On Optimal Power Flow

Transcription

1 On Optimal Power Flow K. C. Sravanthi #1, Dr. M. S. Krishnarayalu #2 # Department of Electrical and Electronics Engineering V R Siddhartha Engineering College, Vijayawada, AP, India Abstract-Optimal Power Flow (OPF) of a power system resulting in optimum generation scheduling is very much needed as it minimizes the Total Fuel Cost (TFC). In classical Economic Dispatch (ED) all the units are committed. OPF is obtained using classical optimization methods. In Unit Commitment (UC) method only the required number of units is committed based on load demand and different optimization techniques like dynamic programming, Lagrange relaxation methods to obtain OPF. A new method, Unit Commitment with Economic Dispatch (UCED), considers the necessary units like in UC and the required OPF is obtained using the classical ED methods resulting in minimum TFC. In this paper, a 15-unit test system with transmission loss is considered. ED and UC are solved by Particle Swarm Optimization (PSO) algorithm including the transmission power loss. UCED method is also applied to the same system along with transmission losses. Results depict that UCED is more efficient. Keywords Optimal Power Flow, Total Fuel Cost, Transmission Power Loss, Economic Dispatch, Unit Commitment, Unit Commitment with Economic Dispatch, Particle Swarm Optimization, Fifteen-unit Test System. I. INTRODUCTION In power systems economy is the main aspect for sustaining the competition of deregulation. Economic Dispatch of electric power is one of the solutions for reducing the operating costs of thermal generating units. Finding the optimum generation schedule that minimizes the total input fuel cost of the steam units is known as OPF. This results in reduced operating cost. Here the OPF should meet the load demand and transmission losses [1]. The two basic methods, Economic Dispatch (ED) commits all the units whereas Unit Commitment (UC) commits the units as per the load demand. Hence UC can supply the load demand with minimum fuel cost with less spinning reserve [1, 2]. The input Fuel Cost of a generator is given by non-linear and complex function. To find the optimal solutions of all the generating units different optimization techniques are already used. For solving the ED problem some iterative techniques like Lagrangian multipliers method, Lambdaiteration method, and Gradient methods are basically used. In UC, there is no necessity of involving all the units. Minimum and efficient units are sufficient based on load demand and transmission losses. The optimization function in UC is more complex than ED. The basic methods like Dynamic Programming method and Lagrange Relaxation method are generally used [1-3]. The new approach to solve the complex nonlinear function TFC is Unit Commitment with Economic Dispatch (UCED), a combination of UC and ED methods. The units are committed such that the switchovers of the units during scheduling are less and the most capable units are committed first. UCED is applied to ten unit power system without considering transmission losses [3]. In this paper UCED is being applied to a fifteen-unit system considering transmission losses. Here Lambda Iteration Method is used to get OPF. Particle Swarm Optimization (PSO) is a popular heuristic optimization method that gives local optimum very quickly [4-10]. In this paper, a 15-unit test system with transmission losses is taken as the case study system. ED and UC methods are also applied to the system and solved using PSO algorithm. II. TRANSMISSION LOSSES In power systems, different generating units are placed at different places. In an interconnected system all the plants are connected by transmission lines. The power consumed by transmission lines is treated as transmission power loss. For an N unit system the transmission loss is represented by [7] where B mn are the loss coefficients. III. ED PROBLEM FORMULATION ED is a constrained optimization problem that involves an objective function and the power constraints. The power constraint balances load demand and transmission losses by optimal power dispatch. The combination of objective function and the power constraint, an augmented cost function is framed by using a Lagrange multiplier [7]. A. Objective Function The objective function in ED is minimizing the Total Fuel Cost (TFC), is sum of fuel costs of all the generating units. In ED, even at lower loads each unit must be in on state at minimum power bound of the generating unit. Considering an N generating unit system the TFC is framed as: $/h (2) ISSN: Page 447

2 where F i, input fuel cost of i th unit, represented in terms of cost coefficients as Objective function to be minimized:, $/h (3) $/h (4) B. Generation Constraints The total power generation of the units/plants must be equal to the total load demand and transmission losses. Hence the constraint is Equality Constraint: where P i is Power output of i th unit, P loss is the total transmission loss and P D is the total load demand of the power system. The constraints of the generating units are given as: IV. UC PROBLEM FORMULATION UC is also a constrained optimization problem which is a combination of objective function and the power constraints involving load demand and transmission losses. Unlike ED, all the units are not committed in UC. Based on the load demand and transmission losses, minimum number of units are scheduled so that TFC is minimum. In UC, the cost of committed (scheduled) units is only considered. The committed units are indicated by a binary matrix [8]. A. Objective Function The objective function in UC is minimizing TFC, is sum of fuel costs of the scheduled generating units. For an N generating unit system the TFC is framed as: B. Generation Constraints The total power generation of the units must be equal to the total load demand and transmission losses. Hence the constraint is The bounds of the generating units are given in MW if they are committed: V. UCED METHOD A new method, Unit Commitment with Economic Dispatch (UCED) is a combination of UC and ED methods. Similar to UC, minimum number of units are committed maintaining load demand and transmission loss. Minimum number of units is selected such that highest capable units are committed first. Units are also selected in a way resulting in minimum switchovers during generation scheduling. Then for the committed units, Lambdaiteration method is applied to solve OPF for minimum TFC. This is one of the methods that give unique solution [2, 3]. VI. PARTICLE SWARM OPTIMIZATION Particle Swarm Optimization (PSO) is one of the population based heuristic and stochastic optimization methods that works based on the behavior of bird flocking and fish schooling. Dr. Kennedy and Dr. Eberhart developed PSO algorithm in PSO involves the terms Particles, Population size, generations/iterations number, Positions and Velocities, Fitness function. Total particle/swarm size is given by Population size. Among the terms fitness function mainly deals with the objective function and the constraints. Positions and velocities of particles are used to obtain the optimal value. The main advantage of PSO is search space is multidimensional. Each particle has its own experience. Particles can also share their experience among neighbors. Based on personal experience and neighbors experience the velocities of particles can change, thereby their positions can change in each generation/iteration which can lead to optimal solution. For n swarm/particles, positions and velocities of the particles are given by vectors and respectively. The positional vector with n population size is given by The i th particle is represented by as where is number of dimensions. In the initial generations the personal best is and for the next generations is the decided from the fitness function of last and present generation. Generally is represented as. The ISSN: Page 448

3 particle which is having the highest fitness value is the global best,. The velocity of the particle required to reach a new position of the particle in PSO is (10) The velocity equation mainly includes two components, Cognitive and Social components. The cognitive component includes an acceleration constant, a random number within (0,1) and personal experience of particles. Similarly, the social component includes an acceleration constant, another random number within (0,1) and neighboring experience of particles. The acceleration constants for greater convergence are selected as. Let is number of iterations/generations and step time. Then the change in position is given by Start Initialize Particles, Population Size, c 1, c 2, Velocities, Iterations, t=1 Read Generator Data Initialize Calculate find and randomly & Modify Velocities by (11) Then the Inertia Weight needs to be changed for each iteration as it provides a balance between global and local best solutions. In general, the inertia weight is given by (12) is linearly decreased from 0.9 (w max ) to 0.4 (w min ), where is maximum number of iterations and is current iteration. Let be number of dimensions and be the number of individuals/ particles. For ED, d equals number of units N. Then the position vector in ED is given by are randomly selected values. Maintain velocities within limits by Modify positions (power generations) by t=t+1 t = maximum iterations? A. Objective Function The fitness function f, to satisfy both objective function and constraint simultaneously is given by (13) Yes Obtain Global best among t iterations Stop In ED all elements of are unity. In UC elements of are unity or zero depending on commitment. No Fig.1. Flowchart for PSO applied to ED and UC methods ISSN: Page 449

4 TFC, $/h Comparison of all Methods-15_unit test system-with transmission losses Hours, h PSO_ED PSO_UC UCED Fig. 2. Comparison of all methods-15-unit test system-with transmission losses (15) Here, to get minimum TFC while satisfying constraint the value of must be maximum. The value of is about 1 and the value is lowest. The flowchart for ED and UC methods using PSO is given in Fig. 1. The terminology used in General PSO and in ED, UC methods is given in Table 1. Table 1. Nomenclature Terminology General PSO PSO applied to OPF methods Birds/particles/swarm -Birds -search points Position of each bird Positions of all birds -Dimensions in multidimensional search space -Initial search points each in dimensions -Number of power generating units Initial Power generations for each search point in N units Rows(size( )) Population Total birdsrows(size( )) Generations Iterations Iterations VII. CASE STUDY AND RESULTS A case study of 15-unit test system is considered where Table 2 and Table 3 show Generator data and 24-hour load data respectively, along with the B mn coefficient matrix as given in Table 4 [6]. For optimal power flow the power generations for each interval of time solved by ED, UC and UCED methods are shown in Table 5, 6 and 7 respectively and together in Fig. 2. Table 8 and Fig. 2 show that UCED is the most economical method than ED and UC. Table 8. Comparison of all Methods-15- Unit test system, with transmission losses Method TFC, $/h PSO_ED PSO_UC UCED VIII. CONCLUSIONS OPF of electrical power including the transmission loss can be solved by ED, UC and UCED methods. ED and UC methods are solved by PSO algorithm. In UCED method, the most capable units are committed first so that minimum number of units can maintain load demand and transmission loss. OPF in UCED is solved by Lambda-iteration method. The other advantage of UCED method is scheduled switchovers of units is less as given in Table 7 and Fig. 2. Fig. 2 depicts that curve of UCED (green line) is smoother than the curve of PSO_UC (blue line) with lesser switchovers. Most importantly, Table 8 and Fig.2 show that UCED is the most economical. This paper along with [3] shows that UCED is most economical with or without transmission losses. ACKNOWLEDGMENT The authors greatly acknowledge Siddhartha Academy of General and Technical Education, Vijayawada for providing the facilities to carry out this research. REFERENCES [1] Allen J. Wood and Bruce F. Wollenberg, Power Generation Operation and Control, second edition, Wiley India, New Delhi, [2] M. S. Krishnarayalu, Unit Commitment with Economic Dispatch, International Electrical Engineering Journal, Vol. 6 (2015) No.5, pp [3] K. C. Sravanthi, M. S. Krishnarayalu, On Economic Dispatch of Electrical Power, International Journal of Computer Applications (IJCA), Vol. 147, No. 6, August 2016, pp ISSN: Page 450

5 [4] Mohammad Reza Salimian and Mohammad Taghi Ameli, HGAPSO based Method for Solving Unit Commitment Problem, International Electrical Engineering Journal, Vol.6 (2015) No.3, pp [5] Cheng-Chien Kuo, A Novel Coding Scheme for Practical Economic Dispatch by Modified Particle Swarm Approach, IEEE Transactions on Power Systems, Vol. 23, No. 4, November 2008, pp [6] Leandro dos Santos Coelho, Chu-Sheng Lee, Solving economic load dispatch problems in power systems using chaotic and Gaussian particle swarm optimization approaches, ELSEVIER transactions on Electrical Power and Energy Systems 30(2008), pp [7] Zwe-Lee Gaing, Particle Swarm Optimization to Solving the Economic Dispatch Considering the Generator Constraints, IEEE Transactions on Power Systems, Vol. 18, No. 3, August 2003, pp [8] T. O. Ting, M. V. C. Rao and C. K. Loo, A Novel Approach for Unit Commitment Problem via an Effective Hybrid Particle Swarm Optimization, IEEE Transactions on Power Systems, Vol. 21, No. 1, February 2006, pp [9] R. Jahani, H. Chahkandi Nejad, A.H. Araskalaei and M. Hajinasiri, A Solution to The Unit Commitment Problem Using Hybrid Genetic and particle swarm optimization Algorithms, Australian Jouranl of Basic and Applied Sciences,5(5), 2011, ISSN: , pp [10] Huseyin Hakam Balci, Jorge F. Valenzula, Scheduling Electric Power Generators using Particle Swarm Optimization Combined with the Lagrangian Relaxation Method, International Jouranl of Applied Mathematics and Computer Science, Vol. 14, No. 3,2004, pp Table 2. Generator data for 15-unit test system Uni Fuel cost coefficients t Table Hour Load data for 15-unit test system Load Load Hour Hour MW MW Table 4. B mn -coefficient matrix-15-unit test system ISSN: Page 451

6 Table 5. PSO applied to ED-15-unit test system-with transmission losses P D P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 10 P 11 P 12 P 13 P 14 P 15 P loss,mw P sum, MW TFC, $/h TOTAL TFC, $ ISSN: Page 452

7 Table 6. PSO applied to UC-15-unit test system-with transmission losses P D P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 10 P 11 P 12 P 13 P 14 P 15 P loss,mw P sum, MW TFC, $/h TOTAL TFC, $ ISSN: Page 453

8 Table 7. UCED-15-Unit Test System-with Transmission Losses P D P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 10 P 11 P 12 P 13 P 14 P 15 P loss, MW P sum, MW TFC, $/h TOTAL TFC,$ ISSN: Page 454