ECONOMIC OPERATION OF POWER SYSTEMS USING HYBRID OPTIMIZATION TECHNIQUES

Transcription

1 SYNOPSIS OF ECONOMIC OPERATION OF POWER SYSTEMS USING HYBRID OPTIMIZATION TECHNIQUES A THESIS to be submitted by S. SIVASUBRAMANI for the award of the degree of DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY MADRAS. SEPTEMBER 2011

2 1 Introduction Power system is a very large, complex and interconnected network having generation, transmission, distribution and loads. Since loads are far away from generating stations, the electrical energy generated has to be carried over by long transmission lines. The main objective of power system is to provide a reliable power supply with cheapest cost. In order to achieve this, the power system must be monitored, analyzed and controlled at every moment. The power engineers need special tool to optimally analyze, monitor and control the power system. These special tools are generally modeled as some sort of optimization problems. Few of them are economic dispatch and optimal power flow. Economic dispatch (ED) is an important optimization problem which aims at scheduling the committed thermal generating units to meet the load demand for minimum operating cost while satisfying the equality and inequality constraints [1]. The operating cost of thermal unit includes fuel cost, labor cost and maintenance cost. The cost curve of thermal unit is generally modeled as a smooth curve. However, the input-output characteristic of large steam turbine generators, which normally have many steam valve openings, are not always smooth. Optimal power flow (OPF) problem is a special tool to obtain the optimal state of the control variables by minimizing the certain objectives while satisfying the equality and inequality constraints [2]. The most commonly used objectives are fuel cost minimization, loss minimization, sum of load bus voltage deviations and voltage stability index. The control or independent variables of power system are real power generation except at slack bus, voltage magnitudes at generator buses, transformer tap settings and reactive power injection due to capacitor/inductor banks. The dependent or state variables are load bus voltage magnitudes and angles, slack bus power generation, reactive power generation at generator buses and transmission line loadings. 2 Motivation Power system optimization problems have complex and non-linear characteristics with several equality and inequality constraints. There are two ways namely traditional and evolutionary methods by which the problems are solved. The traditional optimization methods are linear programming, quadratic programming, Newton s method and interior point method. Similarly, evolutionary optimization methods are classified as genetic algorithm (GA), evolutionary pro- 2 / 16

3 gramming (EP), particle swarm optimization (PSO), differential evolution (DE) and harmony search algorithm (HSA). Conventional optimization methods, which use gradients for the search of optimum, need the function as at least twice differentiable. They also require function to be convex and variables as continuous. However, the power system optimization problems do not have these characteristics [3]. Evolutionary optimization methods have become an alternative to conventional optimization techniques for solving real world problems having non-convexity, nondifferentiability and discontinuity. In this way, evolutionary methods have been successfully applied to power system problems as well. Though, the evolutionary methods find suitable for power system problems, the premature convergence and stagnation make not to use them directly [4]. Therefore, there is a need for new and improved optimization techniques for the solution of power system optimization problems. These factors have motivated to search for an improved optimization algorithm which overcomes the short comings of both conventional and evolutionary optimization methods. In order to achieve this, hybrid algorithm by combining evolutionary and conventional optimization methods is found to be more suitable. 3 Objective and Scope 3.1 Objective of the work The main objectives of this work are 1. Study and application of hybrid algorithm for power system problems like economic dispatch, optimal power flow and hydro thermal scheduling. 2. Study and application of multi-objective evolutionary algorithms for problems such as environmental/economic dispatch and multi-objective optimal power flow 3.2 Scope of the work The scope of this work is limited to form a hybrid algorithm by combining evolutionary and conventional optimization techniques for power system problems. The hybrid method has been validated by using the standard test systems. The proposed hybrid method can, in principle, 3 / 16

4 be used for solving the problems having non-convexity and non-smoothness in the function space. The multi-objective evolutionary algorithm has also been used for handling multiple objectives simultaneously. This multi-objective algorithm has been tested on different power system problems having multiple and conflicting objectives. 4 Summary of Research Work 4.1 Power system optimization The important power system planning and operation problems have been formulated as mathematical optimization problems. These problems are solved for minimum operating cost within a set of equality and inequality constraints. Generally, power system problems are mathematically modeled as min f(x) subject to g(x) = 0 (1) h(x) 0 where x is set of decision variable vector, f(x) is the objective function, g(x) is the equality constraints and h(x) is the inequality constraints. The two important power system problems such as economic dispatch and optimal power flow, have been formulated and solved here. Evolutionary methods and their combination with classical methods have been used to solve those problems. 4.2 Economic dispatch with valve-point effects The generating units with multi-valve steam turbine exhibit a greater variation in the fuel cost function. Since the valve point results in the ripples, a cost function contains higher order nonlinearity. Fig. 1 shows the input-output curve of a thermal generator with valve-point effects. The cost function of the generating units with valve point loadings is represented as follows [5]; F i (P i ) = a i P 2 i +b i P i +c i + e i sin(f i (P i,min P i )) (2) 4 / 16

5 with valve point without valve point e Fuel cost ($/hr) c d a b Power output (MW) Fig. 1 Input-output curve with valve-point effects. a,b,c,d,e - valve points wheree i andf i are the cost coefficients of generatorireflecting valve-point effects and subject to the following equality and inequality constraints. 1. Real power balance constraint: N G i=1 (P i ) P D = 0 (3) where P D is the total system demand. 2. Real power generation limit: P i,min P i P i,max i = 1,2,,N G (4) where P i,min andp i,max are minimum and maximum power output ofith generator. 4.3 Optimal power flow The objective of OPF is to determine the optimal settings of the variables that minimize a certain objective while satisfying several equality and inequality constraints. The problem can be mathematically modeled as f(x,u) (5) Subject to g(x,u) = 0 h(x,u) 0 (6) where g(x, u) is the typical load flow equation, h(x, u) is the system operating constraints, x is the vector of state variables consisting of slack bus power P G1, load bus voltages V L, reactive 5 / 16

6 power generator outputs Q G and transmission line loading S l Hencexcan be expressed as: x T = [P G1,V L1,,V LNPQ,Q G1,,Q GNG,S l1,,s lnl ] (7) wheren PQ,N G andnl are the number of load buses, the number of generators and the number of transmission lines, respectively. u is the vector of control variables consisting of generator real power outputs except at the slack bus P G, generator voltages V G, transformer tap settings T and reactive power injections Q c. Hence,ucan be expressed as: u T = [P G2,,P GNG,V G1,,V GNG,T 1,,T nt,q c1,,q cnc ] (8) where nt is the number of regulating transformers and nc is the number of VAR compensator Objective functions Minimization of fuel cost This objective is to minimize the total fuel costf T of the system. The fuel cost function of the thermal generators is modeled as a quadratic function and represented as: N G F T = (a i Pi 2 +b i P i +c i ) $/hr (9) i=1 where a i,b i,c i are the fuel cost coefficients of the ith generator, P i is real power output of the ith generator. Minimization of real power loss This objective is to minimize the real power transmission line losses P L in the system which can be expressed as follows. nl P L = g k [Vi 2 +Vj 2 2V i V j cos(δ i δ j )] (10) k=1 6 / 16

7 where g k is the conductance of a transmission line k connected between i and jth bus, V i, V j, δ i andδ j are the voltage magnitudes and phase angles of i andjth bus respectively Constraints Equality constraints These constraints are typical load flow equations which can be described as follows N B P Gi P Di V i V j (G ij cos δ ij +B ij sin δ ij ) = 0 i N PQ j=1 (11) N B Q Gi Q Di V i V j (G ij sin δ ij B ij cos δ ij ) = 0 i N G j=1 Inequality constraints There are two types of inequality constraints. 1. Functional inequality constraints u min u u max (12) where u is vector of control variables defined in (8). 2. Parameter inequality constraints x min x x max (13) where x is vector of state variables defined in (7). 4.4 Evolutionary Algorithm (EA) Evolutionary algorithms play an important part in problems having discontinuity, non-smoothness and mixed integer characteristics. Most of the evolutionary algorithms mimic a certain natural phenomenon in its search for an optimal solution. They are genetic algorithm (GA), evolutionary programming (EP), particle swarm optimization (PSO), differential evolution (DE), and harmony search (HS) algorithm. In this work, two recent methods such as DE and HS were considered. 7 / 16

8 4.4.1 Differential evolution (DE) DE was first proposed by Price and Storn [6]. It is also a population based algorithm like other evolutionary algorithms. It is a simple and efficient algorithm. The main difference between DE and GA is in mutation process. It also starts with initial set of population and evolves using mutation, crossover and selection. In DE, the mutation is carried out using arithmetical operations on randomly selected vectors. The evolution process of classical DE are initialization, mutation cross over and selection Harmony search algorithm The harmony search (HS) algorithm, proposed by Geem [7], is a nature inspired algorithm, mimicking the improvisation of music players. The harmony in music is analogous to the optimization solution vector, and the musician s improvisations are analogous to the local and global search schemes in optimization techniques. The HS algorithm uses a stochastic random search, instead of a gradient search. This algorithm uses harmony memory considering rate and pitch adjustment rate for finding the solution vector in the search space. The optimization procedure of the HS algorithm are initialization of harmony memory and parameters, improvisation of new harmony memory and updating the harmony memory. 4.5 Sequential Quadratic Programming (SQP) The SQP method seems to be the best non-linear programming method for constrained optimization problems. It outperforms every other non-linear programming method in terms of efficiency, accuracy and percentage of successful solutions over a large number of test problems [8]. The method closely resembles Newton s method for constrained optimization, just as is done for unconstrained optimization. At each iteration, an approximation is made of the Hessian of the Lagrangian function using Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi Newton updating method. The result of the approximation is then used to generate a quadratic programming (QP) sub-problem whose solution is used to form a search direction for a line search procedure. The SQP used in this work consists of three main stages, as follows: 1) calculation of an approximation of the Hessian matrix of the Lagrangian function using quasi-newton method; 2) 8 / 16

9 formulation of the QP problem; 3) line search and merit function calculation. SQP simulations are done using the MatLab optimization tool box. 4.6 Hybrid Method In this work, SQP is combined with EA to form a hybrid algorithm for solving ED and OPF problem. There are two ways the algorithms are combined. In first approach, EA is used as a global optimizer and SQP as a local optimizer to fine tune the optimum solution obtained from EA. This approach has been used to solve ED problem with valve-point effects. For OPF problem, SQP is first used to solve by relaxing the discrete variables. The solution obtained from SQP is rounded off to nearest integer and given to the initial population of EA algorithm as an individual. Since this individual is nearer to optimal solution even after it is rounded off, the evolution process of EA will be enhanced. Hence the stagnation and premature convergence of EA algorithm can be overcome. This hybrid algorithm is found to be more suitable for the problem having more non-convexity. 4.7 Test Systems ED with valve-point effects This system consists of 40 thermal units with valve-point loading effects. The entire system data and cost coefficients are available in [9]. The system load demand is MW. Table I gives the best, mean and worst values of the proposed method after 30 simulation runs. Table I Convergence result of 40 unit system for 30 runs Method Cost ($/hr) Best Mean Worst Standard deviation HS HS-SQP Table II shows the comparison of the proposed method with other methods reported in the literature. From the results and comparison, the proposed method is able to produce better quality solution than the other methods. 9 / 16

10 Table II Comparison With Other Methods Reported in the Literature Method Cost ($/hr) DE [9] DE-SQP [9] Proposed (HS-SQP) Optimal power flow IEEE 30 bus system This system consists of 6 generating units and 41 transmission lines. The system has 4 tap changing transformers between buses 6-9, 6-10, 4-12, and and two capacitor banks at buses 5 and 24. Fig. 2. shows the one line diagram of IEEE 30 bus system. The system data and cost coefficients with valve-point effects are taken from [10]. It has 17 control variables with both continuous and discrete. In this system, fuel cost is taken as an objective for OPF problem. Two different cases have been studied. They are as follows Fig. 2 One line diagram of IEEE 30 bus system 10 / 16

11 Case 1: Quadratic cost function In this case, the fuel cost function of the thermal generator is modeled as a quadratic function. This case has been solved by DE and the proposed hybrid method. Table III gives the statistical results of best, mean and worst values for robustness after 30 simulation runs. The simulation results have also been compared with the existing methods available in the literature and given in Table V. It is clear from the results and comparison that the proposed hybrid method is giving comparable results than the existing methods. Table III Statistical results for case 1 (30 runs) Method Fuel cost ($/hr) Best Mean Worst Standard deviation DE Proposed Case 2 : Quadratic cost function with valve-point effects Table IV Statistical results for case 2 (30 runs) Method Fuel cost ($/hr) Best Mean Worst Standard deviation DE Proposed Here, the fuel cost function of the generator is modeled as a quadratic with rectified sinusoidal terms to account the valve-point loading effects. Due to the addition of sinusoidal terms, the function becomes non-convex and challenges the gradient based optimization algorithms. The proposed hybrid method has been used to solve the OPF problem for this case. The best, Table V Comparison of results for IEEE 30 bus Method Fuel cost ($/hr) Case 1 Case 2 HPSO [10] DE Proposed mean and worst values for 30 different runs are given in Table IV. Table V shows the comparison of the proposed method with the methods reported in the literature. It is found that the proposed method is giving better solution than the other methods. 11 / 16

12 IEEE 118 bus system This system has 77 control variables including 54 generators, 9 tap changing transformers and 14 capacitor banks. The complete system data is available in [11]. In this test system, transmission line loss has been taken as an objective for the OPF problem. The statistical results of best, mean and worst after 30 independent runs are given in Table VI. From Table Table VI Statistical results for IEEE 118 bus Method Loss (MW) Best Mean Worst Standard deviation DE Proposed VI, the best value and standard deviation of the proposed method are very low compared to DE algorithm. The comparison of the best solution with the other methods reported in the literature is given in Table VII. Simulation results clearly reveal the superiority of the proposed hybrid method over other methods. Table VII Comparison of results for IEEE 118 bus Method Loss (MW) PSO [12] DE [13] MAPSO [14] Proposed Conclusions This work proposed a hybrid method combining SQP and EA for ED and OPF problem. The classical methods, like SQP, are best in finding the global optimum when the function is convex and smooth. However, they usually get struck with local optimum when the function is non-convex and non-smooth. However evolutionary algorithms, like DE and HS, can find the optimum solution irrespective of the function space. The major drawbacks of EA are stagnation and premature convergence. In order to overcome the drawbacks of either methods, a hybrid method combining classical and evolutionary methods has been proposed. The proposed method has been used to solve 12 / 16

13 the two important power system problems such as ED and OPF. The results were compared among themselves and with other methods. From the simulation results and comparison, the proposed hybrid method is able to produce the high quality solutions for both ED and OPF. It is also found that this hybrid method is more suitable for optimization problem with non-convex function. References [1] B. Wollenberg and A. Wood, Power Generation, Operation and Control. John Wiley & Sons, [2] H. W. Dommel and W. F. Tinney, Optimal power flow solutions, IEEE Trans. Power App. Syst., vol. PAS-87, no. 10, pp , Oct [3] M. R. AlRashidi and M. E. El-Hawary, Applications of computational intelligence techniques for solving the revived optimal power flow problem, Electr. Power Syst. Res., vol. 79, no. 4, pp , Apr [4] P. Attaviriyanupap, H. Kita, E. Tanaka, and J. Hasegawa, A hybrid EP and SQP for dynamic economic dispatch with nonsmoothfuel cost function, IEEE Trans. Power Syst., vol. 17, no. 2, pp , May [5] D. C. Walters and G. B. Sheble, Genetic algorithm solution of economic dispatch with valve point loading, IEEE Trans. Power Syst., vol. 8, no. 3, pp , Aug [6] R. Storn and K. Price, Differential evolution a simple and efficient heuristic for global optimization over continuous spaces, J. Global Optim., vol. 11, no. 4, pp , Dec [7] Z. W. Geem, J. H. Kim et al., A new heuristic optimization algorithm: harmony search, Simulation, vol. 76, no. 2, pp , Feb [8] P. Boggs and J. Tolle, Sequential quadratic programming, Acta numerica, vol. 4, no. -1, pp. 1 51, [9] L. S. Coelho and V. C. Mariani, Combining of chaotic differential evolution and quadratic programming for economic dispatch optimization with valve-point effect, IEEE Trans. Power Syst., vol. 21, no. 2, pp , May [10] M. R. AlRashidi and M. E. El-Hawary, Hybrid particle swarm optimization approach for solving the discrete opf problem considering the valve loading effects, IEEE Trans. Power Syst., vol. 22, no. 4, pp , [11] Power system test case archive. [Online]. Available: [12] J. G. Vlachogiannis and K. Y. Lee, A comparative study on particle swarm optimization for optimal steady-state performance of power systems, IEEE Trans. Power Syst., vol. 21, no. 4, pp , / 16

14 [13] M. Varadarajan and K. S. Swarup, Differential evolutionary algorithm for optimal reactive power dispatch, Int. J. Electr. Power Energy Syst., vol. 30, no. 8, pp , Oct [14] B. Zhao, C. X. Guo, and Y. J. Cao, A multiagent-based particle swarm optimization approach for optimal reactive power dispatch, IEEE Trans. Power Syst., vol. 20, no. 2, pp , May Proposed Contents of the Thesis The outline of the thesis is as follows: Chapter 1 Introduction 1.1 Power System Optimization 1.2 Motivation 1.3 Objectives and Scope of the Work 1.4 Organization of the Thesis Chapter 2 Power System Optimization Methods 2.1 Conventional Methods 2.2 Evolutionary Methods 2.3 Survey of Related Literature Chapter 3 Economic Dispatch With Valve-point Effects 3.1 Introduction 3.2 Problem Formulation 3.3 Hybrid Algorithm for economic dispatch 3.4 Simulation Results Chapter 4 Environmental/Economic Dispatch 4.1 Introduction 4.2 Multi-objective Optimization 4.3 Evolutionary algorithm for environmental/economic dispatch 4.4 Case study Chapter 5 Optimal Power Flow 5.1 Introduction 5.2 Formulation of OPF 5.3 Combining evolutionary and classical methods 5.4 Test system and simulation Results Chapter 6 Hydro Thermal Scheduling 6.1 Introduction 6.2 Problem Formulation 6.3 Implementation of Hybrid Method 6.4 Simulation Results Chapter 7 Conclusions 14 / 16

15 7 Publications 7.1 Papers in Refereed Journals 1. S. Sivasubramani and K. S. Swarup, SQP based differential evolution algorithm for optimal power flow problem, IET Generation, Transmission and Distribution, Accepted for Publication (July 2011). 2. S. Sivasubramani and K. S. Swarup, Environmental/economic dispatch using multiobjective harmony search algorithm, Electric Power Systems Research, vol. 81, no. 9, pp , Sep S. Sivasubramani and K. S. Swarup, Multi-objective harmony search algorithm for optimal power flow, International Journal of Electric Power and Energy Systems, vol. 33, no. 3, pp , Mar S. Sivasubramani and K. S. Swarup, Hybrid DE-SQP algorithm for non-convex short term hydrothermal scheduling problem, Energy Conversion and Management, vol. 52, no. 1, pp , Jan S. Sivasubramani and K. S. Swarup, Hybrid SOA-SQP algorithm for dynamic economic dispatch with valve-point effects, Energy, vol. 35, no. 12, pp , Dec Presentations in Conferences 1. S. Sivasubramani and K. S. Swarup, Multiagent based particle swarm optimization approach to economic dispatch with security constraints, in International Conference on Power Systems, 2009, ICPS 09., pp. 1 6, S. Sivasubramani and K. S. Swarup, Multiobjective harmony search algorithm for optimal power flow problem, in National Conference on Power Systems, 2010, NPSC 10., / 16