OPTIMAL POWER FLOW AND NETWORK LOADABILITY USING FEEDBACK-BASED SELF-ADAPTIVE DIFFERENTIAL EVOLUTION AND MULTIOBJECTIVE ALGORITHMS By Fares Theyab A Alharbi A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Electrical Engineering - Master of Science 2018 OPTIMAL POWER FLOW AND NETWORK LOADABILITY USING FEEDBACK-BASED SELF-ADAPTIVE DIFFERENTIAL EVOLUTION AND MULTIOBJECTIVE ALGORITHMS ABSTRACT By Fares Theyab A Alharbi In modern electrical grids, planning and operation processes require efficient optimization tools. Optimal placement and sizing of Flexible AC transmission system (FACTS) devices, re- newable energy resources, and energy storage units, to name a few, are optimization tasks in the planning process. Minimizing the cost of generated power from committed generators in the op- eration process is an important part of power system operations. For power system optimization problems, several optimization algorithms have been proposed and used in the past two decades. However, the need for efficient optimization algorithms customized to power system problems still exists. The research reported in this thesis develops novel evolutionary optimization approaches for two applications: optimal power flow (OPF) and optimal placement and sizing of FACTS to enhance electrical network loadability. For optimal power flow, two new feedback-based self-adaptive differential evolution algorithms are proposed. Prior to applying the proposed methods to the power system test cases, they are tested on standard mathematical benchmark problems. The self-adaptive differential evolution algorithms showed significant improvement in the benchmark problems compared to other algo- rithms. More importantly, in this work, the feedback-based self-adaptive differential evolution algorithms demonstrated good improvement in results and in convergence rate in several power system test cases. To enhance the loadability of an electrical network, a new multiobjective-based frame work is proposed for optimal placement and sizing of FACTS devices. The proposed method has been applied to commonly used FACTS devices, thyristor-controlled series controllers (TCSCs), and demonstrated excellent results in the electrical loading margins as well as the investment costs compared to other available methods. My parents, Theyab Alharbi and Radhyah Alharbi Dedicated To iii ACKNOWLEDGMENTS I am grateful to my advisor Dr. Joydeep Mitra, for his advice, assistance, and support. His encouragement and valuable advice have motivated me to succeed in my graduate studies and to accomplish this thesis. His great support was not only limited to academics, but also in other aspects of life such as personal and family life. I thank Dr. Mitra for his support and solidarity during the time when a member of my family was experiencing serious health challenges. Without his support, this work could not have been done. I am thankful to my committee members Dr. Fang Z. Peng and Dr. Bingsen Wang for their time and effort. I would like to thank all professors who taught me and advised me at MSU and before I joined MSU. I also want to thank all the authors I have cited in this thesis; without their inspiring work I could not accomplish this thesis. Thanks go out to my colleagues in the ERiSe lab, Dr. Mohammed Benidris, Dr. Samer Sulae- man, Dr. Nga Nguyen, Mr. Saleh Almasabi, Ms. Yuting Tian, Mr. Atri Bera, for their help and collaboration. I express my sincere thanks and gratitude reach to my parents, brothers, sisters, and my wife, for their support, encouragement, and love. iv TABLE OF CONTENTS LIST OF TABLES . . LIST OF FIGURES . . . . . . . LIST OF ALGORITHMS . Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . Introduction . . Chapter 2 Feedback Based Self-Adapting Control Parameters for Differential Evolution 3 3 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Differential Evolution (DE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 The self-adapting control parameters in differential evolution (jDE) . . . . . . . . . 6 . . . . . . . . . 2.4 Feedback-based self-adapting control parameters for DE (FBjDE) 7 Feedback-based self-adapting control parameters for DE-I (FBjDE-I) . . . 8 Feedback-based self-adapting control parameters for DE-II (FBjDE-II) . . . 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 Benchmark problems . . 2.6 Simulation . . . 2.7 Discussion . . 2.8 Conclusion . . . 2.4.1 2.4.2 . . . . . . . . . . . . . . . Chapter 3 Feedback-Based Self-Adaptive Differential Evolution Algorithms for Opti- . . . Introduction . mal Power Flow . . . . . 3.1 . . 3.2 Problem Formulation . 3.3 Simulation . . . 3.4 Results and Discussion . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 . Chapter 4 Multiobjective Optimal TCSC Placement and Sizing for Enhancing Net- . . . . . . . . Introduction . work Loadability . . 4.1 4.2 Multi-Objective Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . 4.3 Flexible AC Transmission System devices (FACTS) . . . . . . . . . . . . . . . . . 4.4 Related Work . . . 4.5 Problem Formulation . 4.6 Simulation . . . 4.7 Results and Discussion . . 4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 . 32 . 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 . . . . . . . . . . . . . . Chapter 5 Future Work . BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 v LIST OF TABLES Table 2.1: Comparison between DE, jDE, FBjDE-I, and FBjDE-II in terms of NFE . . . 17 Table 2.2: Comparison between jDE and FBjDE-I in terms of NFE . . . . . . . . . . . . 18 Table 2.3: Comparison between jDE and FBjDE-II in terms of NFE . . . . . . . . . . . 18 Table 3.1: Comparison of Mean Cost of the Population over 100 Experiments IEEE 14-bus 26 Table 3.2: Comparison of the Best Cost of the Population over 100 Experiments IEEE 14-bus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Table 3.3: Comparison of the Mean Cost of the Population over 100 Experiments IEEE 30-bus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Table 3.4: Comparison of the Best Cost of the Population over 100 Experiments IEEE 30-bus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Table 3.5: Comparison of the Mean Cost of the Population over 100 Experiments IEEE 57-bus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Table 3.6: Comparison of the Best Cost of the Population over 100 Experiments IEEE 57-bus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Table 4.1: IEEE 118-bus Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Table 4.2: IEEE 118-bus Maximum Loadability . . . . . . . . . . . . . . . . . . . . . . 43 vi LIST OF FIGURES Figure 2.1: De Jongs Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Figure 2.2: Axis parallel hyper-ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Figure 2.3: Rotated hyper-ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Figure 2.4: Rosenborks valley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Figure 2.5: Griewank Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Figure 2.6: Rastrigin Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Figure 2.7: Ackley Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Figure 2.8: Schwefel Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Figure 3.1: Flow chart of DE, jDE, FBjDE-I, FBjDE-II algorithms for OPF . . . . . . . 25 Figure 3.2: Mean of the Best Cost of the Population over 100 Experiments IEEE 14-bus Figure 3.3: Mean of the Best Cost of the Population over 100 Experiments IEEE 30-bus Figure 3.4: Mean of the Best Cost of the Population over 100 Experiments IEEE 57-bus 27 28 29 Figure 4.1: TCSC Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Figure 4.2: Constructing an Individual in the Population . . . . . . . . . . . . . . . . . 41 Figure 4.3: Flow chart of the proposed approach . . . . . . . . . . . . . . . . . . . . . . 42 Figure 4.4: Pareto Efficient Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Figure 4.5: Number of TCSC Vs. Loadability . . . . . . . . . . . . . . . . . . . . . . . 45 vii LIST OF ALGORITHMS Algorithm 1: Controlling the Limits of the Scale Factor Range . . . . . . . . . . . . . . . . . . . . 9 Algorithm 2: Pseudocode of the FBjDE-II Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 viii Chapter 1 Introduction Optimization is an essential part of electrical grids planning and operation processes. With the current development towards modern electrical grids, the available optimization methods need to be improved and customized for power system problems. In this thesis, two main optimiza- tion problems are considered for improvement namely, optimal power flow (OPF) and optimal placement and sizing of FACTS devices to control power flow and to enhance electrical network loadability. Optimal power flow is an important optimization problem in power system planning and oper- ation. The full AC model of the OPF is a nonlinear complex problem. The complexity of the OPF problem increases as the size of the electrical grid increases. In the first part of this thesis, two feedback-based self-adaptive differential evolution algo- rithms are proposed to solve the OPF. Differential evolution (DE) has been widely used in global optimization due to its effectiveness and simplicity compared to other evolutionary algorithms. However, the performance of evolutionary algorithms is dependent on control parameters of the evolutionary process. Adaptive algorithms have significantly improved the performance of evolu- tionary algorithms. Chapter 2 proposes the feedback-based self-adaptive differential evolution algorithms to im- prove the performance of DE. Two feedback-based self-adapting control parameters algorithms, 1 (FBjDE-I) and (FBjDE-II), are proposed and tested on several mathematical benchmark problems and have been compared to other DE algorithms in the literature. The results show that the pro- posed feedback-based self-adaptive algorithms have successfully improved the performance the DE algorithm. In Chapter 3, the optimal power optimization problem is presented and the pro- posed algorithms in Chapter 2, FBjDE-I and FBjDE-II, are introduced to solve the OPF. In the second part of the thesis, the optimal placement and sizing of FACTS devices is con- sidered to improve electrical network loadability. Enhancing the network loadability is crucial in modern electrical grids because transmission lines are vulnerable to being overloaded beyond their thermal limits in deregulated and highly competitive markets with increasing load demands. Overloading transmission lines will not only increase losses, but also may drive the system into insecure operating conditions. Flexible AC transmission system FACTS devices have been effec- tively utilized to enhance transmission network loadability. Chapter 4 presents a new mutliobjective-based framework for optimal placement and sizing of FACTS devices to improve loadaiblity in electrical grids and to relieve congestion. Additionally, Chapter 4 provides a brief summary of FACTS devices, multiobjective optimization problems, and the application of the proposed method using thyristor-controlled series controllers (TCSCs) devices. The thesis is concluded with future work in Chapter 5. 2 Chapter 2 Feedback Based Self-Adapting Control Parameters for Differential Evolution 2.1 Introduction Differential evolution algorithms have been under extensive research to improve their reliabil- ity, robustness, and performance [1,2]. It has been pointed out that the performance of evolutionary algorithms depend largely on the parameters that are controlling the evolutionary process. Com- pared to other evolutionarily algorithms, DE has fewer parameters [1]. Yet the performance of the differential evolution is affected by the choice of the control parameters due to the fact that finding suitable parameters set is a problem-dependent task [3]. In DE, the control parameters are the population size N P , crossover rate CR, and the mu- tation scale factor F . One of the effective methods that has enhanced the performance of DE is controlling or tuning the scale factor F and the crossover rate CR. According to [4], tuning the control parameter takes place before running the evolutionary algorithm, whereas controlling the parameters takes place dynamically during the run of the evolutionary algorithm. According to [4], the techniques to control the parameter are classified into three types. First, deterministic parameter control in which the parameters are updated based on deterministic rules. 3 Second, adaptive parameter control where the parameters are updated based on feedback from the evolutionary process. Third, the self-adapting Control in which the control parameters are incorporated with individuals’ parameters. The evolution operators are not only applied to the candidate solutions, but are also applied to the control parameters. In JADE algorithm [5], F and CR are generated from Gaussian (normal) and Cauchy distri- butions around the mean of the successful parameters from previous generations .The MDEpBX algorithm in [6] proposed that the control parameters should be controlled based on statistical values of the successful population average population. A self-adapting control algorithm for dif- ferential evolution by J.Brest et al in [3], known as (jDE), is widely used due to its effectiveness and simplicity compared to other adaptation schemes [7]. In this chapter, two adaptive techniques based on the feedback of the evolutionary process are proposed to improve the performance and robustness of jDE, namely FBjDE-I and FBjDE-II. The proposed feedback-based self-adapting control parameters for DE algorithm ,FBjDE-I and FBjDE-II, will be tested on several benchmark problems and compared to DE and jDE algorithms. 2.2 Differential Evolution (DE) The general procedure of the classical differential evolution (DE/rand/1/bin) as in [1] and [2] can be briefly summarized in the following steps. Step 1, the algorithm starts with a population of random points N P in the search space. Step 2, the objective or fitness function values of all the population are evaluated. Step 3, the mutation operator of the differential evolution is performed to generate a donor vector Vi,g+1 for each original vector Xi as in (2.1). Vi,g+1 = Xr1,g + F × (Xr2,g − Xr3,g) (2.1) where i is the vector index i = 1...N P ; g is the generation number; r1, r2, r3 are random indices ∈ N P ; N P is the population size and r1, r2, r3 (cid:54)= i. Step 4, the crossover operator in (2.2) builds a trail vector for each Ui,g+1 individual using the donor vector Vi,g+1 and the original 4 vector Xi,g with a crossover rate CR. vij,g+1, If rand ≤ CR or xi,j,g, Otherwise uij,g+1 = j = jr (2.2) where j = 1, ..., D ; and D is the dimension of the decision variables of the objective function. In [1], to ensure that the crossover takes place in at least one dimension of the donor vector Ui,g+1 an or condition, when the dimension j equals a random dimension index jr, is added with the crossover rate CR as in (2.2). Step 5, the selection operator in (2.3) compares the parent Xi,g and its offspring Ui,g+1 and chooses the one that has a better objective function or fitness value to become the parentXi,g+1 for the next successive generation. In (2.3) and in most optimization algorithms, the optimization problem is considered to be a minimization problem. Ui,g+1, If f (Ui,g+1) < f (Xi,g) Otherwise Xi,g+1 = Xi,g, (2.3) 2.3 The self-adapting control parameters in differential evolu- tion (jDE) Differential evolution (DE) has been widely used in the literature [2]. The DE has three parameters which are the number of population N P , a mutation scale factor F , and a crossover rate CR. The population size N P is usually proportional to the dimension of the problem. The performance of the differential evolution is highly dependent on the choices of the mutation scale factors F and the crossover rates CR. Empirical or recommended values of the parameters have been used in the literature [8] [9]. In practice, however, the proper choice of parameters is a problem dependent task [3]. In the self-adapting algorithm [3], each candidate solution is encoded with a scale factor Fi,g 5 and a crossover rate CRi,g. These parameters are updated to possibly better values as follows: the scale factor Fi,g is controlled using equation (2.4). Fl + rand1 × Fu, If rand2 < τ1 Otherwise Fi,g+1 = Fi,g, (2.4) Fl is the lower boundary of the scale factor F . Fu is the upper boundary of the scale factor F . In [3] the values of the lower and upper limits of F are suggested to be Fl = 0.1, Fu = 0.9. Similarly, the crossover rate associated with each individual is controlled using equation (2.5) rand1, CRi,g+1 = CRi,g, Otherwise If rand2 < τ2 (2.5) τ1 and τ2 are constants ∈ [0, 1]; rand1, rand2 are random numbers, generated uniformly ∈ [0, 1]. In [3], τ1 and τ2 are recommenced to be τ1 = 0.1 and τ2 = 0.1. 2.4 Feedback-based self-adapting control parameters for DE (FBjDE) jDE adaptive algorithm is a simple yet an efficient adaptive algorithm for F and CR and has shown better results compared to DE and other adaptive algorithms [3, 7]. The adaptation of Fi,g+1 and CRi,g+1 is dependent on the choice of τ1 and τ2. These parameters should be defined by the user before running the algorithm. In [3] the authors have used and recommended values of τ1 = 0.1 and τ2 = 0.1 which were successful in wide range of single objective problems. In this work, feedback-based self-adapting control algorithms are proposed to utilize the feed- back information from the evolutionary process to improve the performance of DE. The adaptation of the control parameters Fi,g+1 and CRi,g+1 will be based on the feedback from heuristic rules. The proposed adaptive algorithms are explained in the following subsections. 6 1, 0, Fi,g, F Bi,g = If f (Ui,g+1) < f (Xi,g) Otherwise (2.6) F Bi,g in equation (2.6) is a feedback information of whether the crossover operation in the current generation g is successful or not for each individual Xi. Based on the feedback information F Bi,g, the scale factor is controlled as in (2.7). Fi,g+1 = Fl + rand × Fu, Otherwise If F Bi,g = 1 (2.7) The crossover rate CRi is controlled similarly using the previous information of each individ- ual F Bi,g as in equation (2.8). To compare FBjDE and jDE, the values of the lower and upper limits of F are as suggested in [3] Fl = 0.1 and Fu = 0.9. 2.4.1 Feedback-based self-adapting control parameters for DE-I (FBjDE-I) In the FBjDE-I, the adaptation of the control parameters depends on how the current parame- ters Fi,g and CRi,g have contributed to the evolution of an individual Xi. In other words, it gets feedback from the current generation ofFi,g and CRi,g and uses the information to decide whether to keep the next successive Fi,g+1 and CRi,g+1 or to change them to possible better values. In FBjDE-I, each individual will be encoded with a mutation scale factor Fi,g and a crossover rate CRi,g along with feedback information flag F Bi,g. CRi,g, rand, CRi,g+1 = If F Bi,g = 1 Otherwise (2.8) Compared to jDE, the adaptation of the FBjDE-I does not depend on the parameters τ1 and τ2. It only depends the performance of the parameters Fi and CRi in the evolution of each individual. In FBjDE-I, successful parameters will surviver with their corresponding individuals, and they will most likely contribute in creating successful offsprings. 7 2.4.2 Feedback-based self-adapting control parameters for DE-II (FBjDE- II) In general the performance of the DE depends highly on the mutation scale factor F compared to the crossover rate CR [5]. In FBjDE-II, more emphasis in controlling the scale factor F is applied, while the crossover rate will be controlled as in FBjDE-I. In jDE and FBjDE-I, the limits of the scale factor Fl and Fu are fixed in the evolutionary processes, and Fi is uniformly randomly drawn between the upper and the lower bounds of Fi for each individual. To exploit the information of how an individual has evolved to the current position, the range for generating a scale factor Fi should change according to the performance of the previously generated scale factors. The learning process from the feedback of the scale factor Fi performance can be used as indicator to dynamically control the lower and upper bounds of the scale factor range of each candidate solution in the population. For instance, if the scale factor has increased and the individual has not improved to a better position, it is likely that the scale factor range needs to be moved to a slightly better region. How- ever, if the current solution is better than its parent and the scale factor that generated the current solution is higher than the scale factor that was used to generate its parent, it is more likely that higher scale factors are more suitable for that individual in this region of the search space. This information should be utilized to generate potentially better scale factors for the next successive iteration. That could be achieved by dynamically updating the lower bound Fl,i and the upper bound Fu,i for the scale factor Fi. In [1], the range of the scale factor is Fi ∈ [0, 2]. In FBjDE-II, however, a more flexible range for the scale factor is suggested Fi ∈ [−2, 2]. Allowing a negative scale factor would make the movement of each individual more flexible in the search space. The feedback information for the current individual is given in (2.6). Based on that, the range of generating Fi,g+1 will be updated depending on the improvement that was achieved by pre- ceding values of Fi,g and Fi,g−1 in the previous generations. Controlling the scale factor bounds 8 else Fl,i,g+1 = Fl,i,g + λ Fu,i,g+1 = Fu,i,g + λ Fl,i,g+1 = Fl,i,g − λ Fu,i,g+1 = Fu,i,g − λ Algorithm 1 Controlling the limits of the scale factor range 1: Evaluate F Bi,g using (2.6) 2: if F Bi,g = 1 then if Fi,g ≥ Fi,g−1 then 3: 4: 5: 6: 7: 8: 9: 10: else 11: 12: 13: 14: 15: 16: 17: 18: end if end if if Fi,g ≥ Fi,g−1 then Fl,i,g = Fl,i,g − λ Fu,i,g = Fu,i,g − λ else end if Fl,i,g = Fl,i,g + λ Fu,i,g = Fu,i,g + λ Fl,i,g+1 and Fl,i,g+1 for the following successive generations is illustrated in the pseudocode presented in Algorithm. 1. Fl,i and Fu,i will change continuously based on the evolution of the solution Xi. In Algorithm1, λ defines movement of the scale factor range [Fl,i, Fu,i]. Since the span of the allowed range is 4 units, a reasonable value of λ is in the range ∈ [0.01, 0.25]. As the value of λ gets higher the movement of the range gets faster. A value of λ = 0.1 is used in this work. The complete algorithm for FBjDE-II is depicted in Algorithm 2. To limit the scale factor Fi ∈ [−2, 2], Fl,i can take values ∈ [−1.5, 0.5] and Fl,i ∈ [−0.5, 1.5]. After controlling Fi,l and Fi,u, Fi,g+1 is controlled using (2.7). Note that the crossover rate in FBjDE-II is controlled as in FBjDE-I. 2.5 Benchmark problems In this work the performance of the three adaptive algorithms jDE, FBjDE-I, and FBjDE-II will be tested on well-known numerical benchmark problems [10, 11]. 9 Algorithm 2 Pseudocode of the FBjDE-II Algorithm 1: Initialize: N P Population, Maximum generation gmax, Scale factors Fi,0, and Crossover rates CRi,0 Build a donor vector Vi,g for each individual Xi,g using equation (2.1) Creat a trial vector Ui,g using the parent Xi,g and the donor Vi,g using equation (2.2) Select: Replace the child with the parent if it is better using equation (2.3) Get Feedback information for all individuals F Bi,g using (2.6) Control scale factor Fi,g+1 as in algorithm 1 Control crossover rate CRi,g+1 as in equation (2.8) 2: while Termination Criteria is not met or g < gmax do 3: 4: 5: 6: 7: 8: 9: end while 1. DeJong’s Function: a unimodal function with global minimum f (X∗) = 0; at X∗ = (0, ., 0) D(cid:88) i=1 f (x) = x2 i ; (2.9) − 10 ≤ xi ≤ 10; D = 30 2. Axis parallel hyper-ellipsoid: a unimodal function with global minimum f (X∗) = 0; at X∗ = (0, 0, ., 0) D(cid:88) i=1 i.x2 i ; f (x) = − 10 ≤ xi ≤ 10; D = 30 (2.10) 3. Rotated hyper-ellipsoid: a unimodal function with global minimum f (X∗) = 0; at X∗ = (0, 0, ., 0) D(cid:88) (cid:0) i(cid:88) i=1 j=1 (cid:1)2; xj f (x) = − 10 ≤ xi ≤ 10; D = 20 (2.11) 4. Rosenborks valley:a multimodel function with a global minimum f (X∗) = 0; at X∗ = 10 (1, 1, ., 1). The global minimum is in a narrow flat valley. D−1(cid:88) i=1 f (x) = [100(xi+1 − x2 i )2 + (1 − xi)2], − 2.048 ≤ xi ≤ 2.048, D = 30. (2.12) 5. Griwank Function: highly multimodal function a with a global minimum f (X∗) = 0; at X∗ = (0, 0, ., 0). f (x) = 1 + 1 4000 n(cid:88) i=1 i − x2 n(cid:89) i=1 cos( xi√ i ) − 512 ≤ xi ≤ 512; D = 30. (2.13) 6. Rastrigin Function: highly multimodal function with a global minimum f (X∗) = 0; at X∗ = (0, 0, ., 0). D(cid:88) f (x) = 10D + i=1 i − 10cos(2xi)), (x2 (2.14) − 5.12 ≤ xi ≤ 5.12, D = 20 7. Ackley Function: highly multimodal function with a global minimum f (X∗) = 0; at X∗ = (0, 0, ., 0). f (x) = 20 + e + 20e (cid:113)(cid:80)D −0.2 i=1 x2 i /D − e (cid:80)D i=1 cos(2πxi) D , (2.15) − 30 ≤ xi ≤ 30, D = 20 8. Schwelfle Function : a highly multimodel function with a global minimum f (X∗) = 0; at 11 X∗ = (420.9687, 420.9687, ...., 420.9687). n(cid:88) (cid:113)|xi|)) + 418.982887; f (x) = (−xisin( i=1 − 512 ≤ xi ≤ 512; D = 20. (2.16) 2.6 Simulation In this work, the feedback-based self-adaptive DE algorithms, FBjDE-I and FBjDE-II, are applied to the previously described benchmark problems in Section 2.5. Both FBjDE-I and FBjDE- II are compared with classical DE and the self-adapting algorithm jDE. The performance of the three adaptive algorithms jDE, FBjDE-I, and FBjDE-II with the differential evolution DE will be compared in controlling the scale factor F and the crossover rate CR based on the number of function evaluations needed to reach a desired value, value to reach (VTR) [12]. For all the benchmark problems that have a global minimum of f (X∗) = 0, the VTR is V T R = 10−5; for shifted functions, the global minimum is not at zero, the VTR is V T R = 10−2. The convergence behavior of the three adaptive algorithm jDE, FBjDE-I, and FBjDE-II will be compared based on a fixed number of function evaluation (NFE) .In the simulation, the initializa- tion is as follows: N P = 5 × D, D is the dimension of the problem, Fi = 0.5 , CRi = 0.5. For jDE all the recommended vales will be used: τ1 = τ2 = 0.1 , Fl = 0.1 ,Fu = 0.9. To compare the feedback-based self-adaptive algorithms, FBjDE-I, and FBjDE-II, to jDE, they are initialized with the same upper and lower bounds of the scale factor range Fl = 0.1 ,Fu = 0.9. The adaptive algorithms are used to control the scale factor F and the crossover rate CR. 12 Figure 2.1: De Jongs Function Figure 2.2: Axis parallel hyper-ellipsoid 13 50100150200250300350400450500550600Iteration02004006008001000Function Value f(x)jDEFBjDE-IFBjDE-II50100150200250300350400450500550600Iteration050001000015000Function Value f(x)jDEFBjDE-IFBjDE-II Figure 2.3: Rotated hyper-ellipsoid Figure 2.4: Rosenborks valley 14 1002003004005006007008009001000Iteration050010001500200025003000350040004500Function Value f(x)jDEFBjDE-IFBjDE-II01002003004005006007008009001000Iteration02000400060008000100001200014000Function Value f(x)jDEFBjDE-IFBjDE-II Figure 2.5: Griewank Function Figure 2.6: Rastrigin Function 15 0100200300400500600Iteration0100200300400500600700800900Function Value f(x)jDEFBjDE-IFBjDE-II5001000150020002500300035004000Iteration050100150200250300350400Function Value f(x) F6jDEFBjDE-IFBjDE-II Figure 2.7: Ackley Function Figure 2.8: Schwefel Function 16 05001000150020002500300035004000Iteration012345678Function Value f(x)jDEFBjDEFBjDE205001000150020002500300035004000Iteration0100020003000400050006000700080009000Function Value f(x)jDEFBjDE-IFBjDE-II Table 2.1: Comparison between DE, jDE, FBjDE-I, and FBjDE-II in terms of NFE DE 87618 99405 D BenchMark Problem De Jongs Function 30 Axis parallel hyper-ellipsoid 30 Rotated hyper-ellipsoid Rosenborks valley Griewank Function Rastrigin Function Ackley Function Schwefel Function jDE FBjDE-I FBjDE-II 85245 97419 20 861374 255646 30 1635663 856554 30 120111 116004 20 1735586 169606 20 2077214 200508 20 198102 105530 69873 79782 187768 813555 93132 153186 184488 98674 54921 63036 220314 825255 73938 80096 103282 58316 2.7 Discussion From Table 2.1, it can be clearly concluded that the adaptation algorithms jDE, FBjDE-I and FBjDE-II have outperformed the traditional DE. This shows the importance and the need for controlling the parameters during the evolutionary process. In comparing FBjDE-I to jDE as in Table2.2, the percentage of improvement in the NFE varies from 5% to 26.5%. The overall improvement of the NFE for the eight benchmark problems is 13.9%. The improvement is measured by of the number of function evaluations needed to reach the desired value with above specified VTR values. The results of comparing the FBjDE-II to jDE are summarized in Table2.3. The overall improvement achieved when using FBjDE-II compared to jDE is 33.8%. In comparing FBjDE-I to FBjDE-II, we noticed that FBjDE-II outperformed FBjDE-I except in function 3 and function 5. However, for highly multimodel functions, the performance of FBjDE-II is more efficient than FBjDE-I. Controlling the range of the scale factor has effectively exploited the feedback history of suc- cessful scale factors; as a result, more successful scale factor are generated in the evolutionary process. The crossover rate CRi was controlled in the same manner in both FBjDE-I and FBjDE- II. Figures 1 to 8 show the comparison of the convergence behavior of the average population 17 averaged over 50 independent runs of the eight benchmark problems. The convergence rate of FBjDE-I and jDE is almost comparable in all the benchmark functions. The convergence rate of FBjDE-II is slightly better than the other two algorithms on functions F1 to F5. Figures 6 to 8 show that FBjDE-II has converged faster than jDE and FBjDE-I. Table 2.2: Comparison between jDE and FBjDE-I in terms of NFE Function D jDE FBjDE-I Improvement F1 F2 F3 F4 F5 F6 F7 F8 30 30 85245 97419 69873 79782 20 255646 187768 30 856554 813555 30 116004 93132 20 169606 153186 20 200508 184488 20 105530 98674 18% 18.1% 26.5% 5% 19.7% 9.7% 8% 6.5% Table 2.3: Comparison between jDE and FBjDE-II in terms of NFE Function D jDE FBjDE-II Improvement F1 F2 F3 F4 F5 F6 F7 F8 30 30 85245 97419 54921 63036 20 255646 220314 30 856554 825255 30 116004 73938 20 169606 80096 20 200508 103282 20 105530 58316 35.6% 35.3% 13.8% 3.6% 36.3% 52.8% 48.5% 44.7% 18 2.8 Conclusion In this chapter, feedback-based self-adapting control algorithms for differential evolutions are applied to control the differential evolutionary parameters. FBjDE-I and FBjDE-II algorithms have successfully improved the convergence rate and the results of the DE algorithm. The results show that FBjDE-I is comparable to jDE , while FBjDE-II has demonstrated better performance compared to DE, jDE and FBjDE-I. 19 Chapter 3 Feedback-Based Self-Adaptive Differential Evolution Algorithms for Optimal Power Flow 3.1 Introduction In electrical power systems, power flow analysis is an essential tool in studying the mechanism of electrical power that is generated, transmitted and consumed in an electrical grid. Power flow is the first step in planning, operation, control, and optimization processes. Determining the values of the voltages and angles at each bus enables the operators to calculate the real and reactive power needed to be generated from each dispatchable generator. Since the power flow problem is a highly nonlinear problem, nonlinear methods have been used in the power flow problem such as the Gauss-Seidel and Newton-Raphson methods. According to [13], the Newton-Raphson outperformed the Gauss-Seidel in the convergence rate. The aforementioned methods are well explained in [13]. In practice the committed generators need to be optimally dispatached to meet the load demand and to cover the system losses at optimal generation cost. This problem is known as the optimal 20 Power Flow (OPF). The OPF is a highly nonlinear and constrained optimization problem. In the literature the OPF problem has been addressed using the following methods. A commonly used method is the linear programming LP [14] [15]. The linear programing could be used when the AC power flow model represented in (3.8) and (3.9) is linearized to a linear model known as DC power flow. The DC model reduces the complexity of the problem and the computation time of the OPF problem at the expense of the accuracy. The linear programming method is also used with a piecewise linearized power flow model which has resulted in better accuracy compared the DC power flow model. Classical methods, mathematical-oriented methods, have been widely used to solve the AC model of the power flow, such as Newton’s method, quadratic programming, and sequential quadratic programming, [16] [17] [18] respectively. In the classical methods used for OPF, the optimization problem is solved using gradient-based search methods starting from an initial guess of the solu- tion. Even though classical methods produce acceptable results, they have limitations in handling complex features of objective functions and constraints such as non-convexity, discontinuity, and multimodality. Since the OPF is commonly used with other complex optimization problems such as optimal placement and sizing for energy storage, renewable energy resources, and FACTS de- vices, the use of classical methods is not the suitable choice. Evolutionary methods such as particle swarm optimization (PSO), genetic algorithms (GA), and differential evolution (DE) provide more flexibility in handling the aforementioned complexi- ties in objective functions and constraints. Moreover, evolutionary algorithms could be expanded for multiobjective optimization problems more efficiently as opposed to classical methods [19]. In the literature the evolutionary algorithms have been used for OPF and have achieved good results as in the flowing references [20] [21] [22]. In evolutionary algorithms, the optimization process starts with multiple points, a population of randomly created solutions, in the search spaces. The population evolve with iterations using evolutionary operators such as crossover, mutation and selection. 21 Although evolutionary algorithms provide acceptable results, their performance is highly de- pendent on the parameters of the evolutionary process such as the number of individuals in the population, the crossover rate, and the mutation rate. From practice and based on the no free lunch theorem [23], no specific optimization algorithm is optimal for all optimization problems, it is the case for the control parameters; in other words, within each algorithm, no control parameters set is the best set for all problems. Controlling the parameters of evolutionary algorithms produced better results with faster convergence rates as shown and discussed in Chapter 2. The feedback-based self-adapting control parameters for differential evolution algorithms, FBjDE- I and FBjDE-II, proposed in Chapter 2 will be used for optimal power flow. The optimal power flow studies are carried over the IEEE 14-bus, IEEE 30-bus, IEEE 57-bus test systems. The results of the FBjDE-I and FBjDE-II are compared with the results of DE and the self-adaptive algorithm jDE. The proposed algorithms resulted in better results and faster convergence speed compared to other algorithms. 3.2 Problem Formulation The objective of the OPF in an electrical grid is to minimize the cost of the generated power while meeting the operational and security constraints. The cost of the generated electrical power from the committed generators is modeled in (3.1) subject to the generator constrains in (3.2) (3.3), voltage constraints(3.4) (3.5), and meeting load demands and power losses (3.6) (3.7). G(cid:88) Minimize subject to: i=1 (aiP 2 i + biPi + ci) gm ≤ Pgm ≤ P max P min gm 22 (3.1) (3.2) gm ≤ Qgm ≤ Qmax Qmin gm m ≤ Vm ≤ V max V min m −π ≤ δm ≤ π Pm(δ, V ) + Pdm Qm(δ, V ) + Pdm − Pgm = 0 − Pgm = 0 (3.3) (3.4) (3.5) (3.6) (3.7) The equations of AC power injection at each bus that is used in the solution of the power is given in (3.8) and (3.9). In this work the Newton-Raphson method is used in solving AC power flow model equations. Pm(δ, V ) = |Vm| Qm(δ, V ) = |Vm| N(cid:88) N(cid:88) n=1 n=1 |Ymn||Vn|cos(δm − δn − θmn) |Ymn||Vn|cos(δm − δn − θmn) (3.8) (3.9) 3.3 Simulation In this chapter the feedback-based self-adaptive differential evolution algorithms, FBjDE-I and FBjDE-II, explained in 2 along with the self-adaptive differential evolution algorithms jDE and the original DE are used to solve the OPF in three IEEE test cases. To compare the performance of the optimization algorithms, the following procedure is followed. The optimal power flow using all algorithms is repeated over a number of experiments Emax. In each experiment, all algorithms run for a specified number of generations Gmax. Each algo- rithm starts with a number of population Np individuals randomly initialized in the search space. In each experiment, all the optimization algorithms are started from the same initial population in 23 order to reduce the effect of randomness in the evolutionary process. To illustrate, the flow chart depicted in Fig. 3.1 explains the procedure for all algorithms tested in one experiment. In this study Emax = 100 experiments and Gmax = 100 generations. For each algorithm, the objective function values of all individuals in Np at the Gmax genera- tion over all the Emax experiments are sorted to find the minimum (best), median, and maximum (worst) for the both the average population and the best individual in the population. For instance, if the EM ax = 100, there will be 100 best function values and 100 average population values for each algorithm. From these recorded results we can fairly compare the minimum (best), median, and maximum(worst) values of the algorithms. This procedure is commonly used in comparing optimization algorithms [19]. The average over the total number of experiments is referred to as the mean. In differential evolution, the control parameters are the number of population N P , the mutation scale factor F , and the crossover rate CR. The number of population could be chosen proportion- ally to the number of decision variables in the search spaces to achieve good results. The crossover rate and the mutation scale factor are self-adaptive controlled parameters as explained in Chapter 2. For comparison, all algorithms shown in Fig. 3.1 are initialized with the same number of population, scale factors, and crossover rates. In this case Np = 10 × N, where N is the number of decision variables; F = 0.9 and CR = 0.1 for all individuals. The control parameters τ1 and τ2 for the jDE algorithm are set as recommended in the original paper τ1 = τ2 = 0.1. The ranges of the scale factor are as follows fl = 0.1 andfu = 0.9. As explained in Section 2.4, the feed back based self-adaptive differential evolution algorithms, FBjDE-I and FBjDE-II, control the mutation scale factor Fi and the crossover rate CRi for each individual in the population based on the feed back information about the performance of the previously used the crossover rate for each individual. FBjDE-II takes a further step in controlling the range of the mutation scale factor for each individual as illustrated in the pseudocode presented in Algorithm. 2 in Section 2.4. 24 Figure 3.1: Flow chart of DE, jDE, FBjDE-I, FBjDE-II algorithms for OPF 25 Initialization Calculate objective function(3.1) and constraints violations(3.2-3.7) using Newton-Raphson for AC power flow (3.8)and(3.9) initial population. DE Mutation Operator Using Equation (2.1)DE Crossover Operator Using Equation (2.2)DE Crossover Operator Using Equation (2.3)Calculate Objective Function (3.1) and Constraints Violations (3.2-3.7) Using Newton-Raphson method for AC power flow (3.8) and (3.9) Final Results Is G < G_MaxYesNoIn FBjDE2, Control F and CR for each individual Using Algorithm II in Chapter 2In FBjDE1,Control F and CR for each individual Using (2.7)(2.8)In jDE, Control F and CR for each individual Using (2.4), (2.5)In DE, Control Parameters F and CR are fixed as in the initialization step. 3.4 Results and Discussion The optimization algorithms, DE, jDE, FBjDE-I, and FBjDE-II, explained in the above section are used to solve the optimal power flow in the following IEEE test case systems: IEEE 14- bus system , IEEE 30-bus system, and IEEE 57-bus system. The cost coefficients ai, bi, and ci of the generated power used in equation (3.1) for all IEEE test systems as well as the generator constraint values in equation (3.2) and (3.3) are taken from the Matpower reference in [24]. The voltage security constraints are within 0.4 of one p.u. for all buses. For the IEEE 14-bus, the comparison between the convergence rates of the algorithms is de- picted in Fig.3.2. The IEEE 14-bus system consists of 14 buses, 20 branches, three transformers, and five generators [25] [24]. The convergence rates shown in Fig.3.2 are averaged over 100 exper- iments of the best cost of the population. It clearly shows that DE is outperformed by the adaptive algorithms jDE, FBjDE-I, and FBjDE-II. The feedback self-adaptive DE algorithms FBjDE-I, and FBjDE-II show better convergence rates compared to other algorithms. The mean and the best OPF values of the population over 100 experiments using all algorithms for IEEE 14-bus system are compared in Tables 3.1 and 3.2, respectively. The best, median and worst final results over the 100 experiments of the best cost as well as the average cost are presented in Tables 3.1 and 3.2, respectively. Table 3.1: Comparison of Mean Cost of the Population over 100 Experiments IEEE 14-bus Algorithm Worst of the Mean Median of the Mean Best of the Mean DE jDE FBjDE-I FBjDE-II 8370.9729 8091.7723 8083.0628 8082.3251 8112.2398 8081.5715 8081.5447 8081.5442 8081.8343 8081.5262 8081.5266 8081.5261 26 Figure 3.2: Mean of the Best Cost of the Population over 100 Experiments IEEE 14-bus Table 3.2: Comparison of the Best Cost of the Population over 100 Experiments IEEE 14-bus Algorithm Worst of the Best Median of the Best Best of the Best DE jDE FBjDE-I FBjDE-II 8370.9729 8089.1285 8082.8258 8081.9719 8112.2396 8081.5357 8081.5288 8081.5296 8081.8342 8081.5251 8081.5250 8081.5251 The IEEE 30-bus system consist of 30 buses, 41 branches, four transformers, and six generators [25] [24]. Fig.3.3 compares the convergence rate of the algorithms by plotting the mean best cost of the population over 100 experiments of each algorithm for the IEEE 30-bus. The comparison between the final results of the mean and the best OPF values of all algorithms for this test case system are listed in Tables 3.3 and 3.4, respectively. 27 05101520253035404550Iteration8050810081508200825083008350Objective Function (Cost & Penalties)DEjDEFjDE-IFjDE-II Figure 3.3: Mean of the Best Cost of the Population over 100 Experiments IEEE 30-bus Table 3.3: Comparison of the Mean Cost of the Population over 100 Experiments IEEE 30-bus Algorithm Worst of the Mean Median of the Mean Best of the Mean DE jDE FBjDE-I FBjDE-II 802.9079 802.2823 802.2150 802.2106 802.4129 802.1990 802.1872 802.1871 802.2641 802.1846 802.1843 802.1841 Table 3.4: Comparison of the Best Cost of the Population over 100 Experiments IEEE 30-bus Algorithm Worst of the Best Median of the Best Best of the Best DE jDE FBjDE-I FBjDE-II 802.3179 802.2132 802.1876 802.1919 802.2121 802.1850 802.1843 802.1842 802.1859 802.184108 802.184109 802.184107 28 05101520253035404550Iteration800810820830840850860870Objective Function (Cost & Penalties)DEjDEFjDE-IFjDE-II The last test case is the IEEE 57-bus. It contains 57 buses, 80 branches, 17 transformers, and seven generators [25] [24]. As for the previous test case systems, the comparison is based on the convergence rates as in Fig.3.4 and on the comparison between the final results of the mean and best costs of the population over 100 experiments as Tables 3.5 and 3.6. FBjDE-I and FBjDE-II demonstrated better converges rates as in Fig.3.4 as well as better results as in Tables 3.5and 3.6, respectively. Figure 3.4: Mean of the Best Cost of the Population over 100 Experiments IEEE 57-bus Table 3.5: Comparison of the Mean Cost of the Population over 100 Experiments IEEE 57-bus Algorithm Worst of the Mean Median of the Mean Best of the Mean DE jDE FBjDE-I 42891.7011 41779.3649 41745.2177 FBjDE-II 41750.1667 41820.3454 41742.2904 41739.9688 41738.7995 41745.9749 41738.0285 41738.0823 41737.8638 29 05101520253035404550Iteration4.164.184.24.224.244.264.284.3Objective Function (Cost & Penalties)104DEjDEFjDEFjDE2 Table 3.6: Comparison of the Best Cost of the Population over 100 Experiments IEEE 57-bus Algorithm Worst of the Best Median of the Best Best of the Best DE jDE FBjDE-I 42891.6723 41758.1262 41740.0185 FBjDE-II 41746.4469 41820.3429 41745.9735 41738.6577 41737.8522 41738.2268 41738.1402 41737.8287 41737.8067 3.5 Conclusion By comparing the performance of the adaptive algorithms, jDE, FBjDE-I, and FBjDE-II, to the DE algorithm, it is evident that controlling the parameters of the differential evolution algorithm has resulted in significant improvement not only on the results but also on the convergence rate of the optimal power flow solutions. Feedback-based self-adapting differential evolution algorithms FBjDE-I ,and FBjDE-II for op- timal power flow have demonstrated better performance compared to jDE in terms of convergence rate and results especially in large systems. The performance of FBjDE-I ,and FBjDE-II are very comparable in terms of the convergence rate. In the largest test case used in this study, IEEE 57-bus, the results of FBjDE-II are slightly better than the results of the FBjDE-I. From Chapter 2, it was noted that the performance FBjDE- I and FBjDE-II are comparable in solving benchmark problems. It was also observed that the results of FBjDE-II are better than the results of FBjDE-I in complex multimodel problems. These observations suggest that FBjDE-II should be used in solving complex and large scale problems. 30 Chapter 4 Multiobjective Optimal TCSC Placement and Sizing for Enhancing Network Loadability 4.1 Introduction This chapter presentes a mutiobjective-based approach to optimally size and allocate FACTS devices to enhance electrical network loadability. The proposed method has been applied to com- monly used FACTs devices, hyristor-controlled series capacitor (TCSC). TCSC devices have been utilized for enhancing transmission network loadability. The technical benefit of the TCSC devices comes at the expense of their high investment cost. In this paper, a new approach is proposed to provide a profound insight into the compromises between technical and economical aspects of installing TCSC devices in a transmission network. The proposed approach is a multiobjective optimization based algorithm used to maximize the loadability of the network and to minimize the investment cost of installing TCSCs under secured The content of this chapter has been reproduced with permission from Fares T. Alharbi, Saleh Almasabi, and Joydeep Mitra, Enhancing Network Loadability Using Optimal TCSC Placement and Sizing, 2018 IEEE/PES Trans- mission and Distribution Conference and Exposition (T&D), Denver, CO,2018. 31 operation conditions. The results provide an efficient set of nondominated solutions to maximize the loadability at minimal cost for the decision-making process. The IEEE 118-bus system is used as a test case to validate the effectiveness of the proposed method. Furthermore, the results are analyzed and compared with available results in the literature. The rest of the chapter is organized as follows: the multiplicative algorithms are briefly summa- rized in Section. 4.2. Section 4.3 introduces FACTS devices, describes their effect on the system, and illustrates the modeling of the TCSCs. The objective functions and constraints are presented in Section 4.5. The simulation of the proposed approach is shown in Section 4.6. Results and discussion are in Section 4.7, followed by conclusions in Section 4.8. 4.2 Multi-Objective Optimization Algorithms Evolutionary algorithms (EA), such as genetic algorithm, particle swarm, and differential evolution have been commonly used in power system computation [1, 26, 27]. These algorithms were originally used for optimizing single objective problems. Single objective algorithms were commonly used to solve multiobjective problems by assigning weights to the objective functions. Depending on the weight of each objective a potential optimal solution could be found. There are, however, certain difficulties with the wighted sum approach. First, the weights used to scale the contradicting objective functions need to be properly studied and chosen beforehand. Choosing multiple weights requires more computations for each set of weights. Second, wighted sum-based algorithms are not able to discover the non-convex parts of the Pareto-optimal solutions [19]. For multiobjective problems, evolutionary algorithms are a more suitable in generating well compromised efficient solutions (a set of non-dominated solutions in the objective spaces); their corresponding values in the decision space are called Pareto optimal solutions [19]. A nondominated sorting genetic algorithm (NSGA-II) has demonstrated an efficiency in solv- ing multiobjective functions, i.e. two objective functions. Thus, in this study, the NSGA-II [28] is 32 used to solve the optimal placement and sizing of TCSCs to enhance the loadability of the system at minimum investment cost. In multiobjective algorithms, the dominance principle is used as a selection operator to favor solutions that can have better compromise between the objective functions, which are called non dominated solutions. According to [19], a solution x dominates a solution y, if the solution x is not worse than y in all objectives, and it has a better value than y in at least one objective. The NSGA-II emphasizes the non-dominated solutions, and uses a diversity preservation mech- anism to explore the entire search space. It also has an elitism preservation algorithm [28]. The proposed procedure using NSGA-II is illustrated in the flow chart in Fig. 4.3. In NSGA-II, after creating a uniform random population N in the decision space and evaluating the objective func- tions, the population are sorted according to the level of domination in objective space. That would classify the population as a number of sets called fronts. To maintain the diversity, solutions in each front are ranked according to their crowding dis- tance, which gives higher rank to solutions that are apart from each other in the objective space. Once the population is sorted in non dominated fronts and ranked according to the crowding dis- tance within each front, a tournament selection operator is used to choose parents for the mating pool. The NSGA-II algorithm handles constraints violation in the the non-dominated sorting pro- cess; it favors feasible over all infeasible solutions. In the infeasible regions it gives higher rank to solutions that have smaller constraints violations. This will force the infeasible solutions to con- verge to a feasible area or to the constraints boundary, where most likely the actual Pareto-optimal solutions are. Following this, the selected parents undergo genetic crossover and mutation operators to pro- duce the offspring population. The resulting population of the current parents and their offspring are sorted again based on dominance and ranked according to crowding distance again. The first nondominated sets are then selected to pass to the next generation. When the size of the first nondominated fronts is larger than the population size N, only solutions that have better, 33 higher, crowding metrics are selected to pass to the next generation [28]. 4.3 Flexible AC Transmission System devices (FACTS) FACTS devices are commonly used to enhance the capabilities of the transmission network. Besides enhancing the system loadability, FACTS devices are utilized to improve both transient and steady state stability, to dampen power oscillation, and to limit short circuit currents [29]. The use of FACTS devices can allow controlling the voltage and angle differences between two connected buses as well as regulating active and reactive power flow. Different FACTS devices are used to control different parameters [29]. The thyristor-controlled series capacitor (TCSC) controls the effective reactance of the transmission line; the thyristor- controlled phase shifting transformer (TCPST) adjusts the angle difference between two connected buses; the thyristor-controlled voltage regulator (TCVR) is used to control the voltage difference. The static var compensator (SVC) and the static synchronous compensator (STATCOM) control the reactive power and regulate the voltage magnitudes at the terminals. The unified power flow controller (UPFC) is used to control the active and reactive power flow and to regulate the voltage magnitudes at the same level. FACTS devices could be classified, according to their type of compensation, as shunt con- trolled, series controlled, and combined (shunt-series controlled) [29]. For instance, SVC and STATCOM are shunt controlled, whereas TCSC, TCVR, and TCPST are series controlled FACTS devices. The unified power flow controller (UPFC) is a combined controlled FACTS device. Se- ries controlled FACTS devices are the most commonly used type [30]. In particular, the TCSC is widely used among them due to its performance and relatively low cost [30]. Series controlled FACTS are utilized to control the effective transmission line impedance Xl, which has a direct effect on the real and reactive power transmitted from a sending bus to a receiving bus in the trans- mission network [29]. Compared to the SVC, which is shunt controlled, TCSC has demonstrated more effectiveness and efficiency in maximizing the system loadability [31]. The TCSC was the 34 second best for enhancing the loadability next to the UPFC, but has a better investment cost than the UPFC [32]. Figure 4.1: TCSC Model When a TCSC device is installed on a transmission line, it could be considered as a capacitor or an inductor series compensation depending on its impedance jXT CSC. In networks where the transmission lines are represented as a π model, installing the TCSC device in the line is modeled as variable reactance as depicted in Fig.4.1. This model affects the corresponding elements in the admittance matrix of the system. 4.4 Related Work In a deregulated and highly competitive market with increasing load demands, transmission lines are vulnerable to being overloaded beyond their thermal limits. Overloading transmission lines will not only increase losses, but also may drive the system into insecure operating condi- tions. Flexible AC transmission system (FACTS) devices have been effectively utilized to enhance transmission networks loadability. The benefits of the FACTS devices are realized if they are optimally installed in the network with proper sizes [29]. The problem of optimal location and sizing of FACTS devices is a highly constrained nonlinear problem due to the nonlinearity of the AC power flow, the model of FACTS devices, and the cost function model of the FACTS devices. Linear and nonlinear programming [33] as well as heuristic methods [34] have been used to overcome the complexity of the problem. Heuristic methods are 35 LineLineZRjXTCSCX2LineB2LineB the most popular procedures used in solving the optimal placement and sizing of FACTS devices due to their effectiveness in handling mixed variables, discontinuities, and nonlinear objectives and constraints [19]. The best locations and respective sizes of multi-type FACTS devices were determined using a genetic algorithm (GA) to maximize the system loadability [35] and to enhance system security [36]. In [32], the authors proposed a multiobjective weighted sum approach based on particle swarm optimization (PSO) to maximize the loadability of the system and to minimize the cost of installing multi-type FACTS devices. Optimal placement of multi-type FACTS devices with a graphic user interface was proposed in [34] for a selected number of FACTS devices, by the user, to maximize the power system loadability. A self-adaptive firefly algorithm for multi-type FACTS placement was presented in [37] to min- imize real power loss and to enhance voltage stability. The study in [30] proposed a multiobjective adaptive differential evolution (ADE) algorithm based on a weighted sum approach to minimize the following: the real and reactive power losses over the transmission lines, voltage deviation on the buses, the installation cost of the TCSCs and the number of TCSCs. In [31], the effect of optimizing the locations of the TCSCs and SVCs was studied to maximize the loadability in both normal and contingency operations. A real genetic algorithm was used to optimize the location and the settings of the TCSC and the SVC. According to [31], TCSCs are more effective in improving the loadability under both normal and contingency conditions. In the aforementioned studies, the number of FACTS devices is predetermined as in [31, 32, 34–36], and the optimization is focused on the locations and the parameter settings of the FACTS devices. The multiobjective nature of the problem was addressed using a weighted sum approach, a compromised technique, for this multiobjective problem in [30, 32, 37]. The allocation of FACTS devices was addressed as a multiobjective algorithm in [38] to opti- mally allocate multi-type FACTS to enhance the system security and minimize the investment cost of FACTS devices. However, the optimization was done for a predetermined number of FACTS devices and applied to a small test system. An epsilon-constrained method based on mathematical 36 programming is introduced in [39] to handle the multiobjective nature of the optimal allocation problem with a fuzzy logic procedure for the decision-making process. In all noted references, the loadability was tested on discrete values of loading margins. Considering related work in the available literature, a profound insight is needed into the trade- off between maximizing the system loading margin and the investment cost of the FACTS devices using intelligent computation methods. To this end, this paper proposes a new multiobjective approach to maximize the power system loadability using TCSC devices and to minimize the investment cost of the TCSC units under secured operating conditions. The nondominated sort- ing genetic algorithm II, NSGA-II [28], is used to generate the nondominated set of solutions in this work. The number, locations, and sizing of TCSC units, as well as the loading margin of the system, are optimized simultaneously in order to determine the efficient set of nondominated solutions. The presented method provides a flexible and efficient procedure for handling the con- strained mixed integer decision variables. In this work, the loading margin is treated as both an objective and a decision variable to explore the search space more effectively. 4.5 Problem Formulation The mathematical formulation of the objective functions are as follows. First, enhancing the loadability (λ) of the transmission network is tested by uniformly increasing the active and reactive power demands on all buses as given in (4.1) and (4.2). Pd,i = λ × Pd,i,0 Qd,i = λ × Qd,i,0 (4.1) (4.2) In the above equations, Pd,i,0 and Qd,i,0 are the real and reactive power demands at bus i under normal conditions, respectively. Pg,i = λ × Pg,i,0 37 (4.3) The power produced by each generator is also scaled by λ to distribute the extra load demands and the losses on the generators in proportion to their sizes (4.3); where Pg,i is the power generated at the PV bus i. Even though for small λ load demands could be supplied by the generator at the slack bus, the objective is to inject power from all generators to allow testing all transmission network branches. max λ (4.4) In equation (4.4), λ is the system loadability factor with a base case of λ = 1. Second, minimizing the total investment cost CT of NT CSC TCSC units is minimized as given in equation (4.5) [32, 33]. NT CSC(cid:88) min CT = ST CSC,j × CT CSC,j × 1000 $ j=1 CT CSC,j = 0.0015S2 T CSC − 0.713ST CSC + 153.75 (4.5) (4.6) The investment cost of installing NT CSC units, given by (4.5), depends on the reactive power capacity of each TCSC unit (4.7) and on their total number. ST CSC,j = Im(I2 j × XT CSC,j) (4.7) In equation (4.7), ST CSC,j is the capacity of the TCSC unit in MVAR on transmission line j. The compensation level of the TCSC device is constrained by the reactance of the transmission line as in (4.8) [32, 35]. −0.8 × Xline ≤ XT CSC ≤ 0.2 × Xline (4.8) To maintain secure operation while maximizing the system loading margin, the voltage mag- nitudes are limited within 5% of one per unit of the nominal values (4.9). The power flow over 38 a transmission line is restricted to its rated capacity as in (4.10). The rest of the power flow con- straints are presented in (4.11-4.14); Vi and δi are the voltage magnitude and angle, respectively, at bus i. Pi and Qi are active and reactive power injections at bus i, which are calculated using (4.15) and (4.16), respectively. V min i i ≤ Vi ≤ V max Sl ≤ Smax −π ≤ δi ≤ π l i,g i,g ≤ Qi,g ≤ Qmax Qmin Pi(δ, V ) + Pi,d − Pi,g = 0 Qi(δ, V ) + Qi,d − Qi,g = 0 N(cid:88) N(cid:88) j=1 Pi(δ, V ) = |Vi| Qi(δ, V ) = |Vi| |Yij||Vj| cos(δi − δj − θij) |Yij||Vj| sin(δi − δj − θij) (4.9) (4.10) (4.11) (4.12) (4.13) (4.14) (4.15) (4.16) j=1 4.6 Simulation In this work the simulation is applied on the IEEE 118-bus test system, the data can be found in [40, 41]. The IEEE 118-bus test system has a total number of 596 constraints in the optimiza- tion problem. Some constraints must be handled in the power flow algorithm to make sure that the power flow converges for the suggested solutions; other constraints are handled using the opti- mization algorithm. The total number of equality and inequality constraints of the IEEE 118-bus system are shown in Table. 4.1. 39 Table 4.1: IEEE 118-bus Constraints Constraint Type Optimization Power Flow Inequality constraints Equality constraints Total number of constraints 304 0 304 54 238 292 In the optimization problem, the decision variables are the loading factor λ, the number of TCSC units NT CSC, the optimal location ∈ [1, NL], and respective size XT CSC of each TCSC unit. The locations and the number of the TCSCs units are considered as integer numbers, while the loading margin and the size for each device is represented in real continuous variables. For the continuous variables, λ and XT CSC, the simulated binary crossover operator (SBX) [42] is used to perform the crossover; the crossover probability is Pc = 0.9 and the distribution index of the (SBX) operator is ηc = 20. The polynomial mutation operator is used to perform the mutation over the continuous variables [43] with a mutation probability of Pm = 0.3 and a polynomial order of ηm = 20. The integer variables (i.e the number of TCSC units and their locations) are decoded in the binary space to permit the use of genetic operators, such as crossover and mutation. In the binary decoded space of integer variables, a double site crossover technique is applied with the probability of Pc = 0.9, and a bit wise mutation of a probability of Pm = 0.3. Since the genetic operators have some mutation probabilities, unrealistic offspring solutions could be generated. For instance there can be a candidate location Loci for the TCSC with zero size XT CSC,i = 0, or a size for the a TCSC unit XT CSC,i where there is no location (Loci = 0). To avoid such cases, the recombination of population is modified as follows: 1) The number of TCSC variables NT CSC controls the total number of TCSC candidate locations. 2) There can not be a candidate location Loci, where the size of the TCSC parameter XT CSC,i is zero, and vice versa. 40 3) Only one TCSC device can be installed at a line; no repeated locations in the same individual are allowed. Figure 4.2: Constructing an Individual in the Population The optimization starts with a population of candidate solutions, individuals. Each individual has a loading factor λ, number of TCSC units NT CSC, locations, and respective sizes for all TCSC units in the individual as shown in Fig. 4.2. A full AC power flow using NewtonRaphson algorithm is used to evaluate the power flow for each candidate solution; then the loadability of the system and the investment cost of the TCSCs are evaluated as in (4.4) and (4.6). λmin i = λmax i = 1, i = 1 λmin λmax i−1 + τ, i > 1 i−1 + τ, i > 1 λmin i=1 + τ, i = 1 (4.17) (4.18) Optimizing over a continuous range of λ has resulted in less diverse solutions due to the nature of the mixture and the large diversity of the decision variables. To overcome this problem and better explore the search space, the optimization is controlled over the loadability domain λ . A control variable τ ∈ [0, 1] is introduced to restrict the optimization over the loadability domain. The optimization process is divided into Ni segments, where i is the segment’s index. More computations are needed for the smaller control variable τ. In this work the control variable is 41 Location 1Location 2Location 3Location NTCSC,1XTCSC,2XTCSC,XLTCSCN0.2X0.8TCSCLN0TCSCLNNminmaxiii,3XTCSC Figure 4.3: Flow chart of the proposed approach 42 NoYesIs G