 10.00 5.00 10.00 5.00 0.00 7.00 10.00

X (%) 25.0 10.00 40.00 26.00 30.00 10.00 20.00 20.00 30.00 20.0 30.0

B (%) 6.00 2.00 0.00 5.00 6,00 0.00 0.00 0.00 0.00 5.00 4.00

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Table 3. Power Flow Results - 6 bus system Bus

Type

1 2 3 4 5 6

slack PV PV PQ PQ PQ

V (pu) 1.043 0.955 1.050 0.958 0.935 0,950

δ (0 ) 0.00 -0.03 -6.07 -5.18 -11.16 -10.53

Pg (MW) 340.4 80.2 97.6 -

Qg (Mvar) 110.1 -31.9 122.3 -

Pd (MW) 120.0 190.0 180.0

Qd (Mvar) 16.5 30.0 80.0

References [1] A. Sim˜oes Costa and V. H. Quintana. “An Orthogonal Row Processing Algorithm for Power System Sequential State Estimation”. IEEE Trans. on Power App. and Syst., 100:3791–3800, Aug 1981. [2] N. Vempati, I. Slutsker, and W. F. Tinney. “Enhancements to Givens Rotations for Power System State Estimation ”. IEEE Trans. on Power Systems, 6(2):842–849, May 1991.

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51

[3] J. Nocedal and S. J. Wright. Numerical Optimization. Springer, 1999. [4] J. J. Mor´e and D. C. Sorensen. Computing a Trust Region Step. SIAM J. Sci. Stat. Comput., 4(3):553–572, Sep 1983. [5] S. Pajic and K. A. Clements. Globally Convergent State Estimation via the Trust Region Method. In Proceedings of the IEEE Bologna Power Tech, pages 23–26, Bologna-Italy, Jun 2003. [6] S. Pajic and K. A. Clements. Power System State Estimation via Globally Convergent Methods. IEEE Transactions on Power Systems, 20(4):1683–1689, Nov 2005. [7] A. Sim˜oes Costa, R. S. Salgado, and P. Haas. Globally Convergent State Estimation Based on Givens Rotations. IREP Symposium 2007, Bulk Power System Dynamics and Control VII, Charleston, South Carolina, USA, Jul. 2007. [8] A. Sim˜oes Costa, E. Lourenc¸o, and F. Vieira. Topology Error Identification for Orthogonal Estimators Considering A Priori State Information. In Proceedings of the Power System Computation Conference, volume 1, pages 1–6, Liege-Belgium, Ago 2005. [9] A. Monticelli. “State Estimation in Electric Power Systems: A Generalized Approach”. Kluwer Academic Publishers, 1999. [10] A. Abur and A. G¨ı¿ 21 mez Exp¨ı¿ 12 sito. “Power System State Estimation - Theory and Implementation”. Marcel Dekker, 2004.

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[11] M. W. Gentleman. Least-Squares Computations by Givens Transformations Without Square Roots. Journal of the Inst. Math. Applics., 12:329–336, 1973. [12] A. Sim˜oes Costa and J.P. Gouvˆea. “A Constrained Orthogonal State Estimator for External System Modeling”. International Journal of Electrical Power and Energy Systems, 22(8):555–562, Nov. 2000. [13] D.W. Marquardt. “An Algorithm for Least Squares Estimation of Nonlinear Parameters. SIAM J. Appl. Math., 11(2):431–441, Jun 1963. [14] N. D. Rao and S. C. Tripathy. Power System Static State Estimation by the Levenberg-Marquardt Algorithm. IEEE Transactions on Power Apparatus and Systems, 99(2):695–702, Mar-Apr 1980. [15] University of Washington. Power Systems Test Case Archive. WEB, 2007.

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Chapter 3

THE IMPACT OF DEREGULATION ON MATHEMATICAL MODELS USING OPTIMIZATION TECHNIQUES TO AID SYSTEM PLANNING AND OPERATIONS Narayan S. Rau Life Fellow IEEE, USA

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Abstract In the following, I shall discuss some modeling challenges as affected by the deregulation of the electrical industry in most industrialized countries. I shall focus my attention on models that require optimization principles in one manner or another. The debates about market related issues is very vast and cannot be covered in a short article such as this. Therefore, although market rules and the design of market affect transmission and generation additions, I shall avoid a discussion of market issues, except in a fleeting manner when necessary to make a point about mathematical modeling. Prior to the deregulation of the electric industry, optimization principles, in particular the optimal power flow, were viewed as a researchers play toy. Despite research publications and research supported by the Electric Power Research Institute (EPRI), it was neither employed in actual system dispatch nor did it attract attention of system planners for several years except in a cursory manner. Schweppe’s work on spot pricing, the forerunner to nodal pricing in today’s deregulated markets gathered dust for some time.

The Issue of Reactive Power Before deregulation, all the generators in a system were loaded so that the marginal cost of all generators were the same, and the cost of generation from fully loaded generators were below the cost of marginal generators. After deregulation, pricing withdrawals and paying injections at a value equal to the incremental cost of producing the next unit was introduced via the Locational Marginal Prices (LMP). At each node energy price was set equal to the Locational Marginal Prices thus taking into account the intervening transmission system for energy

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Narayan S. Rau

delivery. In spite of this, real time system dispatch in today’s world is essentially driven by a DC load flow. This means that the reactive vars and node voltages are not taken into consideration. The optimal dc load flow output from the computer is used by the system operator who adjusts the voltage levels by suitable actions such as tap changer positions and var injections. Then the system state as evaluated by the state estimator is used to determine the locational marginal prices. Therefore, the dispatching of reactive vars, and consequently the pricing of vars is not based on any optimization. The market rules as agreed to by the system operator and market participants determine the price of vars. This is hardly a satisfactory state of affairs given that the LMP can change significantly by changing var injections. In my paper [1] examples of this are shown. A proper approach would be to depend on a dispatch based on an ac OPF, which incorporates var optimization as well. Among many possible objectives, one such dispatch objective may be to minimize the Euclidean distance between the nodal LMPs. In [1], examples show how varying var injections at nodes can substantially change LMP. This aspect is not addressed in the present markets and dispatch.

Transmission Rights

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A feature in present day markets is that of granting transmission rights, called by several names such as firm Transmission Rights (FTR) and so on. The amount of transfer capability right given to participants is based on what is called a simultaneous feasibility test. In such tests, bidders nominate the MW transfer required between network points specifying points of injection and withdrawal. When all such bids are stacked to clear an auction1, one uses a DC load flow. Hence there is no direct way to account for and to price reactive power transfer rights. A feasibility test based on an ac load flow to account for vars is a topic for research. Till this day, no one has proposed an alternative to the dc load flow based granting of rights.

Generation and Transmission additions Major areas of concern are those of generation and transmission expansion. The incentives for capital inflow and appropriate additions at the right places in the network depend on the market rules rather than on optimization principles per se. However, the use of optimization techniques will be necessary to determine the appropriate capacities of required additions. I expatiate on these matters below. In the past under the vertically integrated utility world, generation expansion was determined assuming that there was no transmission congestion. The underlying assumption was that the transmission planning organ of a utility would eventually build adequate transfer capacity from a generator sited in a particular location based on availability of fuel, cooling water and other such considerations. Then, an expansion algorithm such as WASP or EPRI’s EGAS would be used to determine capacity additions. In all such algorithms, production cost simulating forced outages of agglomerated generators at one node was calculated and this cost was added to the associated capital cost. For a chosen risk level in each year of the 1

Details of the auction to determine FTRs that can be granted can be seen in the literature. For example, PJM website is one good source to understand the simultaneous feasibility test.

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The Impact of Deregulation on Mathematical Models Using Optimization Techniques… 55 analysis, risk level being the same for all consumers and all nodes, a dynamic program type of optimization was carried out to determine the size and time frame for generator additions. As for transmission additions, as said earlier, adequate transfer capacity was built to transport power from generator additions. Further, more transmission was built than necessary in some instances anticipating future load growth and generator additions. Such a building of excess transmission in order to accommodate future addition resulted in a minimization of life cycle costs because of economies of scale; that is, making the building of a higher voltage transmission possible rather than quantum additions of several lower voltage lines in steps. Now the situation is totally unclear and is rendered more complex because of locational prices at nodes of injection and withdrawal in the grid. The risk is different for different customers at different locations. Given that there is bound to be congestion at least for some hours, let us examine the situation in two illustrative sub regions of the network, one with lower LMP and one with higher LMP. Clearly, the power transfer into the sub-region with higher LMP is at its maximum limit (binding constraint), and the marginal energy in that subregion has to be supplied by a generator located within that sub region itself. This means that unless enough reserve in kept within that sub-region, outage of any generator there will result in interruption. Several questions arise: Since the risk is related to size of generator and their forced outage rates, will the risk in that sub-region be at an acceptable level? This also means that the generator held as reserve within the sub-region is not economical to be dispatched to supply energy. If that is not the case (energy cost from that generator is low but it is held in reserve from operational considerations), the earnings to the generator owner per hour in the reserve market cannot be less than that of supplying energy. When the generators outside the sub-region with a higher LMP suffer an outage, the generator within the high LMP sub-region can export energy out of that high LMP sub-region. The question then is what should be the new paradigm in generation planning? First is the paramount question about proper market signals and future contracts that incentevizes investment on projects to build them to be on line at some future time. I shall avoid a discussion of such matters for one has written volumes, and continue to debate such issues. However, I confine my attention to the challenges required in building appropriate mathematical models to determine “optimum” generation and transmission additions in various sub-regions. In discussing the modeling challenge, we assume that some central authority, such as the system operator, can determine and incentivize the required transmission and generation. The first question is what is to be optimized. Is it the production cost? Certainly, production cost cannot and should not be minimized in all hours. A mathematical model should use a relation between transmission addition and cost, and determine where the curve of such a relation intersects the cost of production at various level of transmission improvements. The IEEE short letter [2] suggests one possible way of computing this. The objective function is augmented by additional terms, which consist of the cost of upgrades of either specified links, or the links that are binding as identified by Lagrange multipliers in real time dispatch. A minimization of this objective, subject to some upper and lower bounds for the upgrades, gives the optimum upgrades vis a vis the benefits due to reduced dispatch cost. In terms of required generation related to risk, one has to agree as to what sort of risk variations between sub-regions can be tolerated. Unfortunately, risk or reliability is not a commodity in the deregulated markets – it is not directly monetized to be transparent to the

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Narayan S. Rau

consumer. Consequently, an objective function related to risk for optimization has to be devised. One possible objective function is that the sum of risks plus the Euclidian distance between risks should be as close to a chosen value as possible – say to a value corresponding to an expectation of one day in ten years. Such models do not yet exist. Alongside with the effort for proper market signals to attract adequate generation and transmission capacity, the development of such models is desirable, if not to mandate construction, but to at least use as a screen to judge transmission and generation adequacy. In [3] one way of identifying required generation in sub-regions of a network is indicated. The underlying principle is that risk in no sub-region should be inferior to a desired value. The central operator through forward auctions or some other mechanism solicits the required generations in sub-regions. However, an important point arises. Addition of generation in any sub-region makes that region’s dependency on the transmission network less. In the ultimate limit, each sub-region becomes autonomous when the generation there balances the demand in that sub-region. On the other side of the coin, the risk in any sub-region can be reduced, not necessarily by adding generation there, but by building additional transfer capability to another sub-region that may have excess capacity. Since generation and transmission are intertwined in reducing risk, should there not be a correspondence between payments for generation additions and transmission enhancements? Unfortunately, such a correspondence is absent in present markets. One possible way of computing such correspondence is shown in [3].

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Relation between Mathematical Models and Markets, and Operating Philosophy In sub-regions that have a large amount of base load (cheaper) generation, more than the subregions demand, the LMP will be low and equal to the bid value of production from such generators. However, in sub-regions those have cheaper base load generation less than the demand of that sub-region, the LMP will be high and equal in value to the marginal cost of supplying energy from other generators in the system. In such a situation, those willing to install generators prefer the latter type of sub-region since it brings higher profits. The analysis of generation expansion conducted over the years in the past has indicated that a proper mix of base load, mid range, and peaking generators are necessary to minimize production cost. Therefore, with LMP pricing, what is the incentive to install peaking generators? Who and how does one decide where and how much peaking generation is required? What should be the payment mechanism in the market rules? Models combining all these issues will be necessary. It is easier said than done to develop such models. One has to agree first what the objectives should be and that there will be interplay between market models and the simulation, or optimization, models. Related to the above issue of proper market signals to attract generation and transmission within and intra sub-regions is the question about curtailment that has to be addressed sooner or later. If there is shortage for whatever reason, be it because market failure, shortsighted policies of deregulation, or natural events such as freezing of coal, what should be the philosophy of curtailment?

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Greenhouse Gases A challenge that will arise in the coming years is that of CO2 emissions. The system dispatch, in addition to minimizing cost has to account for a proxy cost of CO2 emissions. Green resources such as solar and wind generation will, of course be dispatched as and when energy from them is available. Since cleaner dispatchable resources will be located in some subregions gradually, holding non-clean resources as reserves will be necessary. These matters compound the optimal dispatch of the system, particularly those systems where transmission constraints are severe. Additionally, it raises several market and tariff related questions such as how to price withdrawals in sub-regions. Should it be based on LMP when the LMP at nodes is seriously altered by the dispatch of cleaner resources, or should the additional costs (or reduced costs) resulting from their dispatch be spread equally at all nodes, and what is the formula for doing so? While on this matter of CO2 emissions, it is interesting to note that in North America, natural gas is preferable to coal generation. The market for natural gas is deregulated as well; it is complex and interesting. There are nominations in the market for the next day, and for withdrawals and charging storage locations. A detailed discussion of this market is beyond our scope here. However, it is somewhat strange that the two markets – for electricity, and the fuels – are completely decoupled. The generator owner buys fuel, and based on his cost of fuel places a bid to supply electricity. The electrical system operator can be unaware and be insulated from the dynamics of the gas market. If the spot price in the gas market is very high, the generator may sell his gas entitlement to others and forfeit the supply of generation to make higher profit, thus affecting the reliability of the electrical system. From an energy policy perspective, a combined model may be desired, the objective being to enhance societal benefits, however the societal benefit can be defined, such as minimum cost, minimum risk and so on. From a generator’s perspective, models to indicate optimal playing in the two markets in order to maximize profit are the need. It will be interesting to see how researchers and market players combine the two in the future to maximize profit, to reduce green house gases, or to achieve any other evolving objective.

Unit Commitment In some deregulated systems, the central operator does the unit commitment for the next day (next week). Commitment models recognize the minimum up and down times of generators, their ramp rates and so on to decide on the optimal start and stopping of generators in subregions. In my view, this contravenes deregulation philosophy, despite the fact that the commitment programs are interesting exercises in dynamic programming [4]. Why should the central operator be concerned about unit characteristics? The generator owner can factor the characteristics in his bidding strategy. For example, for the hours when a generator has not finished it minimum run time, the bid for energy can be below the expected market clearing price for that sub-region. This assures its dispatch. The same argument applies to centralized dictates of maintenance scheduling. Consequently, the modeling requirements now should shift to the perspective of market players. The question then becomes one of determining optimal bidding strategy as related to maintenance, risk of forced outage and loss of revenues, and decision to start and stop generators.

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Narayan S. Rau

Epilogue In the above paragraphs, I have merely pointed out certain inconsistencies and problems. I apologize for not offering any solution to them. However, I hope that the above provokes ideas and solutions from researchers of the future.

References

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[1] N.S.Rau, “Issues in The Path Toward an RTO and Standard Markets,” IEEE Transactions on Power Systems, Vol 18, No. 2, May 2003, pp 435-443. [2] N.S. Rau, “Transmission Congestion, and Expansion Under Regional Transmission Organizations” IEEE Power Engineering review, Power Engineering Letters, Vol.22, No. 9, September, 2002. [3] N.S.Rau, Zeng. F, “Dynamic Optimization of Monte Carlo Simulation to Assess Locational Capacity, Transmission, and Market Parameters”, IEEE Transactions on Power Systems, vol. 21, pp. 34 - 42, February 2006. [4] A.J.Wood, B.F. Wollenberg, “Power Generation Operation and Control,” John Wiley and Sons, 1996.

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Chapter 4

METAHEURISTIC-BASED OPTIMIZATION METHODS FOR TRANSMISSION EXPANSION PLANNING CONSIDERING UNRELIABILITY COSTS Armando M. Leite da Silva1, Cleber E. Sacramento2, Luiz A. da Fonseca Manso3, Leandro S. Rezende1, Leonidas C. de Resende1 and Warlley S. Sales1

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1

Institute of Electric Systems and Energy, Federal University of Itajubá, UNIFEI, Brazil 2 CEMIG – State Energy Company of Minas Gerais, Belo Horizonte, MG, Brazil 3 Department of Electrical Engineering, Federal University of São João del Rei, UFSJ, MG, Brazil

Abstract Transmission expansion planning (TEP) is a complex optimization task to ensure that the power system will meet the forecasted demand and the reliability criterion, along the planning horizon, while minimizing investment and operational costs. Metaheuristic methods have demonstrated the potential to find good feasible solutions, but not necessarily optimal. These methods can provide high quality solutions, within an acceptable CPU time, even for largescale problems. This chapter presents a comparison among three metaheuristic-based methodologies to solve the TEP problem. The search algorithms for simulation-based optimization are: Evolution Strategies, Tabu Search and Ant Colony. Other heuristics, however, are also used to assist the optimization process. The proposed methodologies include the search for the least cost solution, bearing in mind investments, transmission losses and unreliability or interruption costs. Also, the chronological multi-stage nature of the TEP will be accounted. Case studies on a small test system and on a real sub-transmission network are presented and discussed.

1. Introduction The main objective of the transmission expansion planning is to define where, when and what reinforcements should be placed in the electrical network to ensure an adequate level of

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energy supply to customers, taking into account the load growth and new generator capacities. In a competitive energy market, TEP is a complex optimization task to ensure that the power system will meet the forecasted demand and the reliability criterion, along the planning horizon, while minimizing investment, operation and interruption costs. This practice is the only rational response to conflicting customer and regulatory demands [1], [2]. Traditionally, TEP problems have been solved by using deterministic criteria such as the so-called “N-1” and “N-2”. These criteria are applied to a framework based on worst-case analyses (drawn from single or double contingencies), and on comparisons of candidate transmission expansion plans. From the set of plans that comply with the performance criteria, the planner chooses the one that shows the smallest cost present value. In general, the deterministic approach provides a reference transmission expansion plan. In many cases, however, the obtained reference plan may lead to over-investments and, at the same time, may not guarantee adequate reliability levels for all system areas or buses. Owing to today’s power network dimensions, random behavior of transmission and generation equipment, load growth uncertainties, new generator source types and locations, discrete nature of decisions, market aspects, etc., the TEP problem has become combinatorial, stochastic and highly complex. Even considering only deterministic aspects, it is very difficulty to find the optimal solution for TEP problems, since it requires the use of combinatorial algorithms, which have a great difficulty to solve medium size problems. If probabilistic and chronological aspects are added to these problems, the optimal solution becomes almost inaccessible. The chronological aspect refers to the dynamic nature of the TEP problem. It requires the consideration of multi-time periods, determining possible sequences (i.e. stage-by-stage) of transmission reinforcements. To circumvent the dynamic nature, simplified studies (also known as static analyses) determine, for just one stage, where new transmission facilities should be installed. Even so, the search for the optimal solution is still combinatorial. Several works to solve TEP problems can be found in the literature [2]-[20]. Some of them [2], [15][20] have emphasized the importance of unreliability costs in the planning process. Metaheuristic methods [21]-[30] have demonstrated the potential of finding good feasible solutions, but not necessarily optimal. The success of such methods is related to their ability of avoiding local minima by exploring the basic structure of each problem. These methods can provide high quality solutions, within an acceptable CPU time, even for large-scale problems. Several metaheuristic tools have evolved in the last decade to solve TEP problems, e.g.: Simulated Annealing (SA) [7]; Tabu Search (TS) [8]-[10]; Genetic Algorithms (GA) [11], [13]; Greedy Randomized Adaptive Search Procedure (GRASP) [12], [14]; Evolution Strategies (ES) [19]. Another possibility to solve TEP problems is the Ant Colony Optimization (ACO) [26]-[28]. This metaheuristic is based on the ants’ behavior to find the shortest paths from food sources to the nest, based on indirect communication mediated by pheromones. The same principle can be used to search for the optimum solution of complex problems including power systems [29], [30]. This chapter presents a comparison among three metaheuristic-based methodologies to solve the TEP problem. The search methods for population-based optimization are: ES, TS and ACO. Other heuristics such as the one used in GRASP [12], for the construction of initial solutions, are also used to assist the optimization process. The TEP problem includes the search for the least cost solution, bearing in mind investments, transmission losses and unreliability, representing by the index LOLC, i.e. Loss of Load Cost, [15]-[20], [31]). Also,

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Metaheuristic-Based Optimization Methods for Transmission Expansion Planning… 61 the chronological multi-stage nature of the TEP will be accounted. Case studies on a small test system and on a Brazilian sub-transmission network are presented and discussed.

2. Heuristic-Based Methodologies Several metaheuristic-based methodologies have evolved in the past decades that assist the assessment of optimization problem solutions that were previously difficult or even impossible to decipher. Recently, these new heuristics tools have been combined among themselves and with more classical techniques to solve extremely challenging problems. This section provides the basic concepts of three metaheuristic-based methodologies, i.e. ES, TS and ACO, to be used to solve the TEP problem.

2.1. Evolution Strategies Evolutionary Computation (EC) methods share the principle of being computer-based approximate representations of natural evolution. These algorithms alter the population solutions over sequence of generations according to statistical analogues of the processes of evolution. Although there are currently many varieties of EC, there have historically been three general approaches that fall under the umbrella EC heading: GA, ES, and evolutionary programming. The three approaches differ in the types of generation-to-generation changes and the form of computer depiction for the population elements. However, an iterative blending has occurred such that all classes of EC algorithms now appear quite similar; each area within EC has borrowed and modified ideas from the others.

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2.1.1. Basic Concepts GA concepts have been more popular among power system researchers and engineers [21]. The main differences between ES and GA are in the presentation of population and the types of evolution operators. ES algorithms, instead of binary strings, use real value to present parameters of optimization. Also, contrary to GA that incorporates both crossover and mutation, ES based methods just use mutation. Regarding these differences, it seems that ES are easier to implement and might be faster than GA. ES search for the optimal solution through the evolution of a population (set of possible solutions). At each generation, or iteration, the individuals are evaluated by a fitness function, and the best will be selected to be the parents of the next generation. A new population is then generated by mutating these parents. The new individuals are, in turn, evaluated and, once again, the best are selected for the next generation. This procedure is repeated, generation after generation, until the convergence criterion is achieved. Since in each generation the best individuals are selected and generate offspring, it is expected that future generations are enhanced by good performance individuals. Therefore, in satisfying the convergence criterion, the best individual is taken to be the solution for the problem. Usually, selection and mutation mechanisms are used to simulate the evolution process. These two mechanisms are described as follows.

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Selection – The various versions of ES derive from the model (μ,κ,λ,ρ)-ES [23], whose parameters have the following meanings: μ is the number of parents in a generation; κ is the number of generations that an individual survives; λ is the number of offspring produced in a generation and ρ is the number of parents of an individual. In this chapter, the (μ + λ)-ES model [19], [23] is used. Therefore, the parameters κ and ρ are considered equal to 1. Thus, the new μ parents are selected from the set μ + λ. This mechanism of selection ensures that the best individuals of the next generation are never worse than their parents. Mutation – Mutation allows for diversity in the populations, thus ensuring that different regions, within the search domain, are explored. In ES, mutation consists of adding to each component (gene) of the individual a normally distributed perturbation. Consider that vector xold, with dimension (n×1), represents an individual at certain generation, composed of n real variables or parameters of a specific optimization problem. Also, consider that vector xnew represents a new individual, who has undergone mutation due to a perturbation Δx with dimension (n×1). This can be described as follows:

x new = x old + Δx

(1)

Δx = σ × [N1 (0,1) ,..., N l (0,1) ,..., N n (0,1)]T

(2)

where parameter σ is the mutation magnitude or mutational step; Nl (0,1) corresponds to a Gaussian distribution with zero mean and unit variance. In case each gene has its own mutational step σl, the perturbation vector Δx can be generalized to an n-dimensional normal random vector with zero mean and diagonal covariance matrix σ2. The way of controlling the mutation magnitude σ, during the evolutionary process defines two ES categories: nonadaptive and self-adaptive [24]. In the non-adaptive ES, σ is kept constant during the whole iterative process and only the decision variables of x are subject to mutation and selection. On the other hand, in the selfadaptive ES, the parameter σ is also subject to mutation and selection. The basic idea is that each individual is governed by parameters, including the mutational steps, which are also subjected to evolution. If an individual is selected for the next generation, it takes along its strategic parameters. Thus, the algorithm is capable of learning which the best mutation magnitudes to be used during evolution are.

2.1.2. Application to TEP Problems Different from most works that performed a static and deterministic planning, the present chapter applies all heuristics (i.e. ES, TS or ACO) to the TEP problem considering the chronology of the investments, the reliability worth and transmission ohmic losses. The representation of individuals will be such that each gene (i.e. a transmission branch) will correspond to one of the reinforcement options and, besides that, it will be filled in by an integer number, varying from zero until the maximum number of reinforcements per gene, which is denoted by ne. In this work, the parameter σ will be associated with the individuals.

[

Thus, individuals will have the following aspect, x = x1 , x2 , ..., xl , ..., xn ,σ y

y

y

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y

y

]

T

where

Metaheuristic-Based Optimization Methods for Transmission Expansion Planning… 63

xly represents the number of circuits added to branch l, in the year y, and n the number of genes, i.e. the number of possible and distinct circuit connections between network buses. y

Due to the discrete nature of this problem, each gene xl is represented by integer values (i.e. 0, 1, …, ne) and, therefore, after mutating, it will be discretized by using a rounding function [19]. The ES algorithm should be interrupted as soon as the maximum number of generations (iterations) is achieved. Alternatively, the process can be interrupted when the best solution found is unchanged for a pre-specified number of consecutive generations.

2.2. Tabu Search Tabu Search metaheuristic [25] is an adaptive process used to solve combinatorial optimization problems to modify the behavior of local search heuristics, so the space solution exploration is not interrupted or disturbed when there are no movements that improve the current solution. Through the knowledge acquired during the exploration of the solution space, the method avoids that recently visited local minima have an attractive effect to the search trajectory, ensuring a more intelligent exploration when compared to traditional local search methods.

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2.2.1. Basic Concepts A TS algorithm is based on concepts of neighborhood, movement, tabu list, aspiration criterion, intensification and diversification [25]. They are explained next: Neighborhood and Movement – The neighborhood of a solution s, initial or current, is defined as the set of solutions obtained through elementary modifications in s, called movements. At each iteration, a TS algorithm finds the neighborhood of solution s and then identifies its best neighbor, turning it into the current solution. This procedure is employed even in the case of worsening the solution, which is interesting to avoid the premature imprisonment of the search process in valleys (i.e. local optima). Tabu List and Aspiration Criterion – The strategy to perform a movement from the current solution to its better neighbor is not sufficient to escape from earlier capturing by local optima. Thus, a tabu list is used, which is made up of rules called tabu restrictions. These rules avoid that reverse movements (i.e. opposite the movements previously performed) be made, preventing, thus, the repetition of cycles. The size of the tabu list, usually called tabu time, is a parameter that defines how many iterations a reverse movement will stay in the list. It should be carefully dimensioned for each type of problem. Therefore, the tabu list is of the FIFO type (first in first out), that is, the first element (movement) to come into the list is the first element to leave it. A tabu restriction may be ignored when one identifies that the respective movement will lead to new attractive solutions (for example, when the visited solution is the best one found until that point). This is determined by the aspiration criterion that allows that a solution prohibited by the tabu list be exceptionally visited. Intensification and Diversification – The search for solutions in a given region is called intensification, for the most promising this region is, the more intense the search becomes. On the other hand, the more intense the search process becomes, the greater the probability that it

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will get “stuck” in a promising region, obtaining only local optima. In order to avoid this imprisonment in local minima or maxima, the diversification procedure should be employed, through which the search process is resumed many times, from new initial solutions. Diversification allows that the algorithm make a broader search, visiting regions not yet explored and with a greater probability of involving the whole universe of possible solutions for the problem.

2.2.2. Application to TEP Problems In [8], Tabu Search is applied for the first time in the transmission system expansion planning. The proposed method, intended for the single stage (static) problem, uses the DC power flow model and formulation based on the integer programming 0-1, having as an objective the minimization of line overloads. The process starts with the addition of all transmission reinforcements, producing a highly connected, redundant and uneconomic meshed network. Then, TS is applied in order to obtain, progressively, best expansion schemes until the maximum number of iterations is achieved. Afterwards, other studies applied Tabu Search in a more elaborate form, either employing new steps/heuristics to the search process [9], [10], or performing parallelization of the TS algorithm [9]. As previously stated, in a transmission or sub-transmission system planning, one wishes to find a set of reinforcements along the planning horizon. For a certain year or period y of the

[

horizon, a representation of solution x = x1 , x2 , ..., xl , ..., xn y

y

y

y

y

]

T

is a vector containing y

elements, i.e. branches of the system that may receive reinforcements, where: xl represents

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the number of circuits added to branch l in the year y, and n is the number of branches that may receive reinforcements. This is the same vector used in the previously described ES method, excluding the parameter σ , i.e. the mutation magnitude.

2.3. Ant Colony Optimization The ACO metaheuristic was firstly proposed to solve combinatorial optimization problems such as the traveling salesman [26]-[27], and afterward, several other applications have been carried out, particularly those involving discrete optimization problems [28]. Since ACO is a relatively new optimization technique, particularly in power systems, its basic concepts bearing in mind the TEP problem are described as follows in more details as compared to the previous heuristics.

2.3.1. Basics Concepts The ACO is based on the behavior of real ants, which can manage to establish shortest route paths from their colonies to feeding sources and back. It was found that the media used to communicate among individuals information regarding paths, and used to decide where to go, consists of pheromone trails. A moving ant lays some pheromone on the ground, thus marking the path it follows by a trail of this substance. While an isolated ant moves essentially at random, an ant encountering a previously laid trail can detect it, and decide with high probability to follow it, thus reinforcing the trail with its own pheromone. The collective

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Metaheuristic-Based Optimization Methods for Transmission Expansion Planning… 65 behavior that emerges is a form of autocatalytic behavior, where the more the ants following a trail, the more attractive the trail becomes for being followed [26]-[28]. This process causes the quantity of pheromone on the shorter path to grow faster compared to the long ones, and, thus, the probability that any single ant chooses the path to follow is quickly biased towards the shorter one. The final result is that, very quickly, all ants will choose the shorter path. Bearing in mind the TEP problem, the following analogies can be considered. Firstly, the paths to be chosen represent the possible circuit reinforcements among the existing buses of the transmission power network being planned. Although real ants have no memory associated with the followed paths, this information becomes essential in ACO-based algorithms, since it allows building a set of solutions that satisfy all problem constraints. For instance, in TEP problems, one of these constraints is the maximum number of transmission circuits that can be added to each pair of system buses. Moreover, each artificial ant represents a trial to find a feasible solution, whose search orientation is provided by heuristic information and pheromone trails laid on those transmission circuits to be reinforced. The very first algorithms that were developed, known as Ant System – AS, have presented good quality results only for low dimension problems. They were not able to find better solutions for large combinatorial problems, as compared to other metaheuristics, although they showed fast convergence to good solutions [28]. This hurdle has motivated further research in order to develop a new algorithm able to overcome these limitations. This new algorithm, known as Ant Colony System – ACS [27], has presented better performance for solving computationally intensive combinatorial problems, and that is the reason it will be used for dealing with TEP problems. The main simulation mechanisms of the ACS algorithm, already adapted for the TEP problem, are presented as follows: State Transition Rule – A new state configuration “c” is characterized whenever a circuit reinforcement is added to the network being analyzed for a specific year, within the planning horizon. A searching mission is completed when the corresponding artificial ant provides a set of circuit reinforcements, which defines a feasible solution for the transmission network being planned. A feasible solution is encountered as the generating power accesses the demanded load at all buses, without violating any constraints such as the transmission capacities. During the kth mission, the artificial ant moves from one reinforcement to another by means of transitions. The transition rules are defined through probabilities involving heuristic information and pheromone trails laid on possible circuit reinforcements. These probabilities must point out the most interesting regions of the state-space solution, and, at the same time, be able to avoid fast convergence to local minima. The following equation shows this state transition mechanism known as proportional pseudo-random rule [27]:

{

}

⎧arg max [ τ (i, j ) ][ η (i, j ) ]β , if q ≤ q0 ⎪ c = ⎨ (i , j ) ∈Tk ⎪⎩ C, otherwise

(3)

where τ(i,j) corresponds to the pheromone trail laid on the circuit between buses i and j; η(i,j) represents the value of the heuristic function associated with circuit (i,j); β defines the importance of the heuristic function in relation to the trail left on the corresponding circuit; arg max is a function that selects the maximum value for the result [τ(i,j)][η(i,j)]β ; Tk is the set of transmission circuits not yet selected by the kth artificial ant; q represents a random

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number uniformly distributed in [0,1]; q0 is an adjustable parameter (0≤ q0 ≤1), which indicates the degree of intensity that the search will concentrate on the best reinforcements suggested by both the heuristic function and pheromone trails; C corresponds to a random variable that follows a discrete distribution given by (4), where pk(i,j) is the probability that the kth ant chooses circuit (i,j):

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⎧ [ τ ( i, j ) ][ η( i, j ) ] β ⎪⎪ β pk ( i , j ) = ⎨ ∑ { [ τ ( t ,u ) ] [ η( t ,u ) ] ( t , u )∈Tk ⎪ ⎪⎩ 0 , otherwise.

}

, if (i, j)∈Tk (4)

The proportional pseudo-random rule has two purposes: if q ≤ q0, the ACS takes advantage of the accumulated knowledge about the TEP problem, to select the best circuits. Conversely, the circuit to be added is obtained by further exploring the state-space according to (4). In any case, the state transition rule has to favor circuit reinforcements that bring the best benefit to the network at the lowest cost. Pheromone Updating Rules – These rules play a very important role on the learning process of the ACS algorithm. The numerical values associated with the pheromone trails, laid on the possible circuit reinforcements, indicate how much the search will be intensified in the direction of the previously established solutions. The ACS algorithm has two ways to update the pheromones [27]: local updating rule and global updating rule. The local pheromone trail updating is carried out while building up a solution; i.e. during each ant’s tour or mission. Local updating is intended to avoid making those relevant reinforcements being chosen by all the ants during the same expedition (i.e. a predetermined number of missions). Every time a transmission circuit is chosen by an ant, its amount of pheromone is changed by applying the local updating rule, described by the following equation:

τ ( i , j ) = ( 1 - ϕ ) τ (i, j) + ϕ τ 0

(5)

where ϕ is the local pheromone trail reduction rate, and τ0 is a pheromone trail defined for all possible circuit reinforcements at the beginning of the simulation process. From the optimization point of view, the main idea of (5) is to allow the ants to explore a wider range of circuit reinforcements, avoiding local minima, and, therefore, preventing early convergence. Real ants do not have this ability. Therefore, each ant’s tour is likely to be different from the previous one. The global pheromone trail updating is carried out after finishing an expedition. There are two major strategies to be used considering this trail updating [28]: iteration-best or bestso-far. The best-so-far strategy uses the best solution found among all missions of all expeditions analyzed so far. This strategy seems to give far better results for many applications including the salesman problem. Bearing in mind the TEP problem, however, the iteration-best strategy will be applied to update the pheromones between two successive expeditions. The iteration-best uses the best solution found after finishing each expedition. The way that this strategy is adapted in this work has consistently avoided local minima.

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Metaheuristic-Based Optimization Methods for Transmission Expansion Planning… 67 Every time an expedition is concluded, the pheromone amounts associated with the circuit reinforcements are changed by applying the global updating rule, described by the following equations:

⎧⎛ K pher ⎞ ⎟ n , if ( i , j )∈Solp∗ ⎪⎜ Δτ ( i , j ) = ⎨⎜⎝ V ∗ ⎟⎠ circ ⎪ ⎩ 0, otherwise

(6)

where Kpher is a parameter or factor [29] for adjusting the expedition pheromone trails, which must be greater than any cost achieved during the first expedition; V* represents the best value (i.e. the lowest cost) associated with the best solution Solp* found in the previous expedition; ncirc corresponds to the number of circuits added to the branch (i,j); and Δτ(i,j) is the trail laid on circuit branch (i,j) provided by Solp*. Thus,

⎧⎪if ( i, j ) ∈Solp* →τ ( i, j )new = (1 − ρ )τ ( i, j )old + ρ Δτ ( i, j ) ⎨ ⎪⎩if ( i, j ) ∉Solp* →τ ( i, j )new = (1 − ρ )τ ( i, j )old + ρ τ 0

(7)

where ρ represents the learning rate of the algorithm in relation to the pheromone trails, and it is motivated by the trail evaporation in real ants’ world; τ(i,j)old corresponds to pheromone trails achieved by the last global updating; τ(i,j)new is the new pheromone trail that will be used in the next expedition. Equation (5) is slightly different from that one used by the traditional ACS algorithm [27], since it always tries to keep the memory from the updating of expeditions.

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2.3.2. Application to TEP Problems In the power system literature, some interesting applications of ACO can already be found in different areas, e.g.: distribution planning [29] and generation expansion [30]. However, no ACO based methodology can be found in the technical literature to solve the TEP problem. For a certain year y of the planning horizon, a representation of solution

[

x y = x1y , x2y , ..., xly , ..., xny

]

T

is a vector containing elements, i.e. branches of the system y

that may receive reinforcements, where: xl

represents the number of circuits added to

branch l in the year k, and n is the number of branches that may receive reinforcements. This is the same vector used with the previously shown TS method. Although the previous ACS concepts will be used for developing the TEP solution, for the sake of simplification the name ACO (Ant Colony Optimization) will be employed in the next sections of this chapter.

3. Proposed Methodologies The proposed methodology to solve TEP problems follows some basic principles. Firstly, the algorithms (ES, TS or ACO) are applied to find the nb best solutions associated with the last

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year (NY) of the planning horizon. In order to improve the performance of these algorithms, a special initialization process is obtained through some concepts used by the GRASP [12] heuristic, which will be detailed in the next section. Once the best nb solutions associated with the last year are obtained, the nb best solutions should be found for the load levels of the preceding years, according to what will be described in Subsection 3.3. The basic idea is to coordinate the set of solutions found for each preceding year with the set of reinforcements obtained for the last year. Finally, the interruption costs associated with the nb sequences are evaluated and considered together with the investment and transmission losses costs.

3.1. Initialization Process

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In general, the convergence of metaheuristic-based methodologies depends mainly on the initialization process. In order to improve the performance of the ES algorithm, it is important to use a heuristic to generate good initial solutions. For that reason, this work makes use of the constructive phase of the GRASP algorithm proposed in [12], [14]. Thus half of the initial population is generated randomly, and the other half is generated based on a greedy function obtained from the linear optimization involving the equations of the TEP problem. In the TS technique, it is necessary to employ an initial solution for each step of the diversification. Similarly, with the objective of improving the performance of the TS-based algorithm, it is essential the use of heuristics to build good initial solutions. For such purpose, the same greedy function previously mentioned will be used. Finally, the previously described ACO concepts must be applied to find the nb best solutions associated with the last year of the planning horizon. In order to assist the search of these solutions, the ACO uses the same heuristic information η(i,j) that is based on a greedy function [12], [14]. The initialization process (ES and TS) and the heuristic information (ACO) can be obtained from the following linear optimization, involving the equations of the TEP problem: Minimize

z = αTr subject to:

(8)

g + r + Bθ = d

(8.1)

f ≤ f max

(8.2)

0 ≤ g ≤ g max

(8.3)

0≤r ≤d

(8.4)

where, α represents the load shedding penalty vector; g is the generation bus vector; r is the load not-supplied vector; B represents the susceptance matrix; θ is the voltage angle vector; d is load vector; gmax is the generation limit bus vector; f is the power flow vector; and, fmax represents the power flow limit vector.

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Metaheuristic-Based Optimization Methods for Transmission Expansion Planning… 69 This linear programming problem can be efficiently solved by a dual simplex code and the Lagrange multipliers associated with each constraint are obtained as by-products of the solution. Among these multipliers, those associated with the constraints of (8.1) are those of interest, since they measure the benefit in terms of minimum load not-supplied index concerning changes on the circuit susceptances. Denoting πd this Lagrange multiplier vector, the benefits of susceptance changes can be estimated by [12], [14]:

π ijd = ( π id − π id )( θ i − θ j ) where

(9)

π ijd is the Lagrange multiplier associated with the circuit susceptance connecting buses

i and j. In order to take into account the costs associated with the new circuit reinforcements, the greedy or heuristic function is set as:

η( i , j ) =

π ijd ICij

(10)

where ICij is the investment cost required to add a new circuit between buses i and j. This function takes into account both financial and technical aspects. The financial aspect is represented by the investment cost associated with the circuit reinforcement (ICij). The technical aspect is represented by the angular (θij) and by the Lagrange multiplier ( π ij ) d

differences. Tests were previously carried out on the heuristic information either by

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considering only the financial aspect (i.e.

π ijd = 1) or by considering only the technical aspect

(i.e. ICij = 1). It could be concluded that the proposed ES, TS and ACO algorithms become more robust (high probability of finding very good solutions with less computational effort) when both aspects were simultaneously considered. The greedy function defined by (10) is ready to be used by the ACO algorithm (see Subsection 3.5.2) as being the heuristic information η(i,j) needed in (3) and (4). However, the ES and TS algorithms need, as previously mentioned, a set of initial solutions. From the greedy function (10), it is possible to build a candidate list of reinforcements formed from the most promising variables ranked by this function. In Subsection 3.5.1, a five step algorithm provides a good set of feasible solutions to be used as the initial population for both proposed ES and TS methodologies: Observe that the greedy function must be related with the fitness function to be used by the ES, TS and ACO algorithms. In fact, Eq. (8) is rewritten to include losses and investment costs as it will be shown in the Subsection 3.3.

3.2. Transmission Loss Costs In order to include the costs associated with the transmission losses in the optimization process, a special DC flow model is used. Moreover, a slack is defined for all transmission circuit capacities, and it should be sufficient to accommodate the corresponding amount of losses. This amount associated with the circuit between buses i and j (new and existing ones)

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can be approximated as Pij = ( rij × f ij2 ) , where rij is the resistance of the circuit and fij is the DC flow, and all quantities are in pu. The total transmission loss cost (Closs) is given by:

Closs = 8736 × C kWh × LF × ∑ Pij

(11)

∀ij

where CkWh represents the loss unit cost in US$/kWh and the LF is the loss factor, which modulates the load curve. k

Considering the transmission losses, the expansion cost S y associated with the kth sequence of reinforcements in the year “y” is given by:

⎡ n ⎤ k S yk = ⎢∑ Cinv y ,l × M yk ,l ⎥ + Closs y ⎣ l =1 ⎦

(12)

where Cinv y,l represents the investment cost of the new transmission branch “l” in the year “y”; M yk ,l is the number of units added to the branch “l” in the year “y” for the kth sequence; n is the number of branches that can receive new reinforcements; and Closs ky is the cost of losses in the year “y” for the kth sequence. The use of the value 8736 in Eq. (11) aims at transforming the incremental loss costs into annualized costs. Therefore, both contributions to the total cost are obtained in the same yearly basis, and Eq. (12) is consistently formulated.

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3.3. Chronological Aspects For a given horizon with a certain number of years, NY, the TEP problem can be defined as:

⎧⎪ S yk ⎫⎪ Min S k = Min ⎨ ∑ y ⎬ k k ⎪⎩ y =1,NY ( 1 + t ) ⎪⎭

{ }

(13)

where Sk represents the present value of the total cost (i.e. investments and losses) associated k

with the kth sequence of reinforcements made over the period of analysis. Each S y represents the total cost due to transmission reinforcements and losses of year “y” considering the kth sequence, according to (12) and t is the discount rate. Using the ES, TS or ACO algorithms, one can assess the nb best solutions conditioned to the load level of the last year of the horizon, i.e. L(NY). For each solution, a certain set of reinforcements are found and they are made available for finding the best solution for the preceding year (i.e. NY -1). Therefore, one can find the best solution conditioned to the set of reinforcements found with one of the solutions among nb and conditioned to the load level of this year, i.e. L(NY -1). This process continues for years NY -2, NY -3,…1, and also for all nb best solutions found for the last year. At the end of this process, there will be nb sequences of

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Metaheuristic-Based Optimization Methods for Transmission Expansion Planning… 71 coordinated solutions, and the one whose total investment costs and losses are minimal is chosen as the best option, yet without considering unreliability costs. In the previous description of the methodology, the last year was chosen as being the most important one, since all sequences are generated from it. However, it is possible to choose any year as the most important, for instance, due to some previous knowledge about the system under study. Moreover, the subsequent years to be coordinated in terms of reinforcements can be defined by the planner, creating a much more flexible searching process. In order to search for the best solution for a particular year using the ES, TS or ACO algorithms, Eq. (8) is slightly changed to: Minimize:

z = δ T x + α T r + Closs

(14)

where δ represents the investment unit cost vector and x is the reinforcement decision vector whose components are the genes of the ES process (i.e. the elements/branches of the TS or ACO process), for a particular year or load level. As previously stated, this fitness function is related with the greedy function, and is subject to the same constrains: Eqs. (8.1) to (8.4).

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3.4. Unreliability Costs Usually, if a specific horizon is defined (i.e. NY), the overall optimization process tries to find a good (or even optimal) solution S*, which has very short margins of capacity at the end of the period of analysis. As the expansion planning goes on through the subsequent years, i.e. NY +1, NY +2, …, the solution found tends not to be a good option as it appeared. In order to solve the previous problem, the capacity margins, associated with every good solution found at each stage of the optimization, have to be measured and included into this process. This can be achieved through the evaluation of the LOLC indices [15]-[20], [31]. Therefore, the final optimum solution S* will penalize the lack of reserve margins, so that the effects of the last year will be minimized.

3.5. Proposed Algorithms The following steps summarize the proposed dynamic algorithms, for the ES and TS (Subsection 3.5.1) and ACO (Subsection 3.5.2), considering a given priority order (e.g. NY→ NY-1 → NY-2 → … → 1, or NY-2 → NY→ NY-1 → NY-3 → NY-4 → … → 1):

3.5.1. ES and TS Algorithms (i).

Generate the initial population for the first year of the priority list (e.g. the last year of the horizon, i.e. NY) based on the five step algorithm below: a. For a given load forecasted, run an optimal DC linear power flow with losses, considering the transmission network without any reinforcement (i.e. original system);

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(ii). (iii).

(iv).

(v). (vi). (vii).

b. Evaluate the greedy function, i.e. Eq. (10), and rank the parameters η(i,j); c. Sample, through a Uniform distribution, among the n-best ranked circuits. Choose only one reinforcement to be added to the network and run a new optimal DC flow with losses; d. If the new solution is feasible, then go to step (e), otherwise, go back to step (b); e. If the pre-specified size of the initial population is reached, then the ES (or TS) algorithm is ready to start the search, otherwise, go back to (a). Find all nb best solutions for the first year of the priority list using an ES (or TS) algorithm based on Eq. (14); Define the set of possible reinforcements for the next year to be analyzed, according to the given priority order. A coordination scheme for the reinforcements has to be carried out. For instance, an added circuit can not be removed if the analysis goes in the direction of the last year of the horizon. If the analysis goes in the direction of the zero year, those added reinforcements already established by previous ES (or TS) solutions must be duly considered; Based on the ES (or TS) algorithm used in step (ii), find the nb best solutions for the next year. For this optimization process, as in step (i), the initial population must be reevaluated; Repeat steps (iii) and (iv) until all years of the planning horizon have been analyzed; Calculate the interruption costs for all analyzed years; Calculate the present value of all costs (investment, losses and interruption) using a pre-specified discount rate. Thus, find the optimal solution S*, using Eq. (13), but including the unreliability costs.

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3.5.2. ACO Algorithm (i).

Find all nb best solutions for the first year of the priority list using the following proposed ACO algorithm: a. For a given load forecasted (e.g. for the last year NY), run an optimal DC load flow with losses, considering the transmission network without any reinforcement (i.e. original system). If a feasible solution is found (i.e. loss of load = 0), go to step (d), otherwise go to step (b); b. Evaluate the heuristic function, according to (10), apply the proportional pseudo-random rule, according to (3) and (4), and select the circuit to be added in the system; c. Apply the local updating rule, according to (5), considering the corresponding branch of the added circuit, and run a new optimal DC flow with losses. If the new solution is feasible, go to step (d), otherwise, go back to step (b); d. Store the solution and interrupt the ant’s mission or tour. If the preestablished number of missions or ants (na) within an expedition is

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Metaheuristic-Based Optimization Methods for Transmission Expansion Planning… 73

reached, go to step (e). Otherwise, another ant will start a new mission, then, go back to step (a); e. Choose the best solution based on (14) found among all missions of the last expedition, and apply the global updating rule, according to (6) and (7). If the pre-determined maximum number of expeditions (nmax) is reached, or if the best solution so far has not been improved after a predefined number of consecutive expeditions (ncon ≤ nmax), go to step (f). Otherwise, a new expedition starts, i.e. go back to step (a); f. Select the nb best solutions found among all expeditions based on (14). (ii). Define the set of possible reinforcements for the next year to be analyzed, according to the given priority order. A coordination scheme for the reinforcements has to be carried out, as described in item (iii) of the ES/TS algorithm of Subsection 3.5.1; (iii). Based on the ACO algorithm used in step (i), find the nb best solutions for the next year; (iv). Repeat steps (ii) and (iii) until all years of the planning horizon have been analyzed; (v). Calculate the interruption costs for all analyzed years; (vi). Calculate the present value of all costs (investment, losses and interruption) using a pre-specified discount rate. Thus, find the optimal solution S*, using Eq. (13), but including the unreliability costs.

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Obviously, the previous algorithms (3.5.1 and 3.5.2) do not provide the optimum solution for the TEP problem, but they do offer high-quality solutions for the planning horizon, within an acceptable CPU time. However, the previous algorithms can still be refined to explore several different trajectories within the solution space.

4. Results Case studies on a small test system and on a real sub-transmission network (CEMIG Company, Brazil) are presented and discussed in the next subsections. All CPU times refer to a Pentium IV processor (2.4GHz).

4.1. Small Test System Figure 1 illustrates this test system, which has 6 buses (3 generation and 3 load buses) and 11 double transmission circuits. Considering the reference or starting year, the installed capacity is 260 MW and the load peak is 210 MW. All deterministic, stochastic, bus and circuit data are described in the Appendix. The system expansion horizon is 8 years, and for each year the load and generating capacities are increased by 25%, in relation to the reference year. Therefore, the load and installed capacity will be 630 MW and 780 MW, respectively, at the end of the period of analysis (8th year). New transmission branches can only be added to the existing double circuits and are limited to 3 single circuits per branch.

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Table 1 presents five cases of different priority orders specified by a hypothetical planner searching for the best (i.e. high quality) sequences of transmission reinforcements. The column “Priority Order” shows the order to be followed by the algorithm. In Case 1, for instance, the years are analyzed as follows: 8→7→6→ …→1. For every first year of each case, the five best sets (i.e. nb = 5) of reinforcements are obtained. These solutions are used to create, according to the proposed ES, TS and ACO algorithms, five high quality sequences of reinforcements for each case among the five orders shown in Table 1. The composite reliability assessment is carried out at the end of the searching process. Non-sequential Monte Carlo simulation [31] is used. The ES, TS and ACO algorithms consider only the costs of investments and losses for the fitness function. The simultaneous inclusion of interruption costs would radically increase the computational effort. In order to calculate the costs associated with the transmission losses, the following parameters are used: CkWh = 0.10 US$/kWh and LF = 0.6144. For the LOLC indices, a unit interruption cost of 1.50 US$/kWh is used, and the hourly load curve of the IEEE-RTS [32] is adopted.

Figure 1. Test System.

Table 1. Studied Cases Case

Priority Order

1

8, 7, 6, 5, 4, 3, 2 and 1

2

7, 8, 6, 5, 4, 3, 2 and 1

3

6, 8, 7, 5, 4, 3, 2 and 1

4

5, 8, 7, 6, 4, 3, 2 and 1

5

4, 8, 7, 6, 5, 3, 2 and 1

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Metaheuristic-Based Optimization Methods for Transmission Expansion Planning… 75

4.1.1. ES Results For the proposed (μ+λ)-ES algorithm, a population of 10 individuals (i.e. new sets of reinforcements) is considered, from which 50% are initially generated using the methodology described in Subsection 3.1. The remaining 5 individuals are randomly generated. The mutation step is σ = 0.4 for all tests. The evolutionary process is interrupted if 150 generations is reached, or if the same best solution is repeatedly obtained 20 times. The ES optimization algorithm is run 10 times for each year, within the period of analysis, in order to capture the best solution. A value of 3000 US$/kW is adopted for the load shedding penalty vectorα. Table 2 presents the costs obtained by the proposed ES methodology. It shows the five best sequences for the five cases. Considering Case 1, for instance, the best sequences are A1, B1, C1, D1 and E1. Table 2. Present Value for the 5 Best Sequences – ES Algorithm

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Seq.CASE A1 B1 C1 D1 E1 A2 B2 C2 D2 E2 A3 B3 C3 D3 E3 A4 B4 C4 D4 E4 A5 B5 C5 D5 E5

Invest. (106 US$) 202.00 202.43 204.76 204.53 204.53 204.53 202.43 201.07 204.17 224.09 201.47 201.47 202.00 202.94 202.94 202.10 202.00 202.00 202.38 202.94 202.00 202.66 205.09 203.25 204.80

Losses (106 US$) 27.61 27.53 27.47 27.50 27.52 27.50 27.53 27.72 27.55 27.55 27.67 27.71 27.61 27.68 27.72 27.61 27.60 27.61 27.60 27.57 27.61 27.42 27.58 27.47 27.31

LOLC (106 US$) 63.88 65.49 65.95 67.16 66.02 67.16 65.49 60.92 59.19 55.92 59.55 54.27 63.88 58.81 53.74 60.57 60.22 63.88 57.30 63.61 63.88 66.11 59.76 63.00 65.77

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Total (106 US$) 293.49 295.45 298.18 299.19 298.07 299.19 295.45 289.71 290.92 307.57 288.68 283.45 293.49 289.43 284.40 290.27 289.82 293.49 287.28 294.12 293.49 296.19 292.43 293.72 297.88

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All costs are expressed by their present values, considering a discount rate of 10% per year. The best sequence of each case is shown in bold (gray scale). As it can be observed, if only investment and loss costs were considered, the Sequence C2 would be the winning one for it presents the lowest cost: US$228.79×106, i.e. US$(201.07 + 27.72)×106. However, if interruption costs, expressed by LOLC indices, are also included in the analysis, the sequence B3 becomes the best solution with total costs of US$283.45×106. Table 3 shows the optimal plan of investments for Sequence B3. Only those transmission circuits that have been added are shown and also the associated costs at that particular year. For instance, considering the last year of the planning horizon, i.e. the 8th year, only two circuits are added: one to branch 8 and another to branch 10. These additions have cost US$ 35 millions. All these investment costs are converted into present value (i.e. US$ 201.47 millions). The interruption costs, represented by the LOLC index and the costs associated with transmission losses are also shown in Table 3. For instance, considering the last year, LOLC = US$ 27.26 millions and losses are equal to US$7.39 millions. For this example, it looks like that the optimum condition does not change if losses are included, since these costs are similar for all sequences. However, investments are higher when transmission losses are considered in the optimization process. This can be noticed by comparing the optimum sequence obtained in this study with the one found in reference [19]. This proves the importance of including transmission losses in the expansion analysis. The mean CPU time to obtain the 5 best sequences of each case, with the proposed ES algorithm, was 59 minutes. The total CPU time spent by the methodology to obtain the 25 sequences presented in Table 2, including the reliability analysis, was 340 minutes. The composite reliability evaluation analyzed a total about 50 million states through nonsequential Monte Carlo simulation. Table 3. Optimal Expansion Plan – Sequence B3

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Yr. 8 7 6 5 4 3 2 1 0

2 0 0 1 1 0 1 0 0 0

Added Circuits 3 5 7 8 0 0 0 1 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 6 Total P.V. (10 US$)

9 0 1 0 0 1 0 0 0 0

10 1 0 0 0 0 0 0 0 0

Annual Cost (106 US$) Inv. Losses LOLC 35.00 7.39 27.26 100.00 6.47 18.20 50.00 6.31 11.65 45.00 5.57 11.12 40.00 4.83 10.50 45.00 4.31 4.69 20.00 3.37 5.65 0.00 2.67 3.55 0.00 2.17 0.14 201.47 27.71 54.27

4.1.2. TS Results For the study of the test system, the TS algorithm uses the following parameters: 10 for the number of diversification and 5 for the size of the tabu list. For each one of the 10 search processes (diversifications), an initial solution is obtained through the methodology described

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Metaheuristic-Based Optimization Methods for Transmission Expansion Planning… 77 in Section 3.1. Each diversification step is interrupted when it reaches 80 movements, or if the best solution is kept for 25 consecutive movements. The value of 8000 US$/kW is adopted for the penalty (α) of load shedding, and it has no relation with the interruption costs. Table 4 shows the costs obtained by the proposed algorithm, for the best 5 sequences of each case studied. All of the costs are expressed in terms of their net present value, also considering a discount rate of 10% per year. The best sequence of each case appears in bold. Table 4. Present Value of the 5 Best Sequences – TS Algorithm

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Seq.CASE F1 G1 H1 I1 J1 F2 G2 H2 I2 J2 F3 G3 H3 I3 J3 F4 G4 H4 I4 J4 F5 G5 H5 I5 J5

Invest. (106 Losses (106 LOLC (106 US$) US$) US$) 202.43 27.53 65.49 204.76 27.47 65.95 204.53 27.50 67.16 204.53 27.52 66.02 206.86 27.50 64.30 204.53 27.50 67.16 202.43 27.53 65.49 224.09 27.55 55.92 214.76 27.60 54.98 214.76 27.62 53.41 201.47 27.67 59.55 201.47 27.71 54.27 202.00 27.61 63.88 202.00 27.60 60.22 202.94 27.68 58.81 202.10 27.61 60.57 202.00 27.60 60.22 204.97 27.53 63.03 203.62 27.41 63.92 204.41 27.62 59.76 202.43 27.53 65.49 202.66 27.42 66.11 205.09 27.58 59.76 206.21 27.39 62.17 204.80 27.31 65.77

Total (106 US$) 295.45 298.18 299.19 298.07 298.66 299.19 295.45 307.57 297.34 295.79 288.68 283.45 293.49 289.82 289.42 290.27 289.82 295.53 294.94 291.80 295.45 296.19 292.43 295.77 297.88

As it can be noted, if only the investment and loss costs were considered, the Sequence F3 would be the winning one for it presents the lowest cost: US$229.14×106. However, when the LOLC is also included, Sequence G3 becomes the best solution, presenting a total cost of US$283.45×106. It can be seen that Sequence G3 corresponds to the best investment plan for this system, since the same sequence was also found by the ES heuristic, i.e. Sequence B3, in Table 2. In fact, most sequences of reinforcements shown in Table 4 are exactly the same as those obtained with the ES algorithm and shown in Table 2. To identify these sequences, one

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has to compare the summation of investment cost and loss cost. For instance, Sequence B1 and Sequence H1 are the same, since the summation of investment costs (US$204.76×106) and loss costs (US$27.47×106) are exactly the same (i.e. US$232.23×106). The mean CPU time to obtain the 5 best sequences of each case, with the proposed TS algorithm, was 34 minutes. The total CPU time spent by the methodology to obtain the 25 sequences presented in Table 4, including the reliability analysis, was 215 minutes.

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4.1.3. ACO Results For the ACO algorithm, a number of 15 ants for each expedition is considered (i.e. na = 15). The search is interrupted after 50 expeditions (i.e. nmax = 50) or if the best solution is not improved after 12 consecutive expeditions (i.e. ncon = 12). The previous maximum number of expeditions is considered only for the optimization process of the most relevant year (e.g. the last one NY). For any another year, nmax = 25. The other parameters were as follows: q0 = 0.2; β = 0.7; ϕ = 0.05; τ0 = 1; ρ = 0.55 and Kpher = 4V, where V refers to the investment and loss costs found for any solution of the first expedition and 4 is just a multiplicative factor. Table 5 shows the costs obtained by the proposed algorithm, for the best 5 sequences of each case studied. All of the costs are expressed in terms of their net present value, also considering a discount rate of 10% per year. The best sequence of each case appears in bold. As it can be noted, if only the investment and loss costs were considered, the Sequence M2 would be the winning one for it presents the lowest cost: US$228.79×106. However, when the LOLC is also included, Sequence L3 becomes the best solution, presenting a total cost of US$283.45×106. This sequence is the same found by the ES heuristic (Sequence B3) and by TS heuristic (Sequence G3). In fact, most sequences of reinforcements shown in Table 5 are exactly the same as those shown in Tables 2 and 4. The mean CPU time to obtain the 5 best sequences of each case, with the proposed ACO algorithm, was 32 minutes. The total CPU time spent by the methodology to obtain the 25 sequences presented in Table 5, including the reliability analysis, was 205 minutes. Table 5. Present Value of the 5 Best Sequences – ACO Algorithm Seq.CASE K1 L1 M1 N1 O1 K2 L2 M2 N2 O2 K3

Invest. (106 US$) 202.00 202.43 204.76 204.53 204.53 204.53 202.43 201.07 204.17 214.76 201.47

Losses (106 US$) 27.61 27.53 27.47 27.50 27.52 27.50 27.53 27.72 27.55 27.62 27.67

LOLC (106 US$) 63.88 65.49 65.95 67.16 66.02 67.16 65.49 60.92 59.19 53.41 59.55

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Total (106 US$) 293.49 295.45 298.18 299.19 298.07 299.19 295.45 289.71 290.92 295.79 288.68

Metaheuristic-Based Optimization Methods for Transmission Expansion Planning… 79 Table 5. Continued Seq.CASE L3 M3 N3 O3 K4 L4 M4 N4 O4 K5 L5 M5 N5 O5

Invest. (106 US$) 201.47 202.00 203.43 204.83 202.10 202.00 202.00 202.38 202.29 202.00 202.66 203.25 204.80 203.25

Losses (106 US$) 27.71 27.60 27.54 27.54 27.61 27.60 27.61 27.60 27.57 27.60 27.42 27.47 27.31 27.53

LOLC (106 US$) 54.27 60.22 58.47 55.80 60.57 60.22 63.88 57.30 63.17 60.22 66.11 63.00 65.77 57.84

Total (106 US$) 283.45 289.82 289.44 288.17 290.27 289.82 293.49 287.28 293.03 289.82 296.19 293.72 297.88 288.62

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4.1.4. Performance Comparison of ES, TS and ACO Algorithms For this particular test system, a general comparison based on the quality of solutions and computational effort can be summarized as follows. From the optimization point of view, taking into account only the costs related to investments and transmission losses, one can observe that among the 75 reinforcement sequences shown in Tables 2, 4 and 5, the ES, TS and ACO algorithms have captured 8, 7, and 9 sequences, among the ten best sequences, i.e. those with lowest costs. Moreover, among these 75 sequences, the lowest value is US$ 228.79×106, and the highest one is US$ 251.64×106, corresponding to a variation of 10%. Among the ten best sequences, however, this variation decreases to only 0.5%. Therefore, it is not possible to state that one heuristic is superior to the other, since the objective of such tools is to find a set of good sequences, and not necessarily the optimum. The CPU times spent by the optimization were 295, 170 and 160 minutes for the ES, TS and ACO, respectively. On other hand, when the interruption costs, i.e. LOLC, are also included in this process, the ES, TS and ACO algorithms have captured 7, 4, and 8 sequences, respectively, among the ten best sequences, i.e. those with lowest total costs. It can be noted that the lowest value becomes US$ 283.45×106, while the highest one is US$ 307.57×106, corresponding to a variation of 8.5%. Among the ten best sequences including interruption costs, this value decreases to 2.2%. Therefore, one can concluded that any sequence among the previous ten may be considered as a good option for the system planner. As a general conclusion, for this particular system and specified simulation parameters, one can conclude that the computational performances of the TS and ACO algorithms are similar and superior to the ES algorithm, although the quality of the solutions achieved by the ES and ACO algorithms was slightly better as compared to the TS algorithm.

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4.2. Brazilian Sub-transmission Network

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A small sub-transmission network, named Pirapora 2, which belongs to CEMIG (State Energy Company of Minas Gerais), located in the North Region of Minas Gerais, was used for a second test with the proposed methodology. This sub-transmission system is composed by 12 buses, including 6 loads, 1 interconnection and 1 generation bus. The peak load of this subsystem is 780.05 MW and the maximum local generating capacity 226.76 MW. The remaining power is supplied by the interconnection bus. There are 20 transmission circuits operating with two voltage levels: 138 kV and 345 kV. Fig. 2 shows a basic picture of this sub-transmission system. An expansion horizon of 10 years is considered and the loads in this area will be assumed to grow with an average rate of 5% per year. For sub-transmission reinforcing purposes, all possible interconnections among the 138 kV buses are considered. Thus, 22 reinforces allocation points (branches) are defined. A maximum of three transmission lines per branch is accepted. Therefore, if there is already one existing transmission line between two buses, only two new transmission lines may be added to this branch. The expansion sequences are obtained for each 2 years of the study period. Even for this small sub-transmission network, a total of 3.29 × 1038 chronological sequences of reinforcements are, in principle, eligible for this TEP problem. Finally, for this real sub-transmission system other factors not related with section size and length of transmission lines were also considered; e.g. substation reconfigurations, etc.

Figure 2. Sub-transmission Network, Pirapora 2, North Region of Minas Gerais State, Brazil.

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Metaheuristic-Based Optimization Methods for Transmission Expansion Planning… 81

4.2.1. ES Results A population of 20 individuals is used with the proposed ES algorithm. The other ES parameters used for this system is the same used for the small test system. The CPU time spent to obtain the 30 best sequences with the proposed ES based methodology, including the reliability analysis, was almost 208 minutes. To analyze 6 cases of priority orders, provided by the planner, the ES were run 1300 times. Moreover, the composite reliability evaluation analyzed 43,715,080 states. For this particular sub-transmission system, transmission losses and unreliability costs have proved to be extremely important to achieve the optimum expansion plan.

4.2.2. TS Results A maximum of 50 movements were used for each step of the diversification of the proposed TS algorithm. The other parameters used in this system are the same of the previous test system. The CPU time spent to obtain the 30 best sequences with the proposed methodology, including the reliability analysis, was 156 minutes.

4.2.3. ACO Results

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The ACO algorithm has used 15 ants in each expedition (i.e. na = 15). For the optimization of the most relevant year, nmax = 40 and ncon = 20. For any another year, nmax = 25 and ncon = 10. The other ACO parameters used for this system are the same used for the small test system previously studied. The CPU time spent to obtain the 30 best sequences with the proposed ACO based methodology, including the composite reliability analysis by a non-sequential Monte Carlo algorithm, was about 131 minutes.

4.2.4. Performance Comparison of ES, TS and ACO Algorithms For this particular sub-transmission system all methods achieved the same best solution or sequence of reinforcements. In relation to the computational effort, one can see that again the ACO and TS algorithm had the best performance. Regarding the strategic parameters, however, the algorithms based on the ES and TS heuristics are the easiest to be adjusted. Finally, although the CPU time to get the best chronological sequences of transmission reinforcements, taking into account investment, losses and unreliability costs, can be considered huge, it is much smaller when compared with the number of hours spent by planners in searching for the same good solutions. Undoubtedly, these automatic searching methods do not cover all planning options, but it can considerably help engineers in their decisions.

5. Conclusion In the last few years, several metaheuristic-based approaches have been used to solve static power transmission network design problems, such as the transmission expansion planning (TEP) task. This is an extremely complex problem, even if some simplifications are

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introduced. For instance, the TEP problem is usually separated from the generation and distribution expansion planning. Other simplification is to use linear network representation based on the DC model. The other inherent complexity of the TEP problem is its dynamic nature, which requires the consideration of multiple time periods, determining sequences of transmission expansion plans. Finally, almost all algorithms do not take into account the random behavior of the system, due to generation and transmission equipment unavailabilities as well as load fluctuations. These aspects can only be captured by probabilistic methods. Three methodologies to solve the TEP problem were discussed in this chapter based on the following heuristics: Evolution Strategies (ES), Tabu Search (TS) and Ant Colony Optimization (ACO). Other heuristics were used to assist the initialization of the search process and also to account for the priority orders provide by system planners to deal with the multi-stage nature of the TEP problem. The costs related to investments and transmission losses were considered while searching for the best expansion sequence. Moreover, interruption or unreliability costs were also included into the analyses, but outside the previous searching process. It was clear from the analyzed tests that the effect of reliability on the performance of the expansion sequences is quite significant to decide the best plan for the system. By ensuring a good reliability performance, it will be possible to have a better continuity of the planning process after the specified horizon. The results obtained by all discussed metaheuristic-based methodologies with both systems, and also with other analyzed networks, were very good in terms of solution robustness and computational performance. A more comprehensive study comparing different metaheuristics for large network configurations, including non-linear AC network representation, load uncertainties, etc. are among the future research studies.

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6. Appendix: Simple Test System Data Table 6 shows the deterministic and stochastic generation data for the simple test system of Fig. 1. The maximum capacity per generation unit is defined for the reference year. All loads are equal to 70 MW at the reference year. The failure rate “λ” and the MTTR (i.e. Mean Time to Repair) are also shown per generation unit. Table 7 shows the deterministic and the stochastic transmission circuit data. The values of resistance, reactance, capacity, investment costs and failure rate are for each individual circuit. For instance, between bus 1 and 2, there are already installed (see Fig. 1) two transmission lines (i.e. double circuit), whose combined reactance and capacity are 0.20 pu and 50 MW, respectively. A new additional circuit, to be added between these two buses will have a reactance of 0.40 pu, capacity of 25 MW, failure rate of 1 failure per year, and it will cost US$ 25 × 106. All individual circuits have a MTTR of 10 hours. Table 6. Generation Data – Deterministic and Stochastic Bus 1 2 3

No. of Units 2 1 1

Max. Cap. per Unit (MW) 60.0 70.0 70.0

λ(/year)

MTTR (h)

7.0 10.0 10.0

20.0 40.0 40.0

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Metaheuristic-Based Optimization Methods for Transmission Expansion Planning… 83 Table 7. Circuit Data – Deterministic and Stochastic Buses 1 1 1 2 2 2 2 3 3 4 5

Circuit No.

R (pu)

X (pu)

Cap. (pu)

1 2 3 4 5 6 7 8 9 10 11

0.10 0.10 0.15 0.13 0.05 0.15 0.10 0.13 0.05 0.20 0.15

0.40 0.40 0.60 0.50 0.20 0.60 0.40 0.52 0.20 0.80 0.60

0.25 0.25 0.20 0.20 0.40 0.20 0.25 0.20 0.40 0.15 0.20

2 4 5 3 4 5 6 5 6 5 6

Cost (106 US$) 25 25 20 20 40 20 25 20 40 15 20

λ (/year) 1.00 1.00 1.50 1.25 0.50 1.50 1.00 1.18 0.50 2.00 1.50

References [1]

[2] [3]

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[4]

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[9]

R. Billinton, L. Salvaderi, J.D. McCalley, H. Chao, Th. Seitz, R.N. Allan, J. Odom, and C. Fallon, “Reliability issues in today’s electric power utility environment,” IEEE Trans. on Power Syst., vol. 12, no. 4, pp. 1708–1714, Nov. 1997. A.A. Chowdhury, and D.O. Koval, “Value-based system facility planning,” IEEE Power & Energy Magazine, vol. 2, no. 5, pp. 58–67, Sept./Oct. 2004. L.L. Garver, “Transmission network estimation using linear programming,” IEEE Trans. on PAS, vol. 89, no. 7, Sept. 1970. G. Latorre, R.D. Cruz, J.M. Areiza, and A. Villegas, “Classification of publications and models on transmission expansion planning,” IEEE Trans. on Power Syst., vol. 18, no. 2, pp. 938–946, May 2003. Z. Xu, Z.Y. Dong, and K.P. Wong, “Transmission planning in a deregulated environment,” IEE Proc.-Gener. Transm. Distr., vol. 153, no. 3, pp. 326-334, May 2006. M. Shahidehpour (Guest Editorial), “Global Broadcast – Transmission Planning in Restructured Systems – Special Issue on Trans. Planning,” IEEE Power and Energy Magazine, vol. 5, no. 5, Sept/Oct. 2007. R.A. Gallego, A.B. Alves, A. Monticelli, and R. Romero, “Parallel simulated annealing applied to long term transmission network expansion planning,” IEEE Trans. on Power Syst., vol. 12, no. 1, pp. 181–187, Feb. 1997. W. Fushuan, and C.S. Chang, “Transmission network optimal planning using the tabu search method,” Electric Power Systems Research, vol. 42, no. 2, pp. 153-163, Aug. 1997. R. A. Gallego, R. Romero, and A. Monticelli, “Tabu search algorithm for network synthesis,” IEEE Trans. on Power Syst., vol. 15, no. 2, pp. 490–495, May 2000.

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[10] E.L. da Silva, J.M.A. Ortiz, G.C. de Oliveira, and S. Binato, “Transmission network expansion planning under a tabu search approach,” IEEE Trans. on Power Systems, vol. 16, pp. 62–68, Feb. 2001. [11] H. A. Gil, and E. L. da Silva, “A reliable approach for solving the transmission network expansion planning problem using genetic algorithms,” Electric Power System Research, vol. 58, no. 1, pp. 45–51, May 2001. [12] S. Binato, G.C. Oliveira, and J.J. Araújo, “A greedy randomized search procedure for transmission expansion planning,” IEEE Trans. on Power Syst., vol. 16, no. 2, pp. 247– 253, May 2001. [13] A.H. Escobar, R.A. Gallego, and R. Romero, “Multistage and coordinated of the expansion of transmission systems,” IEEE Trans. on Power Syst., vol. 19, no. 2, pp. 735–744, May 2004. [14] H. Faria Jr., S. Binato, M.G.C. Resende, and D.M. Falcão, “Power transmission network design by greedy randomized adaptive path relinking,” IEEE Trans. on Power Syst., vol. 20, no. 1, pp. 43–49, Feb 2005. [15] W. Li, and R. Billinton, "A minimum cost assessment method for composite generation and transmission system expansion planning," IEEE Trans. on Power Syst., vol. 8, no. 2, pp. 628–635, May 1993. [16] C. Ray, P. Collins, A. Hiorns, and D. Friend, “Probabilistic transmission planning in England & Wales,” in Proc. of the 5th PMAPS - Probabilistic Methods Applied to Power Systems, Vancouver, Canada, 21-25/Sept.1997. [17] L.A.F. Manso, and A.M. Leite da Silva, “Probabilistic criteria for power system expansion planning,” Electric Power Systems Research, vol. 69, no. 1, pp. 51-58, April 2004. [18] W. Li, Risk Assessment of Power Systems – Models, Methods, and Applications. IEEE Press Series on Power Engineering, J. Wiley & Sons Inc. Publication, 2005. [19] A.M. Leite da Silva, W.S. Sales, L.C. Resende, L.A.F. Manso, C.E. Sacramento, and L.S. Rezende, “Evolution strategies to transmission expansion planning considering unreliability costs,” Proc. of the 9th PMAPS – Probabilistic Methods Applied to Power Systems, Stockholm, Sweden, 11-15/June 2006. [20] J.R.P. Barros, A.C.G. Melo, and A.M. Leite da Silva, “An approach to the explicit consideration of unreliability costs in transmission expansion planning,” European Transactions on Electric Power, vol. 17, no. 4, pp. 401–412, Jul.-Aug. 2007. [21] K.Y. Lee, and M.A. El-Sharkawi (Editors), “Tutorial on Modern Heuristic Optimization Techniques with Application to Power Systems,” IEEE PES, IEEE Pub. No. 02TP160, Jan 2002. [22] D.B. Fogel, “An introduction to simulated evolutionary optimization”, IEEE Trans. on Neural Networks, Vol. 5, No. 1, pp. 3-14, Jan. 1994. [23] H.-P. Schwefel, and G. Rudolph, “Contemporary evolution strategies” in F. Morán et al. eds., “Advances in Artificial Life, 3rd Int. Conference on Artificial Life, Vol. 929 of Lecture Notes in Artificial Intelligence, Springer, Berlin, pp. 893-907, 1995. [24] Z. Michalewicz, and D.B. Fogel, How to Solve It: Modern Heuristics, 2nd Ed., Springer-Verlag, Berlin, 2004. [25] F. Glover, and M. Laguna, Tabu Search, Kluwer Academic Pubs., Boston, 1997.

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[26] M. Dorigo, V. Maniezzo, and A. Colorni, “The ant system: optimization by a colony of cooperating agents,” IEEE Trans. on Syst., Man, and Cybernetics-Part B, vol. 26, no. 1, pp. 29–41, 1996. [27] M. Dorigo, and L.M. Gambardella, “Ant colony system: a cooperative learning approach to the traveling salesman problem,” IEEE Trans. on Evolutionary Computation, Vol. 1, No. 1, pp. 53–66, 1997. [28] M. Dorigo, and T. Stützle, Ant Colony Optimization. A Bradford Book, The MIT Press, England, 2004. [29] J.F. Gomez, H.M. Khodr, P.M. De Oliveira, L. Ocque, J.M. Yusta, R. Villasana, and A.J. Urdaneta, “Ant colony system algorithm for the planning of primary distribution circuits,” IEEE Trans. on Power Syst., vol. 19, no. 2, pp. 996–1004, May 2004. [30] S. Kannan, S.M.R. Slochanal, and N.P. Padhy, “Application and comparison of metaheuristics techniques to generation expansion planning problem,” IEEE Trans. on Power Syst., vol. 20, no. 1, pp. 466–475, Feb. 2005. [31] A.M. Leite da Silva, L.A.F. Manso, J.C.O. Mello, and R. Billinton, “Pseudochronological simulation for composite reliability analysis with time varying loads,” IEEE Trans. on Power Syst., vol. 15, no. 1, pp. 73–80, Feb. 2000. [32] IEEE APM Subcommittee, “IEEE reliability test system,” IEEE Trans. on PAS, vol. PAS-99, pp. 2047–2054, Nov/Dec. 1979.

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In: Optimization Advances in Electric Power Systems ISBN: 978-1-60692-613-0 Editor: Edgardo D. Castronuovo, pp. 87-112 © 2008 Nova Science Publishers, Inc.

Chapter 5

A VOLTAGE CONTROL OPTIMIZATION FOR DISTRIBUTION NETWORKS WITH DG AND MICROGRIDS J.A. Peças Lopes1, 2 and André Madureira1, 2 1

INESC Porto – Institute for Systems and Computer Engineering of Porto, Portugal 2 FEUP – Faculty of Engineering of the University of Porto, Portugal

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Abstract In general, distributed generation is not subject to a centralized dispatch and reactive power generation is usually restricted by operation rules defined by the distribution system operators. With the growth of distributed generation and microgrids in distribution networks, the development of voltage control functionalities for these units needs to be investigated. This requires a new operation philosophy to exploit reactive power generation capability of distributed generation and microgeneration with the objective of optimizing network operation: minimize active power losses and maintain voltage profiles within adequate margins. This implies that distributed generation should adjust their reactive power generation, i.e. supply an ancillary service of voltage and reactive power control. In addition to the growth in distributed generation penetration, microgeneration is expected to develop considerably and contribute to the implementation of efficient voltage control schemes. For this new scenario, a hierarchical voltage control scheme must be implemented, using communication and control possibilities that will be made available for microgrid operation. Technical advantages and feasibility of this operation philosophy are investigated in this chapter by analyzing the impact of the proposed control procedures on distribution networks. In addition, the identification of control action needs is assessed by solving an optimization problem, where voltage profiles are improved and active power losses minimized, subject to a set of technical constraints. The solution for this problem is obtained using an Evolutionary Particle Swarm Optimization algorithm. The control algorithm implemented will enable dealing even with extreme situations, where reactive power control is not sufficient to maintain system operation and therefore generation shedding actions must be performed.

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Acronyms and Abbreviations AVR – Automatic Voltage Regulator CAMC – Central Autonomous Management Controller DG – Distributed Generation DMS – Distribution Management System DSM – Demand Side Management DSO – Distribution System Operator EPSO – Evolutionary Particle Swarm Optimization HV – High Voltage LV – Low Voltage MGCC – MicroGrid Central Controller MV – Medium Voltage OF – Objective Function OLTC – On-Line Tap Changing OPF – Optimal Power Flow SVC – Static VAR Compensator VCA – Voltage Control Area

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1. Introduction Following the widespread penetration of Distributed Generation (DG) in distribution networks, electric power systems have been lead up to a point where operating conditions are becoming more stressed, namely regarding voltage profiles. In addition, reliability and security have become a primary concern for power system operation as they affect strongly the quality of supply. Following the restructuring process in power systems, voltage services are considered an important part of ancillary services provision. Hence, the development of control functionalities for DG units concerning voltage control has been a topic for investigation in recent years. In general, DG is not subject to a centralized dispatch and reactive power generation is usually restricted by operation rules defined by the Distribution System Operator (DSO). However, a significant growth of DG penetration requires a new operation philosophy to exploit reactive power generation capability of DG with the objective of optimizing network operation by minimizing active power losses and maintaining voltage profiles. This implies that DG should be capable of adjusting their reactive power generation, i.e. be able to supply an ancillary service of voltage and reactive power control. The feasibility and the technical advantages that result from implementing this operational strategy are investigated in this chapter by analyzing the impact of the proposed strategies for voltage control in distribution networks. The identification of control action needs may be assessed by solving an optimization problem, where active power losses are minimized, subject to a set of technical constraints [1]. The control variables to consider are DG reactive power generation, On-Line Tap Changing (OLTC) transformers settings and capacitors banks settings. In this case, active power generation is not subject to any type of control. In conventional systems, voltage control is usually formulated as a decentralized control problem. This is mainly because voltage is predominantly a local or regional problem.

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However, the characteristics of the future distribution networks may impose severe restrictions to the design of an efficient system for voltage control, namely:

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• • • • •

The use of power electronic interfaces in most generating units The use of new generating technologies with new control systems The possibility of performing Demand Side Management (DSM) The possibility of both interconnected and islanded operation The creation of local markets for ancillary services provision.

In addition to the growth in DG penetration, microgeneration is also expected to develop considerably and therefore contribute to the possibility of implementing an efficient voltage control scheme [2], [3]. For this new scenario, a hierarchical voltage control scheme must be envisaged by exploiting communication and control possibilities available for microgrid operation [3]. A microgrid, as defined in [2], is a Low Voltage (LV) feeder with several microsources, storage devices and controllable loads connected on that same feeder. A microgrid may be operated wither in interconnected mode with an upstream Medium Voltage (MV) network or islanded from the main power system. The full applicability of this concept requires building a control scheme for microgrid operation [2]. A new concept of multi-microgrids is related to a higher-level structure, formed at the MV level, consisting of several LV microgrids and DG units connected on adjacent MV feeders. Microgrids, DG units and MV loads under DSM control can be considered in this network as active cells for control and management purposes. Consequently, and following the widespread diffusion of both microgrids and DG in distribution systems, the vision for voltage control strategies needs now to be revised. In order to be able to implement this strategy, several mechanisms need to be developed, namely the possibility of giving a fair remuneration for voltage control services. This topic is, however, out of the scope of this chapter. Finally, operation must be optimized for both normal and emergency conditions, i.e. interconnected and islanded operation. This chapter comprises the possibility of DG and microgeneration contributing to voltage control in interconnected mode of operation.

2. Hierarchical Voltage Control A hierarchic control system must be established for voltage control in distribution systems with large penetration of DG and microgeneration, similarly to what happens in frequency control, using communication and control possibilities that will be available in future distribution networks. This scheme may be divided into three control levels that are presented next, according to areas of action and deployment time [4]: •

Primary Voltage Control – It keeps the voltage within specified limits of the reference values. Automatic Voltage Regulators (AVRs) of synchronous machines are used to control voltage in primary control and their action takes effect in a few seconds.

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•

Secondary Voltage Control – It has the main goal of adjusting, and maintaining, voltage profiles within an area and of minimizing reactive power flows. Its action can take up to a few minutes. The control actions associated with secondary control include modifying reference values for AVRs, DG units or microgrids, switching Static VAR Compensators (SVCs) and adjusting On-Line Tap Changing (OLTC) transformers. Tertiary Voltage Control – It has the goal of achieving an optimal voltage profile and coordinating the secondary control in accordance to both technical and economical criteria. It may use an algorithm such as an Optimal Power Flow (OPF) routine. The period of this control action is around some tens of minutes. The control variables used are generator voltage references, reference bus voltages and state of operation of reactive power compensators.

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In Fig. 1, a scheme of the hierarchical voltage control architecture is presented.

Figure 1. Hierarchical Voltage Control.

The main aim of the implementation of this method is to ensure an optimized coordinated voltage control strategy between the several voltage levels in the distribution system. From the physical point of view, the implementation of this control strategy requires the implementation of a multi-level management system, similar to the one described in Fig. 2. This system is described in detail in [2], [3] and [4].

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Figure 2. Control and Management Architecture of the Distribution Grid

In this architecture, apart from the conventional Distribution Management System (DMS) responsible for the supervision, control and management of a large distribution system, there are new additional management levels: 1) The Central Autonomous Management Control (CAMC), to be housed in HV/MV substations; 2) The MicroGrid Central Controller, to be housed at the level of each MV/LV substation, which manages the microgrid including the control of the microsources and responsive loads. Voltage monitoring in each LV grid is performed through the microgrid communication infrastructure. Secondary voltage control functionalities can be housed at the CAMC level and tertiary voltage control at the DMS level.

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3. Characterization of the Voltage Control Optimization Problem The hierarchic control scheme presented above will enable dealing even with extreme situations, where microgrid penetration, combined with multiple DG sources, is so high that voltages raise considerably, especially in LV networks, leading to inadmissible voltage profiles. This effect is particularly serious in rural networks, for it is known that the connection of DG fundamentally alters distribution network operation, thus creating several welldocumented impacts, with voltage rise being the dominant one [5]. In these situations, reactive power control will not be adequate to maintain efficient system operation. In order to overcome this, active power management must also be used, by means of generation shedding, of both microgeneration and DG sources. Hence, a new procedure must be developed that includes optimizing operating conditions by using DG, microgrid and OLTC transformer control capabilities. Voltage control in multi-microgrids has a strong hierarchic structure that may imply dealing with voltage sub-areas, due mainly to the size of the electrical distribution systems. Once a solution has been found to the global problem, sub-solutions for individual microgrids are tested in order to evaluate their feasibility. If the voltage profiles are not satisfactory, a generation shedding scheme is used. The optimization procedure then requires a sequence of local sub-problems solutions and global problem solutions in order to converge to a near-optimum solution. For the optimization procedure, the MV and LV networks are both fully modelled but are considered as being decoupled. Some initial assumptions concerning LV microgrid modelling have been made: •

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•

Each LV microgrid was considered as a single bus with an equivalent generator (sum of all micro-source generations) and equivalent load (sum of all LV loads) Q limits for each microgrid consider a tan(φ) = 0,25 based on the microgrid nominal active power generation.

3.1. Mathematical Formulation The mathematical formulation for the voltage control problem is presented next: • •

• •

Type of Problem: Mixed, non-linear minimization problem Control Variables: Reactive power generation in each active cell (microgrid or DG unit), transformer taps position (discrete variable) and capacitor banks level (discrete variable) Objective Function: Minimization of active power losses and/or improvement of voltage profiles and/or reduction of reactive power flows etc. Constraints: Technical (representing operational limits of equipments and other physical limits imposed to system operation) o Voltage limits in all buses o Reactive power generation limits for active cells o Power flow limits in lines and transformers

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Transformer tap ratio limits Capacitor step limits

Mathematically, this may be represented as follows: min OF ( X ) subject to:

Vi min ≤ Vi ≤ Vi max S ikmin ≤ S ik ≤ S ikmax t imin ≤ t i ≤ t imax Qimin ≤ Qi ≤ Qimax

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where: OF – Objective Function X – Control variables Vi – Voltage at Bus i Vimin, Vimax – Minimum and maximum voltage at bus i Sik – Power Flow in Branch ik Sikmin, Sikmax – Minimum and maximum power flows in Branch ik ti – Transformer tap of or capacitor step position timin, timax – Minimum and maximum tap

3.1.1. Defining an Objective Function The definition of the Objective Function (OF) to be used by the algorithm should be the main concern when formulating the voltage control problem. The most common goals for coordinated voltage/reactive power control are: • • • • •

Keep all bus voltages within specified limits Control transformer, line and feeder loading Minimize active power losses Manage reactive power sources Control the power factor

Naturally, some of these objectives are conflicting objectives, so we are in fact dealing with a multi-objective problem. One strategy that may be employed in order to conjugate these objectives is to combine them by creating a general OF with different weights for each individual goal. This procedure turns a multi-objective problem into a single objective problem. The general OF has the following structure:

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OF ( X ) = w1 ⋅ OF ( X 1 ) + w2 ⋅ OF ( X 2 ) + ... + wn ⋅ OF ( X n ) where: OF – Objective Function wi – Weight i Xi – Control variable i In order to build this global OF, it is necessary to normalize the individual OFs. This can be done in a simple way by dividing each value of each individual OF by the maximum value of the corresponding OF at each run of the algorithm. The adjustment of the weights wi allows favouring the objectives that are considered to be more important for the problem. Obviously, all the weights have to have a unity sum. Several studies need to be performed in order to be able to identify adequate values for the weights to use in the generic OF, depending on the test case network chosen. The main objective functions considered in this chapter are: • •

Reducing active power losses Maintaining voltage

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Reducing Active Power Losses Managing adequately reactive power generation may reduce active power losses in distribution feeders. The optimization of reactive power flows to aid voltage support should be one of the main concerns of the algorithm. Maintaining Voltage Profiles In order to ensure the maintenance of voltage profiles within admissible limits, two different approaches may be envisaged. One approach involves monitoring and controlling voltage at all buses in the network. This approach may not be suitable given the dimension of the network, as it may represent a computational burden for the algorithm. Another approach that may be employed is dividing the network into several Voltage Control Areas (VCA) and selecting adequate Pilot Nodes. Pilot nodes are buses that are considered to be representative of the voltage profile in a given area [6]. Thus, controlling voltage at a pilot node enables to control voltage in a particular area of the network. This method is particularly useful if the size of the network is already a problem. In order to divide the network into VCAs, one may use a method such as electrical distance calculation [6], [7]. Consequently, one of these two strategies may be used for maintaining voltage profiles, depending on the information available from the MV distribution system. If it is possible to monitor voltage at all nodes, voltage control may be performed using that information. On the other hand, if voltage values in some areas remain unknown to the DSO, pilot nodes must be selected for a given area in order to enable monitoring and control actions at these specific buses.

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3.1.2. Defining the Control Variables The control variables that may be used are: • Reactive power from DG sources • Reactive power from microgrid buses by means of capacitor banks located at the MV/LV transformer substation of each microgrid • OLTC settings at distribution transformers A generic vector of the control variables may then be:

QμG1 | QμG 2 | ... | QμGn | QDG1 | QDG 2 | ... | QDGm | ttransf 1 | ttransf 2 | ... | ttransfo where: QµGi – reactive power provided by microgrid i QDGi – reactive power provided by DG unit i ttransfi – transformer tap value for transformer i

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3.1.3. Optimization Tool The optimization procedure used here is based on a meta-heuristic called Evolutionary Particle Swarm Optimization (EPSO). This method is a powerful optimization algorithm developed at INESC Porto, which combines traditional PSO strategies, developed by Kennedy and Eberhardt [8], with Evolution Strategies by Schwefel [9]. Therefore, EPSO brings together the best-of-two-worlds: it is a particle swarm algorithm because there is exchange of information among solutions and an evolutionary strategy because solutions are mutated and passed on to the following generations. This algorithm has been extensively tested in recent years, applied to optimization problems in electrical power systems [10], [11]. In addition to the optimization algorithm based on an EPSO strategy, active power generation control is also performed by means of a generation shedding algorithm that acts in extreme situations where the voltage is outside the limits defined for safe operation (typically 10% for LV networks and 5% for MV networks). This generation shedding scheme is performed outside the main optimization function and is performed proportionally to the voltage deviation from the admissible range at each generator bus. EPSO Description In EPSO, each particle (or solution at a given stage) is defined by its position Xik and velocity vik for the coordinate position i and particle k. The general scheme of EPSO is presented next: • • • •

Replication – each particle is replicated r times Mutation – each particle has its weights w mutated Reproduction – each particle generates an offspring according to the particle movement rule Evaluation – each offspring has its fitness evaluated

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Selection – the best particles are selected by stochastic tournament

The particle movement rule for is the following: given a particle Xi, a new particle Xinew can be obtained:

X inew = X i + vinew with

v inew = wi*0 ⋅ v i + wi*1 ⋅ (b i − X i ) + wi*2 ⋅ (b g* − X i ) where Xi is the position of the particle vi is the velocity of the particle wi is a strategic parameter (weight) bi is the best solution of each particle bg is the best solution among all particles The weights (wi) are mutated as follows:

wik* = wik + τ ⋅ N (0,1) where τ is a fixed learning parameter N(0,1) is a random variable with Gaussian distribution, 0 mean and variance 1

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The global best (bg) is also mutated as follows:

bg* = bg + τ '⋅ N (0,1) where τ’ is also a fixed learning parameter More details on the algorithm can be found in [10] and [11].

3.2. Algorithm The voltage support functionality is designed to be a real-time application to aid the DSO in managing the distribution network. Nevertheless, the implementation of this functionality relies on two different stages: •

A preliminary stage where several studies are performed offline to evaluate the performance and the characteristics of the network in order to be able to select adequate parameters for the optimization problem formulation.

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•

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A final stage that comprises the use of the optimization tool in real-time operation, made available to the operator.

As previously mentioned, in order to have an efficient means to control voltage in multi microgrid systems it is necessary to be able to control not only reactive power but also active power. Active power control for volt/VAR control is based on generation shedding whenever voltage reaches inadmissible values due to high DG and microgrid penetration. The methodology employed here is based in a combination of the EPSO algorithm for voltage support based on reactive power control with a generation shedding algorithm, exploiting active power control, if the voltages in the network exceed their admissible range. This methodology is schematically shown in Fig. 2 and consists of two main steps: a V-Q Control Scheme and a Generation Shedding Scheme that are housed at different levels in the control hierarchy. The V-Q Control scheme is a “soft” control based on the tuning of reactive power generation (from DG and microgrids) and on the setting of OLTC transformers and capacitor banks. It uses the EPSO algorithm to optimize operating conditions according to the chosen objective function. This control scheme operates at the CAMC level, as defined previously in Section 2. The Generation Shedding scheme is a “last resort” control strategy based on the shedding of active power generation in DG units and microgrids. The shedding is proportional to the voltage deviation from the admissible limits in DG and microgeneration buses. This control strategy is operated at the MGCC level for the microgeneration shedding, following a setpoint received from the CAMC, or at the CAMC level considering DG shedding (also as defined previously in Section 2). At each main step of the algorithm, it is necessary to assess voltage levels at the MV network and at the LV network, since these two networks are consider decoupled in this analysis in order to relieve the computational burden of the algorithm. Several iterations of the algorithm may be performed until no voltage violations are found and the algorithm is terminated whenever voltage values reach admissible values at all buses or at pilot nodes, depending on the information available from the distribution system.

4. Test Networks In order to test this voltage control approach, two distribution networks were used: a MV distribution network and an LV distribution network and are shown in Fig. 3 and Fig. 4, respectively. The MV network is a rural network with 2 distinct areas with different voltage levels: 15kV and 30kV and a 30 kV/15 kV transformer at node 206. The total load of the MV network is around 6,3 MW. The LV network (microgrid) is also a rural network with a radial structure. All microgrids have the same structure that is shown in Fig. 3. The total load of the microgrid is around 100 kW.

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Figure 3. Algorithm for Volt/VAR Control.

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Figure 4. Rural MV Distribution Network (Multi-MicroGrid).

Scenarios with high DG and microgrid penetration were then developed in order to enable the analysis of the behaviour of the algorithm in extreme situations. The characteristics of the DG and microgrid generation used are summarized in Table 1. Note that a negative value in Table I means that the unit (DG or microgrid) is consuming power from the upstream network and a positive value means that the unit is injecting power into the upstream network. Table 1. DG and MicroGrid Characteristics Node

Type

P (MW)

5 36 37 64 72 199 3 16 61

µG µG µG µG µG µG DG DG DG

0,1 0,1 0,1 0,1 0,1 0,1 1,5 1,0 1,0

Pmax (MW) 0,2 0,2 0,2 0,2 0,2 0,2 1,5 1,0 1,0

Pmin (MW) 0 0 0 0 0 0 0 0 0

Q (MVAR) 0,01 0,01 0,01 0,01 0,01 0,01 0 0 0

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Qmax (MVAR) 0,04 0,02 0,02 0,04 0,04 0,04 0,8 0,4 0,4

Qmin (MVAR) -0,02 -0,02 -0,02 -0,02 -0,02 -0,02 -0,4 -0,2 -0,2

100

J.A. Peças Lopes and André Madureira The control variables used were: • • •

Reactive power at three DG buses (see Table 1) Reactive power at six microgrid buses, i.e. capacitor banks located at the MV/LV transformer substation of each microgrid (see also Table 1) OLTC settings at one transformer (node 206-2063)

Therefore, the control variables used are:

QμG1 | QμG 2 | QμG 3 | QμG 4 | QμG 5 | QμG 6 | QDG1 | QDG 2 | QDG 3 | t transf 1 where: QµGi – reactive power provided by microgrid i QDGi – reactive power provided by DG unit i ttransfi – transformer tap value for transformer i In addition, active power generation at DG and microgrid buses may be also controlled by means of generation shedding.

5. Main Results In this chapter, two different multi-objective functions have been considered:

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•

•

Objective Function 1 (OF1), which combines active power loss reduction (weight w1 = 0,33) with the maintenance of voltage profiles (weight w2 = 0,66) at all buses, assuming that voltage is monitored at all buses in the MV distribution system; Objective Function 2 (OF2) that combines active power loss reduction (weight w1 = 0,33) with maintenance of voltage profiles (weight w2 = 0,66) only at pilot buses, i.e. admitting that it is not possible to monitor voltage at all buses in the MV distribution system.

The weight concerning the maintenance of voltage profiles is dominant, since the main problem regarding this particular case is the high voltage profiles in the MV and LV networks due to the high penetration of DG and microgeneration. Hence, the stop criterion used was that the maximum voltage should not exceed 1,05 p.u. Only one turn of the algorithm was needed in order to achieve the proposed goal. For the EPSO algorithm used in these simulations, 10 particles (solutions) are used in each generation (iteration) and each particle is replicated 3 times, i.e. each parent “gives birth” to two children. The maximum number of generations is 100 generations.

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5.1. Objective Function 1 The objective function considered involves the maintenance of voltage profiles within admissible limits at all buses in conjunction with the reduction of active power losses. This case assumes the ability of monitoring voltage at all buses in the MV network. For this case, the main results obtained are presented in the next tables and figures. Table 2 shows the active and reactive power generation in the nodes with microgrids and DG and the setting of the OLTC transformer, i.e. the values of the control variables for this optimization problem. Note that a negative active power value P in Table 2 means that the microgrid is now consuming active power, instead of injecting active power such as in the base case situation. Table 2. Control Variables (OF1)

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Node No 5 36 37 64 72 199 3 16 61 206

Control Type QµG QµG QµG QµG QµG QµG QDG QDG QDG ttap

Q (MVAR) 0,0013 0,0074 -0,0025 0,0234 0,0010 0,0146 -0,2518 -0,0682 -0,0303 –

Qinitial (MVAR) 0,01 0,01 0,01 0,01 0,01 0,01 0 0 0 –

P (MW) -0,0764 0,1000 0,1000 0,1000 -0,0004 0,1000 1,1066 0,7590 0,8735 –

Pinitial (Mw) 0,1 0,1 0,1 0,1 0,1 0,1 1,5 1,0 1,0 –

t

tinitial

– – – – – – – – – 1,02

– – – – – – – – – 1,00

The V-Q control scheme based on the EPSO algorithm adjusted the reactive power generation and the transformer tap setting but this soft control strategy was not sufficient to ensure efficient operation within admissible voltage limits. Therefore, some generation shedding was also needed in order to aid the voltage support functionality. In total, 1,04 MW were shed (corresponding to approximately 25% of the 4,1 MW total DG and microgeneration). NO1

µG NO16

~ µG NO23

~ ~ µG NO22

Figure 5. Rural LV Distribution Network (MicroGrid).

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~ µG NO17

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The generation shedding occurred at all DG sources (DG1, DG2 and DG3), via a set-point sent by the CAMC to the DG units, and at two microgrids (µG1 and µG5), via the instructions sent by the corresponding MGCC of each microgrid (see grey cells in Table 2). Fig. 5 shows the bus voltage in the buses where the voltage was higher prior to the action of the voltage control functionality. It may be seen that the highest voltages can be significantly reduced with the application of the voltage support scheme. Fig. 6 shows the most relevant statistical data for voltage in the MV network.

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Figure 6. Ten Highest Voltages in the MV Network (OF1).

Figure 7. Voltage Profiles – Maximum, Average and Minimum Voltage – in the MV Network (OF1).

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Fig. 7 illustrates the evolution of the voltage in the most loaded feeder in terms of voltage values, before and after the volt/VAR control scheme. The total generation shed in this feeder was about 0,80 MW. It may also be seen that the voltage increases starting from the injector node (NO119) until the end of the feeder (NO3). This happens because of the high DG and microgeneration penetration.

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Figure 8. Voltage Values in the Worst Feeder in the MV Network (OF1).

Fig. 8 shows the active power losses in the MV network for each step of the control algorithm: V-Q soft control and generation shedding scheme. It may be seen that the losses rise in a first stage because of the improvement of the global voltage profiles. In a second stage, when generation is shed, the losses drop considerably. Apparently, this contradicts the general belief that the penetration of DG and microgeneration contributes to a reduction in active power losses. In fact, this is still valid. Although in this case, given the high initial penetration of DG and microgeneration (with a total of 4,1 MW) in face of the current demand (around 6,3 MW), the losses are higher. The behaviour of the losses as a function of DG capacity (considering the sites for DG and microgeneration defined in Section 4) is shown in Fig. 9. This plot shows that for DG and microgeneration penetration under 2,5 MW, the losses are reduced with the rise of the penetration level, but over that value begin increasing. Concerning the LV network, the voltage profiles in the six microgrids for the two steps of the control algorithm are presented in Table 3. It may be seen that the voltages do not exceed the limit of 1,1 p.u. considered as the maximum admissible value for voltage in LV networks (the most critical cases concerning the microgrids with higher voltage profiles that needed microgeneration shedding are highlighted in the table). In µG1, 24 kW (corresponding to 12%

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of the installed capacity) of microgeneration had to be shed and in µG5, approximately 100 kW (corresponding to 50% of the installed capacity) of microgeneration had to be shed, according to the instructions sent by the MGCC to each microgeneration unit.

Figure 9. Active Power Losses in the MV Network (OF1).

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Table 3. Voltage Values in the LV Network (OF1) LV Network Max V (p.u.) Avg V (p.u.) Initial Min V (p.u.) Max V (p.u.) Avg V (p.u.) V-Q Control Min V (p.u.) Max V (p.u.) Avg V (p.u.) Gen Shed Min V (p.u.)

µG1 1,21 1,14 1,07 1,19 1,12 1,05 1,03 1,02 1,01

µG2 1,11 1,03 0,95 1,10 1,02 0,94 1,10 1,02 0,94

µG3 1,11 1,03 0,95 1,10 1,02 0,94 1,10 1,02 0,94

µG4 1,10 1,02 0,95 1,09 1,01 0,94 1,09 1,01 0,94

µG5 1,19 1,11 1,04 1,17 1,09 1,02 1,10 1,05 1,01

µG6 1,10 1,02 0,95 1,09 1,01 0,94 1,09 1,01 0,94

Table 4 shows the active power losses in the six microgrids. It may be seen that the losses reduced significantly in the microgrids that had microgeneration shedding and increased only slightly in the others. This is once again because microgeneration penetration was too high and the location was not ideal for an efficient operation of the grid.

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Table 4. Active Power Losses in the LV Network (OF1) LV Network Initial V-Q Control Gen Shed

µG1 13,4 13,9 2,0

µG2 16,0 16,0 16,3

Plosses (kW) µG3 µG4 16,1 16,2 16,1 16,2 16,4 16,5

µG5 14,0 14,0 5,0

µG6 16,2 16,2 16,5

5.2. Objective Function 2

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The objective function considered involves the maintenance of voltage profiles within admissible limits at pilot buses in conjunction with the reduction of active power losses. This approach assumes that it is not possible to monitor voltage at all MV buses, so that pilot nodes have been defined, according to the methodology presented in Section 3.1.1, and voltage monitoring and control is possible at those nodes. The location of the pilot buses is marked with the number of the corresponding bus, defined according to the methodology described previously, is shown in Fig. 10.

Figure 10. Evolution of MV Network Active Power Losses as a Function of DG Installed Capacity.

For this case, the main results obtained are presented in the next tables and figures. Again, the V-Q control algorithm adjusted the reactive power generation and the transformer tap setting but this soft control strategy was not sufficient to ensure efficient operation within admissible voltage limits. Therefore, some generation shedding was also needed in order to aid the voltage support functionality. In total, 1,04 MW were shed (corresponding to approximately 25% of the 4,1 MW total DG and microgrid generation).

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The generation shedding occurred at all DG sources (DG1, DG2 and DG3), via a set-point sent by the CAMC to the DG units, and at two microgrids (µG1 and µG5), via the instructions sent by the corresponding MGCC of each microgrid (see grey cells in Table 5).

Figure 11. Pilot Buses in the MV Network (OF2).

Table 5. Control Variables (OF2) Node No 5 36 37 64 72 199 3 16 61 206

Control Type QµG QµG QµG QµG QµG QµG QDG QDG QDG ttap

Q (MVAR) -0,0091 -0,0087 -0,0022 0,0266 -0,0014 0,0116 -0,3099 -0,1049 0,2462 –

Qinitial (MVAR) 0,01 0,01 0,01 0,01 0,01 0,01 0 0 0 –

P (MW) -0,0723 0,1000 0,1000 0,1000 -0,0107 0,1000 1,1236 0,7673 0,8584 –

Pinitial (Mw) 0,1 0,1 0,1 0,1 0,1 0,1 1,5 1,0 1,0 –

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t

tinitial

– – – – – – – – – 1,02

– – – – – – – – – 1,00

A Voltage Control Optimization for Distribution Networks…

107

Fig. 11 shows the bus voltage at the pilot buses. It may be seen that the highest voltages can be significantly reduced but at the expense of the lowest voltages that also have to decrease. This behaviour is expected, as the main problem with this network is the general high voltage profile. Fig. 12 shows the most relevant statistical data for voltage in the MV network.

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Figure 12. Voltages at the Pilot Buses in the MV Network (OF2).

Figure 13. Voltage Profiles – Maximum, Average and Minimum Voltage – in the MV Network (OF2).

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Fig. 13 illustrates the evolution of the voltage in the most loaded feeder in terms of voltage values, before and after the volt/VAR control scheme. The total generation shed in this feeder was about 0,80 MW. Fig. 14 shows the active power losses in the MV network. Again, the losses develop similarly to what happens in the previous case and for the same reason.

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Figure 14. Voltage Values in the Worst Feeder in the MV Network (OF2).

Figure 15. Active Power Losses in the MV Network (OF2).

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Concerning the LV network, the voltage profiles in the six microgrids for the two steps of the control algorithm are presented in Table 6. It may be seen that the voltages do not exceed the limit of 1,1 p.u. considered as the maximum admissible value for voltage in LV networks (the most critical cases concerning the microgrids with higher voltage profiles that needed microgeneration shedding are highlighted in the table). In µG1, 28 kW (corresponding to 14% of the installed capacity) of microgeneration had to be shed and in µG5, approximately 90 kW (corresponding to 45% of the installed capacity) of microgeneration had to be shed, according to the instructions sent by the MGCC to each microgeneration unit. Table 6. Voltage Values in the LV Network (OF2) LV Network Max V (p.u.) Avg V (p.u.) Min V (p.u.) Max V (p.u.) Avg V (p.u.) Min V (p.u.) Max V (p.u.) Avg V (p.u.) Min V (p.u.)

Initial

V-Q Control

Gen Shed

µG1

µG2

µG3

µG4

µG5

µG6

1,21 1,14 1,07 1,20 1,12 1,05 1,04 1,02 1,01

1,11 1,03 0,95 1,10 1,02 0,94 1,10 1,02 0,94

1,11 1,03 0,95 1,10 1,02 0,94 1,10 1,02 0,94

1,10 1,02 0,95 1,10 1,01 0,94 1,09 1,01 0,94

1,19 1,11 1,04 1,17 1,10 1,03 1,09 1,04 1,01

1,10 1,02 0,95 1,09 1,01 0,94 1,09 1,01 0,94

Table 7 shows the active power losses in the six microgrids. It may be seen that the losses reduced significantly in the microgrids that had microgeneration shedding and increased only slightly in the others.

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Table 7. Active Power Losses in the LV Network (OF2) LV Network Initial V-Q Control Gen Shed

Plosses (kW) µG1 13,4 13,8 1,8

µG2 16,0 16,3 16,3

µG3 16,1 16,4 16,4

µG4 16,2 16,5 16,5

µG5 14,0 14,3 4,2

µG6 16,2 16,5 16,5

Comparing the results obtained exploiting the possibility of monitoring voltage at all buses (OF1, presented in Section 5.1) and monitoring voltage only at pilot buses (OF2, presented in Section 5.2), one can conclude that the results are extremely close. In particular, voltage can be brought back to an admissible range and the total amount of generation shedding is the same in both situations. This means that it is in fact sufficient to monitor voltage at only specific nodes, given that the selection procedure is adequate as the one that was proposed here using electrical distance calculation.

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6. Conclusion

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In conclusion, voltage support functionalities in distribution systems integrating DG sources and microgrids may be implemented with great benefits to the DSO. The voltage control problem must be seen in a larger perspective instead of a local perspective (despite the fact that voltage is often considered a local problem) especially in cases where there is high penetration of DG and microgeneration. In these extreme cases, reactive power control is not sufficient to maintain secure operation and therefore drastic measures may have to be taken that involve generation shedding. The algorithm developed comprises two stages: a “soft control” strategy based on reactive power management (through the control of DG and microgeneration reactive power, transformer taps and capacitor banks) and a “last resort” control strategy based on active power generation shedding. The combination of these two strategies was shown to be essential to the operation of these new distribution systems, particularly in the presented case, where DG and microgeneration penetration is extremely high and their location is not conveniently chose. It was also seen that the most relevant objectives for these cases are reducing active power losses and maintaining voltage within admissible ranges. The objectives, however, may be contradictory in some situations. Consequently, these two objectives were weighed in the simulations performed, and a global goal was achieved. Finally, voltage may be monitored and controlled at all buses or only at specific buses, which are chosen by a method such as electrical distance to select pilot nodes to control voltage, depending on the information available regarding voltage profiles in the MV network. By comparing the two strategies, it was seen that the performance is quite similar for both cases, which should mean that the selection of pilot nodes was efficient in order to find buses that are representative of a certain area.

Annex I. Electrical Distance Calculation The concept of electrical distance involves the matrix [∂Q/∂V], which is a part of the Jacobian matrix J, and its inverse [∂V/∂Q], called the sensitivity matrix. The magnitude of voltage coupling between two buses may be quantified by the maximum attenuation of voltage variation among these two buses (using matrix [∂V/∂Q]). A matrix of attenuations between all the buses of the electrical system, whose terms are written as αij is then available and we have ΔVi = α ij ⋅ ΔV j , where

⎛ ∂V ⎞ ⎛ ∂V j ⎟÷⎜ ⎟ ⎜ ∂Q j ⎠ ⎝

α ij = ⎜⎜ i ⎝ ∂Q j

⎞ ⎟ ⎟ ⎠

A step-by-step method to obtain separate Voltage Control Areas is presented next: 1. Calculate the Jacobian matrix J and hence obtain the sub-matrix J4, where

J 4 = [∂Q / ∂V ]

⎡ ∂V ⎤ −1 = J4 ⎥ ⎣ ∂Q ⎦

2. Invert J4. Say B = ⎢

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A Voltage Control Optimization for Distribution Networks… 3. The elements of matrix B are written as bij, where bij =

∂Vi ∂Q j

4. Obtain attenuation matrix, αij, between all the nodes as follows

(

5. Calculate electrical distances Dij as Dij = − log α ij , α ji

111

)

α ij =

bij b jj

Once electrical distances for any couple of nodes are defined, it is possible to define the boundary of the Voltage Control Areas. Voltage Control Areas are then defined by applying the minimax criterion to the Electrical Distances matrix Dij. This method considers one Voltage Control Area per generator node (microgrid or DG bus) and groups each generator node with the load nodes that are “electrically closer” to it.

References [1]

[2]

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[3]

[4]

[5]

[6]

[7]

[8]

[9]

J. A. Peças Lopes, Â. Mendonça, N. Fonseca, L. Seca, “Voltage and Reactive Power Control Provided by DG Units”, CIGRE Symposium: Power Systems with Dispersed Generation, Athens, Greece, April 2005. J. A. P. Lopes, C. L. Moreira, A. G. Madureira, “Defining Control Strategies for MicroGrids Islanded Operation”, IEEE Transactions on Power Systems, vol. 21, no. 2, May 2006. J. A. P. Lopes, C. L. Moreira, A. G. Madureira, et al, “Control Strategies for MicroGrids Emergency Operation”, International Conference on Future Power Systems (FPS2005), Amsterdam, The Netherlands, November 2005. A. G. Madureira, J. A. Peças Lopes, “Voltage and Reactive Power Control in MV Networks integrating MicroGrids”, International Conference on Renewable Energy and Power Quality (ICREPQ’07), Seville, Spain, March 2007. P. N. Vovos, et al., “Centralized and Distributed Voltage Control: Impact on Distributed Generation Penetration”, IEEE Transactions on Power Systems, vol. 22, no. 1, February 2007. P. Lagonotte, J. Sabonnadière, J. Léost, J. Paul, “Structural Analysis of the Electrical System: Application to Secondary Voltage Control in France”, IEEE Transactions on Power Systems, vol. 4, no. 2, May 1989. J. Zhong, E. Nobile, A. Bose, K. Bhattacharya, “Localized Reactive Power Markets Using the Concept of Voltage Control Areas”, IEEE Transactions on Power Systems, vol. 19, no. 3, August 2004. J. Kennedy, R. C. Eberhart, “Particle Swarm Optimization”, IEEE International Conference on Neural Networks, Pert, Australia, IEEE Service Center, Piscataway, NJ, 1995. H. P. Schwefel, "Evolution and Optimum Seeking", Ed. Wiley, New York, 1995.

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[10] V. Miranda, N. Fonseca, “EPSO – Evolutionary particle swarm optimization, a new algorithm with applications in power systems”, IEEE/PES Transmission & Distribution Conference and Exhibition: Asia Pacific, October 2002. [11] V. Miranda, N. Fonseca, “New evolutionary particle swarm algorithm (EPSO) applied to voltage/VAR control”, 14th Power Systems Computation Conference (PSCC'02), Seville, Spain, June 2002.

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In: Optimization Advances in Electric Power Systems ISBN: 978-1-60692-613-0 Editor: Edgardo D. Castronuovo, pp. 113-150 © 2008 Nova Science Publishers, Inc.

Chapter 6

TOOLS FOR THE EFFECTIVE INTEGRATION OF LARGE AMOUNTS OF WIND ENERGY IN THE SYSTEM Jorge Martínez-Crespo, Jorge L. Angarita, Edgardo D. Castronuovo, Hortensia Amaris and Julio Usaola García Department of Electrical Engineering, Carlos III de Madrid University, Av. de la Universidad 30, 28911 Leganés, Madrid, Spain

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Abstract Optimization techniques are used for solving a large amount of engineering problems, and in particular for power system analysis and planning. With the increasing importance of intermittent power generation, these tools must be adapted to take into account the random nature of these new power sources. This chapter proposes new techniques aimed at facilitating their integration into the existing grids. Several issues are addressed, and in the first place the market integration of wind energy, alone or in combination with other kinds of generation, like hydro and pumping plants. The aim of this part is to reduce the cost of the power imbalance due to the low predictability of the wind resource. The association of wind energy with the other mentioned energy sources could reduce this cost and, hence, encourage the wind energy without loss of the system security levels or increase in system costs. Then follows a proposal for optimal management of clusters of wind farms for enhancing the system security. The chapter ends with a strategy for optimizing the system voltage stability when wind generation is present. The technical features of wind generation are taken into account, and measures to enhance the overall stability are also proposed and the improvement quantified.

1. Introduction Wind power systems have massively spread out around the world in the last decade. The growth in Europe has been encouraged by the EU Directive 2001/77/EC [1]. The goal of this directive is to increase the gross energy consumption supplied by renewable energy sources

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up to 12% in 2010. The growth is so significant that in some countries, like Spain, wind power provides around 10% of the power demand −reaching some days up to 23% of the demand−, and is the 15% of the total installed power. This high level of wind energy expansion demands new technical and economic tools. In the present chapter, some of tools used for the effective integration of large amounts of wind energy in the system are presented. In the second section, the optimal bid to be submitted by a wind farm to the market is estimated, considering a hydro-wind ensemble. The results are obtained in a realistic case, showing the benefits of the proposed approach. In the third section, the coordination among wind farms and pumped water stations is analyzed, when facing output restrictions in the wind power generation. Formulation and recent developments in this area are presented. In the following fourth section, delegated dispatches of renewable producers are studied. System Operators (SO) are implementing delegated dispatches to manage settings imposed by the SO in critical situations to the renewable generators of a region. Finally, the last fifth section deals with calculation and enhancement of the voltage stability in systems with large penetration of wind power generation. All the presented tools are formulated as optimization problems, with an exhaustive explanation of the variables, obtained results, etc.

2. Wind Generation Bids in Pool-Based Electricity Markets 2.1. Nomenclature The notation used throughout this section of the chapter is stated as follows:

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Constants: mi ,l Slope of block l of plant i in the hydro unit performance curve UPC [MW/m3/s].

U i ,l

Average slope of plant i in the hydro unit performance curve UPC [MW/m3/s]. Conversion factor, from water discharged to volume, equals 3.6×10-3 [Hm3s/ m3]. Maximum water discharge of block l of plant i [m3/s].

Ui Ui

Maximum water discharge of plant i due to technical constraints [m3/s]. Minimum water discharge of plant i due to technical constraints [m3/s].

mj M

ext

Maximum water discharge of plant i in period t due to external constraints [m3/s].

U i ,t

ext

Minimum water discharge of plant i in period t due to external constraints [m3/s].

Xi Xi

Maximum content of the reservoir associated with plant i [Hm3]. Minimum content of the reservoir associated with plant i [Hm3].

Wi ,t

Forecasted natural water inflow of the reservoir associated with plant i in period t [Hm3/h].

λt

Forecasted energy price in period t [€/MWh].

τ ij

Time required so that the water flows from reservoir j to reservoir i [h].

ψ t , down

Penalty for down deviation, as a percentage of market price in hour t.

ψ t ,up

Penalty for up deviation, as a percentage of market price in hour t.

U i ,t

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Tools for the Effective Integration of Large Amounts of Wind Energy…

γ sci

115

Future value of the stored water [€/Hm3]. Start cost of plant i [€].

Variables EHWP Expected hydro-wind profit [€]. Expected hydro profit in period t associated with outcome w [€]. EHPt , w EWRt , w

Expected wind revenue in period t for outcome w [€].

EPt , w

Expected penalty for the power deviation in period t associated with outcome w [€].

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hpri ,t , w Hydro power generated by plant i in period t associated with outcome w [MW]. Qi , w

Future value of the stored water in the reservoirs associated with plant i and outcome

Rt , w

w [€/Hm3]. Power imbalance in period t for outcome w [MW].

si ,t , w

Spillage of the reservoir associated with plant i in period t for outcome w [m3/s].

ui ,t ,l , w

Water discharge of plant i in period t for block l associated with outcome w [m3/s].

xi ,t , w

Content of the water reservoir of plant i in period t associated with outcome w [Hm3].

xi ,TF , w

Content of the water reservoir of plant i for outcome w and at the end of the

yt ,i , w

scheduling period [Hm3]. Binary variable equals 1 if plant i is started at the beginning of period t.

zt , i , w

Binary variable equals 1 if plant i is shut down at the beginning of period t.

μt , i , w

Binary variable equals 1 if plant i is on-line at the beginning of period t.

hwpdt

Hydro-wind power declared in the bid for period t [MW].

Stochastic Outcomes: ewpt , w Wind power value in period t associated with outcome w [MW]. ewptn

One of the possible values of the expected wind power during period t [MW]

ρw

Probability for outcome w.

Sets: I

Θi

Set of indices of the plants belonging to the same river basin and the same company. Set of indices of blocks of piecewise linearization of the unit performance curve. Set of indices of the periods of the market time horizon. Set of power plants downstream from plant i.

Ωi

Set of power plants immediately upstream from plant i.

L T

Types of variables: ui ,t ,l , w , si ,t , w ≥ 0 ∀i ∈ I , ∀t ∈ T , ∀l ∈ L, ∀w ∈ W yi,t ,w , zi,t , w , μi,t , w ∈{0,1}

∀i ∈ I ,

∀t ∈T , ∀w ∈W

The types of the other variables can be deduced from the variables defined in this list.

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2.2. Justification In order to take part in an electricity market, wind farms will submit price-energy bids and as result of the market, wind generators will have to deliver the arranged amount of energy in each hour of the programming horizon. Nevertheless, random behavior of wind resource makes difficult to predict accurately the wind power output [2], [3] and this uncertainty entails energy imbalances regarding the power committed in pool-based electricity markets. These imbalances usually result in economic penalties [4]-[7]. As a matter of fact, if the actual delivered energy differs from the committed energy in the electricity market, other generating units must also change their schedule in order to keep the generation-load balance. The cost of this re-scheduling must be paid by those that cause it, e. g. wind generators among others. The undispatchable nature of the wind generation makes necessary to forecast the power that the wind farm will deliver in the next hours. One way to reduce the expected imbalance is to use short term wind power prediction programs [4]. The accuracy of these programs depends on the elapsed time between the prediction and the operation; so that the elapsed time is shorter, the accuracy is better. The accuracy of these tools can be also found in [8]. Studies about market integration of wind energy have appeared in literature only recently. In [9], a strategy for wind producers in order to submit bids under BETTA rules is presented. These rules allow the presentation of bids only a few hours before the operation time, making less necessary the prediction tools. Nevertheless, most of the countries follow different rules. The paper [10] analyzes the revenue of Spanish wind farms under conditions slightly different from present ones. In [11], a study using Dutch electricity market rules is presented. In this case, energy bids are presented once a day and are not updated in successive markets. The article [12] carries out a general study via the Spanish market rules under different assumptions, and presenting one bid for each day, with different anticipation times before operation. This paper does not use any actual prediction tool, but it works with average accuracies. In [13], an analysis of the benefits of using short-term wind power prediction tools in an electricity market is presented. Finally, in the work presented here, the optimal wind power bid will be determined for an existent wind farm. One hundred different scenarios of penalty are considered, and the optimal wind bid will be calculated for each penalty scenario. The forecasting tool SIPREOLICO is used for obtaining an estimated wind power value. The results achieved for its forecasted wind power -used as bid- are compared with those obtained for the optimal wind power bid, which is result of the optimization model. As well as the use of prediction tools to improve the wind power bids, another method for reducing the expected imbalance cost is to work together with other types of generating units, like a Hydro Generation Company (HGENCO). Some references use a pure hydro system or a hydro generation/pumping system pooled with a Wind Generation Company (WGENCO) to provide the committed power in the electric power system [14]-[18]. In these cases, the objective is to find the optimal hydro or hydro/pumping operation in order to provide a reliable energy supply. In [19] and [20], a wind and pumping ensemble system is considered in a deregulated market. In both research papers, starting from a deterministic and wellknown wind generation forecasting, optimal hydraulic power generation and water pumping are found. This algorithm is enhanced in [21], where the expected wind generation is considered as a stochastic parameter.

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In this work, the energy imbalances for a deregulated market are dealt in a different way. The WGENCO-HGENCO ensemble (WH-GENCO) tries to define its optimal bid by maximizing the joint profit, so that the company only will reduce imbalances if it increases its profit. In this case, just like the aforementioned individual wind power bid, the market incentive will be defined by the penalty price of imbalances. Therefore, depending on the energy to be traded on the market, the HGENCO plans its power reserve and, consequently, its availability to reduce imbalances generated by wind power. In accordance with the plan, the decision is taken considering the expected wind power distribution probability, which is modeled as a random variable. This bid design strategy sets the optimal energy to be submitted to the market with the aim of getting the maximum profit for the WH-GENCO joint optimal bid. The WH-GENCO ensemble will be considered a price-taker and a detailed hydro model is used. The proposed algorithm is a mixed-integer (0/1) linear problem which has been solved under GAMS mathematical modelling language using the solver CPLEX 9.0 [22]. This section of the chapter is organized as follows. Considerations about wind power forecast and the formulation of the mathematical model for defining the individual wind power bid are shown in Section 2.3. Section 2.4 defines the mathematical model of the combined hydro-wind bid. Section 2.5 provides an application example for the individual wind power bid and analyzes the results obtained for different penalty scenarios. Finally, a new study case is presented in Section 2.6, and the obtained results are evaluated.

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2.3. Mathematical Model for the Individual Wind Power Bid In power systems with a high percentage of wind power, short-term wind power forecast has become a technique used for system operation and for presenting bids in electricity markets. By means of physical and/or statistical models, hourly predictions for a time horizon of about 48 hours are provided. The accuracy of these prediction tools is assessed through the Normalized Mean Absolute Error (NMAEt), defined as shown in (1), where predictions have been made t hours before operation time, N being the number of predictions considered. t=N ∑ actual powert − forecasted powert 1 NMAE = ⋅ t =1 N rated power

(1)

Fig. 1 shows average values of this parameter. Two main features can be observed: first, the higher the time elapsed between prediction and operation is, the higher the forecasting errors; second, the error decreases for an ensemble and with the size of the ensemble. Similar conclusions can be found in [2] and [3]. The results in Fig. 1 have been obtained with SIPREÓLICO, the short term wind power prediction tool running in REE (the Spanish TSO) and developed by Universidad Carlos III de Madrid. This program provides predictions every 15 minutes for more than 14000 MW of wind power connected to the Spanish power system [5].

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45 1 WF 3 WF 5 WF 7 WF

40 35

NMAE [%]

30 25 20 15 10 5 0

0

5

10

15

20 25 Time ahead [h]

30

35

40

Figure 1. NMAE for different wind farm ensembles.

In general, the quality of a prediction depends mainly on two variables, the time between prediction and operation and the forecasting technique. Another important prediction factor is the forecasting process, which has no important bias. Therefore, in general, the probability of over/under prediction might be considered equal. In this document, the hourly wind power output is considered a random variable with a known discrete density function (DDF). The set of values for the DDF at time t is Ω t = ewpt1 , ewpt2 ,..., ewptNt with probability { ρt1 , ρt2 ,..., ρtN } [23]. This hourly information is Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

{

}

t

necessary for calculating the overall time outcomes. Every overall-period outcome is a set that has a possible wind generation value for every hour. This study considers all the hourly sets Ωt to contain N possible outcomes, i.e., Ωt = N t = N ∀t . The assumed hypotheses for this study are the following: •

Four different bidding strategies are going to be compared: o Expected wind power bid (ewpt), which is calculated as follows: ewpt =

o

• •

j=N

∑P j =1

gen , j , t

⋅ ρ ( Pgen, j ,t )

High probability wind power bid (hpwpt), whose value equals: hpwpt = Pgen, j ,t ρ ( Pgen , j ,t ) = max ρ ( Pgen, j ,t )

o Forecasted wind power bid (fwpt). o Best (optimal) power wind bid (bwpt). One hundred hourly penalty scenarios are considered. The optimal hourly wind power bid will be calculated for every penalty scenario. The prediction of the electricity market prices is supposed perfect.

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• • •

119

SIPREÓLICO has been the prediction tool used for performing the prediction. The prediction tool makes new prediction from available data (wind forecasts and realtime production) for every hour. Forecasted wind power bid (fwpt) is provided by SIPREÓLICO. Twenty different generation values are considered in each hour for stochastic wind power generation. Expected wind power bid (ewpt) and high probability wind power bid (hpwpt) do not change with penalty value. Once the optimal wind power bid is known, a penalty calculation process according to the really generated wind power is developed taking into account the four different bidding strategies previously mentioned.

The revenue of a wind farm is calculated using the formula (2). At the time of the electricity market, a prediction is generated for the next hours, and this prediction has an uncertainty estimated from past data. Then, the revenue for a given time could be expressed as a function of the power bid (or traded) in the market, and the power actually generated, as R(bwpt,Pgen), where bwpt, and Pgen are bid vectors and generated powers for a whole period (one day, in our case). The aim of the problem is to obtain the value of bwpt that maximizes the revenue for a given set of energy prices and known penalties, and the uncertainty of the prediction. j=N

{

(

max ∑ R ( bwpt , Pgen , j ) ⋅ ρ Pgen , j Ppred ( Δt ) Plast

j =1

R (bwpt , Pgen , j ) = ( bwpt − Pgen , j ) ⋅ψ up

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R (bwpt , Pgen , j ) = ( Pgen , j − bwpt ) ⋅ψ down

)}

∀t

bwpt − Pgen, j ≥ 0

if if

Pgen, j − bwpt ≥ 0

(2)

(3)

N is the number of bins that have been considered in the uncertainty probability density function, Ppred(Δt) is the prediction of the wind power, that was produced Δt hours before the actual operation time, and ρ(Pgen,j|Ppred(Δt)) is the probability of the generated power to take the value Pgen,j when the prediction Δt hours before has been Ppred. Equation (3) is a very simple way to define the penalty function R(bwpt,Pgen). ψ up , ψ down are the penalty values for long (over production) and short (under production) generations, respectively. Actually, in some energy markets as Nordpool or Spanish market, the penalties are not fixed. They depend on the hourly system position (over or under production). It is necessary to take into account that the revenue maximization is equivalent to the penalties minimization.

2.4. Mathematical Model for the Combined Bid The hydro model is more complex than the wind model. The aim of the short-term hydro scheduling model is to define the optimal generation programming for every plant in the river basin.

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The input-output hydro generation function describes the relationship between discharged water and generated power (Hill Chart). This relationship is strongly non-linear but it can be represented through a concave piecewise linearization [24]-[26], as shown in Fig. 2. A polynomial approximation is considered in reference [27], whereas other references [28], [29] use a non-concave piecewise linearization. Fig. 2 draws the relationship between power and water discharge for a hydro plant i. This relationship has been represented in this work by a piecewise linear approximation with 4 blocks. Every piece has its own slope and water discharge limit. The hydro model considered is based on the model proposed in [29].

Figure 2. Concave piecewise linearization of Hill Chart.

The following equations represent the model for defining a HW-GENCO combined bid. This strategy consists of calculating the optimal combined hourly energy, hwpdt. The model defines the optimal power to be traded, calculating its expected profit for every wind outcome. In the present model, the energy trade price and the penalty prices are perfectly known. The objective function to maximize is: t =T w =W

Max. EHWP = ∑ ∑ ρ w ⋅ { EHPt ,w + EWRt ,w − EPt ,w } t =1 w =1

(4)

subject to: i=I

E H Pt , w = ∑ { λt ⋅ h p ri,t , w − sc i ⋅ y t , i , w + Q i , w } i =1

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∀ t ∈ T , ∀ w ∈ W (5)

Tools for the Effective Integration of Large Amounts of Wind Energy… l=L

hpri ,t , w = ∑ mi ,l ⋅ ui ,t ,l , w

∀i ∈ I , ∀t ∈ T , ∀w ∈ W

121 (6)

l =1

l=L

l=L

h∈Ωi l =1

l =1

xi ,t , w = xi ,t −1, w + Wi ,t + M ⋅ ∑ ∑ {uh ,t −τ ih ,l , w + sh ,t −τ ih ,l , w } − M ⋅ ∑ {ui ,t ,l , w + si ,t ,l , w }

(7)

∀i ∈ I , ∀t ∈ T , ∀w ∈ W

X i ≥ xi ,t , w ≥ X

{

m a x U i ,U

ext i ,t

l=L

} ≤∑u l =1

i ,t ,l ,w

∀i ∈ I , ∀t ∈ T , ∀w ∈ W

i

ui ,t ,l , w ≤ U i ,l ⋅ μt ,i , w Qi , w =

{

≤ m in U i , U

}

∀ i ∈ I ,∀ t ∈ T ,∀ w ∈ W

∀i ∈ I , ∀t ∈ T , ∀l ∈ L, ∀w ∈ W

j=I

⎪⎧ xi ,TF , w ⋅ m j ⋅ γ

j∈Θi

⎩

∑ ⎨⎪

yt ,i , w − zt ,i , w = μt ,i , w − μt −1,i , w

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ext i ,t

⎪⎫ M ⎬⎪ ⎭

∀i ∈ I , ∀w ∈ W

∀i ∈ I , ∀ t ∈ T , ∀ w ∈ W

(8)

(9)

(10)

(11)

(12)

EWRt , w = ewpt , w ⋅ λt

∀t ∈ T , ∀w ∈ W

(13)

EPt , w = Rt , w ⋅ λt ⋅ψ t

∀t ∈ T , ∀w ∈ W

(14)

⎧ ⎫ ⎡ i=I ⎤ ⎪ ψ t ,up if ⎢ ∑ {hpri ,t , w } + ewpt , w − hwpdt ⎥ > 0 ⎪ ⎪ ⎣ i =1 ⎦ ⎪ ψt = ⎨ ⎬ i=I ⎡ ⎤ ⎪ψ ⎪ + − < if hpr ewp hwpd 0 t ,w t⎥ ⎢ ∑ { i ,t , w } ⎪⎩ t , down ⎪⎭ ⎣ i =1 ⎦

(15)

It should be noticed that time periods of one hour are considered, so x MW is equivalent to x MWh and a water discharge k represents a change of k M Hm3 in the volume. Notice also that the problem aims to find the optimal power that provides the maximum expected revenue for a joint optimal operation. Equation (4) shows the objective function. The profit is calculated as the revenue for the energy produced by both companies, HGENCO and WGENCO, minus the penalties due to the energy deviations. Equation (5) sets the HGENCO profit: the revenue due to the produced energy plus the future revenue as a consequence of stored water in the reservoir minus startup costs. Note that the hydro generation model sets a certain water price at the end of the time horizon as a constraint. On the contrary, other references set limits on the final volume [18]. Equation (6) defines the piecewise linear hydro generation model. Constraint (7) establishes the temporal continuity of the hydro reservoir level. Equation (8) sets the lower and upper

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limits on the reservoir level. Constraint (9) fixes the limits on the discharge for every plant which can come from either technical constraints on the power plant discharge or external constraints like environmental considerations, navigability conditions, fishing constraints, etc. The upper discharge limit of each block of the piecewise linear curve is fixed by equation (10). Equation (11) models the future value of the stored water in the reservoirs. Constraint (12) sets a logical relation between power plant start-up, shut-down and on-line status and Equation (13) defines the expected profit for the WH-GENCO. Finally, the expected penalty in the operation is modeled in (14) and (15). The optimal bid maximizes the weighted-probability profit for each overall-period outcome. That is, for every possible wind outcome, the WH-GENCO ensemble decides the best hydro generation. A hydro power generation different from its optimal value (a value obtained when the HGENCO operates individually) always involves a profit reduction for the HGENCO. Therefore, the HGENCO will only support the wind power deviations if the hydro profit reduction is lower than the avoided wind penalty. The optimization problem for defining HGENCO generation is constrained by giving a value to the water remaining at the end of time horizon Qi , w . As is stated in [30], in general, a hydro power station i will consume water to generate power only those blocks that satisfy the following equation:

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λt ⋅ mi ,l ≥ γ ⋅ mi

Figure 3. Optimal hydro schedule for time t.

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(16)

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Equation (16) states that the HGENCO turbines water only when the revenue due to the power generated is higher than the future revenue of storing it. As the Hill Chart is a piecewise linearization, equation (16) is only fulfilled up to a certain block l of that piece-wise linearization, but it is not valid for the next ones. The necessary conditions for defining the optimal hydro power bid hpri*,t , w are shown in Fig. 3 for a three-block linearization. The x and y axes represent the water turbined and the power generated, respectively. With regard to wind power model, in addition to the aforementioned considerations in previous section, it is necessary to consider that the hourly decisions are coupled. Thus, the set Ω , which represents the total number of outcomes for the whole time horizon, will t =T

t =T

t =1

t =1

contain W = ∏ Ωt = ∏ N = N T elements. The probability ρw of every outcome w is defined by multiplying the probabilities associated with every possible hourly power output in the overall-period outcomes.

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2.5. Results for Individual Wind Power Bid It is going to show the results for a wind power plant whose rating power is 15 MW. There are one hundred different penalty factors. The up-deviation penalty ranges from 1 to 6, but its value is equals 1 up to the scenario 50 and from here up to the end, it is changing its value in increasing order. On the contrary, the up-deviation penalty ranges in decreasing order from 5.9 to 1 along all the hourly penalty scenarios. Its value is equal to 1 during the last fifty scenarios. It should be noted that twenty different generation values are considered in each hour for stochastic wind power generation. Their values are (0; 0,05; 0,1; 0,15...;1 } p.u. Given that a single day is an insufficient time period to evaluate the penalty costs versus generated wind power, this exercise is extended a 44-day period. It is supposed that the generated wind power is known for those 44 days and is different for every one. As was mentioned in Section 2.3, four different bidding strategies are going to be compared. These strategies are: expected wind power bid (ewpt), high probability wind power bid (hpwpt), forecasted wind power bid (fwpt) and optimal power wind bid (bwpt). Firstly, the calculation procedure consists of the computation of optimal hourly wind power bid according to the equation (2). It is going to be calculated for every penalty scenario. Once the optimal wind power bid is known, a calculation process of the imbalance costs determined by the penalty factors is developed for each bidding strategy. Take notice that these costs are calculated by comparing the selected bid (ewpt, hpwpt, fwpt or bwpt) with the really generated wind power. Fig. 4 summarizes the results and shows the penalty results (y axis) for the different generation scenarios (x coordinates). The scenarios 1 and 100 correspond to large differences between up-deviation and down-deviation penalties (relation 1:5.9 and 6:1, respectively). The scenario 50 is referred to an up-deviation and down-deviation penalty relation equals 1:1. The results show that the higher the up-deviation deviation is, the higher the optimal wind power bids are. bwpt ranges from 0.1 (up to the scenario 30) to 0.4 (last penalty scenarios in almost all the time periods). As was pointed out in Section 2.3, expected wind

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power bid (ewpt) and high probability wind power bid (hpwpt) always have the same value independently of the penalty value. As can be observed in Fig. 4, the optimal power wind bid defines the lower penalty costs for all the possible penalty scenarios. Nevertheless, when the up and down penalty factors are the same, the importance of using an optimization tool diminishes (the expected wind power value could be offered), but it is not the case for unbalanced penalty scenarios. It is due to the fact that the generation error distribution is centred on zero, that is to say, up and down generation errors are the same.

Penalty costs

450 400 350 ewp

300 250 200 150

hpwp fwp bwp

100 50 0 0

20

40

60

80

100

Penalty scenarios

Figure 4. Penalty costs for all the penalty scenarios and considering the four bid strategies.

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2.6. Results for Combined Hydro-wind Power Bid 2.6.1. Hydro Generation Data The HGENCO considered in this test system has eight reservoirs linked between them. The example has been taken from [29]. Table 1. Blocks Slope and Water Discharge Limits Slope [MW/m3/s]

Umax [m3/s]

block 1

block 2

block 3

block 4

all blocks

Maximal Power Output [MW]

Plant 1

0.9

0.85

0.83

0.78

15.0

28.62

Plant 2

0.75

0.73

0.7

0.67

39.5

69.52

Plant 3

0.7

0.65

0.63

0.6

112.5

139.05

Plant 4

0.82

0.8

0.77

0.75

160.8

116.38

Plant

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Table 1. Continued Slope [MW/m3/s]

Umax [m3/s]

block 1

block 2

block 3

block 4

all blocks

Maximal Power Output [MW]

Plant 5

0.85

0.82

0.8

0.75

29

186.66

Plant 6

0.78

0.73

0.7

0.68

29.5

833.28

Plant 7

0.75

0.72

0.7

0.66

152.5

1159.63

Plant 8

0.8

0.76

0.74

0.72

116.25

550.9

Plant

Table 1 illustrates three items of the test system: maximum water discharge for every plant, slope of every block, and maximum power output. The future value of water is 60 €/MWh. Table 2 shows the hourly power output for the case in which the HGENCO submits a bid considering only its own revenue [29]. This information will be useful in order to compare these data with the results of the optimal combined bid model. Hourly prices (see last row of Table 2) correspond to average prices in the Spanish daily electricity market. As can be checked, the higher the prices are, the higher the generated power is. Table 2. Power schedule (MW) and hourly price (€/MWH)

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Plants

Generation Schedule for all the plants [MW] hour 1 hour 2 hour 3 hour 4 hour 5

Plant 1

0

0

27.69

51.84

14.94

Plant 2

0

0

60.36

114.47

0

Plant 3

0

0

154.58

292.95

0

Plant 4

0

0

0

506.68

0

Plant 5

0

0

0

0

0

Plant 6

0

0

0

0

0

Plant 7

453.09

453.09

0

453.085

453.09

Plant 8

0

0

205.3

375.03

116.95

Total

453.09

453.09

901.01

1794.06

584.98

Price [€/MWh]

50

55

60

65

58

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2.6.2. Wind Generation Data The rated power of the wind farm is 250 MW. This power could be considered as the product of an aggregation of individual wind farms. For stochastic wind power generation, every hour contains three different generation scenarios, Ω t = 3 ∀t . Therefore, the total number of t =5

outcomes for the scheduling horizon are W = ∏ Ωt = 243 . t =1

Table 3 shows hourly wind power values and their probabilities. Two different bid strategies can be used depending on the wind power offered: the highest probability wind power value (HPWPV strategy) or the expected wind power value (EWPV strategy). As it happens in hydro generation data (hourly power output), these wind power values are going to be useful to be compared with the optimal combined bid. Table 3. Wind power outcomes

Time

Power and Probability 1

hour 1

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hour 2

hour 3

hour 4

hour 5

Expected deviation

Outcomes 2

EWPV HPWPV using EWPV using HPWPV 3

Power [MW] 230 200 190 Probability

0.3

0.5

0.2

Power [MW] 250 235 220 Probability

0.4

0.3

0.3

Power [MW] 230 220 200 Probability

0.2 0.45 0.35

Power [MW] 210 190 175 Probability

0.45 0.35 0.2

Power [MW] 190 180 170 Probability

0.5

0.4

0.1

TOTAL

down

up

down

up

207

200

6.90

6.90

2.00

9.00

236.5

250

5.40

5.40

13.50

0.00

215

220

5.25

5.25

7.00

2.00

196

210

6.30

6.30

14.00

0.00

184

190

3.00

3.00

6.00

0.00

1038.5

1070

26.85 26.85

42.50

11.00

The highest probability wind power value (HPWPV) and the expected wind power value (EWPV) are calculated using hourly expected wind power values ( ewptn ). The HPWPV, in column 7 of Table 3, is the highest probability wind power value for every hour (see (17)). ewptHP = {ewptk

ρtk ≥ ρtn

∀n ∈ N t : k ∈ N t }

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The EWPV shown in column 6 is calculated using (18). ewptE =

n = Nt

∑ρ n =1

n t

⋅ ewptn

(18)

The expected deviations, columns 8-11 of Table 3, are calculated as is shown in (19) and (20), where ewpt* takes the values ewptHP or ewptE depending on the expected deviation from what is being calculated in the optimization process. Expected deviationt ( down ) =

Expected deviationt ( up ) =

⎧⎪ ρtn ⋅ ewp*t − ewptj ⎨ ∑ n =1 ⎪ ⎩ 0

n = Nt

⎧⎪ ρtn ⋅ ewp*t − ewptn ⎨ ∑ n =1 ⎪ ⎩0

n = Nt

ewp*t ≥ ewptn ⎫⎪ ⎬ ewp*t < ewptn ⎭⎪

ewp*t ≤ ewptn ⎫⎪ ⎬ ewp*t > ewptn ⎭⎪

(19)

(20)

The total wind expected deviations are shown in the last row of Table 3. They are 53.7 (26.85+26.85) and 53.5 (42.5+11) MW for EWPV and HPWPV, respectively. According to (1), these deviations mean a NMAE value of 4.3% for both cases. The wind data used represent a realistic situation for an ensemble of 5 wind power plant for which forecasts are made one hour in advance.

2.6.3. Results

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With the previous data, the WH-GENCO ensemble must elaborate the combined bid. In this example case, the penalties ψ t ,up , ψ t , down considered for the study will have the same value, i.e., ψ t ,up = ψ t , down = ψ = 0.1 ∀t . Other different penalty factors could have been analyzed but the study conclusions would not be different. The agents have many options for defining their bids, but the two most natural ones would be: 1) Expected wind power value strategy (EWPVS): use the expected wind power value plus the optimal hydro bid (when hydro plant bid independently). 2) Highest probability wind power value strategy (HPWPVS): use the highest probability wind power value plus the optimal hydro bid (obtained when hydro plant bid independently). In each case, the combined bid would be defined by the ewptE and ewptHP , respectively, plus the optimal hydro power, and the traded power would be fixed regardless of the value of the penalties ψ t , down , ψ t ,up . Consequently, these strategies cannot ensure the best bid, as they do not consider the penalties. The main disadvantage of these strategies is that they do not take into account keeping enough reserve for all the wind deviations. The results of these two

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strategies are compared in Table 4 with the optimal bid resulting from the optimization mathematical model presented in this work. Table 4. Comparison of the Power Bid (MW) for the Different Strategies Period

Optimal Hydro Bid [MW]

EWPVS [MW]

HPWPVS [MW]

Optimal Combined Power Bid [MW]

hour 1

453.09

660.09

653.09

653.09

hour 2

453.09

689.59

703.09

688.09

hour 3

901.01

1116.01

1121.01

1101.01

hour 4

1794.06

1990.06

2004.06

1969.06

hour 5

584.98

768.98

774.98

760.04

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Finally, Table 5 demonstrates the advantages of the combined hydro-wind bid by showing the revenue obtained for the different bid strategies. This higher total revenue in combined model is because of the fact that this model allows to compensate the expected wind power deviation (see Table 3). It must be remarked that the total revenue is the result of the maximization of the objective function of the optimization model (see eq. (4)). Take notice that the value of the revenue is so high due to the hydro plant revenue, mainly the future value of water. The hydro revenue (third row of Table 3) considers exclusively the turbined water. Neither start-up costs nor the future revenue of stored water in the reservoir are considered in this hydro revenue term. It can be inferred from hydro revenue that optimal combined power bid (hydro revenue lower than the other cases) allows to manage the power reserve in an optimal way. Furthermore, it can be observed that the wind revenue is the same for all the bid strategies. Finally, the energy imbalances costs involve a little part of the total revenue. Table 5. Total Revenue EWPVS

HPWPVS

Optimal Combined Power Bid

59669.679

Wind revenue (€) Hydro revenue (€)

251760

252800

249400

Total revenue (€)

74358589.847

74358536.595

74358639.234

3. Coordination among Wind Farms and Water Pump Stations Wind farms are sensitive to voltage sags, which may produce the disconnection of the generation units. To limit the possible generation loss facing voltage sags, the SO’s specify local restrictions to the inclusion of this non-controllable generation. Therefore, in some buses of the network the wind power production is restricted by security reasons, requiring the disconnection of turbines to avoid critical situations. In [21], the cooperation between a wind

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park and a water pump station to reduce the active power losses caused by these restrictions is proposed. In the method, three objectives are searched: a) to store energy in the water pump station when the production of the wind farm surpasses the security limit specified by the SO; b) to complement the wind farm generation in low generation periods, assuring a minimum of wind power generation in all the periods; and c) to increment the profit in the combined operation, taking advantage of the different hourly prices for the wind energy in the market. The optimal behaviour of the two plants is calculated for a horizon of 24-hours. In the model, the stochastic characteristics of the wind power prevision are considered, modelling the active power forecast through average and standard deviation values. A MonteCarlo algorithm randomly obtains S samples of available wind power series, each of them representing a wind power scenario. Sample vectors of the available wind power Pv characterize these scenarios. For each scenario, the following optimization problem is solved:

∑( c P − cp Pp ) + n cα α

(21)

s.t. Pi = Pw i + Phi

(22)

Pv i = Pw i + Ppi + PDLi

(23)

⎛ Ph ⎞ Ei +1 = Ei + t ⎜η p Ppi − i ⎟ ηh ⎠ ⎝

(24)

E1 = E1esp

(25)

En +1 = Enesp +1

(26)

α Pi L ≤ Pi ≤ Pi U

(27)

Pg L ≤ ( Pw i + Ppi ) ≤ Pg U

(28)

E ⎞ ⎛ PhL ≤ Phi ≤ min ⎜ PhU ,ηh i ⎟ t ⎠ ⎝

(29)

PpL ≤ Ppi ≤ PpU

(30)

0 ≤ Ei ≤ E U

(31)

0 ≤ α ≤ αU

(32)

PDLi > 0

(33)

Max.

n

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i =1

i

i

i

i = 1,...., n

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Where the variables are vectors describing: P , hourly active powers delivered to the network by the wind-hydro facility; Pw , hourly active powers delivered to the network by the wind generator; Ph , hourly active powers produced by the hydro generator; Pp , hourly average active power consumed by the water pump station; PDL , hourly dumping power loads, i.e.: part of the active energy of the wind power curve not used to generate electricity (equivalent to wind energy to be curtailed or reduced, if technologically possible); E , energy storage levels in the reservoir in each hour. In this formulation, α is a variable that represents a decrease factor in the output lower limit. The following parameters were also defined: P L and P U representing respectively hourly vectors of minimum and maximum power output limits related with market requirements and network restrictions; Pv , hourly vector of available wind power in the considered scenario; c , vector of hourly active power prices; cp , pump operation cost; cα , penalty for generation below the lower output limit; E U , reservoir storage capacity; η p , efficiency of water pump station and water pipes network; ηh , efficiency of the water reservoir and hydro generator; L U E1esp and Enesp +1 , initial and final levels of the reservoir, respectively; Pg and Pg , lower and

upper power capacity limits of the wind park, respectively; PhL and PhU , lower and upper production power limits of the hydro generator, respectively; PpL and PpU , lower and upper physical power limits of the pump station, respectively; α U is the upper limit of α; t , duration of each interval (1 hour in this case); and n is the number of discrete intervals. Objective function (21) has two parts: the first one aims to increase the profit in the energy sale, considering the pump cost. The second part of (21) considers the lower limit of generation delivered to the system. The variable α shows the performance of the lower limit in the simulations: if in the solution α < 1.0, the specified lower limit Pi L can not be

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maintained in all the periods of the scenario; if α = 1.0 and αU = 1.0, the lower limit Pi L is respected in all the intervals; and if α > 1.0 and αU > 1.0, the minimum of energy delivered by the combined operation can be greater than Pi L . The hydro pump production and the portion of the available wind power directly delivered to the grid constitute the hourly active power output of the combined operation, (22). The available wind power in the period may be divided in two parts, as shown in (23): a fraction directly supplied to the grid during the considered interval and other portion stored (by using the hydro components) and delivered in subsequent intervals. In some particular cases, it may happen that a part of the available wind energy could not be used. Equation (24) describes the energy balance in the reservoir. At the beginning of the (i+1)-interval, the energy in the reservoir is the initial level in the i-interval plus the pumped energy, minus the energy supplied to the grid by the hydro generation during that same interval. The initial and final energy levels of the reservoir should be specified in the formulation, (25) and (26). The initial level is known, because it is the final level of the previous day. However, the optimal final level of the current day is unknown and depends on the expected operation strategy to be defined for the next day. In order to obtain an optimal value for the reservoir level, the original study horizon (24 hours) is extended one day ahead, resulting in 48 hourly periods. As the W-H operation is daily performed, only the first 24 periods are used.

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The output power of the combined operation between the wind farm and the water pump station is restricted in the intervals, as shown in (27). These output limits could represent: a) operational restrictions of the network, usually associated to the thermal limits of the critical branch in the grid or stability constraints; b) contractual limitations resulting from the participation of the wind park in market negotiation platforms (daily market or bilateral contracts). Equations (28)-(31) describe the operational restrictions of the wind and hydro generators, pumping units and storage capacity. As shown in (29), the maximum hourly hydro generation level depends on the generation equipment limits and on the available energy in the reservoir for that interval. Equation (32) explicit the limits in the decrease factor of the output lower restriction. Optimization problem (21)-(33) must be solved for each scenario of the Monte-Carlo algorithm. In the presented formulation, the optimal operation is obtained by the solution of a linear optimization problem, using a Predictor-Corrector Primal-Dual Interior Point method [32, 33]. In [21], 150 scenarios are represented. In the performed simulation, the gains obtained by the combination between the wind park and the water pump station are over 20%, when compared with the exclusively wind operation. Moreover, the active power output of the combined operation is enclosed between the imposed limits, complying with the System Operator restrictions and assuring a minimum of wind power injected to the grid. In Figure 5, average, maximum and minimum values for the combined output are shown, providing an envelope of operational condition (the shaded area).

Figure 5. Active Power Output of the Combined Operation between a Wind Farm and a Pump Water Station.

In [19], considerations about the optimal size of the wind farm and the elements of the water pump station are performed. Anagnostopoulos and Papantonis [34] also consider the optimum sizing and design of a pump station unit for the combined operation with a wind farm. The work aims to find the optimal Net Present Value for the investment, in a one-year simulation, varying the number of pumps to be used in the station. The algorithm is based on stochastic optimization software based in evolutionary methods. The results show the

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importance of variable-speed pump units. The utilization of this improved pump units allows enhancing the overall financial prospects of the investment. The active power production in a wind farm can be fairly estimated in a short horizon. However, the participation of the wind farms in the electricity markets requires generation forecasting for, at least, several hours ahead. These forecasts may be communicated by the wind owner to the market, allowing the generations’ schedule. Due the interval between the forecast and the real operation, deviations among the estimated production and the real one may be expected. Koeppel and Korpås [35, 36] analyse the utilization of a generic energy storage device for balancing the differences between forecasted and real productions in a wind farm, when acting in a market environment. The proposed algorithm uses the forecasted values of wind production to evaluate the best storage operation, using data from a wind farm located in northern Norway. The combined operation of the wind farm and the energy storage device may reduce the penalties for production deviations to about 4%, improving the participation of the wind farm in the market. Matevosyan and Söder [37] consider the cooperation between a wind farm and a conventional multi-reservoir hydropower system, in a one-year horizon. In this chapter, wind power and hydropower are assumed to be owned by different utilities. If transmission congestions are expected in the system, the hydropower utility decreases its planned production, storing energy for wind power low-production hours. The algorithm executes linear and stochastic optimization software, calculating the adjusted hydropower production and the reservoir content at the end of each day of simulation. The coordination between the hydropower producer and the wind farm decrease the wind energy curtailments, solving the congestion restriction with an efficient approach.

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4. Coordination among Different Wind Farms, Delegated Dispatches Some European System Operators are implementing Delegated Dispatches (DD’s) to coordinate renewable generation in some regions of the country. Wind energy, like other renewable resources, is generally concentrated in some areas with better resources. Regrettably, these areas are frequently distant to the consumption, producing transmission restrictions and stability difficulties. Therefore, constraints to the injection of these not fully controllable sources can be applied by the System Operator in critical situations. Instead of reducing the production, if necessary, of each wind producer in critical situations, the System Operator may find better to settle down constraints for all the non-controllable producers of a region. DD’s are regional intermediate entities, designed to receive the System Operator restrictions and to calculate the best operational point for the producers of the region, considering actual and forecasted productions, controllability and the economical interest of the producers. In [38], [39] and [40], alternatives for the DD’s operation are considered. In [38], three probable approaches are mathematically formulated and analyzed, separately considering the controllability and the possibility of disconnecting the wind farms. The chapter analyses different options to share the reduction in the total of active power production in the DD: proportional, considering controllability prices and taking in consideration both controllability

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and interruptive prices. The alternatives are evaluated in a realistic case, extracted from the Spanish National grid. In [39], some functions of the DD are analysed, with particular considerations related to both active and reactive optimal actions of the DD, aiming to accomplish the restrictions imposed by the System Operator. In [40], a new methodology for sharing the power reduction among the producers, considering interruptible and control capabilities of the wind parks is formulated. Wind farms are classified according to their control possibilities and the desire of each producer to participate in the control process. The wind farms’ modified schedule is fixed using optimization techniques, minimizing the active power decrease necessary for a safe system operation. The wind farms can be classified in the following groups: •

•

•

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•

Type 1 Wind Parks: wind farms that wish to participate fully in controlling the entire active power production under contingencies. Type 1 wind farms offer to the DD their possible disconnection, the control of their power output, or a combination of both. Type 1 wind farms can be also required to set their reactive production and their power factor between the limits imposed by the reactive power production of the generator. Type 2 Wind Parks: wind power producers that can only control their power factor. These producers cannot control active power generation, and their output is the wind power available. Type 3 Wind Parks: wind parks that cannot, or do not wish, to control both active power generation and power factor. The active power delivered by these wind farms is the available wind power at each instant. The power factor of the wind farm can be specified as a fixed value, according to the existing incentives for power factor, but it can not be modified in real time following the operation profit. Type 4 Wind Parks: those generators that do not belong to the same DD, but that are connected to the same output transmission bus. The Type 4 Wind Parks receive the reduction setting directly from the SO, which applies a proportional rule. Like Type 3 wind farms, the Type 4 wind power producers usually specify a fixed power factor for the production.

The formulation proposed in [40] for the optimal allocation of the reductions required by the SO between the wind farm producers of a DD is shown in (34)-(48). m1

min

∑ ( cp j =1

j

⋅ CR j + ip j ⋅ IR j ⋅ PGjAv ) +

m4

∑ fnp ⋅ ( S j =1

∑

⋅ cos ϕ j − pf ⋅ PGjAv ) + 2

Gj

m 2 + m3 j =1

fnp ⋅ ( SGj ⋅ cos ϕ j − PGjAv ) + ....

m 3+ m 4

∑ j =1

2

fnc ⋅ (ϕ j − ϕ f

)

2

(34)

Max s.t. 0 ≤ Pout ≤ POut

(35)

S G j ⋅ cos ϕ j + C R j = PGavj ⋅ (1 − IR j )

(36)

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Jorge Martínez-Crespo, Jorge L. Angarita, Edgardo D. Castronuovo et al. SGi ⋅ cos ϕi − PDi − Pi (V , α ) − Pout = 0 SGi ⋅ cos ϕi − PDi − Pi (V , α ) = 0

i = iout ⎫⎪ ⎬ i ≠ iout ⎪⎭

(37)

S Gi ⋅ sinϕ i − QDi − Qi (V , α ) − Qout = 0 S Gi ⋅ sinϕ i − QDi − Qi (V , α ) = 0

i = iout ⎫⎪ ⎬ i ≠ iout ⎪⎭

(38)

(39)

α sk = 0 α sk = 0

SGi ⋅ cos ϕi ≥ 0

(40)

cos ϕ j ≥ cos ϕ min j

(41)

Vi min ≤ Vi ≤ Vi max

(42)

−Tikmax ≤ Tik ≤ Tikmax i ≠ k

(43)

Max Max −QOut ≤ QOut ≤ QOut

(44)

⎧ cos−1 ( fpf ) for capacitive fpf ⎪ ϕf = ⎨ ⎪− cos−1 ( fpf ) for inductive fpf ⎩

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pf =

Max POut ng

∑P j =1

(45)

(46)

Av Gj

CR j ≥ 0

(47)

IR j = {0;1}

(48)

i, k = 1...n j = 1...m1

Where cp j and CR j are the controllability price and controllability reduction factor of wind producer j, respectively; ip j is the price of wind producer j for disconnection; IR j is a binary variable, representing the connection of wind producer j; PGjAv is the maximum Max is the admissible forecasted production of wind generator j, in the considered period; POut

wind power, as calculated by the SO; fnp and fnc are the weight coefficients for maintaining the active power production (of Types 2, 3 and 4 wind farms) and the power factor (for Types 3 and 4 wind farms) in the specified values, respectively; pf is the proportional factor, used by Type 4 wind farms to calculate their required active power production; Pout is the total active

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power output of the DD area; PDi and QDi are active and reactive power demands at bus i, respectively; SGj and ϕ j are the values of apparent power generation and production angles of

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the renewable generator j, respectively; QOut and QOutMax are the reactive output and maximum reactive output from the DD area to the system, performed at the output bus; Vi and ϕi are the module and angle of the bus voltage, respectively, at bus i; iout is the output bus; min is the minimum power factor of the j α sk is the voltage angle at the slack bus; cosϕj producer; Vjmin and Vjmax are minimum and maximum limits of the voltage module in bus i; Tik is the apparent power transmission between buses i and k; Tikmax is the maximum apparent power limits of the transmission line between buses i and k; fpf is the specified power factor in Type 3 wind farms; m is the number of wind farms included in the DD action; m1, m2, m3 and m4 are the number of Types 1, 2, 3 and 4 producers; and n is the number of the internal buses of the DD system. Objective function (34) aims to decrease the reduction cost within the DD, considering prices and controllability options of the wind farms. All the Type 1 wind farms participate in an internal market, offering bids to the DD for the reduction and interruption services. These bids are compared in the first part of (34), minimizing the decreasing cost. Non-controllable options of the wind farms (active power generation of Types 2, 3 and 4 wind farms and power factor of Types 3 and 4 producers) are also included in (34), balanced by weight factors fnp and fnc. By adding non-controllable options to the objective function, the robustness of the algorithm is largely increased in critical situations. In (35), the restrictions for the active power production of the DD are specified. The DD should be a producer, but it can not surpass the maximum of injection specified by the System Operator. Equation (36) regulates the relationship between interruption and controllability controls within the wind farm. Only if the wind park is not interrupted, the controllability abilities can be executed. By (37) and (38), the conventional power flow equations are represented in the optimization problem. In the output bus, the DD production (Pout) is characterized as a variable load. As in a conventional power flow, the angle of the voltage in one bus (the slack bus) adopts a null value (39). The active power production in each wind park can only have a positive value, as specified in (40). It is considered that Type 1 wind parks can vary their active power production between 0 and the available wind power. The power factor of a wind turbine has a minimum value, which may be different for each wind park (41). The bus voltages and the apparent power flows in the lines of the internal system controlled by the DD must be within their limits, (42) and (43). It must be stressed that the voltage limits in the output bus are expressly included in (42). The reactive input/output from the DD area to the system is restricted by operative limits, (44). Equation (45) calculates the power angle from the specified power factor of Types 3 and 4 producers. In (46), the proportional factor for the Type 4 wind farms is calculated, using the proportional rule utilized by the System Operator. In the problem, the variable representing the controllability abilities of the producers (CR) is a continuous variable (47), and the interruptions are symbolized by a integer variable (IR, in (48)).

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Optimization problem (34)-(48) is a nonlinear mixed-integer constrained problem. In [40], the solution is efficiently obtained by an exhaustive enumeration algorithm, because the reduced quantity of wind farms included in a DD. The optimization problem is applied to a 9 buses test network (Fig. 6), extracted from the Spanish grid. In this network, 6 wind generators inject their production into the system by the same bus, 1. The generators have different controllability options and controllability prices, as shown in Table 6.

Figure 6. 9-buses Test System.

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Table 6. Wind Farms Producer Name WG1 WG2 WG3 WG4 WG5 WG6

Bus

Type

PGjMax [pu]

6 2 9 5 8 7

2 1 1 1 1 4

2.200 5.000 2.500 2.000 2.100 1.800

Interruption Price ipj [€/MWh] 61 60 58 53 -

Controllability Price cpj [€/MWh] 66 64.7 64 150 -

From Table 6, one wind generator can not control the active power production (WG1), other producer decided not to work with the DD (WG6) and the other four producers offer controllability and interruptive prices to the DD. WG5 can interrupt the production, but this producer can not control the production. Therefore, WG5 offers a high price for controllability actions. In Fig. 7, available power productions of the 6 wind power producers are shown. In Fig. 7, the available active power productions of the wind farms and the output restriction in bus 1 (bold line) are shown. As observed, the productions must be reduced to match the maximum output accepted by the System Operator. Applying optimization problem (34)-(48) to the system, the output restriction is satisfied (Fig. 8).

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10

10

9

9

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

0

0 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Hours

WG1

WG2

WG3

WG4

WG5

WG6

Output Restriction

Active Production [MWh]

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Figure 7. Available Active Power Production and Output Restriction.

10

10

9

9

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

0

0 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Hours

WG1

WG2

WG3

WG4

WG5

WG6

Output Restriction

Figure 8. Active Power Production and Output Restriction after Optimization.

In Fig. 8, the optimal operation of the wind farms in the DD is shown, when facing output restrictions. WG1, without controllability of the production, maintains the generation specified in Fig. 7. WG 6 produces in all the periods, continuously adapting the generation to follow the system operator requirements. WG 2 to 5 will be called to decrease or interrupt the generation, following economical directives in function of the respective submitted bids. The optimization solution satisfies the overall output restrictions imposed by the System Operator in the connection bus.

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5. Voltage Stability in Power Networks with Large Amount of Wind Energy In planning and operation of power systems with large wind energy penetration, the analysis of voltage stability for a given system state involves the fulfilment of two aspects: • •

The network steady-state voltage profile must stay within a fixed predetermined range under normal and post contingency situations. Wind farms should react adequately under grid disturbances in order to avoid or limit the system voltage collapse.

Both issues are related to Voltage Stability. According to IEEE/CIGRE Power System Stability definitions [41], it can be said that Voltage Stability refers to the ability of a power system to maintain steady voltages at all buses in the system after being subjected to a disturbance in a given initial operating condition. Voltage Collapse is the process by which the sequence of events accompanying voltage instability leads to a blackout or abnormally low voltages in a significant part of the power system [42]. Main factor causing voltage instability is the inabilities of a power system to maintain an adequate reactive power management in the network and voltage level. Usually, the driving force of voltage instability is the load. For that reason voltage stability is called load stability [43].

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5.1. Two bus system with a Wind Farm Consider a Wind Farm connected to an infinite bus through a lossless transmission line as shown in Fig. 9.The wind farm is injecting active (PWT) and reactive power (QWT) at BUS 2. A single load, connected to BUS 2, demands active power (PD) and reactive power (QD).

BUS1 E∟0

BUS 2 V∟θ

PWT, QWT WT

I LOAD

PD,Q D Figure 9. A single two bus system with a Wind Farm.

For the sake of simplicity the load is assumed to behave as an impedance, where power consumed by the load does not depend on frequency or voltage variations in BUS 2. The infinite bus (BUS 1) is represented by an ideal voltage source E where voltage and frequency are constant. We assumed 3-phase and steady-state sinusoidal operating conditions, consequently, the phasor voltage source is E = E ∟0. Voltage instability can be produced when loads try to draw more power that can be delivered by the transmission and generation system [44]. With the load increases the voltage in the load bus decreases and reaches a critical value that corresponds to the maximum power

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transfer. This maximum power transfer is related to voltage instability. Beyond this point voltage stability is lost and voltage collapse can occur. From Fig. 9:

V =E − jXI

(49)

The complex power absorbed in the bus 2 is:

S = P + jQ = VI* = V

E* − V * j = ( EV cos θ + jEV sin θ − V 2 ) −j X

(50)

Where:

P = PD − PWT Q = QD − QWT

(51)

EV sin θ X V 2 EV + Q =− cos θ X X

(52)

(V 2 ) 2 + (2QX − E 2 )V 2 + X 2 ( P 2 + Q 2 ) = 0

(53)

This decomposes into:

P= −

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Eliminating θ gives:

This is a second-order equation with respect to V2. The condition to have at least one solution is:

(2QX − E 2 ) 2 − 4 X 2 ( P 2 + Q 2 ) ≥ 0

(54)

The two solutions are given by:

V =

E2 E4 − QX ± − X 2 P 2 − XE 2Q 2 4

(55)

If the wind farm is not operating, the active and reactive power injected by the wind farm are PWT=QWT=0, in this situation active and reactive power in BUS 2 corresponds only to the consumed power load, P=PD and Q=QD.

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Reactive power consumed by the load depends on the active power and the load power factor by Q=P.tan φ. With this expression it is possible to obtain curves of load voltage as a function of active power (PV) for various tan φ. These curves are known as PV curves and they allow calculating the relationship between voltages and load in a region as the load increases.

voltage

VmaxP

λ increase

Pmax

Active load

Figure 10. PV curves.

We can define the quantities Pmax and VmaxP using the PV diagram (Fig. 10). Pmax is the maximum deliverable power and VmaxP is the voltage in which this maximum occurs. This Pmax is often called the point of collapse where the voltage drops rapidly with an increase of load. We can consider different scenarios that can produce voltage instability:

5.1.1. Increasing the Demand

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In this situation, to consider the load change scenarios, PD and QD can be modified as:

PD (λ )= PD 0 (1 + λK )

QD (λ )= QD 0 (1 + λK ) Where: PD0, QD0 K λ

(56)

original load, base case. multiplier designating the rate of load change. load parameter. (λ =0 corresponds to the base case).

The critical point where the load characteristic becomes tangent to the network characteristic defines the maximum loadability limit of the system. When this point is reached the load parameter λ= λcritical. A load increase beyond this limit results in loss of equilibrium, and the system can no longer operate.

5.1.2. Loss of Transmission and Generation Equipments These kinds of disturbances correspond to an increase in the reactance of the transmission admittance and a decrease in E voltage.

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These instabilities are usually local area voltage problems due to lack of reactive power [45].

5.2. Voltage Stability Enhancement in Network with Wind Farms When the Wind Farm connected to BUS 2 has reactive power injection facility, QWT, the maximum deliverable power Pmax is increased, and the stability margin can be increased. VQ curves can help determine the amount of reactive compensation at a given bus to obtain a desired voltage in that bus. When the wind Farm is operating and it has reactive power capability, the load flow equations become:

EV sin θ X V 2 EV =− + cos θ X X

P= −

(57)

If we assume a constant power load for each value of the voltage V, θ is first obtained, then the reactive power QWT is computed. Three VQ curves are shown in Fig. 11 for different load conditions (P=0.5 p.u.; P=3.0 p.u.; P=5.3 p.u.). The two intersection points with the V axis correspond to no-reactive compensation (QWT=0). It can be seen that the load situation P=3.0 p.u. corresponds to a more loaded situation than P=0.5 p.u. In this case the intersection point O’, with no reactive compensation, gives a voltage at the PCC around 0.95 p.u. In this situation, it will be necessary to inject a certain amount of QWT (around 0.5 p.u.) in order to restore the nominal voltage. The third curve represents a load condition (P=5.3 p.u.) where the system cannot operate without reactive power injection. 1.5 1 P = 5.3

0.5 QWT Reactive Power

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Q − QWT

O'

0

O

-0.5 -1 P = 3.0

-1.5 -2

P = 0.5

-2.5 -3 0.4

0.5

0.6 0.7 0.8 0.9 V voltage, X=0.1pu, E =1.0pu, cos φ =1.0

1

1.1

Figure 11. Relationship between voltage and reactive power at BUS 2.

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The main function of reactive compensation devices is to provide voltage support to avoid voltage instability or a large scale voltage collapse. Reactive power capability in fixed speed wind turbines can be done in different ways: connecting shunt capacitors, SVC (static var compensator) and STATCOM (static synchronous compensator).Variable speed wind turbines offer voltage control capability at the Point of Common Coupling (PCC) by utilising its reactive power injection capability. Incorporating the STATCOM functionality in their control, variable speed wind turbines can be seen as controllable reactive power sources similar to STATCOM. In Fig. 12 is shown the relationship between voltage and the consumed load power (PD) in BUS 2 when the wind farm is operating and offers reactive injection capability as a STATCOM. The initial situation corresponds to the point with QWT=0 (no reactive power injection). In this case, in order to keep the voltage at the wind farm at the value of 0.95 p.u., the maximum load that can be connected is 3.0 p.u. It can be seen that the voltage stability margin is increased from the initial situation to maximum injection (QWT=1.5 p.u.) while maintaining 0.95 p.u. voltage constant. It can be noted that at higher reactive power injection level the equilibrium operating point progressively approach the nose point of the PV curve.

1

0.95 pu

0.9

QWT = 0

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V v oltage

0.8

QWT = 1.5

0.7 0.6 0.5 0.4 0.3

0

1

2

3 4 5 PD , X=0.1pu, E =1.0pu, cos φ =1.0

6

7

Figure 12. PV curves in the presence of a wind farm/STATCOM.

5.3. Optimal Power Flow (OPF) Formulation We have defined loadability limit as the point where demand reaches a maximum value. If the load is considered as a constant power demand, the loadability limit corresponds to the maximum power deliverable in buses. Voltage stability limits are difficult to obtain using classical load flow methodologies, because load flow calculations fail to converge close to the point of voltage collapse. The problem of alleviation of voltage limit violations is formulated as an optimisation problem [46], [47], [48]. This optimisation problem is implemented to represent properly the

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system security to the maximum distance to the voltage collapse. The λ parameter to maximized drives the system to its maximum loading condition:

PD = PD 0 (1 + λ )

QD = QD 0 (1 + λ )

(58)

Where the multiplier designating the rate of load change is constant in all nodes. The objective function to be minimised is:

f ( x ) = −λ ⎧ g (x ) = 0 subject to ⎨ ⎩ h( x ) ≤ 0 Minimize

(59)

The constrains are classified as the conventional power flow equality constraints: physical limits in the control variables, physical limits in the state variables, and other limits such as power generation limits: Power flow constraints:

Pgi − PDi − Pi ( x ) = 0 Qgi − QDi − Qi ( x ) = 0

(active power balance)

(reactive power balance)

(60)

Control variables limits:

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Qgi

min

Vi

min

≤ Qgi ≤ Qgi ≤ Vi ≤ Vi

max

max

(reactive power generation ) (bus voltage limits)

(61)

5.4. Modified IEEE 14-bus System The OPF formulation is applied to a modified IEEE-14 bus power system [49] where the original synchronous generator in bus 8 is removed and a wind farm is installed with a rated power PWT = 30MW. Two different scenarios have been considered: ƒ

ƒ

Scenario 1: The wind farm in bus 8 is adjusted to maintain a constant power factor, cos φ = 1 (QWT=0). Scenario 2: The wind farm in bus 8 offers reactive power capability with SVC/STATCOM function. It can also be seen as a controllable reactive power source. The maximum reactive generation is QWT=30⋅1.5MVAR.

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Figure 13. Modified IEEE 14-bus test system. Voltage Profile. Initial Condition.

1.05

1 voltage (pu)

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1.1

0.95

0.9

0.85

0.8

1

2

3

4

5

6

7 8 bus no.

9

10

11

12

Figure 14. Initial Voltage profile at buses.

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14

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Fig.14 shows the voltage profile at IEEE-14 buses considering the initial condition, which correspond to a wind farm connected in bus 8 with unity power factor (QWT=0). It can be seen that voltage levels in all buses are within ±10% of nominal voltage. Bus 3 is the bus with the lowest voltage level. Optimal Power Flow formulation is applied to both scenarios and results are shown in Table 7, 8. Table 7. Optimal loadability in scenario 1 Scenario 1

λ= λOPF_MAX= λcritical

Initial condition OPF solution

0 (Not OPF) 0.45

Maximum loadability (MW) 259 374

Critical Bus #

---14

Table 8. Optimal loadability in scenario 2 Scenario 2

λ= λOPF_MAX= λcritical

Initial condition OPF solution

0 (Not OPF) 0.66

Maximum loadability (MW) 259 430

Critical Bus #

---14

scenario 1 1.1 bus bus bus bus

1 voltage (pu)

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1.05

14 3 13 10

0.95

0.9

0.85

0.8

0

0.1

0.2

0.3

0.4

λ

Figure 15. Voltage versus lambda (scenario 1).

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0.6

146

Jorge Martínez-Crespo, Jorge L. Angarita, Edgardo D. Castronuovo et al. scenario 2 1.1 bus bus bus bus

1.05

14 3 13 10

voltage (pu)

1

0.95

0.9

0.85

0.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

λ

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Figure 16. Voltage versus lambda (scenario 2).

Fig. 15 shows the voltage evolution versus load parameter λ for scenario 1, where critical buses are 3, 10, 13, 14. It can be seen that the maximum loadability of the system to keep the voltages at all buses within ±10% of nominal voltage corresponds to λ= λOPF_MAX= λcritical =0.45. Which means maximum loadability of 374 (MW). It can be noted how bus 14 is the most critical one reaching the limits established in the OPF formulation. The maximum loadability in scenario 2 (Fig. 16) is increased. It can be seen how the load parameter, λ= λOPF_MAX= λcritical =0.66. This means a maximum loadability of 430 (MW). This second scenario involves the reactive injection capability of the wind farm. It must be remarked how the reactive power injection at the wind farm terminals improve the voltage stability of the system. From these numbers, it may be deduced that, if there is more available capacity in the power system to handle disturbances, the voltage stability is improved. As conclusion, it can be noted how wind energy penetration can be increased by incorporating reactive power capability at wind farms and using optimisation algorithms in order to determine the maximum amount of power that can be integrated into the system keeping steady-state voltage profile within a fixed predetermined range under normal and post contingency situations.

6. Conclusion In this chapter, different tools aimed at improving the operation of power systems with large penetration of wind power are analyzed. Two different applications of optimization techniques in order to submit wind power bids to electricity markets are considered. In one of

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them, the optimal wind power bid will be calculated taking into account that the wind plant submits bids individually to the market. The proposed model has been tested for different realistic forecasting conditions and penalties in a programming horizon of 44 days. In the second option, a combined bidding strategy to submit hydro-wind bids in pool-based electricity markets is analyzed. Wind generation is considered a stochastic parameter and the input-output hydro generation function is represented through a concave piecewise linearization. The test system is composed of a HGENCO with eight reservoirs linked between them and a wind power plant with rating power of 250 MW. In this study case, the programming horizon is 5 hours and every hour contains three different wind generation scenarios, so that the total number of outcomes for the scheduling horizon is 243. The results of the optimal combined bid are compared with the obtained results by using other bid strategies. Facing limitations to the injection of wind power to the system (due to critical situations or preventing abnormal circumstances) two alternatives are considered: to store the wind power energy in a pump water station or to share the regional restrictions in a Delegated Dispatch of renewable production. Both approaches are fully formulated in the chapter, showing results in a realistic power system. The inclusion of large amounts of wind energy can produce also problems of voltage stability. In the chapter, the advantages of the optimization model to maintain an adequate voltage profile are demonstrated. The reactive power generation abilities of the wind farms, when utilized according to the analyzed method, could increase the quantity of the wind energy injected to the power system without producing stability collapses.

References

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European Parliament, Directive 2001/77/EC of the European Parliament and of the Council of 27 September 2001: on the promotion of electricity produced from renewable energy sources in the internal electricity market, Official Journal of the European Communities, vol. L283/33. M.C. Alexiadis, P.S. Dokopoulos, H.S. Sahsamanoglou, I.M. Manousaridis, “Shortterm forecasting of wind speed and relaxed electrical power”, Solar Energy 63 (1) (1998) 61-68. U. Focken, M. Lange, K. Mönnich, H. Waldl, H.G. Beyer, A. Luig, “Short-term prediction of aggregated power output of wind farms -a statistical analysis of the reduction of the prediction error by spatial smoothing effects”, Journal of Wind Energy Engineering and Industrial Aerodynamics 90 (2002) 231-246. I. Sánchez, “Short-term prediction of wind energy production”, International Journal of Forecasting 22 (2006) 43-56. Spanish Electricity Market Activity Rules (English version) [online]. Available: http://www.omel.es The Office of Gas and Electricity Markets (OFGEM). An Overview of the New Electrical Trading Arrangement [online]. Available: http://www.ofgem.gov.uk The Common Nordic Power Market (Nordpool). Available: http://www.nordpool.com G. Kariniotakis et al. “Evaluation of Advanced Wind Power Forecasting Models – Results of the Anemos Project”. EWEC 2006. Athens 2006.

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[27] H. Brännlund, D. Sjelvgren, J.A. Bubenko, “Short-Term generation scheduling with security constraints”, IEEE Trans. Power Syst. 3 (1) (1988) 310-316. [28] O. Nilson, D. Sjelvgren, “Variable splitting applied to modeling of start-costs in shortterm hydro generation scheduling”, IEEE Trans. Power Syst. 12 (2) (1997) 770-775. [29] A. Conejo, J.M. Arroyo, J. Contreras, “Self-scheduling of a hydro producer in a poolbased electricity market”, IEEE Trans. Power Syst. 17 (4) (2002) 1265-1272. [30] J. Angarita, Integración de energía eólica en mercados competitivos de energía eléctrica, PhD thesis, Universidad Carlos III de Madrid, 2007. [31] J. Parkes, J. Wasey, A. Tendal, and L. Muñoz. Wind energy trading benefits through short-term forecasting. Available: http://www.garradhassan.com/downloads/reports [32] S.J. Wright, Primal-dual interior-point methods, Philadelphia, USA, SIAM , ISBN 089871382X, 1997 [33] E.D. Castronuovo, J.M. Campagnolo and R. Salgado, “New versions of nonlinear interior point methods applied to the optimal power flow” in Proc. IEEE T&D 2002 Latin America, São Paulo, Brazil, April 2002. [34] J.S. Anagnostopoulos, D.E. Papantonis, “Pumping Station Design for a PumpedStorage Wind-Hydro Power Plant”, Energy Conversion and Management, vol 48/11, pp 3009-3017, November 2007. [35] G. Koeppel, M. Korpås, “Increasing the Network In-Feed Accuracy of Wind Turbines with Energy Storage Devices”, in Sixth Word Energy System Conference, pp 365-370, Torino, Italy, July 10-12, 2006. [36] G. Koeppel, M. Korpås, “Using Storage Devices for compensating uncertainties caused by Non-Dispatchable Generators”, Proceedings of the 9th. PMAPS, Stockholm, Sweden, June 2006. [37] J. Matevosyan, L. Söder, “Short-term Hydropower Planning Coordinated with Wind Power in Areas with Congestion Problems”, Wind Energy, vol. 10, pp 195-208, February 2007. [38] E.D. Castronuovo, J. Usaola, “Alternatives of revenue for corrective actions of wind generators in a Delegated Dispatch”, Proc. of the IEEE International Conference on Clean Electrical Power, pp. 567-573, Capri, Italy, May 2006. [39] E.D. Castronuovo, J. Martínez-Crespo, J. Usaola, “Optimal controllability of wind generators in a Delegated Dispatch”, Electr. Power Syst. Res., vol. 77, n. 10, pp. 1442 – 1448, Aug. 2007. [40] E.D. Castronuovo, J. Usaola, Álvaro Jaramillo, “Delegated Dispatches of Wind Farms: An Optimal Approach considering Continuous Control and Interruption Capabilities”, Wind Energy,to print. [41] Kundur, P.; Paserba, J.; Ajjarapu, V.; Andersson, G.; Bose, A.; Canizares, C.; Hatziargyriou, N.; Hill, D.; Stankovic, A.; Taylor, C.; Van Cutsem, T.; Vittal, “Definition and classification of power system stability IEEE/CIGRE joint task force on stability terms and definitions” IEEE Transactions on Power Systems. Vol 19 Aug. 2004 Page(s):1387 – 1401 [42] P. Kundur, Power System Stability and Control. New York: McGraw-Hill, 1994. [43] V. Ajjarapu, Computational tecnhiques for voltage stability assessment and control, Springer, ISBN 0387260803, 2006. [44] T.V.Cutsem, C.Vournas, Voltage stability of electric power systems, Kluwer academic press, ISBN 079238139, 1998.

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[45] D.Chattopadhyay, B.B. Chakrabarti, “Reactive power planning incorporating voltage stability”, Electrical Power and Energy Systems, vol. 24, pp 185-200, 2002. [46] J.Qiao,Y.Min,Z. Lu “Optimal Reactive power flow in wind generation integrated power system”. 2006 International conference on Power System Technology. Powercon 2006.pp 1-5, 2006 [47] W.Shang, F.Li, L.Tolbert “Review of reactive power planning: objectives, constraints and algorithms”. IEEE Transactions on Power Systems. Vol 22.no4. November 2007. pp 2177-2182. [48] L. Mariotto, H.Pinheiro, G.cardosi, M.R.Muraro “Determination of the static voltage stability region of distribution systems with the presence of wind power generation”. Proc. of the IEEE International Conference on Clean Electrical Power, pp. 556-562, Capri, Italy, May 2007. [49] IEEE-14 bus power flow test case (http://www.ee.washington.edu/research/pstca/pf14/ pg_tca14bus.htm)

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In: Optimization Advances in Electric Power Systems ISBN: 978-1-60692-613-0 c 2008 Nova Science Publishers, Inc. Editor: Edgardo D. Castronuovo, pp. 151-179

Chapter 7

A PPLICATION OF C OST F UNCTIONS FOR L ARGE S CALE I NTEGRATION OF W IND P OWER U SING A M ULTI -S CHEME E NSEMBLE P REDICTION T ECHNIQUE Markus Pahlow1,∗, Corinna M¨ohrlen2,† and Jess U. Jørgensen2 1 Department of Civil and Environmental Engineering, Institute of Hydrology, Water Resources Management and Environmental Engineering, Ruhr-University Bochum, 44780 Bochum, Germany 2 WEPROG, Eschenweg 8, 71155 Altdorf, Germany

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The use of ensemble techniques for wind power forecasting aids in the integration of large scale wind energy into the future energy mix and offers various possibilities for optimisation of reserve allocation and operating costs. In this chapter we will describe and discuss recent advances in the optimisation of wind power forecasts to minimise operating costs by using a multi-scheme ensemble prediction technique to demonstrate our theoretical investigations. In recent years a number of optimisation schemes to balance wind power with pumped hydro power have been investigated. Hereby the focus of the optimisation was on compensating the fluctuations of wind power generation. These studies assumed that the hydro plant was dedicated to the wind plant, which would be both expensive and energy inefficient in most of today’s and expected future electricity markets, unless the wind generation is correlated and has a very strong variability. Instead a pooling strategy is introduced that also includes other sources of energy suitable to balance and remove the peaks of wind energy, such as biogas or a combined heat and power (CHP) plant. The importance of such pools of energy is that power plants with storage capacity are included to enable the pool to diminish speculations on the market against wind power in windy periods, when the price is below the marginal cost and when the competitiveness of wind power as well as the incentives to investments in wind power become inefficient and unattractive. ∗ †

E-mail address: [email protected] E-mail address: [email protected] and [email protected]

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Markus Pahlow, Corinna M¨ohrlen and Jess U. Jørgensen It will also be shown that the correlation of the produced wind power diminishes and the predictability of wind power increases as the wind generation capacity grows. Then it becomes beneficial to optimise a system by defining and applying cost functions rather than optimising forecasts on the mean absolute error (MAE) or the root mean square error. This is because the marginal costs of up and down regulation are asymmetric and dependent on the competition level of the reserve market. The advantages of optimising wind power forecasts using cost functions rather than minimum absolute error increase with extended interconnectivity, because this serves as an important buffer not only from a security point of view, but also for energy pricing.

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1.

Introduction

Wind power is considered one of the most important renewable energy sources in the near future. In the past years the number of wind farms world wide increased strongly, with a total installed capacity of more than 94 GW by the end of 2007 [1], with about 50% distributed in Germany, Spain and the USA. This high concentration suggests that from a global perspective there is space for ample additional wind power. The issue is therefore not whether it is feasible, but rather what does it cost? The main focus in wind power integration in the past has been on producing the most accurate forecast with minimal average error, but experience has shown that this is not necessarily optimal from a cost perspective. Balancing costs will increase with increasing volumes of wind power. The larger the forecasted error (in MW), the higher the balancing costs. Another aspect is the concentration level of wind power, which has a side effect from a forecasting perspective. This is the correlated generation and forecast error and is mainly relevant for large amounts of offshore wind power, as it is planned e.g. in the North Sea in order to reduce transmission costs. Such scenarios make it unfeasible to run an electricity network with a forecast optimisation target of minimal mean absolute error, because it implies short periods during which GW of fossil fuel based power plants will have to be started with short notice. Germany is such an example, where it is the large error that dominates the balancing costs of wind power. Skewness of the balancing costs of negative and positive reserve gives advantage to conservative forecasts in such areas and requires curtailment or other scheduled plants to stop generation with short notice. Such scenarios are also regularly experienced in Spain. An EU 6th-Framework project is assisting the Transmission System Operators (TSO’s) in developing tools to handle such cases with a so-called cluster management [2]. From these experiences, it appears that cost functions will not only benefit the market prices and balancing costs, but also act as a means to increase system security. For those countries, where the installed capacity has gone beyond the 10 GW level, it has become an important consideration in the daily operation. Whether intermittency poses technical limits on renewables in the future is certainly also of concern for other forms of renewable energy sources [3], since OECD (Organisation for Economic Cooperation and Development) Europe and IEA’s (International Energy Agency) World Energy Outlook [4] project up to 23% market share of non-hydro renewable energy by 2030. Natural variations of resource availability do not necessarily correspond with the (also varying) need of the consumers. Balancing supply and demand is therefore a

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critical issue and potentially requires backup by other means of energy supply. The variations can occur at any time scale: hourly changes in output require balancing of short-term fluctuations by the so-called operational reserve, while days with low output require balancing of longer-term output fluctuations by so-called capacity reserves. Conversely, exceptionally windy days or rainy seasons can produce a surplus of supply and there might arise issues of handling excess capacity, where grids are not sufficiently interconnected. TSO’s buy balance or reserve capacity in advance to ascertain secure grid operation. However, in addition to the anticipated cost savings, reliable forecasting aids in reducing the aforementioned problems, enabling high wind energy penetration and at the same time ensuring power system security and stability. The development of wind energy is country dependent and the development structure is a function of the wind resource, the political environment, the electrical grid and the market. This means that different strategies are required to solve forecasting and, in a broader perspective, optimisation of energy systems in different countries. Therefore, it seems to be natural to focus efforts on optimisation targets that can reduce the required reserve of the growing intermittent energy generation.

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2.

The Optimisation Problem

Large amounts of wind power can not be integrated seamlessly into the electrical grid. There is a need for a combination of wind energy forecasting, interconnectivity and storage capacity to ascertain smooth operation. Today’s market participants are not delivering this mixture of services and there are costs associated with each of these services. Forecasting is the cheapest solution and may in fact be sufficient to get a mix of power implemented in an optimal way. The application of wind power forecasting can generally be divided into two different kinds. The first kind is to reduce the need for balancing energy and reserve power, i.e., to optimise the power plant scheduling. The second kind is to provide forecasts of wind power feed-in for grid operation and grid security evaluation. Wind power forecasting may well be one of the most direct and valuable ways to reduce the uncertainty of the wind energy production schedule for the power system. Therefore, the objectives of a wind power forecast depend on the application [5]. In fact one has to differentiate between the following targets: • For optimised power plant scheduling and power balancing, an accurate forecast of the wind power generation for the whole control area is needed. The relevant time horizon depends on the technical and regulatory framework; e.g., the types of conventional power plants in the system and the trading gate closure times. • For determining the reserve power that has to be held ready to provide balancing energy, a prediction of the accuracy of the forecast is needed. As the largest forecast errors determine the need for reserve power, these have to be minimised. • For grid operation and congestion management, the current and forecast wind power generation in each grid area or grid connection point are needed. This requires a forecast for small regions or even single wind farms.

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2.1.

Markus Pahlow, Corinna M¨ohrlen and Jess U. Jørgensen

Energy Prices and Market Structures

Competitive energy prices in a globalised world require a mix of multiple sources of energy. This is in fact a prerequisite in order to keep the energy generation process cost efficient. Shortage of components or resources will increase the price of a given energy generation method, if there is major concurrent demand. Renewable energy systems are therefore likely to be more expensive to install, whereas they are cheaper to operate. This is also due to fear of increasing prices of fossil fuel, further destruction of our environment and climate change. The different marginal costs for different generation is another argument that points towards highly mixed energy systems. The future of energy will therefore most likely contain a more complex mixture of energy generation and development, possibly with the exception of regions with excessive resources of one particular energy source. Consequently, we have to expect that the electricity systems are going to become more complex to manage. There is a strong trend that the society is in favour of supporting new developments of renewable energy and in particular non-scheduled intermittent sources such as wind energy. We therefore focus on how to keep a sustainable price for wind power. The optimisation strategies that will be presented hereafter may result in higher energy prices in the shortterm, but a relatively lower energy price in the long-term compared to non-cost optimisation scenarios. The optimisation target here is therefore to the benefit of the entire society, but first of all to the current wind farm owners in markets, where wind power is frequently the price maker. There may be other valid optimisation targets such as minimal emissions. This would however favour nuclear energy and lead to the dependency on one natural resource, unknown issues regarding the nuclear waste and therefore risk of high energy prices. In addition, nuclear energy is not a flexible energy generation mechanism, which could effectively complement the intermittent energy units. Either the nuclear or the intermittent generator would have to generate less power than they potentially could, each one at their standard marginal cost.

3.

Optimisation Objectives

Production incentives naturally help to increase the amount of renewables. These incentives are typically defined on the national level. The incentives are in most cases not important for the optimisation problem, but they may encourage a phase shift of the generation by using a storage unit and thereby maximising the production while the market price is highest instead of delivering power when the intermittent power generation is highest.

3.1.

Market Considerations

Differences in the market structures, politically defined support systems, demand profiles and the inertia level of the scheduled power generation does make the optimisation problem specific for each region, if not even for each end-user. However, there are some basic difficulties, which all markets will experience once the volume of intermittent energy resources reaches during certain time periods “price maker levels”. The physical nature of

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the difficulties are in fact independent of how and by whom the energy is traded. Our aim is therefore to present an optimisation strategy that works equally well in systems, where wind power is handled centralised or de-centralised, regardless whether it is managed by a TSO or not. The basic strategy is the same. The fundamental problem is how to trade the varying amount of wind power on the market and to determine the value. The value depends on how well the intermittent energy satisfies the demand, the predictability of the weather and also how eager the scheduled generators are to deliver power. The difficulty increases with the ratio between intermittent generation and demand, apart from the eager scheduled power generators such as CHP plants that also need to generate heat and therefore have very low marginal costs for electricity generation.

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3.2.

Transition from Fixed Prices to the Liberalised Market

The incentive to operate wind optimal from an economic perspective might however not always be given, if there is a fixed price policy and the consumers are enforced to pay over tariffs. It is very difficult to clarify in such cases, whether the system can be optimised. The alternative approach is then to leave it up to the wind generation owners to optimise the system. Fixed price policies may also be limited to either a number of years or are valid only until a certain number of MW-hours has been produced. Wind farm owners are usually thereafter enforced to trade their energy on market terms. The transition to market terms can be difficult, if there is a public monopoly that dumps the price with wind energy and takes the loss back over tariffs. This is especially the case on markets that operate with the pricecross principle on the day ahead spot market (e.g. the Nordic states). This leaves almost no possibility to get a good price for the energy produced, if the monopoly has enough volume to meet the demand. Wind power that has to be traded on market terms could in such cases in the future also be traded outside the market directly to other participants, which may be capable to absorb the energy in their energy pool. There are in fact also initiatives in Germany to get the possibility to trade wind energy directly on the market, although wind energy has by law been traded by the TSO’s and paid by a fixed price tariff. In the near future, the market rules will allow renewable energies to be traded under market terms by entirely or temporarily stepping out of the fixed price schemes [6]. The introduction of this scheme is a clear signal that there is an interest by the society in optimising the cost of clean (wind) energy. Even though price dumping, as described above, is neutral for the end-users if there is no or little export, it nevertheless significantly lessens the value of renewables and in particular wind energy for the wind turbines/farms that operate on the market without the possibility to get a fixed price. If price dumping leads to export, then the importing party is receiving energy not only as clean energy, but also for a very low price, partially because it is paid over governmental subsidy in the area where it was produced. It is expensive for a society in the producing area to practise such export in the long run. It also means that such a country either has to produce the bulk of energy from renewables, or otherwise it will create a negative imbalance in the country’s allowance of carbon emissions when exporting renewables. In other words, price dumping prevents wind energy from becoming competitive and to develop to a non-subsidised energy source, at least in Europe. This is in the longer term not

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in the society’s interest. To conclude, a single wind power plant on its own will not be able to compete with a pool of many plants, unless it is located in unusual predictable weather conditions and hence could be considered a semi-scheduled plant. However, there are also means to operate as an individual on the market by outsourcing the handling of the wind power trading to parties that are specialised in this and may pool the energy of their aggregated customer’s power to a larger portfolio. Sufficiently large wind generators may in the future also consider to forecast and trade themselves in order to get the market price in addition to some incentives, but also the penalties. Nevertheless, this is expected to be the most profitable way forward for wind energy in the future.

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3.3.

The Skew Competition in the Trading of Wind

The wind generators are some of the weakest parties on the energy market seen from a trading perspective. Everybody with a good weather forecast and with a reasonable approximation and experience on the market can predict the wind energy generation in windy periods and therefore also the impact on the market price from wind. This is not so much a problem as long as the fraction of wind energy of the total generation is less than ∼ 10%. Above this level, this becomes a more serious problem, since the wind power traders are forced to bid on the energy to a relatively low price or the traders risk to be forced to sell the energy to a pool for a lower price after gate closure. As previously mentioned, it may be the most beneficial way to trade wind energy with a pool that contains sufficient storage to create additional uncertainty on the market regarding the available energy than what the weather forecast provides. In such cases, the market can no longer predict the energy generation from wind power and speculate as aggressively against the wind power trader. The market can as an example not know, if the wind energy pool decides to store energy for some hours instead of delivering the power and thereby take the peak off the wind power generation. The scheduled generators can in time periods of strong wind speculate against the wind power trader by setting a price that lies below the marginal costs, because they know that the wind power will be available for a marginal price, if the wind generation was bid in with a higher price than their own. If the scheduled generators know that wind generation may not be available for a near zero price, because it may be withheld from the market for storage, then the risk becomes too high and the scheduled generator will stop such speculations. Pooling can hence be regarded as a security measure against unfair speculation. The pool will in that case be used in daily operation to phase shift wind generation such that the correlation between the demand and total output of the pool is highest.

3.4.

Uncertainty Considerations

Another factor that has impact on the market price and competition level is the uncertainty of the weather forecast. Uncertain weather requires that more energy generation is active, either as primary energy or as reserve. This again increases the average energy price, because a larger fraction of the available generation will be required for balancing. Steep ramping of wind power is an indicator of uncertainty in the weather and often

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even a cause of such. Nevertheless, steep ramping requires more scheduled capacity online, because the efficiency of the ramping generators is lower during the ramp. On the other hand, there are also times of low uncertainty and high wind power generation. In these times, the traders of wind energy have to do something extraordinary to get rid of the energy. Although it may still be possible to bring the energy into the market by using dumping prices, it may not cover the marginal costs anymore. However, once a certain percentage of the total consumption is exceeded, other methods will have to be applied. These include energy pooling, export, methods to increase the demand and other trading strategies.

4.

Optimisation Schemes

In this section we describe a number of optimisation schemes that are either already in operation, or are likely to become part of the near future’s energy trading at the markets.

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4.1.

Pooling of Energy

The optimisation problem, when looking at pools of energy, is relatively complex. Therefore, an example will be used in the following to illustrate the optimisation targets and possibilities of this scheme. Let us introduce the Balance Responsible Party (BRP) as the party that ascertains that the wind together with other generation units in a pool follows the planned schedule for the pool. The BRP may not be a generator, but some party that has the required tools to effectively predict and manage the different pool members in an optimal way seen from the pool members point of view. The first milestone for the BRP is to bring together a sufficient amount of uncorrelated wind power in the pool. The lower the correlation on timescales greater or equal than an hour, the lower the forecasting error and the higher the energy price. Low correlation means that the area aggregation over individual sites gives a smoother output signal, which is easier to forecast. The same smoothing is then implicitly applied to the forecast. The net result is that the phase on the shortest timescale of the individual wind farms is invisible on the aggregated power generation. A large volume of wind power also helps to level out negative impact from problems due to restrictions on the grid or reduced turbine availability, although these may only appear as small errors in a large pool. One of the key parameters for a BRP is to secure a permanently high level of uncertainty of when and how much energy the pool is delivering to the network. As discussed earlier, this is because the wind power trading becomes vulnerable to speculation from stronger market participants, if it can easily be derived how much wind energy needs to be traded on the market. This is not a trivial task for the BRP, because there are also other constraints, which limit the possible ”confusion” level introduced to the market. Hence, the major tools that the BRP needs apart from the wind farm capacity are: • A storage unit • An ensemble prediction system for the difference between demand and wind power

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• A short-term prediction system for wind power and demand

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• An optimisation tool An optimisation tool hence should combine forecasts with grid constraints, market trends and the physical constraints of the storage unit. Congestion on the grid is a typical reason for higher prices, but mostly to the disadvantage of the energy pool, if it is dominated by wind power. The optimisation problem is in theory global, but in practise a limited area problem, because the global problem extends to political decisions that would have to be modelled by stochastic processes. Instead, optimisation should be applied on the local grid. For this, time dependent boundary conditions are required. These should in theory contain the large-scale global trends, but at some stage approach the average trends for the season. The optimisation tool will have to comprise a set of partial differential equations including the storage unit, a portion of the grid and the intermittent energy sources. The partial differential equations describe the total output and exchange between storage units and the intermittent source. The accuracy of the numerical solution is partially determined by the extent of the domain of dependence for the partial differential equations. The domain increases with forecast horizon and can well reach part of the boundary, if the boundary values are accurately predictable by a simple function or some other prediction tool. An example of a predictable boundary condition could be the large-scale electricity demand, which could be approximated with a periodic function to simulate the diurnal cycle. The large scale demand will after some hours have an influence on how the storage system should be scheduled, but the time derivatives of our intermittent energy is then likely to be the dominant forcing term on the storage equation. The domain of dependence for the solution increases with the forecast horizon, partly because the dependency domain of the weather forecast increases. However, such an energy system has as a good approximation no feedback on the weather, thus this system can be one way coupled. Also, trading of oil and the transmission of gas have both an influence on the pricing. Conflicts between employers and their labour (e.g. strikes), unavailability of multiple plants and extreme weather at offshore platforms can cause peaks in the spot market prices of gas and oil and consequently also electricity. Such peaks can have a dominant negative impact on the average market value of wind power, if there would be imbalance in the pool during an interval with high prices. An objective optimisation process would hence require an algorithm that carries out simulations in a closed system, but with the possibility to control the time dependent boundary conditions. Typically, a strike would be known in advance and the time dependent boundary conditions could be manually adjusted by the user to take account for such effects. Although the optimisation problem should ideally encompass the entire globe, it appears that the impact from far distances on the next couple of days can be modelled equally well by subjective boundary conditions, as with attempts to objectively model such effects. The conditions that take place at far distances are rather consequences of unpredictable events that have spurious nature. Any objective algorithm would have to be tuned to discard observations that conflict with the present state of the system to prevent that the large scale numerical solution would become unstable. Large scale waves would propagate

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through the system and trigger new waves on the local scale and the final solution would destabilise the energy system and the result would be higher balancing costs. It is important that objective systems only accept observations that are likely to be correct. They must lie within an a priori defined uncertainty interval or be discarded. Spurious waves in an objective system are worse than no waves, even if they would be correct, because the user does often not know of the source of the observations and it may be difficult to trace back. It is therefore most important in the design of the optimisation tool that the end-user can work with and define a robust set of boundary conditions and can explore the sensitivity of the solution to a number of incidents that are each relatively unlikely events.

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4.2.

The “Price Maker” Optimisation Problem

If the optimisation target is to predict an optimal price in an area between two market systems with a price difference due to a limited interconnection capacity between the two markets, it is considered a “price maker problem”. This is because the energy would flow from the low price area to the high price area and the BRP would try to sell the energy in the direction of the high price area like other participants and there would be some likelihood of congestion on the line. The “price maker” must therefore continuously predict the price of the neighbouring areas with more inertia and try to trade in the direction of the highest price. This involves usage of weather forecasting on larger scales and computation of demand and intermittent energy generation. In this case, it is no longer sufficient to just predict what may be correct for the BRP itself. Other market participants might have access to different forecast information and may conclude very different or similar scenarios. This is thus an application, where ensemble forecasting is helpful. Depending on the ensemble spread, the likelihood of high or low competition can be determined and therefore also the price level on the market. The weather determines the upper limit for what the BRP can sell. Before gate closure a decision has to be taken on the basis of weather forecast information. However, the BRP may decide to sell less or more than he expects to produce, depending on the likelihood of the weather and the expected balancing costs. The “price maker” is likely to cause the bulk of the imbalance and therefore also the bulk of the balancing costs. The “price makers” sign of the error will correlate with the sign of the total imbalance. This means that the error in every settlement interval counts as a cost, while this is not so for the “price taker” party whose sign is maybe 50% opposite to the ”price makers” sign. The BRP has therefore an incentive to keep the balancing costs at zero and let other parties carry the balancing costs. The BRP can achieve this by using the ensemble minimum as a safe base generation. However, this principle leaves some excess energy at times, that needs to be traded with short notice on average under the sport market price scheme.

4.3.

The “Price Taker” Optimisation Problem

A small pool (in MW) can be traded and optimised with a so called “price taker” policy. This means, that the pool does not necessarily need price predictions, but only needs to keep the generation profiles according to their schedule. The “price taker” can assume a

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diurnal cycle of the demand and a pricing that follows this pattern. As a refinement, the “price taker” can try to predict the generation profile of competing intermittent generation and from this compute a new price profile of the trading interval. This would in some cases encourage to reschedule generation to achieve the highest possible price. Thus, instead of predicting the price for the bid, the “price taker” predicts the intervals with highest prices and schedules the generation, so that most hours are delivered during the high price time interval. This strategy secures that the ”price taker” will get rid of the energy at the highest possible price. However, it also means that the “price taker” needs a rather flexible pool of generating units.

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4.4.

The Combi-Pool Optimisation Scheme

The lower the predictability of the energy pool, the better. Pumped hydro energy is one of the key storage units along with combined heat and power (CHP), and in smaller portions with biogas (e.g. [7], [8], [9]). In a CHP plant the energy can be stored as heat, if the energy regulations allow to do this on market terms. It is however useful to introduce the concept of scheduled demand, which is a better description of what such a “combi-pool” needs to include. Some markets do not allow direct coupling of generation and demand. It is nevertheless the most efficient way to level out differences between demand and generation. Generally, the system operators often export imbalances and there seems to be a trend that larger markets work with increased import and export. This is on the one hand levelling prices out and helping non-scheduled generation and it may in many cases even be a better alternative to use scheduled demand. This would mean that heavy industry could benefit from low prices. Using both would allow for more intermittent energy on the grid. What is going to increase the efficiency of a BRP is therefore a number of inventive solutions. The difficulty in predicting the price and output from a BRP increases with the amount of negative scheduled MWh and MW in the pool. However, information of the pool needs to be kept highly confidential for maximum competitiveness, which is under strong debate in Europe at present [10], if the BRP are TSO’s. Last but not least, the question has to be raised whether it is scientifically correct to optimise a system with a mechanism, where the primary target is to generate confusion for the market participants? The answer is yes, because this is the primary principle of the free market to maintain fair competition. Wind power does not operate under such fair competition, because it is exposed to the world via weather forecasts, which is strictly speaking against market principles.

5.

Wind Power Forecasting Methods

Now that various optimisation strategies have been discussed, it is important to get an understanding of the forecasting methodologies and possibilities to set up optimisation functions for the trading of wind energy. Therefore, the following sections will provide various approaches of wind power forecasting and discuss the error that the forecasting process is subject to. An error decomposition is used to give insight into the forecasting problem, but also into the limitations of forecasting. These limitations are the prerequisite to build up

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an optimisation system that benefits from this information to predict the required reserve to balance the intermittent energy source. Following these principles, two different types of optimisation schemes for existing BRP’s are described and demonstrated. These are discussed and conclusions are drawn to allow other optimisations scenarios to be set up as described above.

5.1.

Different Approaches to Forecast Power Output

The aim of a wind power forecast is to link the wind prediction of the Numerical Weather Prediction (NWP) model to the power output of the turbine. Three fundamentally different approaches can be distinguished (e.g. [5], [11], [12]): • The physical approach aims to describe the physical process of converting wind to power and models all of the steps involved. • The statistical approach aims at describing the connection between predicted wind and power output directly by statistical analysis of time series from data in the past.

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• The learning approach uses artificial intelligence (AI) methods to learn the relation between predicted wind and power output from time series of the past. In practical applications these methods are sometimes combined or mixed. The different types of wind power forecasting methods and systems currently in use worldwide were summarised by Giebel et al. [11]. It should be noted that, whenever possible, dispersion of wind power over large areas should be performed, as the aggregation of wind power leads to significant reduction of forecast errors as well as short-term fluctuations. In countries with a longer tradition and fixed feed-in tariffs this seems to be the natural way wind power is deployed. However, in countries, where wind power is relatively new and where larger single wind farms have been and are being built, developers, TSO’s, authorities and policy makers will have to consider in the future pooling of energy sources not only to it’s technical and economic feasibility, but also to allow for and set rules for such approaches, if efficient and environmentally clean deployment of renewables, especially wind energy, is a target.

5.2.

Ensemble Prediction Systems

The use of ensembles is intended to provide a set of forecasts which cover the range of possible uncertainty, recognising that it is impossible to obtain a single deterministic forecast which is always correct [13]. An ensemble prediction system (EPS) is one that produces a number of numerical weather forecasts, as opposed to a single, deterministic forecast. Ensemble techniques have been employed for some time in operational medium-range weather forecasting systems [14]. Three approaches dominate the field: (1) Ensemble Kalman Filter (EnKF) approach (e.g. [15], [16], [17]), (2) Singular vector (SV) approach (e.g. [18], [19], [20]) and (3) Breeding approach (e.g. [21], [22]). There are two other ensemble methods, the multi-model approach (e.g. [23], [24], [25], [26], [27], [28]) and the multi-scheme approach. These are discussed and tested in several studies of their feasibility (e.g. [27], [29], [30], [31], [32]). The multi-model method results in independent forecasts, but the exact

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reason of the independent solutions will never be understood. Similar arguments do not apply to other ensemble prediction methods.

Figure 1. Schematic of the MSEPS ensemble weather & wind power prediction system.

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5.3.

The MSEPS Forecasting System

The Multi-Scheme Ensemble Prediction System (MSEPS) that has been used for our studies and which will be described later in this chapter, is a limited area ensemble prediction system using 75 different NWP formulations of various physical processes. These individual “schemes” each mainly differ in their formulation of the fast meteorological processes: dynamical advection, vertical mixing and condensation. The focus is on varying the formulations of those processes in the NWP model that are most relevant for the simulation of fronts and the friction between the atmosphere and the earth’s surface, and hence critical to short-range numerical weather prediction. Meng and Zhang [31] found that a combination of different parameterisation schemes has the potential to provide better background error covariance estimation and smaller ensemble bias. Using an EPS for wind power prediction is fundamentally different from using one consisting of a few deterministic weather prediction systems, because severe weather and critical wind power events are two different patterns. The severity level increases with the wind speed in weather, while wind power has two different ranges of winds that cause strong ramping, one in the middle range and a narrow one just around the storm level (the cutoff level). Wind power forecasting models therefore have to be adopted to the use of the ensemble data. In general, a wind power prediction model or module, that is directly implemented into the MSEPS is different from traditional power prediction tools, because the ensemble approach is designed to provide

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Figure 2. Error decomposition example generated with 1-year of data from the western part of Denmark.

an objective uncertainty of the power forecasts due to the weather uncertainty and requires adaptation to make use of the additional information provided by the ensemble. Figure 1 shows the principle of the MSEPS wind power forecasting system. Apart from the direct implementation (e.g. [33], [34], [35]), some effort has been made in recent years by adopting traditional wind power prediction tools to ensemble data from the MSEPS system in research projects and studies (e.g. [36], [37], [38]).

6.

Aspects of the Forecasting Error

There is still a prevailing opinion in the wind energy community that the wind power prediction error is primarily generated by wrong weather forecasts (e.g. [11], [39]). From a meteorological perspective, this is a statement that may cause misunderstanding, because part of the error is due to a complex mixture of weather related errors. The weather forecast process itself can only be blamed for the linear error growth with forecast length. We have therefore conducted an error decomposition in order to quantify the different error sources with a large ensemble of MSEPS weather forecasts. Traditionally, increased spatial weather prediction model resolution has been said to provide better forecasts (e.g. [40], [41]), but the shortest model waves may anti-correlate with the truth and cause double punishment in the verification (e.g. [32], [42]) and thereby additional model error. An ensemble prediction approach is another way to improve forecasts with fewer anti-correlation hours and the possibility to predict and understand forecast errors.

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6.1.

Markus Pahlow, Corinna M¨ohrlen and Jess U. Jørgensen

Wind Power Error Decomposition

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In order to understand the forecasting error in wind power, we have carried out a decomposition of the influencing components that have let to misunderstanding in the past. The analysis to do this includes forecasts generated in 6 hour frequency. This means that one particular forecast horizon will verify at four different times each day. The forecast error was accumulated in 6 hour bins with centre at 3, 9, ..., 45 hour prediction horizon for additional smoothing. This hides the disturbing diurnal cycle arising from verification once per day. Thereby a linear error growth with prediction horizon is achieved. Figure 2 shows such an error decomposition for a 1-year verification over the forecast length 3-45 hours with data from the western part of Denmark. By generating a decomposition of the forecasting error it can be illustrated, which parts of the error can actually be due to the weather forecast process. In Figure 2, we have split the error into a background error and a prediction system error with a linear error growth. We illustrated the potential improvements from the weather part with “good forecast”, “average forecast” and “poor forecast”, which differ to a certain degree. However, when looking at the background error, then the differences between a “good forecast” and a “poor forecast” is less significant. The background error is not directly “felt” by end users, because this initial error is in the daily operation either recovered by short-term forecasts with use of online measurements or if this is not available, by extrapolating the online measurements a few hours ahead. In the day-ahead trading, which takes place in most countries approximately 12 hours before the time period at which the bids have to be given on the market, the first few hours of the forecast are also irrelevant. The light gray line represents a typical pattern of an online forecast. This forecast has a steeper error growth, but starts from zero. A short-term forecast 1-2 hours ahead is typically close to the persistence level except in time periods, where the wind power ramps significantly. The linear error growth indicates that the weather forecast is responsible for about 1/3 of the error in the forecast for the next day and the remainder is a background error originating from different sources. These additional error sources were found to be due to: (i) the initial weather conditions; (ii) sub grid scale weather activity; (iii) coordinate transformations; (iv) the algorithm used to compute the wind power; (v) imperfection of turbines and measurement errors. The question remains, which fraction of the background error is caused by imperfect initial conditions of the weather forecast and which fraction is due to erroneous wind power parameterisations. By extrapolating the linear forecast error growth from 9-45 hours down to the 0 hour forecast, the background mean absolute error (MAE) could be estimated to be just under 4% of installed capacity. Therefore, we added an additional fixed uncertainty band of +/- 4% of the installed capacity to the native MSEPS ensemble uncertainty to account for the background error that exists in addition to the weather forecast generated error. With this band, we achieved that 8120 hours out of 9050 hours or 89.7% of the hours are covered by the predicted uncertainty interval. The remaining 10% have numerically large errors that are only partially covered by the MSEPS uncertainty prediction. Figure 3 shows a scatter plot of this test. The x-axis shows the measured wind power [MWh] and the y-axis the mean absolute error (MAE) in % installed capacity. The black crosses are those forecasts that deviate less than +/- 4% from the measurements. Here 8120 hours are equivalent to 89.7% of the time. The gray crosses towards the top show those

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errors that are greater than +4% and are measured for 576 hours, equivalent to 6.4%. The gray crosses towards the lower boundary are 354 hours and equivalent to 3.9% of the time. A 4% constant background error is a poor approximation and probably the explanation why 10% of the error events are unpredicted. Part of the background error is due to the computation of the wind power. The inherent error from the conversion of wind to power of course also has an impact on the error, not only the weather forecast. The difference of different methodologies for different forecast problems can be quite large. We will therefore demonstrate this difference in the next section. The following list shows the methodologies that have been used in this demonstration. The power prediction methods can be distinguished as follows: 1. Method: Direction and time independent simple sorting algorithm. 2. Method: Time dependent and direction independent least square algorithm. 3. Method: Direction dependent least square algorithm. 4. Method: Direction dependent least square algorithm using combined forecasts. 5. Method: Same as method 4, but including stability dependent corrections.

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6. Method: 300 member ensemble forecast of method 5 - all farms are handled individually and 6 parameters for each one of the 75 members are used to compute the power. Note, when considering the results from investigating these different power prediction methods, all values are given as the improvement in % to the mean absolute error (MAE) for the day-ahead forecasts based on only one daily (00UTC) forecast performed with the basic power prediction (method 1). To verify the impact of the conversion from meteorological parameters to wind power on the forecasting error, we use these 6 different methods to convert wind and other weather parameters into wind power output. Table 1 shows the results of the different power prediction approaches based on the same weather data. Each result differs only in the level of detail of the statistical computation of wind power. The verification for Ireland took place for a period of 1-year (2005) for 51 individual wind farms of a total capacity of 497 MW, the Danish verification of aggregated wind power was up-scaled from 160 sites and took place for the period 09/2004-10/2005 and in Germany the verification was conducted for Germany as a whole and three individual TSO’s for a 10month period with the up-scaled online measurements valid for approximately 19 GW from estimated measurements published by the three TSO’s (01/2006-10/2006). It can be seen from Table 1 that the forecast quality has been improved significantly for all investigated areas from a relatively simple power curve conversion method (i.e. method 1) to a more complex method (e.g. method 5 or 6), when making use of the additional information from the ensemble. The relatively low improvement in Australia is due to the fact that the background error is higher for only 6 wind farms in comparison to the areas with a large number of farms. Three major results can be drawn from from this investigation:

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Figure 3. Scatter Plot of the mean absolute error (MAE) with a constant background error of 4% added to the native forecast of the EPS mean. The black crosses are EPS mean forecasts +/- 4% error, while the gray crosses display the errors that lie outside this band.

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• The advanced methods, using several parameters and EPS information are providing superior results to using simple methods. • The forecasting error is dependent on the location, the size and the load of the farms/areas. • Detailed measurements can significantly reduce the forecast error.

7.

Reserve Prediction and Optimisation

The economic value of wind power is related to the predictability of the weather and hence the wind power. The required forecast horizon depends on the market structure and the inertia of the conventional power plant. An economic benefit from uncertainty predictions can in the most static markets only be achieved, if the prediction ranges up to 48 hours ahead. The trading of up-regulation and down-regulation can then be synchronised with the trading of the total generation and therefore the handling costs of wind power on the grid can be optimised. This creates not only an economic advantage for the system operator, but also a fair strategy for the intermittent energy source, as it prevents dominant generators with large market shares gaining exclusive contracts caused by the imbalance from wind generation.

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Table 1. Statistics of the investigation of different wind power forecast methodologies in various countries. Method Cap [MW] Load [%] 1 2 3 4 5 6

Ireland 497 33 1.00 2.18 9.34 14.44 15.29 17.23

Australia 388 30 1.00 3.01 1.51 4.38 6.58 9.18

Denmark 1830 21 1.00 8.52 9.87 12.41 13.60 n/a

Germany 19030 16 1.00 1.40 12.32 12.32 15.41 n/a

E.ON 7787 16 1.00 1.09 8.30 8.30 11.57 n/a

VE 7486 15 1.00 2.11 12.05 12.05 14.80 n/a

RWE 3464 16 1.00 2.26 15.44 15.44 16.95 n/a

In addition to the economic benefit, there is a security benefit of trading reserve capacity in advance, as the capacity can then be scheduled with the focus on grid security.

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7.1.

Optimisation of Reserve Predictions: Example Denmark

Next an example of a reserve prediction scenario is presented, based on a situation in which there is knowledge of the uncertainty of the wind power forecast in advance. This assumption is reflected by the statistical scores. The up-regulation and down-regulation has been counted separately, because of the asymmetry in the market pricing structure. Note however, that for convenience, the computation of the optimal forecast in the example below is for simplicity based on the assumption that the price of up-regulation and down-regulation is the same, which would normally not be the case. It should in most cases be more efficient to trade with a lower prediction of future production of wind power than expected and arrange for down-regulation, when the wind power production increases above the predicted power, because down-regulation is in many markets on average only 25% of the price for up-regulation. In our example, we compared different forecasting scenarios: 1. Scenario: Use of the optimal forecast and reserve prediction for next day’s wind generation. 2. Scenario: Use of the optimal forecast for next day’s wind generation, where reserve is only allocated on the next day according to the demand. 3. Scenario: Use of the Ensemble average forecast for next day’s wind generation, where reserve is only allocated on the next day according to the demand. The scenarios in this example were constructed for a fully competitive market, where reserve capacity is traded day-ahead. In this example, we used the same single forecast in both scenario no. 1 and no. 2. In scenario no. 1 we allocated reserve capacity in a market with competition on reserve. In case no. 2 and no. 3 all regulation was traded according to the demand on the short-term spot market.

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Table 2 shows the results of the 3 scenarios. Note, that all values are in units percent of installed capacity. The numbers are taken from prioritised production in the western part of Denmark in the period January to April 2005. The forecast horizon is 17 to 41 hours until daylight saving starts, then the forecast horizon changes to 16 to 40 hours. The total prioritised production is rated at 1900 MW in this period, which is almost equivalent to the minimum consumption in this area. Table 2. Results of the 3 scenarios of using and not using spinning reserve predictions in the wind power production forecasts for the western part of Denmark in the period January to April 2005. All statistical quantities are given as [%] of installed wind power capacity of Western Denmark.

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Scenario no. EPS configuration Number of Hours Forecasted Mean Bias Mean Absolute Error Stdev Error Correlation

1 optimal* 2331 27.0 -0.77 2.72 5.39 0.977

2 optimal 2331 26.6 -1.19 5.30 7.60 0.953

3 average 2331 26.7 -1.02 5.83 8.32 0.946

It can be seen in Table 2 that using the optimal forecast and reserve prediction for next day’s wind generation (scenario no. 1) almost halves the mean absolute error (MAE) of both scenario no. 2 and scenario no. 3 and hence the costs for expensive spinning reserve for unexpected events (errors). The root mean square scores are slightly lower, but still of the order 30% and 40% better for scenario 1 than for scenario 2 and 3, respectively. The lower improvement in the root mean square (RMSE) is related to the fact that the prediction of the forecast error removes the smaller errors in a band-like way around the optimal forecast. Since the RMSE is more sensitive to the larger errors, the improvement is lower. Table 3 shows the percentage of reserve of installed capacity that is required, predicted and non-predicted in scenario 1. It can be observed that the predicted up-regulation provided better results than the down-regulation. This is the result of optimising the forecast towards the less costly reserve. The average of the ensemble has a score of 5.83%. This is approximately the same score as the best single ensemble member. The optimal forecast gives for the same period 5.30% absolute error. In the above example we found that if we add the uncertainty band and trade the predicted amount of reserve for both up-regulation and down-regulation on the market, we can reduce the error to 2.72%. This means that we have to trade regulation for only 2.72% of the capacity instead of 5.30% in the short-term spot market with high prices. In this computation, it is assumed that there is no error when the error of the optimal forecast lies within the predicted uncertainty band. This increases the correlation signifi-

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Table 3. Results of the verification of the predicted regulative power on a long-term basis for the western part of Denmark in the period January to April 2005. The regulation magnitude is given as % of installed wind power capacity. The results correspond to scenario no. 1.

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Regulation type Average Up Average Down Predicted Up Predicted Down Unpredicted Up Unpredicted Down Unused

Regulation magnitude 2.40 3.43 1.43 1.68 0.98 1.75 0.91

description required regulation required regulation traded day ahead traded day ahead traded on the day or handled by flexible contracts traded on the day or handled by flexible contracts unnecessary regulation capacity

cantly, which indicates that a large fraction of the forecast errors within the uncertainty band are completely unpredictable. We suggest that these errors are handled most efficiently by being balanced with pre-allocated constant reserve. To be able to estimate the impact of such a pre-allocation, we considered 5 cases, with different percentages of pre-allocation and computed the amount of hours, where the preallocation accounts for the forecast error. With these results, it becomes feasible to set up a cost analysis of the optimal amount of pre-allocation. Our intention here is however merely to demonstrate how to design an optimised prediction system for a specific problem. In the following we have therefore constructed 4 cases, where a fixed fraction of the installed capacity is pre-allocated as reserve with a long term contract and additional capacity is assumed to be allocated in a market with competition according to the predicted requirement (see scenario 1 above). 1. Case: Additional long-term contract for reserve of 0.8% of the inst. capacity 2. Case: Additional long-term contract for reserve of 1.6% of the inst. capacity 3. Case: Additional long-term contract for reserve of 3.2% of the inst. capacity 4. Case: Additional long-term contract for reserve of 6.4% of the inst. capacity 5. Case: Additional long-term contract for reserve of 12.8% of the inst. capacity The following equation has been used to compute the number of hours when the preallocated reserve accounts for the forecast error: R = max(Rpre , astability x + bstability )

(1)

where Rpre is the pre-allocated reserve from Table 4 and x is the ensemble spread. The sum of the hours, where the resulting pre-allocated reserve fully covers the forecast error is shown in Table 4. It can be seen in the table, how well the ensemble spread covers

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the forecast error for different levels of pre-allocation. The validation also reveals that only 3 hours out of 2331 hours (97 days) had forecast errors that were not covered by the reserve given by equation 1 when using a 12.8% pre-allocation. Table 4. Number of hours out of 2331 hours where no additional reserve to the day-ahead and pre-allocated reserve is required for different levels of constant pre-allocation.

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case no. 1 2 3 4 5

pre-allocated reserve [%] 0.8 1.6 3.2 6.4 12.8

no reserve required [hours] 1049 1439 1696 1990 2228

percent coverage [%] 45% 62% 73% 85% 96%

A pre-allocation of nearly 13% to reach 96% coverage by using this type of reserve is quite a large amount compared to the mean absolute error (MAE) of the forecast (< 6%). It is therefore recommended to study and evaluate from case to case and with real market data, if more than 10% of pre-allocation is a reasonable result of the optimisation. However, since there exists a pronounced price difference between up-regulating and down-regulating reserve, this amount could still proof as the most cost efficient. It should also be noted, that this procedure does not eliminate the forecast error. Nevertheless, the reserve is traded more competitively than without using the predictions and hence the balancing costs can be reduced significantly. Another aspect that may be of relevance in other countries is the combination of wind power and large power plants with respect to reserve requirements. In the western part of Denmark for example, the largest plant is 640 MW and corresponds to 33% of the prioritised wind generation. When this plant is in operation, then reserve requirements for wind power will never exceed the reserved 640 MW of up-regulation, which is required for this plant. A combination of wind power forecast failure and a 640 MW plant failure could of course cause a problem for grid security. However, it becomes clear from this example, that in a grid with large plants, it is not the amount of installed wind power that is responsible for the maximum reserve requirement and often also not for the large balancing costs. However, the shown methodology may also only provide a first step in how to combine the reserve allocation of larger power plants and wind power within an acceptable risk management structure and well working interconnections to cover parts of missing power in case of correlated failures.

7.2.

Optimisation of the Reserve Prediction: Example Canada

As described above, the reasoning for using cost functions rather than standard statistical measures is that the uncertainty in the weather is random. Converted to wind power, the weather uncertainty is sometimes found to be very low and sometimes found to be very

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high. In the long-term and if a reasonable amount of wind power is installed in an area, a constant reserve would consequently be inefficient and expensive. Additionally, full coverage of all forecast errors is expensive, also because there is always a risk of non-weather related accidents during operation. The following is an example of an area, where the weather is highly variable and so is the wind power production. Within Alberta Electric System Operator’s AESO’s wind power pilot study [43] it has also been found that strong ramping is not seldom. In such areas larger amounts of constant reserve allocation are very expensive. The investigated scenarios are therefore based on this experience and the next step in benchmarking and adopting a forecasting system no longer to a low mean absolute error (MAE), but rather to the conditions under which the wind power integration takes place and optimise the forecasting system with this knowledge. 7.2.1. Optimisation Scenarios 1. Scenario: Static Regulation In this scenario the upper limit for reserve allocation from the error statistic of one forecast over 1 year has been determined. The actual reserve allocation is limited by the forecast with the following restriction: FC + R < UB

(2)

F C − R > LB

(3)

FC − R < G < FC + R

(4)

and

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with

Here, F C is the forecast, R is the reserve, G is the actual generation, U B is the upper bound of full generation and LB is the lower bound of no generation. R is chosen to secure that the generation is always lower than the sum of the forecast and reserve and higher than the forecast without reserve, which is supposed to historically always be valid (here: 90%). 2. Scenario: Deterministic Forecast Regulation In this scenario the reserve R is chosen as a fixed reserve allocation for upward and downward ramping. We chose +/-11% of installed capacity, which is equivalent to the mean absolute error (MAE). 3. Scenario: Security Regulation In this scenario, the reserve R is computed from the difference between minimum and maximum of the ensemble in each hour of the forecast. There may however be areas, where it will be necessary to adopt the difference of minimum and maximum in such a way that single outliers are not increasing the spread unnecessarily. 4. Scenario: Economic Regulation In this scenario, an optimisation of scenario 3 is used, where the unused reserve allocation is reduced for economic reasons. As a first approximation the 70% quantile

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Markus Pahlow, Corinna M¨ohrlen and Jess U. Jørgensen of the level in Scenario 3 was used. In a future optimisation, the skewness of the price of positive and negative reserve also would have to be considered.

The data that were used to simulate different scenarios was 1-year (2006) of forecasting data for two regions in Alberta, the South-West region and the South-Centre region, where 5 wind farms with a combined installed capacity of 251.4 MW were used. The simulation was carried out with 12-18 hour forecasts issued every 6th hour. The forecasting model system was run in 22.5 km horizontal resolution and the wind power forecast was a probabilistic forecast generated with 2 combinations of different power conversion methods each using 300 weather parameters from the 75 member MSEPS ensemble system. In Table 5 statistics of the aggregated forecasts for the 5 wind farms are displayed. Table 5. Statistics of wind power forecasts without reserve prediction. Statistics Parameter Bias (FC) Mean Absolute Error (FC) RMSE (FC) Correlation (FC)

% rated capacity -2.10 11.60 16.80 0.85

In Table 6, where the optimisation results are summmarised, all parameters are given in % of installed capacity. The relative cost of various types of reserve was not taken from market reports, but estimated as a percentage relative to urgent reserve on the spot market, which was set to 1.0. This gives the following estimates:

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• Urgent reserve = 1.0% • Unused passive reserve = 0.1% • Allocated reserve = 0.6% The “allocated reserve” is the reserve that is allocated according to the forecasts and assumed to be bought day-ahead. The statistical parameter in the first rows of Table 6 are based on the forecasts including a reserve allocation. This means that the estimated error is added or subtracted from the forecast according to the used optimisation scheme. Therefore, these parameters have an index FCR (forecast + reserve), while the statistical parameters in Table 5 are based on the raw forecasts and indexed FC. These FC forecasts were optimised on the mean absolute error (MAE) to the observations. There are furthermore output results of three types of reserve in the table, the required reserve, the predicted reserve and the unpredicted reserve, respectively. These are named UpReg for up-regulation or DownReg for down-regulation of the power on the electricity grid because of incorrect forecasts. Although the difference in price for up-regulation and down-regulation, where down-regulation is often a factor of 3-5 cheaper than up-regulation or even generates revenue, was not accounted for in these simulations, the cost function is more favourable towards down-regulation than up-regulation. This can be seen in scenario 2 and 4, which are more cost efficient and use less up-regulating expensive reserve.

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Table 6. Optimisation Scenarios for the AESO area. Optimisation

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Reserve Predictor Bias (FCR) MAE (FCR) RMSE (FCR) Correlation (FCR) Required UpReg (7) Required DownReg (8) Predicted UpReg (9) Predicted DownReg (10) Unpredicted UpReg (7-9) Unpredicted DownReg (8-10) Unused Regulation Effective cost Hours covered by reserve

Scenario 1 Static Reserve [% rated cap.] 75% reserve 0.00 0.00 0.00 1.00 4.70 6.80 4.70 6.80 0.00 0.00 32.30 15.90 100.00

Scenario 2 Determin. FC Reserve [% rated cap.] FC+/-11% -0.85 4.54 10.15 0.95 4.70 6.80 2.84 4.10 1.80 2.70 3.30 10.80 64.50

Scenario 3 Security Reserve [% rated cap.] max-min -0.50 1.40 5.40 0.99 4.70 6.80 4.20 5.80 0.50 1.00 9.30 11.40 85.10

Scenario 4 Economic Reserve [% rated cap.] 0.7*(max-min) -0.80 2.50 7.22 0.97 4.70 6.80 3.83 5.13 0.87 1.67 5.67 10.83 76.00

Table 6 also shows the significant difference in effective costs for the first scenario, the purely static reserve allocation, in comparison to the other 3 scenarios. However, when comparing the hours covered by the reserve, then it becomes clear that the coverage and the effective cost are cross-correlated for this type of optimisation. Scenario 2 is for example equally cost efficient to scenario 4, but covers only 64.5% of the hours, while scenario 4 covers 76% of all hours. Although the security scenario (no. 3) is slightly less cost efficient, the covered hours are quite significantly higher than for scenario 2 and also scenario 4 (85.1% versus 76% and 64.5%). Looking at the mean absolute error (MAE) or the root mean square error (RMSE), scenario 3 seems to also outperform scenario 2 and 4. However, when comparing the unused regulation, then the security scenario has almost double the amount of scenario 4, and three times as much unused regulation as scenario 2. Dependent on the pricing structure of the market, which was excluded in this experiment, this could even change the effective cost levels of the scenarios, The results of the Canadian example demonstrate once again that the statistical error measures are not capable of providing a complete answer for an optimisation target. However, the results do provide an insight for the complexity of the optimisation of cost functions of reserve to end-users requirements.

8.

Summary and Discussion

The ensemble prediction method has a number of applications in wind energy integration and in energy in general. From the discussion in this chapter it can be concluded that energy markets can benefit from using ensemble predictions. Table 7 shows a general description of which forecasts from an ensemble forecasting system should be chosen for minimal costs. We have assumed that the capacity of wind power is sufficient for wind to be the “price maker” when the wind conditions are optimal. It can be seen in Table 7 that the forecast with the lowest mean absolute error (MAE) is not always the forecast that will

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generate the lowest cost of integration. Although the original scope of ensemble prediction was to be able to conduct risk analysis of severe weather, it appears that the application in energy is not limited to grid security, but extends to trading and management of weather dependent energy generation systems. This includes all generation methods except nuclear power after the Kyoto protocol has become effective. Nuclear power generators do not have a CO2 problem and have therefore the least incentive to participate in the balance of wind power. In the future, even nuclear plants may however need forecasts to be able to give bids on the market and to operate efficiently. Table 7. Summary of the cost optimised forecast selection.

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Predicted load factor in [%] 0-10 70-100 20-70

Competition on regulation Good on down-regulation Good on up-regulation Good for up and down

Forecast choice EPS minimum EPS maximum Best forecast

Reserve allocation Downward Upward Down and up

The electricity price has a high volatility level because of the limited storage capacity and the strong relationship to oil prices, political disputes and not to forget the uncertainty in the weather development. An increasing number of people around the world make their living on trading and because of the automatisation fewer people are required in today’s production processes. This means that in the future, increasing volatility of stocks and energy can and have to be expected. However, the volatility of the energy pricing may increase more than that of stocks for two reasons: • The amount of intermittent renewables will increase more than the available storage capacity. • The energy markets are developing slowly with new trading options. Increased volatility on pricing will result in increased volatility on the generation as well, and consequently lower efficiency and higher costs. Increased volatility can also trigger instabilities on the grid. A typical example could be two competing generators that have to ramp with opposite sign to stay in balance. Increased volatility implies that the frequency of ramps will increase. Such ramps are not dangerous, but certainly do not add to the system security. The generators will bare the loss during the ramp, because of the higher average price. The optimisation strategies that we have presented here serve to dampen volatility and the intermittent energy price. The main ingredients in this optimisation is ensemble forecasting, which increases the robustness of the decision process. Decisions will be taken on the basis of many results that are generated by some kind of perturbation. The market participants will, with the help of ensemble predictions, in the future know in which

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range competing parties plan to set their bid on the market. There is also more continuity in time by using ensemble forecasting, because the decision process changes slowly hour by hour. This leads to a more stable decision process. Ensemble forecasting makes market participants aware of the risk of any speculation, although it may not be enough to prevent speculations. A cost efficient way to dampen volatility is to allocate reserve in advance according to the uncertainty forecast for the intermittent energy source. The pre-allocation of reserve secures that a high level of competition can be achieved and last-minute volatility will be reduced. A next step to reduce volatility is to create energy pools that allow phase shifting in time for intermittent energy. This will cap some of the price off the intermittent energy, because the balancing parties in the pool will demand a higher price than the intermittent generators. The more uncertainty there is on the pool’s output for other market participants, the better for the pool. There is an incentive to make the pool members dynamic, so the other market participants cannot guess how much the pool is dependent on the intermittent generation pattern. An additional step is required to secure fair prices during periods, when the intermittent energy is in excess. The market will often know the periods, when the weather is well predictable and the market price drops. Trading the intermittent energy several days in advance will allow scheduled generators to reduce their emissions and withhold their own generation. Their incentive to do so increases, if they can schedule the production well in advance and thereby cut the marginal costs down. Trading of intermittent energy requires therefore a special effort and forecasts with an ensemble technique along with temporary pooling with other energy sources will aid in achieving efficiency and stable prices.

9.

Conclusion

The major benefit of ensemble prediction methods is that weather dependent energy generation can be classified as certain and uncertain. It has been tradition to not separate between certain and uncertain generation in the trading, because day-ahead markets were used to sell according to one best possible forecast only. A creative trader could still trade strategically on the basis of a deterministic forecast and pool the deterministic forecasts with ancillary services and bid in the sum of the intermittent and ancillary service generation with a higher price. Ensemble forecasting however opens up possibilities for more creative and efficient trading strategies. Additionally, the trading process is going to become more complex and in fact too complex for a subjective decision process in the future. The objective decision process then has to be done on a computer, based on predefined criteria. The analysis in this chapter suggests that the model for the objective decision process has to be kept small. This secures fast convergence of the solution as well as the possibility for the trained user to redefine boundary conditions and test the solution’s sensitivity to various likely and less likely events. Another result that can be derived from our analysis is that a balance responsible party for intermittent energy should issue regular tenders on pool participation. This will allow

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an energy pool to deliver power according to a different profile than the weather allows for. Typically the generation should be phase-shifted to match the demand better. The net result is then that the market can no longer force the intermittent generator to bid in with low prices. The intermittent generator thereby gains freedom with the dynamically changing “cocktail” of pooled energy, weather uncertainty and advanced trading according to the ensemble minimum. The more degrees of freedom, the higher the price for the intermittent energy, which is critical for the success rate in the future energy markets, because it can not be expected that the required investments in renewable energy will increase or even be kept at today’s rate without an economic incentive in addition to the environmental benefits.

References [1] Global Wind Energy Council (www.gwec.net), Jan. 2008. [2] B. Lange, M. Wolff, R. Mackensen, R. Jursa, K. Rohrig, D. Braams, B. Valov, L. Hofmann, C. Scholz, and K. Biermann, ”Operational control of wind farm clusters for transmission system operators,” in Proc. European Offshore Conference, Berlin, 2007. [3] IEA, ”Variability of wind power and other renewables: Management options and strategies”, IEA/OECD, Paris, 2005. [4] IEA, ”World energy outlook 2004”, IEA/OECD, Paris, 2004.

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[5] B. Ernst, B. Oakleaf, M. L. Ahlstrom, M. Lange, C. M¨ohrlen, B. Lange, U. Focken, and K. Rohrig, ”Predicting the wind”, IEEE Power and Energy Magazine, vol. 5, no. 6, pp. 78-89, 2007. [6] Bundesnetzagentur, BK6-07-003 Eckpunkte f¨ur die Direktvermarktung von EEGStrom, (http://www.bundesnetzagentur.de/media/archive/11971.pdf), Dec. 2007. [7] T. Acker, ”Characterization of wind and hydropower integration in the USA,” in Proc. AWEA Conference Windpower, Denver, May 2005. [8] E. D. Castronuovo and J. A. P. Lopes, ”Optimal operation and hydro storage sizing of a wind-hydro power plant,” El. Power Energy Sys., vol. 26, pp. 771-778, 2004. [9] M. Pahlow, L.-E. Langhans, C. M¨ohrlen, and J. U. Jørgensen, ”On the potential of coupling renewables into energy pools,” Z. Energiewirtschaft, vol. 31, pp. 35-46, 2007. [10] www.bundesnetzagentur.de, Nov. 2007. [11] G. Giebel, G. Kariniotakis, and R. Brownsword, ”The state- of-the-art in short-term forecasting of wind power - a literature overview,” Position paper ANEMOS Project, 38 pp., 2005. [12] G. Hassan, ”Short-term wind energy forecasting: Technology and policy,” Report for the Canadian Wind Energy Assocoation, 2006. Optimization Advances in Electric Power Systems, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[13] T. P. Legg, K. R. Mylne, and C. Woolcock, ”Use of medium-range ensembles at the Met Office I: PREVIN - a system for the production of probabilistic forecast information from the ECMWF EPS,” Met. Appl., vol. 9, pp. 255-271, 2002. [14] R. Buizza, P. L. Houtekamer, Z. Toth, G. Pellerin, M. Wei, and Y. Zhu, ”A comparison of the ECMWF, MSC, and NCEP global ensemble prediction systems,” Month. Weather Rev., vol. 133, pp. 1076-1097, 2005. [15] G. Evensen, ”Ensemble Kalman Filter: theoretical formulation and practical implementation,” Ocean Dynamics, vol. 53, pp. 347-367, 2003. [16] T. M. Hamill and C. Synder, ”A hybrid ensemble Kalman Filter-3D variational analysis scheme,” Month. Weath. Rev., vol. 128, pp. 2905-2919, 2000. [17] P. L. Houtekamer, L. Herschel, and L. Mitchell, ”A sequential Ensemble Kalman Filter for atmospheric data assimilation,” Month. Weath. Rev., vol. 129, pp. 123-137, 2001. [18] T. N. Palmer, ”A non-linear dynamical perspective on model error: A proposal for non-local stochastic-dynamic parameterisation in weather and climate prediction models,” Q. J. R. Meteorol. Soc., vol. 127, pp. 279-304, 2001. [19] R. Buizza, M. Miller, and T. N. Palmer, ”Stochastic representation of model uncertainties in the ECMWF Ensemble Prediction System,” Q. J. R. Meteorol. Soc., vol. 125, pp. 2887-2908, 1999.

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[20] J. Barkmeijer, R. Buizza, T. N. Palmer, K. Puri, and J.-F. Mahfouf, ”Tropical Singular Vectors computed with linearized diabatic physics,” Q. J. R. Meteorol. Soc., vol. 127, pp. 685-708, 2001. [21] Z. Toth and E. Kalnay, ”Ensemble forecasting at NMC: the generation of perturbations,” Bull. Americ. Meteorol. Soc., vol. 74, pp. 2317-2330, 1993. [22] M. S. Tracton, K. Mo, W. Chen, E. Kalnay, R. Kistler, and G. White, ”Dynamic extended range forecasting (DERF) at the National Meteorological Center: practical aspects,” Month. Weath. Rev., vol. 117, pp. 1604-1635, 1989. [23] P. L. Houtekamer and J. Derome, ”Prediction experiments with two-member ensembles”, Month. Weath. Rev., vol. 122, pp. 2179-2191, 1994. [24] P. L. Houtekamer and L. Lavaliere, ”Using ensemble forecasts for model validation,” Mon. Weath. Rev., vol. 125, pp. 796-811, 1997. [25] R. E. Evans, M. S. J. Harrison, R. J. Graham, and K. R. Mylne, ”Joint medium-range ensembles form the Met. Office and ECMWF systems,” Month. Weath. Rev., vol. 128, pp. 31043127, 2000. [26] T. M. Krishnamurti, C. M. Krishtawal, Z. Zhang, T. LaRow, D. Bachiochi, and C. E. Williford, ”Multimodel ensemble forecasts for weather and seasonal climate,” J. Climate, vol. 13, pp. 4196-4216, 2000.

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[27] D. J. Stensrud, J. W. Bao, and T. T. Warner, ”Using initial condition and model physics perturbations in short-range ensemble simulations of mesoscale convective systems,” Month. Weath. Rev., vol. 128, pp. 2077-2107, 2000. [28] P. J. Roebber, D. M. Schultz, B. A. Colle, and D. J. Stensrud, ”Toward improved prediction: High-resolution and ensemble modelling systems in operations,” Weath. Forecasting, vol. 19, pp. 936-949, 2004. [29] K. Mylne and K. Robertson, ”Poor Man’s EPS experiments and LAMEPS plans at the Met Office,” LAM EPS Workshop, Madrid, October, 2002. [30] K. Sattler and H. Feddersen, ”EFFS Treatment of uncertainties in the prediction of heavy rainfall using different ensemble approaches with DMI-HIRLAM,” Scientific Report 03-07, Danish Meteorological Institute, 2003. [31] Z. Meng and F. Zhang, ”Tests of an Ensemble Kalman Filter for Mesoscale and Regional-Scale Data Assimilation. Part II: Imperfect Model Experiments,” Month. Weath. Rev., vol. 135, pp. 1403-1423, 2007. [32] C. M¨ohrlen, ”Uncertainty in wind energy forecasting,” Ph.D. dissertation, University College Cork, Ireland, 2004.

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[33] C. M¨ohrlen and J. U. Jørgensen, ”Forecasting wind power in high wind penetration markets using multi-scheme ensemble prediction methods,” in Proc. German Wind Energy Conference DEWEK, Bremen, Nov. 2006. ´ Gallach´oir, and E. J. McKeogh, ”Fore[34] S. Lang, C. M¨ohrlen, J. U. Jørgensen, B. P. O casting total wind power generation on the Republic of Ireland grid with a MultiScheme Ensemble Prediction System,” Proc. Global Windpower, Adelaide, Sept. 2006. ´ Gallach´oir, and E. J. McKeogh, ”Appli[35] S. Lang, C. M¨ohrlen, J. U. Jørgensen, B. P. O cation of a Multi-Scheme Ensemble Prediction System for wind power forecasting in Ireland and comparison with validation results from Denmark and Germany,” in Proc. European Wind Energy Conference, Athens, Feb. 2006. ¨ Cali, K. Rohrig, B. Lange, K. Melih, C. M¨ohrlen, and J.U. Jørgensen, ”Wind power [36] U. forecasting and confidence interval estimation using multiple numerical weather prediction models,” in Proc. European Wind Energy Conference, Milan, 2007. [37] C. M¨ohrlen, J. U. Jørgensen, P. Pinson, H. Madsen, and J. Runge Kristofferson, ”HRENSEMBLEHR - High resolution ensemble for Horns Rev: A project overview,” in Proc. European Offshore Wind Energy Conference, Berlin, 2007. [38] P. Pinson, H. Madsen, C. M¨ohrlen, and J. U. Jørgensen, ”Ensemble-based forecasting at Horns Rev,” DTU Technical Report, 2008. [39] J. Jackson, ”A cry for better forecasters in Denmark,” Wind Power Monthly, vol. 12, pp. 40-42, 2003. Optimization Advances in Electric Power Systems, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[40] C. M¨ohrlen and J. U. Jørgensen, Verification of Ensemble Prediction Systems for a new market: wind energy. ECMWF Special Project Interim Report, 3, 2005. [41] C. Keil and M. Hagen, Evaluation of high resolution NWP simulations with radar data,” Phys. and Chem. of the Earth B, vol.25, pp. 1267-1272, 2000. [42] R. N. Hoffman, Z. Liu, J.-F. Louis, and C. Grassotti, ”Distortion representation of forecast errors,” Mon. Wea. Rev., vol. 123, pp. 2758-2770, 2005.

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[43] Alberta Electric System Operator (AESO) wind (http://www.aeso.ca/gridoperations/13825.html), 2006.

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power

pilot

study

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In: Optimization Advances in Electric Power Systems ISBN: 978-1-60692-613-0 Editor: Edgardo D. Castronuovo, pp. 181-223 © 2008 Nova Science Publishers, Inc.

Chapter 8

SECURITY OPTIMIZATION OF BULK POWER SYSTEMS IN THE MARKET ENVIRONMENT Alberto Berizzi, Cristian Bovo, Maurizio Delfanti and Marco Merlo Dipartimento di Energia, Politecnico di Milano, Milano, Italy

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1.

Abstract

The Chapter describes some applications of optimization procedures in the area of power system control, with attention to security in the market environment. In particular, the focus of the Chapter is on the reactive/voltage security. In the first Section, the authors will focus on the use of optimization methods to manage the reactive resources: the Optimal Reactive Power Flow (ORPF) problem will be introduced and analysed in detail, taking the minimization of power losses as reference. The goal of such procedures is to define voltage profiles suitable for the operation of bulk power systems in order to optimize both economy and security. In the market environment, due to a change of perspective, the minimization of power system losses could be no longer adequate; therefore, the authors developed some ORPF procedures adopting different objective functions (OFs) dealing with security. In the Section, such objective functions will be explained and the relevant issues described: the voltage collapse distance, different formulations of voltage stability indices derived from the Power Flow Jacobian, the role of the Secondary Voltage Regulation. These new approaches are dealt with adopting modern optimization techniques. The second Section will present the application of optimization technique based on the Interior point and Artificial Intelligence (in particular, Genetic Algorithms). Finally, the third Section will explain how, in a market environment, different points of view can be taken into account by the use of Multiobjective methodologies. They allow improving the security and economy of power systems at the same time during the short-term planning and the operation. The possible use of several objective functions, the methods adopted to take into account the different (non inferior) solutions and the procedures that can be implemented to compare the non inferior operating points will be described. In particular, once the determination of the Pareto set of non inferior solutions is performed, some different methodologies are available in order to make the ultimate choice and to determine the final operating point for the system. This Section will present a review of these methodologies, with particular attention to the Worth Trade Off analysis.

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2. Introduction Recent blackouts which took place in several Countries demonstrated the need for available resources (including reactive resources) to be managed carefully. In the past, transmission engineers were used to decouple the problem of the steady-state security into a real and a reactive subproblem. For the transmission system, in particular, the above mentioned subdivision allows the separation of the problems of the thermal limits of the grid branches and of the voltage support. The first one is mainly related to the real power flows, thus resulting strongly connected to the electric energy market. The voltage support, on the contrary, deals with the reactive subproblem; usually, it is considered not connected to the electricity market due to its low economic impact. In this Chapter, the focus will be on the reactive subproblem, the relevant responsibility being completely on the system operator, while we will assume that the thermal limits will be dealt with by a zonal approach of the electricity market. Nowadays, a non-optimal management of reactive power resources could result particularly critical, especially because of the increasing exploitation of transmission resources due, on one side, to the increasing demand and, on the other side, to the greater and greater difficulties in building new transmission facilities and generation plants. These needs are forcing the system operators to revise the management procedures of reactive power resources, in order to set new control strategies, which could help maintaining an adequate security level in a market framework. In vertically integrated utilities, system operators aimed at minimizing network real losses, in order to reduce operating costs [1][2][3]; the security was implicitly guaranteed by the respect of operational limits and by the naïve consideration that this type of optimization results in high voltage profiles, and this is usually considered in favor of the security. In a market perspective [4], the minimization of losses is no longer the priority of transmission operators although the efficiency of the operation is still one of the criteria that can be used [5]; on the contrary, the role of system operators is based on the need of maintaining the system security, regardless of any other issue. Therefore, the system security has become, from the philosophical point of view, the focus of the optimization, while previously it was embedded in another objective function. A first topic to be covered is the definition of a suitable objective function able to describe mathematically the concept of security. Beside the minimum real losses, other objective functions have been proposed in order to optimize in a more direct way the power system security. In particular, [5] and [6] present an objective function aimed at a better distribution of the reactive power margins over the system. In [7], a similar objective function is described, used in the Belgian electric system, in order to improve network security. Actually, the wide meaning of security makes the problem of its maximization difficult to model and solve: for example, the comparison presented in [5] between two objective functions (namely, minimum real losses and maximum reactive power margins over the system) shows that in some cases – e.g. for the Italian power system – the minimization of losses performs better, as far as the overall security is concerned. After deregulation, the research on reactive power optimization and electricity market has increased. Objective functions aimed at maximizing security can be defined according to different approaches. In [8], the authors exploited the theory of bifurcations and proposed an algorithm aimed at the maximization of the system loadability (i.e., the distance, in terms of

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real load, to the next bifurcation). In [9], a suitable voltage collapse indicator was proposed, leaving to the Transmission System Operator (TSO) the responsibility of the choice of a suitable threshold; a formula aimed at splitting the operating costs related to either the production of reactive power or to other factors related to system security is also proposed. Dealing with the voltage collapse, other mathematical indicators have been defined, such as the maximum singular value of the inverse Power Flow (PF) Jacobian matrix and its sensitivities [10]. Along with the choice of the objective function representing security, the power engineer should choose the best mathematical method to solve the modeled problem. This twofold approach is necessary to obtain the best results. In general, to calculate the optimal solution of an ORPF problem, different optimization procedures are available. Many of these are based on the Newton method [11], for example the Han-Powell [12] or the Interior Point [13] methods: these techniques require the explicit calculation of the gradient of both the objective function and the constraints and also, in the case of the Interior Point method, of the Hessian. For many power systems and scenarios, these algorithms are efficient and robust but, in some cases, the computation of the derivatives or the features of the optimization problem can make other methods preferable. For example, if discrete variables or discontinuities are to be managed, in case the computation of the needed derivatives becomes intractable or when the goal is to test many different objective functions, Genetic Algorithms (GAs) can be considered faster and more efficient [14]. GAs are efficient tools to solve optimization problems, based on the natural selection principles. The optimization procedure follows an evolutionary strategy to find the best solution for a search problem: starting from an initial population of individuals, each one representing a possible solution, the evolution takes place and modifies the population to form the next generation, until a convergence criterion is fulfilled. The most important characteristics of GAs are: they do not require the calculation of the gradients and the Hessians (therefore, they apply also for problems where the objective functions and constraints are not derivable); thanks to their implicit parallelism, the risk of being trapped in local minima is greatly reduced; finally, the different objective functions are usually very easy to implement and modify. With regards to the complexity of the ORPF problem, recent publications propose the use of Multiobjective optimization, which could be able to carry out a global optimization of the system, taking into consideration both security, according to different formulations, and its costs. Such approach allows easily the definition of costs associated to the displacement of reactive control variables. A first step is described in [15], where some voltage security constraints are introduced and operating costs are minimized. In [16], a Multiobjective (MO) approach is fully adopted and several objective functions are compared on the basis of the impact on the system operation in terms of costs. In [17] and [18] two further Multiobjective functions are proposed, which optimize several goals at the same time: • • •

real losses; costs related to the installation of new compensators; voltage security in terms of deviation of bus voltages from the relevant set-points.

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In the following subsections, the above mentioned issues will be covered and some results will be presented.

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3. Optimization of the Voltage Profile in the Electricity Market Environment In a liberalized framework, it is in general necessary to assess the security of the operating conditions resulting from the market. As a general approach, it is possible to assume that the market is organized as follows: a Day-Ahead Energy Market (DAEM), one or more Adjustment Markets (AM), and the Ancillary Service Market (ASM). The role of the DAEM is the definition of the generation pattern of the different power plants in terms of Unit Commitment (UC) and dispatching, complying with transmission constraints, often modeled adopting a simplified approach (e.g., a zonal representation, as for the Italian system) to approximate the actual power flows among the zones. Since the power factor of the flows through the network branches is quite high, and assuming that the limits of the power transmission (TTC – Total Transfer Capability) from area to area have been correctly taken into account in the DAEM by the relevant congestion management, it seems reasonable to assume that the constraints related to branch currents are satisfied by the zonal approach (see, e.g., [19][20] for the Italian case). On the contrary, for the definition of the voltage-related control variables, an ORPF should be run, assuming the real power generation profile fixed and determined by the electricity market (typically, the DAEM). Its goal is the definition of the generator voltage profiles, according to a calculation model based on the Alternate Current Power Flow (AC PF) equations. Within this very common scheme, voltage control and voltage security have to be set by the TSOs outside the electricity market, in order to maintain the overall security. At the end of the DAEM session, the TSO has to evaluate preliminarily the security conditions of the network for the next day, taking into account that there is still some uncertainty about both the final power dispatch and the possible changes of the generators in operation during the following market sessions and the real-time operation. Therefore, the market environment changes the operation practices adopted for the security enhancement in a twofold way: •

•

on one side, it becomes necessary to highlight security within the optimization model, filtering all the issues that are covered within the electricity market only; this deals with the choice of the objective function; on the other side, the increased uncertainty on the generation profile makes more difficult to select the scenario the ORPF can be applied to, i.e., to identify the best set of the control variables able to face operating conditions that can be different, in the real time, from the optimized scenario. The latter problem is linked to the choice of the best time period to run the ORPF to update the control variable set points.

Both problems are deeply investigated in the following subsections.

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3.1. Objective Functions for ORPF Problems In the present Section, the main objective functions studied suitable for the optimization of power systems in an electricity market environment are described and the relevant mathematical models are discussed. All the optimization problems can be applied also to power systems where the hierarchical voltage control is in operation. The hierarchical voltage control is currently studied and in operation in several Countries [21]. In this subsection, we will refer to the structure adopted for the Italian hierarchical voltage control [22][23][24], which is shortly explained in the following. The Italian hierarchical voltage control is subdivided in a primary level (Primary Voltage Regulation – PVR), given by the Automatic Voltage Regulators of generators, a Secondary Voltage Regulation (SVR) and a Tertiary Voltage Regulation (TVR). The SVR provides a network subdivision into electric areas around the so-called pilot nodes. The pilot nodes are the load busses with the largest short circuit ratio in a given region. The generators controlling each pilot node are chosen through the analysis of the pilot node voltage sensitivities with respect to the generating unit reactive power: the largest entries in the sensitivity matrix define the generators most suitable for the control of each pilot node. In the choice of control areas, attention is paid to verify their reciprocal electrical decoupling, necessary to avoid oscillations due to undesired interactions among the area regulators. In the SVR, the pilot bus voltage of each area is controlled by the relevant controlling generators; they change their reactive power according to the reactive power production in pu (reactive level) computed by the Secondary Voltage Regulator. All the generating units belonging to the same SVR area have to produce the same amount of reactive power in pu (alignment constraints). Generally, in presence of SVR, the ORPF problem can be defined as follows:

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min or max OF s.t. P − P(V, δ) = 0 Q − Q(V, δ) = 0

Q g ≤ Q g (V, δ ) ≤ Q g V≤V≤V Q g ,i = qa Q g , max, i

where: OF P

∀ i ∈ Aa ,

∀ a = 1… N a

(1)

Q

is the objective function chosen; is the vector of real power injections; is the vector of reactive power injections;

Q g and Q g

are the vectors of lower and upper bounds for the reactive power

Q g (V, δ )

generated; is the vector of reactive power generated;

V and V

are the vectors of lower and upper bounds for the bus voltages;

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qa Aa Na

is the vector of bus voltages magnitudes; is the vector of bus voltage angles; is the reactive level for the area a in presence of SVR; is the set of controlling units for the area a; is the number of SVR control areas.

The first and second set of constraints represent the power flow equations, the third set represents the capability limits of generating units; the fourth set takes into account voltage limits. The constraints in (1) are different in the case of PVR or SVR: in the presence of PVR only, the control variables of the ORPF model are the voltage magnitudes of all PV busses. The PF equations are:

∑V

⋅ Yik ⋅ cos ( δi − δk − ϑik ) = 0

∀i ∈ APQ

∑V

k

⋅ Yik ⋅ cos ( δi − δk − ϑik ) = 0

∀i ∈ APV

∑V

⋅ Yik ⋅ sin ( δi − δ k − ϑik ) = 0

∀i ∈ APQ

∑V

⋅ Yik ⋅ sin ( δi − δ k − ϑik ) = 0

∀i ∈ APV

⎧ Pi − Vi ⋅ ⎪ ⎪ ⎪ Pi − Vi ⋅ ⎪ ⎨ ⎪ Qi − Vi ⋅ ⎪ ⎪ Q −V ⋅ i ⎪ i ⎩

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where: Pi

k

k

k

k

k

k

k

is the real power injection in the NPQ PQ busses and in the NPV PV busses;

Qi

is the reactive power injection in PQ busses. Note that for PV busses this value

Yik θik APV APQ

is not defined a priori; therefore, the reactive power is considered a dependent variable; is the magnitude of the i-k element of the admittance matrix; is the angle of the i-k element of the admittance matrix; is the set of PV busses; is the set of PQ busses.

The last equality constraint of (1) is to be written only in the presence of SVR. In this case, it is also necessary to define two new types of busses: the control bus (P-type) and the pilot bus (PVQ-type) [25][26][27]. The P-type bus models a bus in which a SVR control generator is connected. In this case, only the real power is assigned a priori. The voltage (magnitude and phase) is determined by the PF solution. For each P-type bus it is possible to write two equations: the first regards the real power, while the second one represents the alignment condition typical of the SVR: the reactive production of control generators has to be equal to the reactive level of the area. Therefore, if NP is the total number of control busses, 2NP equations can be written. The PVQ-type bus models a pilot bus (which is a load bus) where not only the real and reactive power, but also the magnitude of the voltage is assigned. The number of PVQ busses equals the number of control areas, Na. Therefore, it is possible to define 2Na equations; in this case, the unknown variables are the Na voltage phases only.

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Generators not involved in the SVR are treated as PV-type. PV and PQ busses are dealt with as in a standard PF. In this PF formulation, the reactive level of each control area is unknown: it follows that further NP unknown variables are introduced in the problem. Table 1 shows the number of equations and unknown variables for each type of bus. Table 1. Equations and unknown variables for the SVR PF Type

Unknown

PQ PV PVQ P Reactive Level Alignment equations

E, δ δ, Q δ E, δ

Number of unknown 2NPQ 2NPV Na 2NP

Equations P, Q P, Q P, Q P

Number of equations 2NPQ 2NPV 2Na NP

Na NP

The SVR model can be adopted for any optimization problem described in the following.

3.1.1. Minimization of Real Losses

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As above mentioned, especially in the past, TSOs were very familiar with the minimization of power losses, in order to optimize the voltage profiles. The mathematical model of this ORPF problem is the following: min Ppa

(2)

s.t. P − P(V, δ) = 0 Q − Q(V, δ) = 0

Q g ≤ Q g (V, δ ) ≤ Q g V≤V≤V

where Ppa are the real power losses; It is well-known that the losses of the transmission network show a quadratic dependence on the currents flowing through lines and generators. These currents are a function of bus voltages. Therefore the minimization of losses can be achieved increasing the set point values of controlling generator voltages; generators are thus forced to operate with voltage values close to the maximum values allowed. Usually, higher voltage profiles result in larger security margins. However, such general increase of the network voltages is not always well accepted by the power plant operators due to the increase of both the stress on the machine

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insulation (particularly in case of contingencies) and the unit losses (rotor, stator, step-up transformer).

3.1.2. Modified Minimum Losses A possible compromise between the real loss minimization and the requirements of the power plant operators can be obtained by reducing the maximum voltage limits allowed for the above mentioned generators. On the other hand, this approach is more limiting in terms of enhancing the security of the system, as the feasible search space is reduced. An alternative solution, proposed in [28], is the introduction of suitable penalty factors in the objective function. This approach does not reduce the solution space and, at the same time, it allows the operation of the generators close to the maximum voltage values only if this situation leads to an adequate loss reduction. The modified minimum real losses OF is particularly suitable in markets where the market rules allow explicitly the possibility of exceptions on the generators maximum voltage requirements. Furthermore, the minimization of losses is relevant to an economic management of the system, as losses are paid for at market prices. Let us define GR as the set of generators i for which an operation very close to the rated value Vni is not well accepted by the relevant power plant operator. In this case, the optimization model can be used to define different economic schemes, either tariffs or market-based, depending on whether they belong to GR or not. The mathematical formulation of the proposed objective function is as follows:

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⎡ 2⎤ min ⎢ Ppa + α ⋅ ∑ (V i − V ni ) ⎥ i ∈ GR ⎣ ⎦ s.t.

(3)

P − P(V, δ) = 0 Q − Q(V, δ) = 0

Q g ≤ Q g (V, δ ) ≤ Q g V≤V≤V

where: α is a penalty factor. The modeling of the optimization problem depends on the choice of α that requires several runs in order to be tuned. It is worth noticing that this approach does not imply any opportunity cost, as the real power levels are not control variables. Upper and lower limits of reactive power are calculated from the capability curves with reference to the real power value assigned to the generator, for example after the DAEM.

3.1.3. Minimum Reactive Power Produced The system operation requires a suitable margin of reactive power in order to be able, in case of contingencies or of sudden load changes, to keep the system secure. Therefore, a possible

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goal of the optimization is the minimization of the reactive power produced by generators, to keep the margins for the voltage regulation high. In a market environment, if an economic value ci is assigned to the reactive power, the objective function can be also explained as a minimization of the reactive economic costs. Ng

min

∑cQ i =1

i

gi

s.t. P − P(V, δ) = 0 Q − Q(V, δ) = 0

Q g ≤ Q g (V, δ ) ≤ Q g V≤V≤V

(4)

3.1.4. Proximity to the Voltage Collapse and Voltage Control Other objective functions, more directly related to the system security, are available in literature. In particular, in [15], the distance from the point of collapse (λC) is maximized, assuming a given load ramp pattern: max λc s.t.

f (δ, V, QG , PG , PL ) = 0

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fc (δc , Vc , QG c , PGc , PDc ) = 0 0 ≤ PG ≤ PG QG ≤ QG ≤ QG

0 ≤ PGc ≤ PGc QG ≤ QGc ≤ QG

V≤V≤V

V ≤ Vc ≤ V

(5)

VG c = VG

PGc = PG ⋅ (1 + K G )

where: f, fc δ, V, δc, Vc PG, PGc QG, QGc PL, PLc KG

are the set of PF equations in the base operating point and in the collapse point, respectively; are the angle and voltage magnitude, in the current operating point and in the collapse point respectively; are the real power generated, in the current operating point and in the collapse point respectively; are the reactive power generated, in the current operating point and in the collapse point respectively; are the power required by the loads, in the current operating point and in the collapse point, respectively; is a participation vector for the generators.

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Solving this problem requires the definition of the vector KG and of the load ramp pattern. In [16] the problem of the voltage security is managed by a different approach: in facts [10], the distance of the voltage collapse can be maximized by the minimization of the maximum singular value (σmax) of a suitable matrix, derived from the PF equations; this index tends to infinite at the voltage collapse: min σ max s.t. f (δ, V, Q G , PG , PL ) = 0 PG ≤ PG ≤ PG

(6)

QG ≤ QG ≤ QG

V≤V≤V

An important issue is the choice of the above mentioned suitable matrix, as this index is characterized by a high sensitivity with respect to the control variables and depends also on the power system modeling. In particular, it can be computed with reference to the complete PF Jacobian, to the reactive PF Jacobian, to a reduced Jacobian, and with reference to the presence of PVR or SVR.

3.1.5. Discussion on the Use of σmax

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The most desirable features of the best possible voltage collapse index to be used in an ORPF are: • • •

significant variations with respect to the loading condition; regular (linear, if possible) behavior with respect to the loading condition; high dependence on the voltage profile in the network.

In the following, some features of only the σmax objective function are presented, because its application is relatively recent, while system operators are very familiar with the other objective functions mentioned. In order to find the most suitable form of σmax to be used in the ORPF, a deep study was carried out on the Italian power system, by the simulation of a load ramp and the investigation of the properties of the different indices computed along the load ramp (in the operating points that are solution of the PF equations). In the following, some results are presented, relevant to the computation of σmax based on the reactive Jacobian. In the presence of PVR only, the reactive sub matrix of the Jacobian has the following structure: ⎡ ∂Q PQ ⎢ ∂ Q ⎡ ⎤ ⎢ ∂VPQ = ⎢⎣ ∂V ⎥⎦ ⎢ ∂Q PV ⎢ ∂VPQ ⎣

∂Q PQ ⎤ ⎡ ∂Q PQ ∂Q PV ⎥ ⎢ ∂VPQ ⎥=⎢ ∂Q PV ⎥ ⎢ ∂Q PV ∂Q PV ⎥⎦ ⎢⎣ ∂VPQ

⎤ 0⎥ ⎥ − I⎥ ⎥ ⎦

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If a SVR scheme is adopted, the structure of this matrix changes according to the equations defined for a SVR PF: ⎡ ∂Q PQ ⎢ ∂V PQ ⎢ ⎢ ∂Q PV ⎡ ∂Q ⎤ ⎢ ∂VPQ ⎢⎣ ∂V ⎥⎦ = ⎢ ∂Q PVQ ⎢ ⎢ ∂VPQ ⎢ ∂q ⎢ area ⎣⎢ ∂VPQ

∂Q PQ ∂VP ∂Q PV ∂VP ∂Q PVQ ∂VP ∂q area ∂VP

∂Q PQ ∂Q PV ∂Q PV ∂Q PV ∂Q PVQ ∂Q PV ∂q area ∂Q PV

∂Q PQ ⎤ ⎡ ∂Q PQ ∂q area ⎥ ⎢ ∂VPQ ⎥ ⎢ ∂Q PV ⎥ ⎢ ∂Q PV ∂q area ⎥ ⎢ ∂VPQ ⎥=⎢ ∂Q PVQ ⎥ ⎢ ∂Q PVQ ∂q area ⎥ ⎢ ∂VPQ ∂q area ⎥ ⎢ ∂q ⎥ ⎢ ∂q area ⎦⎥ ⎣⎢ ∂VPQ

∂Q PQ ∂VP ∂Q PV ∂VP ∂Q PVQ ∂VP ∂q ∂VP

⎤ 0 0⎥ ⎥ I 0⎥ ⎥ ⎥ 0 0⎥ ⎥ ⎥ 0 I⎥ ⎦⎥

(8)

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Model 1

Model 2

Figure 1. Singular value index behavior, calculated with respect to different control strategies and to different Jacobian matrices. Model 1 and Model 2.

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Model 3

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Model 4

Figure 2. Singular value index behavior, calculated with respect to different control strategies and to different Jacobian matrices. Model 1 and Model 2.

Different types of matrices can be considered: in [29] a reduced Jacobian has been defined. This matrix has been widely studied and used and therefore its behavior is very well known. In the following, new models are considered and presented [27]: • •

•

•

with Model 1 we will refer to the features of the index σmax computed on the reactive submatix (7) and (8) for PVR and SVR respectively; in Model 2, the singular value is computed on the same submatrices built neglecting rows and columns relevant to the P and PV busses: this can result in sudden changes when a PV (or P) bus switches to PQ bus due to capability limits hitting; in Model 3, the SVR PF equations are solved; then, the Jacobian is built considering only PVR, i.e., the P control busses are now modeled as PV and their voltage is set equal to the voltage given by the SVR PF. Eventually, the computation of σmax is made as in Model 1; finally, Model 4 computes the maximum singular value of the matrix of the sensitivities of the reactive power generated with respect to the generator voltages, computed at the solution point of the SVR PF.

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The different computations of σmax described above are tested on an complete model of the EHV (380/220 kV) Italian Transmission System, characterized by about 1000 busses and 1400 branches, both considering and neglecting the presence of the SVR; in the first case, a SVR model is considered where the Italian power system is divided into 13 SVR areas. The results are shown in Figure 1 and Figure 2, where blue lines are relevant to the PVR, while red lines are related to the SVR control strategies. The behavior of the singular values of the different matrices is plotted versus the load ramp, until the voltage collapse is reached, for Models 1 to 4. Model 2 depends significantly on the discontinuities due to the capability limits of the generators (switching from PV to PQ), while Model 1 and 3 show a regular behavior; however, under SVR, they show a very flat behavior. These results are related to the particular structure of the Jacobian matrix developed for the SVR PF problem. Furthermore, as the hierarchical voltage control regulates the network at a constant voltage set point (up to the capability limit), this results in a strongly non-linear behavior of the voltage collapse indices. The adoption of Model 4 overcomes this problem: it allows a regular behavior either in PVR or in SVR, while maintaining a high sensitivity with respect to the load conditions. For this reason, such an indicator is considered the best candidate for the application in an ORPF problem.

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3.2. Integration of the ORPF into the Electricity Market Operation This subsection deals with the best way to exploit the ORPF features taking into account the typical operation of a zonal electricity market. In this framework, the robustness of the solution with respect to different real generation profiles is very important: from a technical point of view, the best choice should be to carry out ORPF procedures only after the final real power productions are determined by the electricity market. However, the possibility that generation profiles can be modified in the real-time must be taken into account. Therefore, a real-time ORPF would be the best option, although it is time consuming and computationally expensive; moreover, the direct application of this approach would imply a complex structure for the data acquisition, communication and remote control, as well as a huge computational burden for data validation (state estimation) and computation (optimization). The ORPF carried out on the state estimation output can be adopted as a benchmark for any off-line approach; the goal is to define an off-line procedure that results in a power system operation as robust as possible with respect to real-time changes in the real power dispatch. The latter is possible if some particular control structures are available on the power system. SVR is one of such control structures: it allows the control of the overall voltage profile of a large power system regulating the voltage only at some particularly important busses (pilot busses), defined for each area, by means of the coordinated action of the most influent generators (controlling generators) of the relevant area [21][22]. The aim of this subsection is to assess whether the information related to the real power generation profile, defined by the DAEM or by the AM, is sufficient to maximize the voltage security, also depending on the type of power system voltage control in operation [28]; in particular, the results presented are relevant to the system loadability. In order to find the best way to make use of ORPF procedures, different scenarios will be compared. Given the generation pattern coming from the electricity market, it is necessary to compare the performances of the on-line ORPF (characterized by high costs in terms of computational and

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hardware requirements) with the results of the optimization, that can be carried out on few scenarios only, on the forecasted operating conditions. The evaluation of performances is carried out taking as a reference a peak load condition of the Italian power system (Figure 3). In order to give a measure of the security level, the maximum singular value of the complete PF Jacobian is plotted versus the load ramp of the electric system, in terms of the MW distance from the collapse point. A load ramp was applied to the base case, with and without PVR or SVR (squares). The voltage set points have been determined according to different approaches: the solid line is relevant to the load ramp applied on the base case where voltage set points were not changed, i.e., not optimized. Dashed lines are relevant to the same load ramp applied to the base case assuming the presence of an on-line ORPF (in particular, in these examples, the minimization of real losses was adopted), while dotted lines are relevant to the same scenario, with the optimization carried out on the forecasted case (i.e., the voltage set points were optimized the day ahead on a scenario different from the base case, and eventually kept constant during the load ramp).

Figure 3. Analysis of a load ramp applied to the system operating conditions, with reference to the adopted voltage profile.

The results show that the use of on-line optimization can give the best results as expected; however, as already mentioned, the cost can be high, in terms of computational effort and signal transmission hardware. A lighter version of the on-line optimization can be the adoption of voltage profiles derived from the optimization of the forecasted model (which represents the data available after the DAEM and AM); however, the results show that this brings to a security improvement, as compared to the solid line, only in case the system is operated with SVR, that makes the power system more robust. On the contrary, simulations with only PVR in operation show a noticeable decay of system security, measured by the loadability reduction. Therefore, the implementation of SVR makes the power system more robust against changes in real power generation that can occur after the closing of the dayahead electricity market.

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4. Modern Approaches for Solving ORPF Problems Optimization methods are well known in the power engineering community. However, in the last years, new approaches have been developed. They can be essentially divided into two categories. First, methods based on a numerical gradient based approach: this kind of methods needs some assumptions on the continuity and differentiation of both the objective function and the constraints, and can be difficult to program. Second, methods based on the Artificial Intelligence (AI): they usually are able to perform very well in case the assumptions above mentioned for numerical methods are not satisfied. The latter methods are typically very easy to implement, as they only need the computation of the objective function value. Therefore, the present Section is divided into two subsections: the first one describes the numerical approach mostly adopted in the last years to solve power system problems, while in the second subsection some applications of a particular AI tool, the Genetic Algorithms (GAs), are presented and discussed.

4.1. The Interior Point Method

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Together with the choice of the objective function representing security, described and discussed in the previous Section, the power engineer should choose the best mathematical method to solve the problem, which is dependent on how it has been modeled. In general, to calculate the optimal solution of an ORPF problem, different optimization procedures are available. Many of these are based on the Newton method [12][30], for example the HanPowell [31][32] or the Interior Point [13][33][34][35] methods: these techniques require the explicit calculation of the gradient of both the objective function and the constraints; in the case of the Interior Point method, also the computation of the Hessian is necessary. In this Section, a brief analysis of these techniques is discussed. In particular, we consider an optimization problem defined as follows:

min

f (x original )

s.t.

g eq (x original ) = [0]

(9)

g ineq ≤ g ineq (xoriginal ) ≤ g ineq l original ≤ x original ≤ u original

where:

x original

g eq (x original )

g ineq (x original )

is the vector of the independent variables; is the vector of the equality constraints; is the vector of the inequality constraints;

g ineq , g ineq

are the vectors of lower and upper bounds for the inequality

l original and u original

constraints, respectively; are the vectors of lower and upper bounds for the independent variables, respectively.

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The first step to solve this problem is to eliminate the inequality constraints by introducing the vector of independent slack variables s: min

f (x original )

s.t.

g eq (x original ) = [0]

(10)

g ineq − s = 0 g ineq ≤ s ≤ g ineq l original ≤ x original ≤ u original

Now, if we define the vectors x, l and u as follows:

⎡l original ⎤ l=⎢ ⎥ g ⎣⎢ ineq ⎦⎥

⎡x ⎤ x = ⎢ original ⎥ ⎣ s ⎦ we obtain the problem (11):

⎡u original ⎤ u=⎢ ⎥ ⎣ g ineq ⎦

min f (x ) s.t. g (x ) = [0] l≤x≤u

(11)

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To solve the problem (11), it is necessary to define the vectors of slack variables y and z: this leads to problem (12), where two inequality constraints associated with the slack variables are present: min f (x )

s.t. g (x ) = [0] x+y =u x−z =l

(12)

y≥0 z≥0 To eliminate the inequality constraints, a penalty logarithmic function (logarithmic barrier) is introduced, resulting in a problem dependent on μk: N

min

f (x ) − μ k ⋅ ∑ (log( yi ) + log( zi )) i =1

s.t.

g(x ) = [0] x+y =u x−z =l

where N is the number of independent variables.

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Under suitable assumptions, discussed in [11][12][13], if x(μ k ), y (μ k ), z (μ k ) is the solution of the problem (13), it is possible to write the following equations:

lim x(μ k ) = x*

μ k →0

lim y (μ k ) = y *

(14)

μ k →0

lim z (μ k ) = z*

μ k →0

where x* includes the solution of the problems (10) and (11). Therefore, the Interior Point method is characterized by a particular strategy to reduce the parameter µk during the iterative solution process. In any case, to solve problem (13), it is necessary to define the Lagrangian function L(x, λ ) : L ( x,λ ) = f ( x ) − μk ⋅

N

M

N

N

∑( log ( y ) + log ( z ) ) + ∑ λ ⋅ g ( x ) + ∑ r ⋅ ( x + y − u ) + ∑ s ⋅ ( − x + z − l ) i

i =1

i

i

i =1

i

i

i

i

i =1

i

i

i

i

i

i =1

(15)

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where M is the number of equality constraints. Then, the introduction of the Karush-Kuhn-Tucker (KKT) conditions gives the following system of equations:

⎧ ∂L T ⎪ ∂x = ∇f (x ) + ∇g(x ) ⋅ λ + r − s = 0 ⎪ ⎪ ∂L = g (x ) = 0 ⎪ ∂λ ⎪ ⎪ ∂L = x + y = u ⎪⎪ ∂r F (t, μk ) = ⎨ ∂L ⎪ =x−z =l ⎪ ∂s ⎪ ∂L μ = − k + ri = 0 ⇔ ri ⋅ yi = μk i = 1… N ⎪ yi ⎪ ∂yi ⎪ ∂L μ = − k + si = 0 ⇔ si ⋅ zi = μk i = 1… N ⎪ zi ⎩⎪ ∂zi

(16)

To find the solution of the nonlinear system (16), it is possible to use a Newton Method. It requires the calculation of the second order sensitivities of both the objective function and the constraints; in case the Han-Powell method is used to solve the problem, the Hessian of the Lagrangian is approximated numerically by using only the first-order sensitivities.

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Generally, the Interior Point method is very robust, efficient and takes a very low computation time; however, it can require huge resources for the implementation. The Interior Point method is currently widely adopted for many optimization problems. For example, dealing with the objective functions described in the previous Section, some applications exist for certain OFs: minimum losses, minimum reactive power produced, maximum margin etc. On the contrary, in some cases, e.g., if security has to be maximized through the Jacobian minimum singular value, the computation of the derivatives can make the problem intractable. Also in case discrete variables are to be taken into account, the adoption of other optimization methods, not based on derivatives, is justified. In the following subsection, one of such approaches is described, based on Genetic Algorithms, and some results are discussed.

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4.2. An Artificial Intelligence Approach: Genetic Algorithms Genetic Algorithms (GAs) are efficient tools to solve optimization problems, based on the natural selection principles [14][36][37][38]. The optimization procedure follows an evolutionary strategy to find the best solution of a search problem. Starting from an initial population of individuals, each one representing a possible solution, the evolution takes place and changes the population to form the next generation, until a convergence criterion is fulfilled. At this point, the solution of the initial optimization problem is given by the best individual of the current generation. Each individual is represented by a chromosome, which holds all the information on the control variables associated to the point in the search space represented by that individual. Each chromosome is made by a string of numbers that codes the information hold in the chromosome; generally, the binary conversion is adopted, as it is demonstrated to be efficient. The optimisation process tries to optimize an OF by assigning each individual a fitness value strictly related to that function; the fitness must always be maximised during the evolution. The process begins with the choice, random or guided by heuristics, of a starting population. It is important that this population holds a high variety of chromosomes, since diversity is one of the most powerful tools of GAs. The more diverse each generation is, the more efficient the search of the best solution is, as the GA intrinsic parallelism is fully exploited. A classical GA proceeds using the following three operators: 1. Selection:

2. Crossover:

3. Mutation:

Some individuals are selected in order to mate and to transmit their genetic code to the next generation. The choice of these individuals is made random, but the probability of an individual to be selected is proportional either to its fitness or to its position in a merit order (fitness ranking [14][39]): a “not best” solution is not necessarily rejected, as it could hold useful information. Each couple of selected individuals (parents) can experience the crossover with a probability pc (typically pc=0.6-0.9) defined by the user. In the simplest form, a cut point is randomly chosen in the selected individuals and the two couples of substrings are swapped. Each bit of a chromosome can be modified (from 0 to 1 or vice versa) with a very low probability pm (pm=0.001-0.01) set by the user. This introduces new

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features in the next generation and allows the GA search capability to be increased. The above three operators are applied to the current population to determine the next generation, which has a high probability to be made of better individuals. The search process can be stopped (a) (b)

when a maximum number of generations, defined according to the complexity of the problem and the average convergence speed observed, has been reached, or when several individuals (e.g., 95%) are equal in the same generation.

It is worth pointing out that GAs can easily determine the optimum of any OF, even if it is discrete or its derivatives are not defined, because the computation of the OF value only is needed. Moreover, their implicit parallelism [14] makes GAs a very powerful search tool that prevents from convergence to local optima. On the other hand, GAs can present difficulties in convergence or early convergence (when, in the first generations, the population is too uniform); in this case, it may be necessary to introduce new operators or to modify suitably the GA parameters. In the following subsections, a GA implementation used for ORPF applications is described in detail.

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4.2.1. An ORPF Based on GA: Implementation In this Subsection, the implementation of a specific ORPF based on GAs developed for the Italian power system is described and the main features are highlighted. The ORPF above mentioned is able to optimize different objective functions, according to the discussion made in Section 3. It is worth noticing that, in some cases, the optimization could be carried out efficiently by traditional optimization methods (e.g., by the Interior Point method); on the contrary, for the optimization of the security through the Jacobian singular value, the computation of the Hessians would be very burdensome: in this case, the use of GAs is therefore necessary, and suitable also for large power systems. The ORPF here described takes into account also the presence of a Hierarchical Voltage Control, and in particular the SVR. In this case, the control (independent) variables are the pilot node voltages only, while in case of PVR, all the generator voltages should have been chosen as control variables, resulting in an increased size of the optimization problem. 4.2.1.1. Coding Each string of a population holds, suitably coded, the value of the voltage magnitude for all the NPVQ pilot busses. In this GA, a binary coding is chosen. In particular, the range of each input is divided in 2Nbit intervals, where Nbit is related to the accuracy accepted for each input value. In this case, 8 bits are used to code each parameter: therefore, if the nominal voltage of a pilot bus is 400 kV, the optimal solution found by GA is affected by an error of 0,196 kV, that can be considered negligible. As in the Italian power system model adopted the pilot busses are 13, the bits necessary for each string are 104.

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4.2.1.2. Initial Population Generally, the initial population is created randomly to guarantee a wide variety of schemes necessary to find the optimal solution. In this application, such choice is not suitable, because it is necessary that a chromosome represents a convergent power flow solution. Therefore, only a part of the initial population is created randomly: over an initial population of 100 individuals, N1 elements of them has been created by applying the mutation operator, with a 20% probability, to the initial voltage profile derived by the state estimator. Further N2 elements are created in the same way by modifying the nominal voltage profile. Finally, a third set N3 of elements is created randomly (see Figure 4). In the example, the following values were adopted: N1=40, N2=30, N3=30. Pilot Bus 1 Set of N1 elements derived by initial voltage profile

Pilot Bus 2

…

Pilot Bus NPVQ

ind 1 ind 2 ... ind N1

Set of N2 elements derived by nominal voltage profile

ind N1+1 ind N1+2 ... ind N2

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ind N2+1 Set of N3 elements created randomly

ind N2+2 ... ind N3 Figure 4. Structure of the initial population.

4.2.1.3. Fitness Function and Ranking The fitness function associated to a string is directly related to the objective function to be optimized and defines the probability that this string is selected for the next population: the strings with high fitness have a larger probability of selection. When the objective function is flat, i.e., the differences among the string fitness values is small, convergence problems can arise due to a reduced evolutionary pressure. In this case, the ranking technique can be applied [39], which can be adopted for many OFs used for ORPF. In the presented application, the following non linear ranking has been employed [40]:

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FR( p ) =

201

N POP ⋅ X p −1 N POP

∑ X (i ) i =1

where: NPOP p

X (i )

is the number of strings belonging to a population; is the ranking of the string fitness; is a solution of the following equation:

(SP − 1) ⋅ X N

POP −1

+ SP ⋅ X N POP − 2 +

+ SP ⋅ X + SP = 0

where SP is a parameter equal to 10: if SP is too high, the population tends to be too uniform. 4.2.1.4. Selection The selection technique adopted is a classical Roulette Wheel Selection [14]: each element is selected with a probability depending on its value of FR. Moreover, the elitism has been adopted: this means that the best string of the previous generation is always selected for the next one, in order to avoid losing the best information held in the previous population.

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4.2.1.5. Crossover The crossover operator acts, with probability pc, on each couple of selected strings [14][39]. After many tests, a multi-point crossover was chosen for this application: both parent strings are randomly divided in a number of points equal to the number of the pilot busses and then the crossover takes place as described in Figure 5 for three pilot busses. In this way, two offsprings are generated.

Figure 5. Crossover operator.

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This operator is applied with pc=80% probability; therefore, 20% of the strings remain unchanged after crossover. 4.2.1.6. Mutation and Diversity The mutation [14][39] allows introducing diversity. In particular, it is applied with probability pm to each bit of the binary strings, and changes one bit value from 0 to 1 or vice versa. The probability pm is assumed equal to the inverse of the number of elements in the population. Moreover, to preserve the diversity through the convergence process, new elements are created randomly when the best solution remains the same during a given number of generations. In the ORPF case, many tests performed showed that the following strategy is suitable: 10 new individuals created at random are introduced if the best element remains the same for 20 successive generations. 4.2.1.7. Penalty Coefficient The GA above described guarantees that voltage constraints are implicitly taken into account in the binary coding and thus satisfied in the solution point only for the pilot busses; the constraints related to P-type busses – voltage and reactive generation - are dealt with by the limited SVR PF used to compute the OF [27]; the alignment constraints are then solved by the same PF. However, it is necessary that voltage constraints are satisfied also for all PQ busses. This can be forced by introducing a penalty factor PEN in the OF (let us assume that the maximization of the OF is the goal of the optimization):

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OFnew = OF ⋅ PEN If any violation is present in the solution point, PEN < 1; otherwise PEN = 1. PEN can be defined in many ways: if it is too low in presence of a constraint violation, the relevant strings can be discarded in the first iterations and the population can lose useful schemes that could be important to reach the final optimum point. For example, in case of minimization of losses, usually the voltages at many busses are very high: if PEN is too low, many strings with high voltage values (and consequently with schemes useful to find the best solution) will be eliminated at the beginning of the selection process and the GA will be pushed in an area of the search space characterized by low voltages. Many different strategies have been tested to determine PEN, and the following scheme has been chosen:

PEN = e where: NBviol kVviol,i β

−β

NBviol

∑ i =1

kVviol ,i

is the number of busses that present a voltage violation; is the absolute violation value (in kV) for the bus i; is a coefficient that changes during the convergence process according to Table 2.

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Security Optimization of Bulk Power Systems in the Market Environment Table 2. Rule to update the

β coefficient

β

Generation Number in presence of voltage violation

30 35 40 45 50

5 10 20 25 30

For example, according to Table 2,

203

β increases (to make PEN more powerful) during

the iterations when the total number of voltage violations does not decrease.

4.2.2. Tests of the GA ORPF on the Italian Power System The GA above described has been applied on the detailed model of the EHV Italian Transmission System (380/220 kV) with SVR in operation already used in 3.1.5: Figure 6 shows the structure of the secondary voltage control areas and the pilot busses adopted for this study. Different OFs have been tested, as described in the following subsections. For each voltage profile obtained by the ORPF procedure, the loadability margin of the corresponding network condition has been calculated [24] to compare the results of the different OFs in terms of security.

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4.2.2.1. Real Losses Table 3 shows the pilot bus voltages at the beginning of the optimization, the optimized values and minimum and maximum values. The last column shows the SVR area reactive level (when the area reactive resources are fully exploited, Qmin or Qmax are indicated). Table 4 compares, for the optimum point determined minimizing the real losses, the behavior of real losses and of the Jacobian maximum singular value, before and after the optimization. Table 3. Pilot bus voltage values Bus Name Baggio Brindisi Casanova Dolo Ostiglia Poggio a Caiano Rossano S. Lucia S. Massenza Villanova Redipuglia S. Sofia S. Fiorano

Vin [kV] Vmin [kV] V [kV] Vmax [kV] 395.9 370.0 406.6 420.0 404.4 370.0 416.8 420.0 398.1 370.0 404.3 420.0 401.2 370.0 407.4 420.0 398.9 370.0 411.7 420.0 399.4 370.0 404.3 420.0 410.6 370.0 420.0 420.0 401.5 370.0 410.5 420.0 218.0 200.0 219.8 240.0 400.5 370.0 404.3 420.0 399.4 370.0 406.6 420.0 391.9 370.0 401.1 420.0 393.9 370.0 406.6 420.0

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qa 0.7194 0.3791 0.2489 0.5817 0.8579 0.1527 0.3939 0.1574 Qmax 0.0049 0.8375 0.7139 0.9833

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204

Figure 6. Secondary voltage control areas.

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Table 4. Comparison between initial and final values Real losses [MW] – initial value Real losses [MW] – optimal value σmax - initial value σmax - final value Loadability [MW]

441 427 863.7070 418.1825 5289

In this ORPF problem, the real power produced by each generating unit is fixed (only the reactive problem is considered); therefore, the control variables are only the voltages of pilot busses. The minimization of the real power is obtained by the minimization of the power produced by the slack bus. In the south of Italy, some busses are characterized by high values of the optimal voltages (in particular the Rossano bus reaches the upper bound); this can result in a possible overvoltage in case of contingency (i.e., in N-1 conditions). 4.2.2.2. Modified Real Losses This objective function gives optimal values that are closer to the nominal value. Therefore, the optimal voltages are generally lower than in the previous case (Table 5), but in the optimal point the real losses are higher (Table 6). Moreover, it is worth noticing that there are no more nodes at the voltage upper bound; generally the voltages set point are lower than in the previous case and that in any case the loadability is substantially the same.

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Table 5. Voltage value for the pilot busses Bus Name Baggio Brindisi Casanova Dolo Ostiglia Poggio a Caiano Rossano S. Lucia S. Massenza Villanova Redipuglia S. Sofia S. Fiorano

Vin [kV] Vmin [kV] V [kV] Vmax [kV] 395.9 370.0 401.1 420.0 404.4 370.0 416.8 420.0 398.1 370.0 401.1 420.0 401.2 370.0 411.0 420.0 398.9 370.0 405.8 420.0 399.4 370.0 407.4 420.0 410.6 370.0 412.9 420.0 401.5 370.0 412.1 420.0 218.0 200.0 219.6 240.0 400.5 370.0 403.7 420.0 399.4 370.0 402.2 420.0 391.9 370.0 400.9 420.0 393.9 370.0 400.9 420.0

qa 0.5681 0.4319 0.2639 Qmax 0.3900 0.3742 0.2288 0.1469 Qmax Qmin Qmin 0.8871 Qmax

Table 6 - Comparison between initial and final values Real losses [MW] – initial value Real losses [MW] – optimal value σmax – initial value σmax – final value Loadability [MW]

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4.2.2.3. Minimum Reactive Power Produced This objective function allows the reduction of the voltage profile in the network, and it is particularly interesting in a market environment, because it reduces the probability that opportunity costs occur due to the capability constraints of the generators. In this example, only the reactive power produced by the SVR controlling generators is minimized. Table 7 and Table 8 show respectively the values of the pilot voltage profile and the values of the objective function before and after the optimization process. The voltage set points are lower than in previous cases but with some exception, e.g. for the bus of Brindisi (south of Italy) due to its relevance in the corresponding area: keeping the voltage high brings to a greater reactive power production by the line capacitances, reducing the stress on the generators. The GA reduces of about 500 Mvar the total reactive power produced by the control generators; moreover, the loadability decreases in comparison to the previous cases. It is a TSO responsibility to decide whether the improvement in terms of reduction of the reactive power produced can justify the decreased loadability margin.

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Table 7. Voltage value for the pilot busses Bus Name Baggio Brindisi Casanova Dolo Ostiglia Poggio a Caiano Rossano S. Lucia S. Massenza Villanova Redipuglia S. Sofia S. Fiorano

Vin [kV] Vmin [kV] V [kV] Vmax [kV] 395.9 370.0 395.0 420.0 404.4 370.0 416.8 420.0 398.1 370.0 405.4 420.0 401.2 370.0 395.0 420.0 398.9 370.0 395.0 420.0 399.4 370.0 402.7 420.0 410.6 370.0 407.6 420.0 401.5 370.0 413.7 420.0 218.0 200.0 216.6 240.0 400.5 370.0 402.9 420.0 399.4 370.0 395.6 420.0 391.9 370.0 395.8 420.0 393.9 370.0 391.9 420.0

qa 0.3526 0.5274 0.5503 0.5082 0.1673 0.3166 0.1762 0.2475 Qmax 0.0792 Qmin 0.0256 Qmax

Table 8 - Comparison between initial and final values Reactive power produced [Mvar] – initial value Reactive power produced [Mvar] – optimal value σmax – initial value σmax – final value Real losses [MW] – initial value Real losses [MW] – final value Loadability [MW]

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4.2.2.4. Objective Function: σmax (Model 4) The last example reported in this Section regards the σmax objective function calculated according to Model 4. The results (Table 9) show that if the optimization is carried out on a scenario not particularly stressed, due to the very flat feature of the OF chosen, the optimization results in a homogenous voltage profile which is not efficient in terms of security (due to the reduced loadability, Table 10). On the contrary, this OF is suitable when the power system is closer to the collapse point. Figure 7 shows the best value of the OF (inverse of σmax) for each iteration during the GA convergence process. The analysis performed brings to the conclusion that the classical minimization of real losses can in any case give an improvement in terms of security, while other OFs work well either when specific goals have to be attained or in particular operating conditions. Table 9. Voltage value for the pilot busses Viniziale [kV]

Vmin [kV]

V [kV]

Vmax [kV]

qa

Baggio

395.9

370.0

407.0

420.0

0.8711

Brindisi

404.4

370.0

417.8

420.0

0.4447

Casanova

398.1

370.0

403.3

420.0

0.3077

Dolo

401.2

370.0

411.3

420.0

Qmax

Ostiglia

398.9

370.0

407.4

420.0

0.5785

Poggio a Caiano

399.4

370.0

391.6

420.0

Qmin

Rossano

410.6

370.0

420.0

420.0

0.4485

S. Lucia

401.5

370.0

416.0

420.0

0.4235

S. Massenza

218.0

200.0

219.0

240.0

Qmax

Villanova

400.5

370.0

404.9

420.0

Qmin

Redipuglia

399.4

370.0

402.4

420.0

Qmax

S. Sofia

391.9

370.0

396.4

420.0

Qmin

S. Fiorano

393.9

370.0

404.3

420.0

0.7995

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Bus Name

Table 10. Comparison between initial and final values σmax - initial value

863.7070

σmax - optimal value

368.0259

Real losses [MW] – initial value

441

Real losses [MW] – final value

440

Loadability [MW]

4587

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Obj ective fun cti on

208

Itera ti on Numb er

Figure 7. Best value of the OF during the iteration process.

5. Multiobjective Optimization

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5.1. Multiobjective Methodologies As described in the previous Sections, new tools that allow managing the system in the long and short–term environment, complying with the rules of the electric market, must be available to TSOs. The possibility to take decisions is more difficult than in the past because several objectives, that are likely to be in mutual conflict, should be taken into account. In this scenario, the use of Multiobjective Optimisation (MO) [41] techniques brings the following advantages: a) it allows the management of different objectives; b) it makes it easier to take a decision; c) it gives indications on the consequences of the decision with respect to all the objective functions considered. In this way, the TSO has a better knowledge of the electric system and therefore can make a better and transparent choice for market participants. The MO methodologies give a solution conceptually different from that obtained by standard optimization methods: first of all, their intermediate results are not unique, but provide a very large set of optimal solutions: the Pareto set. Each point belonging to the

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Pareto set is such that the improvement in one of the objectives results in the worsening of at least another objective. Therefore, to solve a MO problem, it is necessary to: a) define the objectives (e.g., according to the issues presented in Section 3); b) find the Pareto set; c) select a solution from the Pareto set. Step c) is the most important, because the final solution depends on the point of view of the decision maker [41][42], who has to take into account the relative importance of the conflicting objectives. Let us define the MO optimization problem as: min ⎡⎣ f1 ( x ) , f 2 ( x ) ,

, f j (x) ,

, f p ( x ) ⎤⎦

s.t. g ( x) ≤ 0 x∈X

where:

f j (x )

is the j-th OF considered (to be minimised);

g (x )

is the constraint vector (m elements);

x

is the control variable vector (n elements) and X is the feasibility region.

A point x* belongs to the Pareto set if, for any other x ' ∈ X ,

f j ( x * ) ≤ f j ( x ' ) for all j

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and, at least for one k, f k ( x * ) < f k ( x ' ) . The Pareto set can be not convex if X is not convex in the objective function space. This Section presents the features of the MO techniques than can be of interest for modern TSOs in the short-term environment. In particular, the following items will be discussed in–depth: 1. the methods used to determine the Pareto set. In particular, two different methodologies are considered: the Weights Method (WM) and the ε-Constraint Method (CM) [42][43]. These methods allow finding each point of the Pareto set using traditional optimization techniques. Usually, the CM shows better performances than the WM, especially when a good knowledge of the problem is already available. 2. Once the Pareto set is found, the decision maker has to choose the final operating point. The decision maker is the TSO and the solution depends on his point of view. The TSO can take the final decision using his experience and any additional information given by the knowledge of the Pareto set. Some techniques to organize and characterize the properties of each point of the Pareto set are available. In particular, the trade–off technique [44] and the concordance analysis seem to be the most suitable for the application to the power system operation.

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MO procedures are suitable when it is necessary to know as much as possible the consequences of a decision. In fact, they can provide the sensitivities of any OF with respect to the other OFs, and this can give an idea of the robustness of the decision taken, in order to face possible perturbations. There is another class of MO methodologies that include all the information necessary to make the final choice in a main optimization problem and that can provide directly the final point: in this case, the determination of the Pareto set is not needed. The goal programming method [45][46] is one of such methods that are easier to manage, and need lower computation times, but that can not give any other information except the final point.

5.1.1. The Weight Method A first method to determine the Pareto set is the Weight Method. It converts the MO optimization problem into a traditional problem: the different OFs are weighted and added to form a single OF to be optimised. Therefore, a traditional optimization method can be used. The optimization problem P(w) can be defined as follows: p

min

∑w k =1

k

f k (x )

s.t. g (x ) ≤ 0 x∈X wk ≥ 0 p

∑w

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k =1

k

=1

It is worth noticing that the OFs are usually expressed in different units and the weights are used to compare them and to give the relative importance of each OF. The geometric interpretation of the WM (Figure 8) is that, in the p-dimensional space of the OFs, the weights define an hyperplane, and the solution of each problem P(w) is represented by the intersection of the tangent plane defined by w with the Pareto surface. For this reason, it is possible to generate the entire Pareto set by means of the WM only if the OFs and the constraints are convex. Otherwise, some non inferior solutions could not be discovered by the WM (whereas all solutions can be found by the CM): this is called duality gap. There are two sufficient alternative conditions, for a solution x* of P(w) to be in the Pareto set: 1. x* is in the Pareto set if wk>0 for every k or 2. x* is the unique solution of P(w). The simplest way to draw the Pareto set by the WM is therefore to systematically varying w and solve each optimization problem P(w). A drawback of this method (at least from the mathematical point of view) is that, if wk=0 for at least one k, it is not always easy to check condition 2, and therefore another test of non inferiority is needed (see 5.1.3). The

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computational burden increases exponentially with the number of OFs and the number of the points computed for the Pareto set.

Figure 8. Duality gap in non convex problems.

Another feature of this method is that a good knowledge of the power system is required for a smart definition of the Pareto set: actually, it is not easy to define a priori the weights to obtain a good distribution of the points of the Pareto set: in this case, sometimes it can be difficult for the TSO to have a complete idea of the shape of the Pareto set. The WM allows also the determination of the relationships among the different OFs:

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w ∂f i = j ∂f j wi This result is useful because it gives information on the changes in an OF consequent to an attempt to improve another one, that is given by the slope of the Pareto surface in the considered operating point. This method requires the solution of a high number of optimization problems to trace the Pareto set, but sometimes only a few of these problems are actually necessary, because only an approximated idea of the shape of the Pareto set is needed by the power system engineer, and the remaining points of the Pareto set can be approximated by interpolation.

5.1.2. The ε-Constraint Method The second method that can be used to build the Pareto set is the ε-Constraint Method (CM). It converts the MO problem into a traditional optimization problem choosing an OF as the main OF and taking into account the presence of the other OFs as constraints in the main optimization problem: The main problem Pk(ε) is defined as follows:

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Alberto Berizzi, Cristian Bovo, Maurizio Delfanti et al. Pk (ε ) = min f k (x ) s.t.

g (x ) ≤ 0 x∈X f j (x ) ≤ ε j

where ε = ⎡⎣ ε 1

ε k −1

ε k +1

j = 1, … , p

j≠k

ε p ⎤⎦ T .

A sufficient condition C1 for a solution point x* of the problem Pk(ε*) to be in the Pareto set is that x* is the only solution of Pk(ε*) for at least one k. Moreover, a solution point x* of the problem Pk(ε*) belongs to the Pareto set if and only if:

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1. 2.

x* solves Pk(ε*) for every k=1,.., p (condition C2) or the optimal solution of Pk(ε0) is strictly greater than fk(x*) for every ε0≤ ε* (condition C3).

Therefore, the Pareto set can be calculated by solving Pk(ε) for different values of ε and verifying any of the above mentioned condition C1 to C3. C1 generates the candidate solution for the Pareto set by solving only one optimization problem Pk(ε), but the uniqueness of the solution can be difficult to check. C2 is actually not suitable, with several OFs, because it requires, for each candidate, the solution of Pk(ε) for every k. C3 is usually difficult to check numerically. In particular, if the objective functions and the constraints are convex, other tests are unnecessary only if the primary OF is strictly convex. If the problem is differentiable, but the primary OF is not strictly convex, it is necessary to evaluate second order sufficiency conditions. Problem Pk(ε) can be solved by using any traditional optimization method that associates a Lagrange multiplier to each active constraint. Its value is related to the variation of the main OF with respect to the active constraint, that is, with respect to the other OFs of the MO problem. Therefore, if λ*kj is the Lagrange multiplier associated to the j-th function when the k-th objective function is the main objective function, the following holds:

λ*kj = −

( )

∂f k x * ∂ε j

The Lagrange multipliers are quite valuable for further analyses, because of their meaning as local trade-offs: this property is very useful for the decision maker to choose the final operating point from the non inferior solution set. When the problem is convex, the sign of the Lagrange multipliers can be useful also to verify if the considered point is in the Pareto set: a solution x* belongs to the Pareto set if all the Lagrange multipliers associated to the secondary OFs are strictly positive; further tests are not necessary.

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The CM allows always (even when the Pareto set is not convex) the determination of the entire Pareto set, given that significant values of the component of ε are available. The latter are easier to assign than the weights in the WM, because they are more related to the power system physics.

5.1.3. The Validation of the Pareto Set As described in the previous subsections, it is often necessary to verify whether the solution point found by the WM or the CM belongs actually to the Pareto set. To this goal, the following test can be exploited:

δ = max

n

∑α i =1

s.t. g (x ) ≤ 0 x∈X

i

(17)

( )

f j (x ) + α j = f j x *

i = 1,

,p

αj ≥0 Its solution can ensure that x* belongs to the Pareto set or, if not, it can provide a new point that is in the Pareto set: if x0 is the optimum point of problem (17) and δ0 the corresponding value of δ, x* is in the Pareto set if δ0 =0. Moreover, this tests problem is very important because, if δ0≠0, i.e., if x* does not belong to the Pareto set, x0 is in the Pareto set.

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5.1.4. The Choice of the Best Solution Using the Surrogate Worth Trade off Analysis Once the Pareto set is determined, the decision maker has to choose the ultimate solution, which represents the best compromise. Different methodologies are proposed in the literature for selecting the compromise solution (CS) [41][44][45]; basically, they can be divided in two classes. The methods of the first class find the CS automatically, once defined a single criterion of choice. For example, a criterion often adopted for this goal is the distance from the utopia point; the utopia point is the point whose co-ordinates in the objective space are given by the objectives optimized one at a time. The methodologies of the first class have two important drawbacks: a) the choice of the solution depends on the units used for the objective functions; b) the point of view of the decision maker is considered only once, for the choice of the criterion and of the units. The methodologies in the second class are more fit when the decision maker likes to follow closely the decision process. They are in facts interactive procedures, and the decision

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maker is stimulated to give his opinion on each alternative, in order to select the CS closest to his thinking. The disadvantage of these techniques is • •

the need to calculate several points of the Pareto set and the complexity of the decision structure.

However, once the Pareto set is determined, they are simple from the computational point of view, although the decision maker must analyse a huge amount of information. The Trade Off (TO) analysis is an interactive methodology that belongs to the second class above and makes it possible for the decision maker to manage complex MO problems in a transparent way, i.e., publishing in advance the necessary criteria that will be used to take the ultimate decision. This makes the TO analysis suitable for application in a market environment. After the Pareto set is determined, it is possible to evaluate the TO among the objective functions. The TO is defined as the change of a benefit, moving from a solution point to another, in order to improve a different objective, regarded as more desirable. Given two feasible alternatives x0 and x*, the corresponding values of the objectives are:

and

f x 0 = f1 x 0 , f 2 x 0 ,

( ) [ ( ) ( )

, f p x0

( ) [ ( ) ( )

, f p x*

f x * = f1 x * , f 2 x * ,

( )]

( )]

It is possible to define the partial TO T kj (x 0 , x * ) involving fk and fj between x0 and x* if

( )

( )

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f l x 0 = f l x * for all l=1, .., n, l≠k or j:

(

)

T kj x 0 , x * =

( ) ( )

( ) ( )

fk x0 − fk x* f j x0 − f j x*

On the other hand, if f l (x 0 ) ≠ f l (x * ) for at least one l=1,..,n (l≠k or j), then Tkj (x 0 , x * ) is called total TO [41]. The TO can also be defined locally, when x0 → x*. In this case, using the CM, the Lagrange multipliers of the slave functions are the partial TOs in the point in the Pareto set:

λ*kj = −

( )

∂f k x * = Tkj ∂ε j

Using the WM, the partial TO between f1 and f2 in x* is: Tkj =

wj wk

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The TOs can be used in an interactive procedure to better define the CS: the Surrogate Worth Trade Off technique (SWT) can successfully be used [41][46]. It consists of four steps: a) generate a representative subset of the Pareto set (S points); b) determine the TOs for the generated solutions, in particular the TOs among the master and the slave functions using the CM; c) interact with the decision maker to obtain information about his preference on each generated solution; d) determine the CS based on the interactive process. During step c), the decision maker has to define a worth Wkj for each generated solution, answering to the following question: “How much would you like to improve fk by Tkj (x 0 ) units versus a one–unit degradation of fj, when all other objectives l are fixed at f l (x 0 ) ?” The

decision maker has to give a mark (from +10 to –10) to each solution point: +10 means that, looking at that solution point, the decision maker would like to improve fk by Tkj (x 0 ) units even if this requires a worst value (one unit) for fj; conversely, -10 means that the decision maker does not like to have a one–unit degradation of fj, even if it implies improving fk by Tkj (x 0 ) . The mark “0” represents the indifference of the decision maker. For each non inferior solution x0, p-1 worth values Wkj have to be defined. The total number of marks he has to give is therefore S × ( p − 1) .

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For example, if the MO problem is characterised by two objective functions, and the non inferior solutions are S=5, it is possible to depict the above solution methodology as in Figure 9, where the worth values are plotted versus the values of the f2 (supposed to be the slave function using the CM). The CS for f2 is obtained graphically where the worth W12 is zeroed.

Figure 9. Example of the SWT.

The marks for each candidate are assigned in the SWT based both on the TO values and on the objective values. They give the direction of change preferred by the decision maker, which is dependent on the operating point considered: therefore there is an implicit double criterion of choice. The criteria used to assign the marks can be decided a priori and can be of

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public domain and known in advance by the market operators: for this reason, the TSO behaviour can be transparent. In general, step d) allows the determination of the CS by building, for a given Wkj (fk is the master function), a system of (p-1) equations: ⎧W k 1 ( f 1 , f 2 , , f k −1 , f k +1 , , f p ) = 0 ⎪ ⎪ ⎪⎪W k ,k −1 ( f 1 , f 2 , , f k −1 , f k +1 , , f p ) = 0 ⎨ ⎪W k ,k +1 ( f 1 , f 2 , , f k −1 , f k +1 , , f p ) = 0 ⎪ ⎪ ⎪⎩W k , p ( f1 , f 2 , , f k −1 , f k +1 , , f p ) = 0

The

[f , f 1

2

,

solution , f k −1 , f k +1 ,

for , fp

]

f

T

(18)

determined, given by a vector of the objectives , represents the indifference (mark 0) of the decision maker to

improve (degrade) an objective degrading (improving) the other objectives. The system (18) is usually solved using numerical interpolation techniques. At this point, the complete CS, i.e., the missing value of f k is not yet available. The complete knowledge of the CS is obtained by the CM by solving the following optimisation problem: max f k (x ) s.t.

(19)

g (x ) ≤ 0 x∈X

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f j (x ) ≤ f j

∀j ≠ k

The CS for the MO problem is therefore obtained considering the point of view of the decision maker, by means of the TO information. For the step d) of the SWT, the use of the CM is mandatory. In order to determine a suitable CS, it is necessary to know accurately the shape of the curve in Figure 9 and this implies the determination of several points of the Pareto set: this can be cumbersome from the computational point of view. The SWT method makes use of the TO to choose the best solution, but it does not consider any other criterion. In many applications, although the TO is important because it is related to the value of the objectives and their variations, it is necessary to take into account other features. In [45], a possible way to solve this problem is presented: the marginal rate of substitution (similar to a TO between a point in the Pareto set and the utopia point) is compared with the local TO. In this way, the interaction with the decision maker is lost and the results of the method depend again on the units used; however, the process is in some way more automatic.

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5.2. The Multiobjective Approach for the ORPF Problem In this Section, a couple of practical examples are provided on the application of the methods presented in 5.1 to the optimization of electric power system, with particular attention to the influence of the electricity market.

5.2.1. Minimization of Real Losses and Minimization of the Reactive Power Produced Adopting as OFs the real loss (hereinafter MRL) and the reactive power produced (MRP), both to be minimized, it is possible to develop a MO approach. The reactive power weight β can be introduced through a penalty factor of the MRL approach or as a new term, to take into account also the economic value of the reactive power. In particular, it is possible to define this ORPF problem using two different formulations. In the first case, the absolute value of the reactive production is minimized, while in the second case its square value is minimized:

⎡ ⎤ min ⎢α ⋅ Ppa + ⋅β ∑ | QGi |⎥ i∈NG ⎣ ⎦

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⎡ 2⎤ min ⎢α ⋅ Ppa + β ⋅ ∑ QGi ⎥ i∈ NG ⎣ ⎦ The ORPF problem with the first formulation cannot be solved with the optimization technique derived by the Newton methods due to the discontinuity given by the absolute value. In this case, it is necessary to use methodologies such as the Genetic Algorithms. Figure 10 shows the Pareto set obtained by an implementation of the second formulation, where the weights used in the simulation are reported in the following Table 11. Table 11. Weights used for MRL-MRP problem α β

1 0

0.9 0.1

0.8 0.2

0.7 0.3

0.6 0.4

0.5 0.5

0.4 0.6

0.3 0.7

0.2 0.8

0.1 0.9

0 1

Figure 10 shows an important characteristic of this MO ORPF. Whit α=1 and β=0 the MO ORPF problem is reduced to a standard ORPF problem. Therefore, in this optimal solution point the real losses are minimized (point A). Point B in Figure 10 is calculated using α=0.9 and β=0.1: this point shows that by a small reduction of the weight associated to the real losses α, it is possible to obtain a significant reduction of the reactive power produced by generators. In particular, the reactive power is reduced of 11%, while the losses increase of 6%. In point B, the electric system is characterized by higher resources available to face possible contingencies than point A. A possible interpretation of the presented test, linked with the comments made in previous Sections, is that, adopting a MO approach, an increase in the voltage profiles is correlated to the loss reduction through the Pareto set and the TSO can decide which solution to adopt.

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Figure 10. Pareto Set with objective function MRL-MRP.

5.2.2. Security and Cost In this subection, a significant example of the MO features is presented, applied to a problem slightly different from those previously described: the optimization of the power system is carried out by controlling both the real power generated and the generator voltages, assuming as objective functions the generation cost and the security [47]. In particular, for security, the above described σmax is used. First, the determination of the entire Pareto set is carried out, so that various possible alternatives can be evaluated. The Pareto set is calculated with the WM or the CM. With the WM, the problem can be formulated as follows:

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min

α ⋅ Ppa + (1 − α ) ⋅ σ max

s.t.

(20)

g (x ) ≤ 0 x∈X P≤P≤P

where P , P are respectively vectors of the minimum and maximum real power generated. With the CM, a possible formulation of the problem is the following, where the cost is chosen as the main objective function, and σmax is transformed into a constraint: min

Pk (ε ) = C (P )

s.t. g (x ) ≤ 0

σ max ≤ ε x∈X P≤P≤P

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(21)

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Choosing a value for ε is usually easier than selecting appropriate values for α in (20), as ε defines a security margin, which has a more “physical” meaning. A similar approach is proposed in [48], where voltage security if accounted for in an OPF formulation by adding a minimum distance to collapse constraint. This OPF problem is solved using the Han-Powell procedure, which requires a secondorder approximation of both the objective functions and the constraints. In this case, a critical aspect in the solution of (20) and (21) is the computation of the singular value and its derivatives at every iteration, which can be done using the Hessian of the power flow equations as described in detail in [49]. Finally, Figure 11 shows the complete results of solving the OPF formulation (21) for different values of ε, i.e., the complete Pareto set at different loading levels, λ. Notice that operating costs and security levels are conflicting goals, i.e., improving security, i.e., lower σmax values or higher λ, result in higher costs. Thus, one can observe that as the system loading increases, similar security levels can only be obtained at higher costs. The effect of system security on operating costs can be obtained from the slopes of the Pareto sets, or from the analysis of the weights in problem (20) or of the Lagrange multipliers in problem (21); all these values indicate how the cost changes at different security levels.

Figure 11. Pareto sets as determined by the solution of the Multiobjective problem for different values of α and of λ when applied to the IEEE 118-bus test system.

6. Conclusions The Chapter deals with the advances of the optimization techniques in power systems, with particular attention to the Optimal Reactive Power Flow problem and its changes in the presence of an electricity market.

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The Chapter is divided into three main Sections. First, a detailed discussion on the comparison of standard and new objective functions is presented; in particular, the security issues are highlighted with reference to the reactive/voltage subproblem, assuming that the issues related to the thermal limits associated to the real power flows are solved within the electricity market. Second, some optimization methods for the solution of the above optimization problems are discussed: traditional Newton-based numerical optimization methods are compared to some Artificial Intelligence tools, in terms of computational efficiency, convergence properties and flexibility with respect to different objective functions. In particular, some applications of the Interior Point method and of the Genetic Algorithm are shown. Finally, a multiobjective approach is presented: this approach can be particularly useful in case the optimization must be carried out adopting transparent criteria. Actually, in a market environment, the transmission operator must take decisions based on transparent criteria because its decisions can affect the electricity market results and the overall market efficiency. The multiobjective approach is demonstrated to be a suitable tool. The features of all the methods and the procedures presented are shown by numerical tests and examples with reference to the Italian electricity market and transmission system.

References [1]

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[2]

[3] [4]

[5] [6]

[7] [8]

L. Franchi, M. Innorta, P. Marannino, C. Sabelli, "Evaluation of economy and/or security oriented objective functions for reactive power scheduling in large scale systems", IEEE Transaction on power systems, Volume PAS-102, No. 10, pp. 34813488, October 1983. J.P. Paul, J.L. Lèost, J.M. Tesseron, “Survey of the secondary voltage control in France: present realization and investigations”, IEEE Transaction on power systems, Vol. 2, No.2, pp.505-511, May 1987. A. J. Wood, B. F. Wollenberg, Power generation operation and control, 2nd edition, New York: John Wiley and Sons, 1996. E. Lobato, L. Rouco, T. Gomez, F. Echavarren, M.I. Navarren, R. Casanova, G. Lopez, “Solution of daily voltage constraints in the Spanish electricity market”, in Proc. 14th Power System Computation Conference, Sevilla (Spain), June 2002. A. Capasso, E. Mariani, C. Sabelli, “On the objective function for reactive power minimization”, in Proc. IEEE PES Winter Meeting, New York, 1980. A. Garzillo, M. Innorta, P. Marannino, F. Mognetti, “How to supply appropriate VAR compensation to the planning of an electric network by the solution of linear inequality systems”, in Proc. 9th Power System Computation Conference, Cascais (Portugal), September 1985. J. V. Hecke, N. Janssens, J. Deuse, F. Promel, “Coordinated voltage control experience in Belgium“, in Proc. CIGRE Session, 38-111, Paris (France), 2000. M. L. Latorre, M. L. Oliveria, J. O. Soto, S. Granville, “Voltage collapse and the optimal power flow problem in power systems”, in Proc. Bulk Power Systems Dynamics and Control IV, Santorini, August 1998.

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[10]

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S. Kim, T. Y. Song, M. H. Jeong, B. Lee, Y. H. Moon, J. Y. Namkung, G. Jang, “Development of voltage stability constrained optimal power flow (VSCOPF)” in Proc. IEEE PES Summer Meeting, Vancouver (Canada), 2001. A. Berizzi, P. Finazzi, D. Dosi, P. Marannino, S. Corsi, “First and second order methods for voltage collapse assessment and Security Enhancement”, IEEE Transactions on Power Systems, Vol.13, No.2, pp.543-551, May 1998. R. Fletcher, Practical Methods of Optimisation, 2nd Edition, New York: John Wiley & Sons, 1990. D. G. Luenberg, Linear and nonlinear programming 2nd Edition, Boston: Kluwer Academics, 2003. S. Wrigth, Primal-Dual Interior-Point Methods, SIAM, 1996. D. E. Goldberg, Genetic algorithms in search, optimization, and machine learning, London : Addison-Wesley, 1989. W. Rosehart, C. Canizares, V. Quintana, “Costs of voltage security in electricity Markets”, in Proc. IEEE PES Summer Meeting, Seattle, July 2000. C. Canizares, W. Rosehart, A. Berizzi, C. Bovo, “Comparison of voltage security constrained optimal power flow techniques”, in Proc. IEEE PES Summer Meeting, Vancouver (Canada), July 2001. B. Baran, J. Vallejos, R. Ramos, U. Fernandez, “Multi-Objective reactive power compensation”, in Proc. IEEE PES Summer Meeting, Vancouver (Canada), July 2001. Y. T. Hsiao, H. D. Chiang, C. C. Liu, Y. L. Chen, “A computer package for optimal multi-objective VAR planning in large scale power systems”, IEEE Transactions on Power systems, vol. 9, No.2, pp.668-676, May 1994. A. Berizzi, C. Bovo, M. Delfanti, E. Fumagalli, M. Merlo, “Simulation of a bid-based dispatch subject to inter-zonal transmission constraints”, 2003 IEEE Bologna PowerTech, Bologna, Italy, June 23-26, 2003. A. Berizzi, C. Bovo, M. Delfanti, M. S. Pasquadibisceglie, “Impact of bilateral contracts on the Italian electricity market”, in Proc. Bulk Power System Dynamics and Control - VI, Cortina d’Ampezzo (Italy), August 2004. CIGRE Working Group C4.602, Coordinated voltage control in transmission network, Paris: CIGRE, February 2007. S. Corsi, P. Marannino, N. Lo Signore, G. Moreschini, G. Piccini, “Coordination between the reactive power scheduling function and the hierarchical voltage control of the EHV ENEL system”, IEEE Transaction on Power Systems, Vol. 10, No.2, pp. 686694, May 1995. S. Corsi, “The secondary voltage regulation in Italy”, in Proc. IEEE PES Summer Meeting, Seattle (USA), July 2000. A. Berizzi, P. Marannino, M. Merlo, M. Pozzi, F. Zanellini, “Steady-state and dynamic approaches for the evaluation of loadability margins in the presence of secondary voltage regulation”, IEEE Transactions on Power Systems, Volume 19, Issue 2, pp.1048-1057, May 2004. D. S. Popovic, M. S. Calovic, V. A. Levi, “Voltage reactive security analisys in power system with automatic secondary voltage control”, IEE Proc.-Gener. Transm. Distrib., Vol. 141, No. 3, pp.177-183, May 1994.

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[26] A. Conejo, M. J. Aguilar, “Secondary voltage control: Nonlinear selection of pilot busses, design of an optimal control law, and simulation results”, IEE Proc.-Gener. Transm. Distrib., Vol. 145, No. 1, pp.77-81, January 1998. [27] A. Berizzi, C. Bovo, M. Delfanti, M. Merlo, F. Tortello, “Singular Value Decomposition for an ORPF formulation in presence of SVR”, in Proc. IEEE Mediterranean Electrotechnical Conference, Sevilla (Spain), May 2006. [28] A. Berizzi, C. Bovo, C. Bruno, M. Delfanti, M. Merlo, M. Pozzi, “ORPF Procedures for Voltage Security in a Market Framework”, in Proc. IEEE Power Tech, S. Petersburg (Russia), June, 2005. [29] B. Gao, G. K. Morison, P. Kundur, “Voltage stability evaluation using modal analysis Power Systems”, IEEE Transactions on Power Systems, Volume 7, Issue 4, pp. 1529 – 1542, Nov. 1992. [30] D.I. Sun, B. Ashley, B. Brewer, A. Hughes and W. F. Tinney, "Optimal Power Flow by Newton Approach", IEEE Transactions on Power Apparatus and Systems, Vol. PAS103, No. 10, pp.2864-2880, Oct. 1984. [31] M. Innorta, P. Marannino, G. P. Granelli, M. Montagna, A. Silvestri, “Security constrained dynamic dispatch of real power for thermal groups”, IEEE Transactions on Power Systems, Volume 3, No. 2, pp.774-781, May 1988. [32] B. Cova, G. Granelli, M. Montagna, A. Silvestri, M. Innorta, P. Marannino, "Large scale application of the Han-Powell algorithm to compact models of static and dynamic dispatch of real power", Electrical Power & Energy Systems, Vol. 9, No. 3, pp. 131141, July 1987. [33] G. Astfalk, I. Lustig, R. Marsten, D. Shanno, “The interior-point method for linear programming”, IEEE Software, Volume 9, No. 4, pp.61-68, July 1992. [34] V. H. Quintana, G. L. Torres, J. Medina-Palomo, “Interior-point methods and their applications to power systems: a classification of publications and software code”, IEEE Transactions on Power Systems, Volume 15, No. 1, pp.170-176, Feb. 2000. [35] H. Wei, H. Sasaki, R. Yokoyama, “An application of interior point quadratic programming algorithm to power system optimization problems”, IEEE Transactions on Power Systems, Volume 11, No. 1, pp.260-266, Feb. 1996. [36] K. Y. Lee, Xiaomin Bai, Young-Moon Park, “Optimization method for reactive power planning by using a modified simple genetic algorithm”, IEEE Transactions on Power Systems, Vol. 10, No. 4, pp.1843-1850, Nov. 1995. [37] C. M. Fonseca and P. J. Fleming, “An overview of evolutionary algorithms in multiobjective optimization,” Evol. Comput., vol. 3, No. 1, pp. 1–16, 1995. [38] M. A. Abido, “Multiobjective Evolutionary Algorithms for Electric Power Dispatch Problem”, IEEE Trans. on evolutionary computation, vol. 10, No. 3, pp. 315-329, June 2006. [39] A. Berizzi, C. Bovo, “The use of genetic algorithms for the localisation and sizing of passive filters”, in Proc. 9th International Conference on Harmonics and Quality of Power (ICHQP), Orlando (USA), October 2000. [40] H. Pohlheim, Evolutionary algorithms: Overview, methods and operators. GEATBX. [Online]. Available: http://www.geatbx.com/, 2005. [41] V. Chankong, Y. Y. Haimes, Multiobjective decision making: theory and methodology, New York: North Holland series in science and engineering, vol. 8, 1983.

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[42] C. L. Wadhwa, N. K. Jain, “Multiple objective optimal load flow: a new perspective”, IEE Proceedings-Generation, Transmission and Distribution, Vol. 137, n. 1, pp 13-18, Jan. 1990. [43] U. Nangia, N. K. Jain, C. L. Wadhwa, “Optimal weight assessment based on a range of objectives in a multiobjective optimal load flow study”, IEE Proceedings – Generation, Transmission and Distribution, Vol. 145, n.1, pp. 65-69, Jan. 1998. [44] U. Nangia, N. K. Jain, C. L. Wadhwa, “Surrogate worth trade – off technique for multiobjective optimal power flow”, IEE Proceedings – Generation, Transmission and Distribution, Vol. 144, n.6, pp. 547 – 553, November 1997. [45] F. W. Gembicki, Y. Y. Haimes, "Approach to performance and sensitivity multiobjective optimization: the goal attainment method", IEEE Transaction on Automatic Control, Vol. AC-20, No.6, pp. 769 – 771, pp. 1885 – 1890, December 1975. [46] Y. L. Chen, "Weak bus oriented optimal multi-objective Var planning", IEEE Transactions on Power Systems, Vol. 11, No. 4, pp.1885-1890, November 1996. [47] A. Berizzi, A. Silvestri, D. Zaninelli, P. Marannino, O. Bertoldi, N. Di Gaetano, "Changes in power system planning and operation software tools in a deregulated environment", in Proc. CIGRE Symposium on "Impact of open trading on power systems", Tours (France), June 1997. [48] T. Van Cutsem, “A method to compute reactive power margins with respect to voltage collapse,” IEEE Transactions on Power Systems, Vol. 6, No. 1, pp. 145-156, February 1991. [49] A. Berizzi, P. Bresesti, P. Marannino, M. Montagna, S. Corsi, and G. Piccini, “Security Enhancement Aspects in the Reactive-voltage Control,” in Proc. IEEE Stockholm Power Tech, Vol. on Power Systems, Stockholm (Sweden), pp. 674-679, June 1995.

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In: Optimization Advances in Electric Power Systems ISBN: 978-1-60692-613-0 Editor: Edgardo D. Castronuovo, pp. 225-251 © 2008 Nova Science Publishers, Inc.

Chapter 9

OPTIMAL PLACEMENT IN POWER SYSTEM Gabriel Olguin1 and Tuan A. Le2 1

2

Transelec S.A., Chile Chalmers University of Technology, Sweden

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Abstract This chapter deals with the optimal placement of power quality meters and flexible alternating current transmission systems (FACTS) devices in power systems. The chapter presents two recent applications of optimization techniques to power system planning. The first example addresses optimal monitoring programs of power system for voltage sags assessment. Voltage sags are short duration reductions of the rms voltage caused by the occurrence of remote short circuit faults. A meter placement method for voltage sags monitoring of a large transmission systems is presented. An integer programming based modeling is proposed for choosing the locations of a limited number of power quality meters. A branch and bound type algorithm is used to solve the optimization problem. A large transmission network is used to validate the proposed method and stochastic assessment of voltage sags is applied to the network to obtain simulated monitoring results. Voltage sag system indices are calculated from monitoring programs designed according to the optimization method. Comparisons with the system indices obtained from a full monitoring program show the applicability of the method. The second example considers the optimal placement of flexible alternating current transmission systems (FACTS) devices to reduce the transmission congestion problem in the deregulated power systems. The example deals with the congestion problem in the deregulated electricity market using an optimal power flow framework (OPF). The congestion management method considered is based on a constrained re-dispatch of generation schedule which are formed by the market. From the re-dispatch, the congestion management cost can be evaluated. The installation and operation of the flexible alternating current transmission systems (FACTS) devices in the transmission network is considered in order to alleviate congestion and minimize the amount of active power which has to be re-dispatched. A costbenefit analysis to evaluate the economical justification of using FACTS for congestion management is proposed. The IEEE 14-bus system is used to simulate the market and illustrate the proposed method. The study results show that when FACTS devices are included in the network, the amount of re-dispatched power is greatly reduced resulting in an optimal operating point closer to that dictated by the market settlement and that FACTS is a viable option for congestion management, from a technical as well as economical point of view.

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Optimal Monitoring Program for Voltage Sag Characterization of Power Systems Nomenclature Z: zij :

general entry of the bus impedance matrix, Z.

vkf :

residual phase voltage at bus k during a fault at node f.

Vdip :

matrix formed by vkf .

bus impedance matrix.

MRAk ( p ) : monitor reach area (set) of bus k for threshold p. MRA p :

monitor reach area matrix for a threshold p.

X:

binary row vector in which a component i indicates if a monitor is needed at bus i.

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Introduction Voltage sags (also known as voltage dips) are short duration reductions in the rms voltage [1]-[3]. A single voltage sag event is characterized by its magnitude (the residual or remaining voltage during the event) and its duration (the time during which the rms voltage stays below a given threshold, usually 0.9 pu). Voltage sags are of stochastic nature as most of them are caused by remote short circuit faults occurring in transmission and distribution systems and involving several random factors as fault position, fault type, pre-fault voltage, etc. Equipment used in modern industrial plants is becoming more sensitive to voltage sags as the complexity of the plant increases, hence voltage sags are considered a power quality problem of equal or more importance as interruptions [4],[5]. Since voltage sags can stop an entire industrial plant, compatibility between the sensitive load and the power supply must be assessed [6]. In order to ensure compatibility the sensitivity of the load to voltage sags and the expected performance of the system must be conveniently described and compared. Equipment sensitivity is usually presented by means of voltage tolerance or power acceptability curves like CBEMA [7], whereas system performance is usually described by means of site and system indices. Sag indices can be used to indicate the different performance levels experienced at the transmission, subtransmission, and distribution circuit levels. The voltage sag assessment at a particular site in a system provides the information about frequency of sag occurrence and its characteristics like magnitude and duration. The number of sags with a residual voltage less than a given value (0.9pu as example) is usually used to describe the site performance index. System indices are calculated from the site indices obtained for a number of sites [8],[9]. Two methods can be distinguished for calculating system indices. The system index is defined as a weighted average of the site indices. The system index is defined as the value not

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exceeded for 95% of the sites (the 95 percentile of the site indices). To be able to determine a 95 percentile a rather large number of sites has to be monitored. Voltage sag indices can be obtained over all nodes by using stochastic prediction methods. The method of fault positions is applied here to obtain simulated monitoring results. Power quality monitoring can be used to obtain the average power quality of the network as well as the 95 percentile. Ideally all load busbars should be monitored over a long period of time but resource constrains do not allow such a monitoring program. Optimal decisions regarding the number of meters and their locations are needed so that the number of meters is minimized without missing any essential information. To be effective the monitoring program must be representative in time and space. At least four questions need to be answered:

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1) 2) 3) 4)

How many monitors need to be installed? Where the monitors should be installed? What voltage threshold should be set? How long the monitoring program should be? [11]

Here we address questions 1, 2 and 3. An integer programming model is introduced that allows minimizing the number of meters needed to characterize a large transmission network in terms of voltage sags [12]. The optimal monitoring program identifies optimal locations for meters so that the complete network is monitored. The optimization method is implemented using a 87-bus network and solved using a branch and bound (BB) type algorithm, ensuring optimality. The integer-programming problem may have multiple solutions meaning that the minimum number of meters can be spread out over the network in different forms satisfying constraints. The solution space is explored using a genetic algorithm (GA). A number of different meter arrangements are found that keep optimality. Optimal solutions are then used to calculate system indices. Comparisons with system indices obtained through a hypothetical full monitoring program shows the applicability of the method.

Assessing Voltage Sags Performance This Section briefly describes the method of fault positions for stochastic assessment of voltage sags [13]-[15]. The method will be used to create (pseudo) measurements so that sag system indices can be calculated on the basis of a full monitoring program. Consider a transmission network with N buses and its impedance matrix Z built so that the common generator node is chosen as reference and labeled zero. In order to apply the method of fault positions a number of fictitious nodes (fault positions) have to be created in order to simulate faults along lines. If the pre-fault voltages are assumed 1 pu, which is a reasonable assumption for stochastic assessment, then the retained voltage in pu at an arbitrary observation bus k during a three-phase fault at any fault position f is given by (1) where zkf is the transfer impedance between the observation bus k and the fault point f, and zff is the driving point impedance at the fault point.

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vkf = 1 −

zkf

(1)

z ff

The magnitude of residual voltages (during-fault voltages) can be stored in a so-called dip-matrix. The dip-matrix contains residual voltages at every bus due to faults at every fault position. The dip-matrix can directly be determined as indicated in (2) where Z is the impedance matrix, diagZ is the diagonal matrix of Z and ones is a matrix full of ones of the same dimension of Z.

Vdip = ones − Z ⋅ inv(diagZ)

(2)

In order to obtain a probabilistic assessment of the number of sags and their characteristics, the dip-matrix needs to be combined with the fault rate corresponding to each fault position. Let k be an arbitrary observation bus and λ be the vector containing the corresponding fault rate of each one of the Fp fault positions as shown in (3). Then Table I, which shows row k (transposed) of the dip-matrix and vector λ, presents the yearly frequency of occurrence of these fault related events, sags.

[

λ = λ1 , λ2 ,...λk ,...λ N ,...λ Fp

]

(3)

Each fault in the system will cause a drop in voltage at bus k, however only part of these fault-caused events will be counted as sags (those for which vkf < 0.9 pu). A large part of these events will result in residual voltages above 0.9 pu.

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Table I. An Arbitrary Row of the Dip Matrix and the Corresponding Fault Rate Retained voltage at bus k (vkf)

Frequency (λ)

vk1

λ1

vk 2

λ2

…

…

vkN

λN

…

…

vkFp

λFp

A common way to present site and system performance is by using cumulative frequency tables or histograms. Such tables or histograms present the number of events up to a given residual voltage. In order to obtain the cumulative frequency table the residual voltages contained in Table I need to be grouped according to the ranges of interest. Table II illustrates this procedure.

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The method of fault positions has been applied in [14] to the transmission system shown in Figure 1 that corresponds to a simplified model of the Colombia National Interconnected System 230kV and 500kV in December 1999. The system has 87 buses and 164 lines with a total length of 11651 km. Buses 63, 16, 15 and 68 form part of the 500 kV system. Table II. Cumulative Dip Frequency for a General Bus k Dip bins (pu)

Frequency (events/year)

∑λ : v

0 − 0.55

i

ki

≤ 0.55

i

0 − 0 .6

∑λ

: v ki ≤ 0.6

∑λ

: v ki ≤ 0.65

i

i

0 − 0.65

i

i

:

:

∑λ

0 − 0 .9

i

: v ki ≤ 0.9

i

73

76

12

79

30

23

21

51 28

77 80

86

24

62 63

52 16 85

84

14

22

19

7

29 6

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47

25

11

35

44

78

65

5

10 43

54 26

2

81

32

59

49

53

20

46

60 8

71

75

48

15

68

31

56

3

74

67 61

27

38 40

50

69

17

45 41

4

13

82

70 33

87 34

39

18 58

1

37 72

55

57

83

42

9 64 66 36

Figure 1. Simplified Model of the National Interconnected Systems of Colombia, 230 kV and 500 kV.

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Gabriel Olguin and Tuan A. Le

Symmetrical faults were simulated at 781 fault positions (buses and lines). Fault positions were chosen so that no more than 15 km of line separated two near fault positions. The fault rate for lines is 0.0134 faults per year-km and is the same for all lines. The fault rate for buses is 0.08 faults per year. It is important to realize that the likelihood of faults is given by the fault rate and not by the number of fault positions. Although 781 faults were simulated the rate of occurrence assigned to each fault position corresponds to a fraction of the line fault rate. See [14] for more details on the simulations. Table III shows the number of sag with a residual voltage less than 0.9 pu, SARFI-90 according to [8],[9], for every bus of the system. Table IV presents system indices calculated with basis on the site performance presented in Table III. Note that a full monitoring program is used here to calculate system indices. The average numbers of events with a residual voltage less than 0.9 as well as the 95 percentile have been chosen as system indices. The average number of events seen at load buses is 20.5 sags per year. The 95 percentile resulted 43.7 events indicating that less than 5% of the buses may be exposed to more than 43.7 sags per year with a residual voltage less than 0.9 pu. In practice, however, a full monitoring program would be prohibitively expensive and rather redundant in the sense that a fault caused event may be seen by several monitors. Therefore only a limited number of buses must be selected for locating power quality meters.

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Table III. Site Dip Frequency V