Optimal Compensation of Reactive Power in Transmission Networks using PSO, Cultural and Firefly Algorithms

Transcription

1 Volume 114 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Optimal Compensation of Reactive Power in Transmission Networks using PSO, Cultural and Firefly Algorithms Subhash Shankar Zope 1 and 2 Dr. R.P. Singh 1 Sri Satya Sai University of Technology & Medical Sciences. Opposite OILFED, Bhopal Indore Highway, Pachama, Sehore (India) sszope@rediffmail.com 2 Sri Satya Sai University of Technology & Medical Sciences. Opposite OILFED, Bhopal Indore Highway, Pachama, Sehore (India) prof.rpsingh@gmail.com Abstract Increased demand for electrical energy and free market economies for electricity exchange, have pushed power suppliers to pay a great attention to quality and cost of the latter, especially in transmission networks. To reduce power losses due to the high level of the reactive currents transit and improve the voltage profile in transmission systems, shunt capacitor banks are widely used. The problem to be solved is to find the capacitors optimal number, sizes and locations so that they maximize the cost reduction. This paper is constructed as a function of active and reactive power loss reduction as well as the capacitor costs. To solve this constrained nonlinear problem, a heuristic technique, based on the sensitivity factors of the system power losses, has been proposed. The proposed algorithm has been applied to numerous feeders and the results are compared with those of the 367 authors having

2 treated the problem. The optimal location SVC is studied on the basis of Heuristic methods; Particle Swarm Optimization (PSO), Cultural Algorithm (CA), and Firefly Algorithm (FF) to minimize network losses. Validation of the proposed implementation is done on the IEEE-14 and IEEE-30 bus systems. Key Words andphrases:ca, FDL, FF, PSO, SVC. 1 Introduction The rapid growth of the population as well as the industry are primarily the factors influencing the consumption of electrical energy which is on the other hand continuously increasing. Since storing the electrical energy is a challenging task, it requires a permanent balance between consumption and production for that it is at first sight necessary to increase the number of power stations, and of the various structures (Transformers, Transmission lines, etc.), this leads to an increase in cost and a degradation of the natural environment [1]. 1.1 Classification of the Variables of Power Flow Equations In an electrical network each busbar is connected with four fundamental magnitudes: The modulus of the voltage, the phase of the voltage, the active power injected and the reactive power injected. It is very important to note that for each busbar; two variables must be specified beforehand and the other two must be calculated. These different variables each have a name following their role in the electrical network. The state of the system is determined only after calculating the values of the state variables [2]: The disturbance variables: These are uncontrolled variables representing the power demands of loads, the perturbation variables are: and. The state variables: Modules and phases are called unknown state variables which characterize the state of the system, these variables are: and. Control variables: In a generic way, the active and reactive powers injected are called control. It is also possible, depending on the case, to consider tensions at the generation nodes or the transformation ratios of transformers with load adjuster as control variables, these variables are: [2]. 1.3 Types of busbars In the power flow analysis, the busbars in the system are classified in three categories: 368

3 Reference busbar (slack bus): It is also called the swing bus or swing bus, it is a fictional element created for the study of the distribution of power, and it has for role to provide the additional power necessary to compensate for transmission losses, as these are not known in advance. In general, by convention, this busbar is identified by the set of busbarsn = 1 connected to a voltage source from which the module and phase (δ) of the voltage are known, these values are taken as references = 1pu and δ = 0. The active powers ( ) and ( ) are therefore unknown and must be calculated after solving the problem of the power flow [2] and [3]. Busbars (control bus): Also known as generator or voltage controlled busbars; they may include games of barriers to which generators, capacitor banks, static compensators or transformers with adjustable plug are connected to control the voltage. The parameters specified here are: the active power ( ) and the voltage modulus ( ) from which the term: busbars therefore the remaining parameters must be calculated ( ) and (δ) [2] and [3]. Loadbars (Loadbus): Also called the busbars (P Q), the specified values are the active powers ( ) and ( ), the values to be computed are the modulus and the phase (δ) of the voltage [2] and [3]. This paper is organized into nine sections. Section 2 gives analytical methods. Then in 3, relative unit (Static VAr Compensator) is explained. Section 4 details about modeling branches and loads. Section 5 explains the optimization of the reactive energy compensation by fixed batteries using heuristic method. Section 6 describes load flow analysis using Newton- Raphson method. Section 7 provides optimal location using heuristic techniques. Results and analysis is described in section 8 and finally, a conclusion summarizes the contributions of the paper. 2 Analytical Methods The pioneer in the field is Cook [4]. In 1959, he studied the effects of capacitors on power losses in a radial distribution network where the charges are uniformly distributed. It considered the reduction of power losses as an objective function by considering a periodic reactive charging cycle. Cook then developed a network of convenient curves to determine the most economical power of the capacitor bank and the location of the capacitor bank on the line. The equation giving the optimal location to be assigned to a specified size battery is given by: 2.1 Methods of Digital Programs The development of methods has led researchers to become increasingly interested in the optimization 369 of reactive energy

4 compensation. Therefore, they have developed numerical methods for the analysis of the electrical network. Baran and Wu [5] in 1989, presented a method for solving the problem of placing capacitor banks in distribution networks. In this problem, the locations of the batteries, their sizes, their types, the stresses of the voltage and the variations of the load are taken into account. The problem is considered to be a nonlinear programming problem where load flow is explicitly represented. To solve this problem, it is broken down into a slave problem and a master problem. They exploit for this purpose the property of optimization where: u { u } (1) 3 Relative Unit (Static VArCompensator) The normalization of the resistance of the line is obtained by relating it to a calculated basic resistance by means of the voltage and the power. If the base voltage is given in kv and the power in kva then, this resistance is given by: The normalized resistance is then: Standardized load ratings are obtained by: (2) (3) { (4) 4 Modeling Branches and Loads 4.1 Modeling Branches The distribution networks have a radial configuration and consist of a set of branches. Each branch of this network is modeled as a series resistor with a pure inductance. The impedance of any branch "i" of this network (see Fig. 1) is given by: (5) 370

5 International Journal of Pure and Applied Mathematics i Fig. 1. Single line diagram of a branch The shunt admittances are negligible because the line is of medium voltage. 4.2 Modeling of Loads The loads are generally modeled as being voltage dependent. We write for the active and reactive powers of a charge placed at the node "i" the following expressions: ( ) (6) Where, ( ) (7) and are the nominal active and reactive powers. is the nominal voltage. and are the active and reactive power of the load at node for a voltage equal to. The coefficients and determine the character of the load. If the coefficients and are both zero, the load is considered to be constant power. If, on the other hand, and are equal to 1, the load is considered to be constant current. When they are equal to 2 the load is considered to have a constant impedance. In the remainder, and will be zero, i.e. consider constant power loads. The apparent power of the load connected to the node is in this case: (8) 5 Method of Solution The voltage drop method is an iterative method. Its principle consists in calculating, first and for each section of the line, the powers at the end of the branch, the losses of active and reactive powers and the powers at the beginning of the branch. From these, the currents of the branches are determined by raising the line to the source. These currents are calculated from the estimated values of the voltages, the powers at the beginning of 371

6 the branch and the values of the impedances of each line section between two successive busbars. 5.1 Formulation of the Problem Power losses, low power factor and degradation of the voltage profile are the result of strong current flow in power systems.these phenomena are more pronounced in distribution networks where the branch currents are stronger compared to those circulating in the transport networks. The problem is therefore to decide the number of batteries, their powers and their locations which would make an objective function "F" maximum. This objective therefore makes the problem of reactive energy compensation an optimization problem. However, owed to the discrete nature of the battery sizes and their locations, this problem is non-linear with constraints. It is generally modelled as follows: u u { u u u (9) Where, : is the objective function to maximize. : is the equality constraint. It is the set of equations of the power flow : is the control variable vector u: is the state variable vector. 5.2 Objective Function The objective function on which all the authors who have dealt with the problem of optimization of reactive energy compensation is the so-called "economic return" function or cost reduction noted as " S". Mathematical expression is given by: Where, (10) : is the total number of batteries installed. : is the cost of kw produced ( /kw). : is the annual price of kvar installed depreciation and life included. : is the size of the installed battery at node " ". : is the reduction of the active power losses. 372

7 5.3 Reduction of Active Power Losses The reduction of the power losses due to a battery "k" is equal to the difference of the losses of active power in the network before and after the installation of the said capacitor bank. It is given by: Where, : are the active power losses in line before compensation. : are active power losses in line after compensation. 5.4 Reduction of Reactive Power Losses (11) The reduction of the reactive power losses due to a battery installed at node " " of the distribution line is defined by the difference between the losses before and after the installation of batteries in question of capacitors. It is given by: Where, (12) : are the losses of reactive power in line before compensation. : are the losses of reactive power in line after compensation. 5.5 Reactive Power Losses The losses of reactive power in a distribution network line composed of n branches are given by the following formula: Where, is the reactance of branch (13) is the line current of the branch. As with the active power losses, the active and reactive components of the branch current thus allow to write the losses of reactive power as follows: (14) The losses of reactive power when a capacitor bank is placed on a node are given by: (15) 373

8 The reduction of reactive power losses by calculating the difference between equation (14) and equation (15), will be equal to: 5.6 Heuristic Method (16) Heuristic methods are based on experience and practice. They are easy to understand and simple in their implementation. They use sensitivity factors which they incorporate into optimization methods in order to achieve qualitative solutions with small computational efforts. Since the problem of determining the suitable battery locations has been separated from that of optimum power determination since the locations are determined by the sensitivity factors then the size calculation is generally modeled as follows: 5.7 New Modeling of the Problem { u (17) By substituting the constraint on the tension with that made on the branch current, the new mathematical model of the problem becomes: 5.8 Optimal Operation of Batteries { u (18) The reduction in power losses for a given node "k" is defined as the difference between the power losses before canceling the reactive current of the load at node " " and after the latter has been canceled. It is given by: (19) The power losses before the cancellation of the reactive current of the load at the node " " are given by: (20) The power losses after the cancellation of the reactive current of the load at the node " " are given by: (21) 374 After simplification, the reduction in power losses will have the

9 following expression: (22) 5.9 Determination of Optimal Sizes To calculate the optimum sizes of the batteries, the currents they generate are first determined. This current is calculated so as to make the objective function the maximum cost reduction. This current is acquired by undertaking the accompanying condition: The expression of the current is then given by: (23) (24) The initial optimum power is calculated by the following expression: The maximum value of the cost reduction in this case: (25) * + ( ) (26) The value of the equivalent power loss reduction is given by: ( ) ( ) ( ) [ ] ( ) 5.10 Solution Strategy (27) By optimally compensating the reactive energy we expect the battery locations to be busbars in the network and that the optimum battery power is available commercially or multiple of these batteries. If the constraint on the locations, which can only be busbars of the network, has found a solution by means of sensitivity factors, the optimum powers of the batteries remain to be determined by solving the problem with the following constraints: 375

10 u (28) { The detailed solution algorithm of the problem of determination will be detailed in the following section Calculation Algorithm The algorithm for solving the overall problem, i.e., the suitable locations of the capacitor banks and their sizes is detailed in what follows. A MATLAB 14a environment program has been developed for this purpose. Step 1: Read the network data. Step 2: Perform the program of the power flow before compensation to determine the active and reactive power losses, branch currents, node voltages and their phases at the origin. Step 3: Initialize reduction of power losses and cost. Step 4: as long as the reductions in power losses and cost are positive. Step 4.1: Determine the sensitivities of the nodes according to equation (22) and rank them in descending order. Step 4.2: If the most sensitive node considered has already received a battery, ignore it. Step 4.3: Calculate the initial value of the optimum size of the battery to be placed there, the reduction of the cost and the reduction of the power losses. Step 4.4: Perform load flow to update electrical quantities (voltage, current, power). Step 4.5: Adjust the optimal size of the battery. Step 4.6: If the battery size is negative, smaller than the smallest standard battery or greater than the total power and the reduction of negative power losses then: Step 4.6.1: Remove the battery. Step 4.6.2: Give the voltages at their origin and branch currents the values d before the battery. Step 4.7: Otherwise, take as the optimum size of the battery, the lower standard size where higher giving the greatest cost reduction. 376

11 Step 4.8: Re-calculate the load flow. Step 4.9: If the battery produces overcompensation then: Step 4.9.1: Replace the standard battery with a smaller one that does not overcompensate. Step 4.9.2: Test if the battery is not smaller than the smallest standard battery. Step 4.9.3: Perform load flow and calculate the reduction in power losses and cost based on the actual installed kvar power. Step : Verify that the reduction in power loss and cost reduction are positive. Step 4.10: End if Step 4.11: Go to Step 4 Step 5: Display the results. 6 Load Flow Analysis using Newton- Raphson Method This method requires more time per iteration where it does not requires only a few iterations even for large networks. However, it requires storage as well as significant computing power. Let us assume: (29) (30) (31) We know that: Equation (29) then becomes: (32) By separating the real and the imaginary part, one obtains: { (33) 377

12 Positions: (34) (35) Where, { Then, the equation (33) becomes: { Where, (36) { (37) It is a system of nonlinear equations. The active power and the reactive power are, and the real and imaginary components of the voltage and are unknown for all Bus bars except the reference bus bar, where the voltage is specified and fixed. The Newton-Raphson method requires that non-linear equations be formed of expressions linking the powers and the components of the voltage. [ ] [ ] [ ] (38) Where the last set of bars is the reference bar. The outline of the matrix is given by: Or * + * + * + (39) [ ] [ ] (40) Where [J] is the Jacobian of the matrix. are the differences between the planned values and the values calculated respectively for active and 378 reactive powers.

13 Equation (37) can be written as follows: { (38) From where, one can draw the elements of the Jacobian: The diagonal elements of : The non-diagonal elements of The diagonal elements of The non-diagonal elements of The diagonal elements of The non-diagonal elements of The diagonal elements of The non-diagonal elements of (39) (40) Because of the quadratic convergence of the Newton-Raphson method, a solution of accuracy can be achieved in just a few iterations. These characteristics make the success of the Fast Decoupled Load Flow and the Newton-Raphson. 7 Optimal Location using Heuristic Techniques In the proposed system, the location of SVC in a particular bus system is decided by PSO, CA, FF algorithms. The objective function is minimized using the abovementioned techniques. 7.1 Particle Swarm Optimization (PSO) James Kennedy and Russell C. Eberhart proposed a PSO approach in This approach is a heuristic method [6]. The evaluation of candidate solution of current search space is done on the basis of iteration process (as shown in Fig. 2). The minima and maxima of objective function is determined by the candidate s solution as it fits the task s 379requirements. Since PSO

14 International Journal of Pure and Applied Mathematics algorithm do not accept the objective function data as its inputs, therefore the solution is randomly away from minimum and maximum (locally/ globally) and also unknown to the user. The speed and position of candidate s solution is maintained and at each level, fitness value is also updated. The best value of fitness is recorded by PSO for an individual record. The other individuals reaching this value are taken as the individual best position and solution for given problem. The individuals reaching this value are known as global best candidate solution with global best position. The up gradation of global and individual best fitness value is carried out and if there is a requirement then global and local best fitness values are even replaced. For PSO s optimization capability, the updation of speed and position is necessary. Each particle s velocity is simplified with the help of subsequent formula: (41) Start Initialization on early searching points of all agents Assessment of searching points of all agents Amendment of every searching point using state equation No Extent to extreme iteration Stop 7.2 Cultural Algorithm Fig. 2. Flow chart of PSO algorithm [6] Cultural algorithm corresponds to modeling inspired by the evolution of human culture [7]. Thus, just as we speak of biological evolution as the result of a selection based on genetic variability, we can speak of a cultural evolution resulting from a selection exercising on the variability Cultural development. From this idea, Reynolds developed a model whose cultural evolution is considered as a process of transmission of experience at two levels: a micro-evolutionary level in terms of transmission of genetic material between individuals of a 380 population and a macro level -evolutionary in terms of

15 knowledge acquired on the basis of individual experiences. The following figure presents the basic CA framework. As Fig. 3 shows, the population space and the belief space can evolve respectively. The population space consists of the autonomous solution agents and the belief space is considered as a global knowledge repository. The evolutionary knowledge that stored in belief space can affect the agents in population space through influence function and the knowledge extracted from population space can be passed to belief space by the acceptance function. Belief Space Update Acceptance Function Influence Function Communication Protocol Evaluation Population Space Inherit Adaption, Reproduction Fig. 3. CA framework [8] 7.3 Firefly Algorithm Fireflies are small flying beetles capable of producing a cold flashing light for mutual attraction. In the common language between fireflies, they are also used synonymous lighting bugs or glow worms. These are two beetles that can emit light, but fireflies are recognized as species that have the ability to fly. These insects are able to produce light inside their bodies through special organs located very close to the surface of the skin. This light production is due to a type of chemical reaction called bioluminescence [9]. Principle of operation of the algorithm of Fireflies The algorithm takes into account the following three points: All fireflies are unisex, which makes the attraction between these is not based on their gender. The attraction is proportional to their brightness, so for two fireflies, the less bright will move towards the brighter. If no firefly is luminous that a particular 381

16 firefly, the latter will move randomly. The luminosity of the fireflies is determined according to an objective function (to be optimized). Based on these three rules, the Firefly algorithm is as follows: Define an objective function Generate a population of fireflies Define the intensity of light at a point by the objective function Determine the absorption coefficient As long as ( Max Generation) For to For to If Move the firefly to the firefly End if Vary the attraction as a function of the distance via p Evaluation of new solutions and updating light intensity End For Fig. 4. Pseudo code for Firefly Algorithm 8 Results Analysis On the IEEE-14 and IEEE-30 bus test systems (shown in Fig. 5 and Fig. 6) the proposed heuristic techniques (PSO, CA and FF) have been tested. G Gener C Synchro nous G 1 1 C C G Three Winding Transformer 9 7 C 8 4 C 3 Fig. 5. Single line diagram of the IEEE-14 bus test system 382

17 Power Loss (MW & MVar) International Journal of Pure and Applied Mathematics Fig. 6. Single line diagram of the IEEE-30 bus test system Active & Reactive Power Losses in IEEE Bus System P Q NRPF NRPF with SVC (PSO) Fig. 7. Active & Reactive power losses in IEEE-14 bus system using PSO 383

18 Power Loss (MW & MVar) Power Loss (MW & MVar) International Journal of Pure and Applied Mathematics 70 P Q Active & Reactive Power Losses in IEEE Bus System NRPF NRPF with SVC (PSO) Fig. 8. Active & Reactive power losses in IEEE-30 bus system using PSO Active & Reactive Power Losses in IEEE Bus System P Q NRPF NRPF with SVC (CA) Fig. 9. Active & Reactive power losses in IEEE-14 bus system using CA 384

19 Power Loss (MW & MVar) Power Loss (MW & MVar) International Journal of Pure and Applied Mathematics Active & Reactive Power Losses in IEEE Bus System P Q NRPF NRPF with SVC (CA) Fig. 10. Active & Reactive power losses in IEEE-30 bus system using CA Active & Reactive Power Losses in IEEE Bus System P Q NRPF NRPF with SVC (FF) Fig. 11. Active & Reactive power losses in IEEE-14 bus system using FF 385

20 Power Loss (MW & MVar) International Journal of Pure and Applied Mathematics Active & Reactive Power Losses in IEEE Bus System P Q NRPF NRPF with SVC (FF) Fig. 12. Active & Reactive power losses in IEEE-30 bus system using FF Table 1. Comparative analysis for IEEE-14 bus system Active Power Loss Heuristic Method Reactive Power Loss PSO Cultural Algorithm Firefly Algorithm Table 2. Comparative analysis for IEEE-30 bus system Heuristic Method Active Power Loss Reactive Power Loss PSO Cultural Algorithm Firefly Algorithm Conclusion In our work in this paper, we presented a solution for the problem of the circulation of strong reactive currents in balanced distribution networks. A heuristic solution technique based on a loss-of-power sensitivity factor has been proposed. In this method, the choice of the candidate nodes to receive the capacitor banks is arbitrated by the sensitivity of the power losses of the entire electrical system studied to the reactive load 386

21 current of each node. The most sensitive node is therefore the one whose reactive current of charge produces the most loss reduction. During this work, the problem of the power flow in the distribution networks, which is a prerequisite for the conduct of the reactive energy compensation, is also taken care of, the calculation of the power flow is imperative. An iterative method has been developed for this purpose where a technique specific to us has been given to recognize the configuration of the network. Load flow analysis is also done using Newton-Raphson method. Three Heuristic methods are used to optimize the location of SVC using the MATLAB model; Particle Swarm Optimization, Cultural Algorithm and Firefly algorithm. The tests were performed taking SVC as the FACTS device. It was found that the Cultural Algorithm has less power losses as compared to other methods. References [1] Preedavichit, P. and Srivastava, S.C., Optimal reactive power dispatch considering FACTS devices. Electric Power Systems Research, 46(3), pp [2] Park, J.B., Lee, K.S., Shin, J.R. and Lee, K.Y., A particle swarm optimization for economic dispatch with nonsmooth cost functions. IEEE Transactions on Power systems, 20(1), pp [3] Driesen, J. and Katiraei, F., Design for distributed energy resources. IEEE Power and Energy Magazine, 6(3). [4] Cook, R.F., Analysis of capacitor application as affected by load cycle. Transactions of the American Institute of Electrical Engineers. Part III: Power Apparatus and Systems, 78(3), pp [5] Baran, M.E. and Wu, F.F., Optimal capacitor placement on radial distribution systems. IEEE Transactions on power Delivery, 4(1), pp [6] Kennedy, J., Particle swarm optimization. In Encyclopedia of machine learning (pp ). Springer US. [7] Reynolds, R.G., 1994, February. An introduction to cultural algorithms. In Proceedings of the third annual conference on evolutionary programming (Vol ). Singapore. [8] Reynolds, R.G. and Peng, B., 2004, November. Cultural algorithms: modeling of how cultures learn to solve problems. In Tools with Artificial Intelligence, ICTAI th IEEE International Conference on (pp ). IEEE. [9] Yang, X.S., Firefly algorithm, stochastic test functions and design optimization. International Journal of Bio-Inspired Computation, 2(2), pp

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