Solution of Optimal Power Flow Subject to Security Constraints by a New Improved Bacterial Foraging Method

Transcription

1 Solution of Optimal Power Flow Subject to Security Constraints by a New Improved Bacterial Foraging Nima Amjady, Senior Member, IEEE, Hamzeh Fatemi, Member, IEEE and Hamidreza Zareipour, Senior Member, IEEE Abstract 1 - In this paper, a new solution method, which is an improved version of bacterial foraging (BF) technique, is proposed for optimal power flow with security constraints (OPF-SC) problem. The OPF-SC is a nonlinear programming optimization problem with complex discontinuous solution space. BF is a recently developed advanced stochastic search technique owning high exploitation and local search capabilities to search the promising areas of the solution space with high resolution. Saving the advantages of BF, the exploration capability and diversity of the search process of BF are enhanced in the proposed improved version of BF (IBF) to cover different areas of the solution space avoiding being trapped in local minima. The effectiveness of the suggested solution strategy to solve the OPF-SC problem is extensively illustrated on some wellknown test systems. Moreover, the proposed IBF is compared with many recently published OPF and OPF-SC solution methods. Keywords: Improved Bacterial Foraging, Exploration, Exploitation, Search Diversity, Optimal Power Flow with Security Constraints menclature Pgi: Active power output of unit i Fi(Pgi): Complete fuel cost function of unit i Fi, k ( P gi ): Fuel cost function of unit i using fuel option k aik, bik, cik, eik and fik: Fuel cost coefficients of F, ( P ) P and min gi, k max P gi, k : Minimum and maximum power generation limits of i th unit with fuel option k, respectively. nfi: Number of fuel options for unit i U(i,k): Binary variable, which is 1 if unit i uses fuel option k and zero otherwise NG: Number of thermal units of the power system Vgi: Voltage magnitude set-point of unit i Ti: Tap setting for i th transformer Qi: Reactive power injection for i th capacitor/reactor bank PSi: Phase setting for i th phase shifter NT: Number of tap-changing transformers NC: Number of capacitor/reactor banks NPS: Number of phase shifters DV: Vector of decision variables of the OPF-SC problem NP: Number of decision variables of the OPF-SC problem NS: Number of bacteria (user defined parameter) for BF/IBF AOF: Augmented objective function CF: Cost function ACF: Aggregated cost function JCC: Additional cost function of BF/IBF Δ(.,.,.,.): NP dimensional vector of random numbers in the range of [-1,+1] such that all elements of Δ(.,.,.,.) are not simultaneously zero C(i): Step size for bacterium i in BF algorithm; C(i) parameters by the number of NS (1 < i < NS) are user defined parameters of BF Nswim: Swim length (user defined parameter) for BF/IBF algorithm Nch, Nre and Ned: Number of iterations of chemotaxis, reproduction and elimination-dispersal loops (user defined parameters) for BF/IBF algorithm Ped: Probability value (user defined parameter) for eliminationdispersal process of BF algorithm β: Scaling factor of DE mutation operation βmax: Maximum value of β dattract, ωattract, hrepelent and ωrepelent: Parameters of BF/IBF algorithm for computing JCC 1N. Amjady and H. Fatemi are with the Department of Electrical Engineering, Semnan University, Semnan, Iran ( amjady@tavanir.org.ir and s.hamzehfatemi@gmail.com) H. Zareipour is with the department of Electrical and Computer Engineering, Schulich School of Engineering, University of Calgary, Calgary, Alberta, Canada ( h.zareipour@ucalgary.ca) i k gi I. Introduction Electric power system is usually considered as the most complicated man-made system due to its large size, different static/dynamic behaviors and complex inter-connected equipment. For operation of such sophisticated system, engineers require qualified tools for optimally determining its different settings and control actions. An important operational function in this regard is optimal power flow (OPF) considered as the backbone tool that has been widely researched since introducing its concept by Carpentier [1]. The OPF optimizes a power system operating objective function (such as the fuel cost of thermal generators) while satisfying the constraints of the components and system [1]. After the electricity sector restructuring, OPF has been used to assess the spatial variation of electricity prices and as a congestion management and pricing tool [2]. In its most general formulation, the OPF is a nonlinear, non-convex, large-scale, static optimization problem with both continuous and discrete control variables [3]. By adding nonlinear security constraints of the power system to the OPF formulation, leading to OPF-SC, even a more complex optimization problem is obtained. Thus, OPF and OPF-SC problems challenge the numerical optimization techniques. Earlier OPF solution methods include mathematical programming approaches, such as nonlinear programming (NLP) [4], quadratic programming (QP) [5], linear programming (LP) [6], Newton method [7], mixed integer programming [1] and interior point (IP) methods [8]. However, these methods usually suffer from some disadvantages such as convergence to local solutions instead of global ones if the initial guess is in the vicinity of a local solution, applicability to a specific OPF problem based on its mathematical nature and some inherent theoretical assumptions (such as convexity, differentiability, and continuity) which are inconsistent with the actual OPF formulations [9]. Several stochastic search techniques such as genetic algorithms (GA) [3,10], evolutionary programming (EP) [11], tabu search (TS) [12], particle swarm optimization (PSO) [13], simulated annealing (SA) [14], differential evolution (DE) [15], bacteria foraging (BF) algorithm [16] have been proposed to solve the OPF problem without any restriction on the shape of the cost curves. The results reported were promising and encouraging for further research in this direction. Moreover, some researchers have recently combined different stochastic search techniques with each other or with analytical solution methods to enhance their performance for the solution of OPF and OPF-SC problems. A hybrid technique of EP and Newton-Raphson method [17], hybrid TA/SA approach [18], evolving ant direction hybrid differential evolution (EADHDE) algorithm [19] and combination of GA and SA [20] have been presented for the solution of OPF. Furthermore, a hybrid tool of EP and SQP for the solution of OPF-SC is proposed in [21]. Despite the performed research works in the area of OPF, more efficient OPF solution methods are still demanded due to its importance and complexity. Furthermore, OPF-SC, considered in this paper, is a more comprehensive and more recent concept than OPF and fewer research works on it can be found in the literature. Contributions of this paper can be summarized as follows: 1) A new stochastic search technique, named improved bacterial foraging (IBF) method, is proposed. New search mechanisms and evolution procedures are incorporated into the classical BF in the proposed IBF to enhance its exploration capability, search diversity and convergence behavior. 2) The proposed IBF has been formulated for the solution of OPF- SC problem. For this purpose, decision variables of the problem are coded and objective function and constraints of OPF-SC are modeled within the proposed IBF. The remaining parts of the paper are organized as follows. In the second section, the formulation of OPF-SC problem is briefly

2 introduced. The proposed IBF and its application for the solution of the OPF-SC problem are presented in section three. Obtained numerical results from extensive testing of the proposed solution approach on different case studies are presented in section four and compared with the results of several other recently published methods. Section five concludes the paper. II. OPF-SC formulation Objective function, constraints and decision variables of OPF-SC problem are briefly presented in this section. More details about formulation of this problem can be found in [9,22,23]. II.A. Objective Function The most common objective function of OPF and OPF-SC problems is fuel cost of thermal generating units [9], which is conventionally modeled by a quadratic cost function. However, large steam units usually have a number of steam admission valves that are opened in sequence to obtain ever-increasing output of the unit [1]. By considering valve loading effects of thermal units, which introduces rippling effects to the actual input/output curve, an additional sine term representing the valve effects is added to the quadratic fuel cost function [24]. Moreover, usually there are many units in a practical power system supplied with multiple fuels. The fuel cost function of thermal unit i with valve loading effects and fuel type changes is as follows: n fi F ( P ) F ( P ). U, i 1,..., NG (1) i gi i, k gi ( i, k ) k 1 Where F, ( P ) is as below: i k gi F ( P ) a b P c P e sin( f ( P P )) 2 min i, k gi ik ik gi ik gi ik ik gi, k gi min max if Pgi, k Pgi Pgi, k, k 1,..., nfi, i 1,..., NG (2) Sum of Fi(Pgi) terms over all NG thermal units should be minimized. The other objective functions such as gaseous emissions, transmission real loss, bus voltage deviation, and security margin index for regulated power systems and social welfare and congestion management cost for deregulated electricity markets have also been proposed for the OPF and OPF-SC problems in the literature. A review of these objective functions can be found in [9]. II.B. Constraints The employed OPF-SC formulation includes the following constraints [24,25]: 1- nlinear AC power flow constraints 2- Generators real and reactive power outputs limits 3- Prohibited operating zone (POZ) constraints of units 4- Bus voltage magnitude and branch flow limits (static security constraints) in the steady state and post-contingent state of credible contingencies [23] 5- Limits of discrete transformer tap and phase shifter settings 6- Limits of the discrete reactive power injections of capacitor/reactor banks AC power flow constraints can be handled in two ways. In the first approach, these constraints are simply added to the constraints of the optimization problem and the solution method solves the problem subject to all constraints. Definition of slack bus is not required in this approach. In the second approach, AC power flow is executed in the solution process instead of adding its equations as equality constraints to the optimization problem. For the second approach, slack bus should be defined so that the AC power flow can be executed. This approach is used in this research work due to the following reasons. In the second approach, the tight equality constraints of the OPF-SC problem are processed by an AC power flow solution algorithm and so the optimization method should only handle inequality constraints. Moreover, the proposed IBF has been implemented in the MATLAB software package. There are efficient computer codes in MATLAB for the solution of AC power flow. By the second approach, we can employ high efficiency of these computer codes for the constraint handling of the OPF-SC problem. Furthermore, the main users of OPF-SC are power system operators and dispatchers, which are usually familiar with the concept of slack bus, since slack bus is actually defined in most of power systems. Thus, an OPF-SC solution method using the concept of slack bus is more sensible for the power system operators and dispatchers. II.C. Decision Variables The vector of decision variables of the employed OPF-SC formulation includes both continuous and discrete decision variables as follows: DV1 DV=[ P,, P, V,V,,V, T,T,,T, Q,Q,,Q g2 DV 5 gng DV2 g1 g2 gng DV3 1 2 NT DV4 1 2 NC, PS,PS,,PS ] (3) 1 2 NPS The first sub-vector of DV includes active power generation of all units except the generation of slack bus (Pg1), since generation of the slack bus is a dependent variable or state variable [26] determined based on the power system load and generation of the other units. The first and second sub-vectors contain continuous decision variables, while those of the third, fourth and fifth subvectors are discrete decision variables. Observe from (3) that the OPF-SC formulation includes NP decision variables as follows: NP = (NG 1) + NG + NT + NC + NPS (4) The real OPF-SC model considered in this paper is a mixed-integer, nonlinear, non-convex, non-differentiable and non-smooth optimization problem with discontinuous solution space. This kind of optimization problem is very hard, if not impossible, to solve using traditionally deterministic optimization methods. Moreover, very large number of research works performed on the area (such as those reported in [3], [9]-[21], [23], [24], and [26]) further confirm that OPF-SC is sufficiently complex and has enough room for the application of stochastic search techniques. III. The Proposed IBF In this section, the BF algorithm is presented and its application for the solution of the OPF-SC problem is described. Then, the proposed IBF is introduced. BF algorithm is an efficient population based stochastic search technique recently developed by Passino [27]. BF has found an increasing interest in the recent years as an optimization technique due to its high ability to search the promising areas of the solution space. The idea of BF algorithm is based on the foraging mechanism of E. coli bacteria that are present in human intestines. The performance of BF algorithm as an optimization technique is briefly illustrated in the flowchart of Fig. 1 and can be described as the following step by step procedure: Step 1) BF algorithm has three main nested loops, namely chemotaxis, reproduction and elimination-dispersal (the algorithm also has some other internal loops). In the first step, the counter of the three main loops (denoted by j, k, and l, respectively) are initialized to zero as shown in Fig. 1. Moreover, the initial bacterial population should be produced. For the solution of the OPF-SC problem, each bacterium or individual has the structure shown in (3), containing the NP decision variables. Therefore, the bacterial population including NS bacteria becomes as follows: { DV ( i, j, k, l )} i 1,..., NS { DV (1, j, k, l ), DV (2, j, k, l ),..., (5) DV ( NS, j, k, l )} To generate the initial population, i.e. 1,..., { (,0,0,0)} i NS DV i, the NP decision variables of each bacterium are randomly generated within their allowable ranges. Step 2) Set the counter of bacteria, denoted by i, to one (i=1) to implement the chemotaxis loop. Step 3) For bacterium i, the cost function of BF, denoted by CF(i,j,k,l), is computed as follows: CF(i,j,k,l) = AOF(i,j,k,l) + JCC(i,j,k,l) (6) where AOF(i,j,k,l) represents augmented objective function of the OPF-SC problem including its objective function (e.g., the fuel cost) and penalty terms (penalizing deviations from the OPF-SC constraints as defined in [25]) for i th bacterium. The constructed

3 START Step 1 (Initialization): Set j=0 (counter of chemotaxis loop), k=0 (counter of reproduction loop), l=0 (counter of elimination dispersal loop). Moreover, randomly generate NS bacteria of the initial population within the allowable ranges. The BF population is as shown in (5). Step 2 (chemotaxis loop): Set i=1 (counter of bacteria) Step 3: Compute the BF cost function for bacterium i, based on (6)-(8), using the parameters of (9) Step 4 (Tumble): Update position of bacterium i, based on (10) Step 5: Update the cost function of bacterium i, considering its new position, according to (11) Step 6 (swim): Set m=0 (counter of swim loop) Step 6-1: Better cost function? Yes Step 6-2: Bacterium i proceeds one more step in the direction of the tumble as shown in (12) Step 6-3: Increment the counter of swim loop (m=m+1) Step 6-4: m < N swim? Step 7: i < NS? Yes Step 8: i=i+1 Step 9: Increment the counter of chemotaxis loop (j=j+1) Step 10: j < N ch? Step 11 (reproduction loop): Set i=1 END Yes Step 11-1: Compute ACF for bacterium i as shown in (13) Step 11-2: i<ns? Yes Step 11-3: i=i+1 Step 11-4) S r bacteria with the highest ACF values are replaced by S r bacteria with the lowest ACF values Step 12: Increment the counter of reproduction loop: k=k+1 Step 13: k < N re? Step 15 (elimination-dispersal loop): Set i=1 Step 15-1: The elimination-dispersal process is executed for bacterium i Step 15-2: i<ns? Step 16: Increment the counter of elimination-dispersal loop (l=l+1) Step 17: l < N ed? Step 19: Return the best bacterium with the lowest value of the BF cost function as the final solution Yes Yes Fig. 1) Flowchart for the BF algorithm Yes Step 14: j=0 Step 15-3: i=i+1 Yes Step 18: j=0, k=0 AOF should be minimized. In (6), JCC(i,j,k,l) is defined as follows: NS in, cc (7) J ( i, j, k, l) = J DV ( i, j, k, l), DV ( n, j, k, l) CC cc n1 where i, n J DV ( i, j, k, l ), DV ( n, j, k, l ) d exp. DV ( i, j, k, l ) DV ( n, j, k, l ) attract attract h exp. DV ( i, j, k, l ) DV ( n, j, k, l ) repelent repelent In the above equation,. denotes the Euclidean norm. In other words, the additional cost function JCC(i,j,k,l) for bacterium i is i, n J DV ( i, j, k, l ), DV ( n, j, k, l ) composed of NS terms cc measuring attracting and repelling effects between two bacteria i and n, illustrated in the right hand side of (8), respectively. In the original version of BF proposed by Passino [27], the parameters of dattract, ωattract, hrepelent and ωrepelent are set as follows: ωattract=0.2, ωrepelent=10, dattract=hrepelent (9) Considering the above parameters, each bacterium will try to move toward other bacteria to decrease JCC(i,j,k,l), but not too close to them, which is called swarming effect enhancing the exploitation capability of BF. Step 4) The position of bacterium i is updated (or equivalently bacterium i moves), known as tumble, as follows: DV ( i, j 1, k, l ) DV ( i, j, k, l ) C ( i ). (10) This results in a step of size C(i) in the direction of the tumble (i.e., ) for bacterium i. Step 5) The cost function of bacterium i for the next iteration of the chemotaxis loop (j+1) is computed similar to (6) as follows: CF(i,j+1,k,l) = AOF(i,j+1,k,l) + JCC(i,j+1,k,l) (11) where AOF and JCC are as defined for (6) except that DV(i,j+1,k,l) instead of DV(i,j,k,l) should be considered in (11). Step 6) This step is known as swim. At first, an inner counter m is initialized to zero (m=0) and a parameter Jlast is set as Jlast = CF(i,j,k,l). Then, an inner loop is executed as follows. Step 6-1) If CF(i,j+1,k,l) < Jlast, go to step 6-2; otherwise exit the inner loop (go to step 7). Step 6-2) Set Jlast = CF(i,j+1,k,l) and update position of i th bacterium in iteration j+1 once more as follows: DV ( i, j 1, k, l ) DV ( i, j 1, k, l ) C ( i ). (12) Also, execute (11) once more with new DV ( i, j 1, k, l ) obtained from (12). In other words, if moving in the direction of the tumble (i.e., ) results in a better position with lower cost function for bacterium i, then the bacterium i should move one step ahead in this direction. Step 6-3) Increment the counter m (m=m+1). Step 6-4) If m < Nswim, go back to step 6-1. Otherwise; exit the inner loop. Step 7) If i<ns, go to step 8; otherwise go to step 9. Step 8) Increment i (i=i+1) and go back to step 3. Step 9) Increment j (j=j+1). Step 10) If j < Nch, go back to step 2. Otherwise, go to the next step. Step 11) Set the counter of bacteria to one (i=1) to implement the outer loop, i.e. reproduction. Step 11-1) For bacterium i, ACF(i) is computed as follows: N ch ACF ( i ) CF ( i, j, k, l ) (13) j 1 where ACF(i) is a measure of how successful the bacterium i was during its lifetime for solving the optimization problem. Step 11-2) If i<ns, go to step 11-3; otherwise go to step Step 11-3) Increment i (i=i+1) and go back to step Step 11-4) Sort all bacteria in the terms of ACF(i) such that a lower aggregated cost function indicates a more successful bacterium (it is also referred to as a healthier bacterium). Then, Sr bacteria with the 2 2 (8)

4 highest ACF(.) values are discarded. Instead, Sr bacteria from the remaining ones, owning lowest ACF(.) values, are copied and occupy the place of the discarded bacteria. Sr is usually set to NS/2. Step 12) The counter of reproduction loop is incremented (k=k+1). Step 13) If k < Nre, go to step 14; Otherwise; go to step 15. Step 14) Set j=0 and go back to step 2. Step 15) Set i=1 to implement the outermost loop of BF. Step 15-1) Elimination-dispersal process for each bacterium i is executed. For this purpose, a random number, uniformly distributed in the interval [0,1], is generated. If this random number is lower than Ped, the bacterium i is eliminated and instead a new bacterium is randomly generated within the allowable limits of the decision variables (similar to step 1). Otherwise; the bacterium i is retained. Step 15-2) If i<ns, go to step 15-3; otherwise go to step 16. Step 15-3) Increment i (i=i+1) and go back to step Step 16) The counter of elimination-dispersal loop is incremented (l=l+1). Step 17) If l < Ned, go to step 18; otherwise got to step 19. Step 18) Set j=0 and k=0 and go back to step 2. Step 19) The BF algorithm is terminated and the best bacterium of the population owning the lowest value of the cost function CF is returned as the final solution of the optimization problem. Exploration and exploitation capabilities are two important aspects for a stochastic search technique. Exploration is the algorithm s ability to cover and explore different areas in the feasible search space. Exploitation, on the other hand, is the ability to concentrate only on promising areas in the search space and to enhance the quality of the potential solution in the promising region. BF algorithm has high exploitation capability due to its swarming behavior and nested loops. However, to solve nonlinear OPF-SC problem with complex solution space, the stochastic search technique should also have high exploration capability to search different areas of the solution space. In this paper, a new version of BF algorithm, named improved BF (IBF), is presented to enhance the exploration capability and diversity of the search process of BF, such that it can obtain good solutions for complex optimization problems. For this purpose, step 4 (tumble), step 6-2 from swim loop, steps 11-1 and 11-4 from reproduction loop, step 15-1 (elimination-dispersal process) and step 19 (selection of the final solution) of BF are improved in the IBF. Differential evolution (DE) is a relatively recent stochastic search technique that finds an increasing interest in the recent years as an optimization technique due to performing a global search of solution space. In [28], it has been discussed that DE, by computation of difference between two randomly chosen individuals from the population in its mutation operation, determines a function gradient in a given area (not in a single point), and therefore prevents sticking the solution in a local extremum of objective function. We came to this idea from DE mutation operation to design the proposed IBF. To better illustrate the idea, DE mutation operation is first introduced in the framework of BF: DV ( i, j 1, k, l ) DV ( i, j, k, l ). DV ( i, j, k, l ) DV ( i, j, k, l ) (14) 1 2 where i1 and i2 are two randomly selected individuals from the current population such that i1 i2 i; β controls the amplification of the differential variation. Theoretically, β (0, ), but it is usually taken from the range (0,1] [29]. However, DE mutation operation only uses one gradient vector. To avoid generating similar search directions and enhance the diversity of the search process, this operation is modified here as follows: DV ( i, j 1, k, l ) DV ( i, j, k, l ). DV ( i1, j, k, l ) DV ( i 2, j, k, l ) (15) i1, i 2NS In (15), each pair of the individuals of the population is randomly selected as (i1, i2) and their difference is considered in the summation. In this way, the effect of all possible difference vectors by the number of [NS/2] (instead of a single difference vector) is considered to construct the search direction. It is noted that the randomly selected pairs (i1, i2) for each individual DV(i,j,k,l) are different from those of the other individuals. In other words, the whole capability of the population is employed to generate diverse search directions as much as possible. To further enhance the exploration capability of the IBF, β is randomly generated for each bacterium along the IBF iterations with a uniform distribution in the interval (0, βmax]. In other words, we have: DV ( i, j 1, k, l ) DV ( i, j, k, l ). DV ( i1, j, k, l ) DV ( i 2, j, k, l ) (16) i1, i 2NS In step 4 of the proposed IBF, the suggested mutation operation of (16) is used for tumble instead of (10). BF based on (10), can only search different directions, while the step size C(i) for each bacterium i is a fixed value. On the other hand, the proposed IBF can benefit from both variable search directions and variable tumble steps (with high diversities) leading to higher search capability of the IBF. Furthermore, βmax is adaptively changed along the iterations of the reproduction loop of the IBF to enhance its convergence behavior: 1 max ( k, l) (17) l N re k 1 10 The IBF begins with βmax=1. Then (17) is executed to update it at the end of step 11-4 such that βmax becomes 1/10, 1/20, 1/30, etc. after successive executions of the reproduction loop. Thus, the proposed IBF begins with a high value of βmax(k,l) to search different regions of the solution space with high exploration. After a number of executions of the reproduction loop, when the bacteria enter the promising area, βmax(k,l) is adaptively reduced, limiting the range of variations of β(i,j,k,l), to search the area with higher resolution. Another advantage of (16) compared with (10) is that the proposed mutation operation does not require setting a large number of user defined parameters C(i), 1<i<NS. Similarly, for step 6-2, C( i). in (12) of BF is replaced by. DV ( i1, j, k, l ) DV ( i 2, j, k, l ) in the IBF. i1, i 2NS A bacterium may have poor positions in the initial iterations of the chemotaxis loop, but it can reach good positions in the last iterations of the loop. Thus, considering the cost function values in all iterations may be misleading for judgment about the quality of solutions presented by the bacteria. Hence, in step 11-1 of the IBF, (13) is changed as follows: ACF(i) = CF(i,Nch,k,l), 1 < i < NS (18) In (18), the obtained value of i th bacterium for the cost function CF in the last iteration of the chemotaxis loop is considered as ACF(i). Another modification that is applied to step 11-4 is that instead of copying Sr bacteria with the lowest ACF(.) values, Sr individuals are generated from them using the proposed mutation operation of (16). Therefore, the reproduction loop of the IBF saves the good solutions of the population (like the reproduction of BF) and, at the same time, adds some new offspring individuals generated from these good solutions to the population, which increases its information content. In this way, the diversity of the population generated from the IBF reproduction increases and thus the bacteria can search more areas of the solution space compared with BF. The next improvement of the IBF is related to step 15-1 with the aim of improving its convergence behavior. In the eliminationdispersal process of BF a good solution may be eliminated and replaced by a poor one, considering that each execution of this process is after completing a series of all reproduction loops (i.e., when the population becomes more mature). To remedy this problem, in step 15-1 of the proposed IBF, a new bacterium is generated from each individual i based on the proposed mutation operation of (16). Considering adaptively reduced value of βmax in (17), the newly generated bacterium will be in the vicinity of the i th matured individual and so better performance may be expected from it compared with a randomly generated bacterium within the entire domain. Moreover, the new bacterium is compared with the i th individual. If it has a lower CF value, it replaces the i th individual; otherwise the i th individual survives and the new bacterium is discarded. In this way, the IBF saves its good solutions and can benefit from a local search around the matured individuals to find possible better optima.

5 Finally a simple but important improvement is applied to step 19 of BF in the IBF. In the step 19, when the IBF algorithm is terminated, the bacterium with the lowest value of AOF (instead of CF) is selected as the final solution of the IBF. Its rationale is discussed in the next section. IV. Numerical Results The first test case of this section is an optimization problem with two decision variables θ1 and θ2, presented in [27]. In Fig. 2, the objective function of the problem, which should be minimized, in terms of the decision variables θ1 and θ2 is shown. In this figure, different areas from the worst area to the best one are indicated by white, light grey, heavy grey and black colors, respectively. As seen, it is a nonlinear optimization problem with multiple local minima. Thus, the enhanced exploration capability of the IBF can appropriately be illustrated on it. Sample results of BF algorithm for this optimization problem in the two dimensional space of the decision variables are shown in Fig. 3. Trajectory of each bacterium of the BF population is indicated by dashed line. As seen, each bacterium moves toward the other bacteria, due to the swarming effect, and all bacteria try to concentrate on the promising areas indicated by black or heavy grey circles. The results of Fig. 3 have been obtained after the first iteration of the reproduction loop (the first execution of chemotaxis loop). The BF results after the first, second, third and fourth iterations of the reproduction loop are shown in Fig. 4(a), 4(b), 4(c) Fig. 2) Objective function of the first test case with respect to its two decision variables θ 1 and θ 2 Fig. 3) Obtained results of BF algorithm for the optimization problem of Fig. 2 after the first iteration of the reproduction loop (l=0, k=1) Fig. 4) Obtained results of BF algorithm for the optimization problem of Fig. 2 for l=0, k=1 (a), l=0, k=2 (b), l=0, k=3 (c) and l=0, k=4 (d) and 4(d), respectively. After the second iteration, all bacteria of BF concentrate on the global optimum and four local optima of the optimization problem. After the fourth iteration, all bacteria only concentrate on the global optimum. For this example, Nre=4 and thus each iteration of elimination-dispersal loop contains 4 iterations of reproduction loop. This example illustrates high exploitation capability of BF algorithm for the local search of the promising area. However, the good convergence behavior of BF algorithm to solve this problem is obtained when the bacteria of the initial population are generated within the promising area (the ranges of 0<θ1<30 and 0<θ2<30, shown in Fig. 3), enabling the BF algorithm to locally search the area and find the global optimum. However, there is no guarantee that the random generation produces the initial population within the promising area, especially for OPF-SC problem with complex solution space and much more dimensions (decision variables). To better illustrate this matter, we changed the initial population of BF for the optimization problem of Fig. 2 such that the initial population is randomly generated within the ranges 0<θ1<3000 and 0<θ2<3000 (100 times larger than the ranges of the promising area) and executed the BF algorithm with the same user defined parameters. Its obtained results after the first, second, third and fourth iterations of the reproduction loop are shown in Fig. 5 similar to those of Fig. 4. However, Fig. 5 shows that no bacterium can even enter the promising area in this case. We even tripled the number of iterations and executed the BF with 3 4=12 iterations of the reproduction loop (3 iterations of elimination-dispersal loop), but again no bacterium enters the promising area. Thus, no useful solution is obtained at all with the new initial population even with the increased number of iterations. The proposed IBF is tested on this problem in the two cases. The user defined parameters of the IBF (NS, Nch, Nswim, Sr, Nre, and Ned) for this problem are set the same as the BF. With the initial population generated within the promising area, the proposed IBF can find the global optimum after the first iteration of the reproduction loop. More importantly, when the initial population is generated within the ranges 100 times larger than the ranges of the promising area (0<θ1<3000 and 0<θ2<3000), two bacteria of the proposed IBF can enter the promising area after 2 iterations of the reproduction loop, as shown in Fig. 6(b), due to its high exploration capability. Then, one of these bacteria reaches the global optimum in the third iteration of the reproduction loop and some other IBF bacteria also enter the promising area as shown in Fig. 6(c). It is noted that when even one bacterium reaches the global optimum, the problem is solved and the global optimum is found, since this bacterium is returned as the final solution of the IBF. Here, the importance of the last improvement, applied to step 19 of the IBF, is revealed. In step 19 of BF, the bacterium with the lowest value of CF is determined as the final solution of the optimization problem.

6 Fig. 5) Obtained results of BF algorithm for the optimization problem of Fig. 2 with the initial population randomly generated within the ranges 100 times larger than the ranges of the promising area after iteration 1 (a), iteration 2 (b), iteration 3 (c) and iteration 4 (d) of the reproduction loop Fig. 6) Obtained results of the proposed IBF algorithm for the optimization problem of Fig. 2 with the initial population randomly generated within the ranges 100 times larger than the ranges of the promising area after iteration 1 (a), iteration 2 (b), iteration 3 (c) and iteration 4 (d) of the reproduction loop Since the BF bacteria concentrate on the optimum point, provided that it can be found (like the results shown in Fig. 4), the difference between CF and AOF of the best individual, reaching the optimum point, is very small and thus CF can be considered instead of AOF to judge about the best individual. However, considering the higher search diversity of the IBF, it is possible that only one or a few bacteria find the global optimum like the results shown in Fig. 6. Thus, considering CF for selecting the final solution may be misleading for the IBF, since its best individual may have considerable distances with the other bacteria leading to high values of the virtual cost function JCC(i,j,k,l) in (7)-(8) and so high values of CF in (6). Therefore, although CF is employed in the IBF to save its exploitation ability along the evolution process, at the end of the algorithm, the final solution of IBF is selected based on the AOF. If we further continue the IBF algorithm in this numerical experiment, more individuals enter the promising area and also more bacteria reach the global optimum as seen from Fig. 6(d). For this optimization problem, we even randomly generated the initial population within the ranges 300 and 500 times larger than the ranges of the promising area and the proposed IBF found the global optimum as well, further indicating its high exploration capability. In the following, obtained results for the OPF-SC test cases are presented and discussed, respectively. Obtained continuous values from the IBF for the discrete decision variables of the OPF-SC (presented in subsection II.C) are rounded up/down to the nearest discrete values in these test cases. Test Case 1: 26-bus test system (reference of data: [30]). Obtained results from the proposed IBF for this test system are shown in Table I and compared with the results of seven other solution methods. All reported results for the other solution methods in the OPF-SC test cases of this paper are directly quoted from their respective references. The best, average and worst results of the IBF among 20 trial runs are represented in Table I (in all numerical experiments of this paper, 20 trial runs are considered for the IBF). The OPF-SC model employed for the seven benchmark methods of Table I in their respective references has fuel cost of thermal units with valve loading effect as the objective function subject to constraints mentioned in subsection II.B except POZ constraints. Moreover, test case 1 has 6 generators, 7 tap-changing transformers and 9 capacitor banks without any phase shifter leading to NP= =27 decision variables based on (4). For the sake of a fair comparison, this OPF-SC model with the same data is also adopted for the proposed IBF in this numerical experiment (solution methods compared in each test case of this paper have the same OPF-SC model and data). Observe from Table I that the best, average and worst results of the proposed IBF are better than the best, average and worst results of all seven other methods of Table I. Even the worst result of the IBF is better than the best result of all other methods illustrating its capability for the solution of the OPF- SC problem. Test Case 2: IEEE 30-bus test system (references of data: [24] and [31]). Objective function and constraints of this test case are as described for the previous one. However, it has NP= =17 decision variables according to (4). Obtained best, average and worst results from the IBF for this test case are represented in Table II and compared with the results of three other methods. It is seen that the IBF outperforms the other methods of Table II. Compared with the first two benchmark methods of Table II (PSO [24] and HPSO [24]), even the worst result of the IBF is better than their best results. Moreover, the average and worst results of the IBF are better than the best and average results of the third benchmark method (RDEA [33]), respectively. As a sample, the details of the best solution of the IBF for this test case are also represented in Table III. However, due to space limitation and for the sake of conciseness, only the final results (like Tables I and II) are presented for the other test cases of the paper. Test Case 3: IEEE 30-bus test system (reference of data: [26]). This test case, reported in Table IV, includes 3 subcases, wherein the proposed IBF is compared with seven other OPF-SC solution methods. The objective function of this test case is fuel cost function (subcase 3.1), fuel cost function with multi-fuel option (subcase 3.2) and fuel cost function with valve loading effect (subcase 3.3), respectively. Constraints of all subcases of this test case are as described for the test case 1. Moreover, the decision variables of the three subcases are the same with NP= =15. In Table IV, Table I) Obtained results for test case 1 Generation cost ($/hr) Best Average Worst EP [30] MIGA [30] EP [32] GA [32] PSO [23] MIPSO [32] RDEA [33] IBF Table II) Obtained results for test case 2 Generation cost ($/hr) Best Average Worst PSO [24] HPSO [24] RDEA [33] IBF

7 Table III) Obtained values for the variables of test case 2 in the best solution of the proposed IBF Variable Value Variable Value P g V g P g V g P g V g P g T P g T P g T V g T V g Q c5 16 V g Q c24 2 Generation cost ($/hr) Table IV) Obtained results for the three subcases of test case 3 Fuel cost ($/hr) Subcase 3.1 Subcase 3.2 Subcase 3.3 EP [34] TS [34] TS/SA [34] ITS [34] IEP [34] SADE_ALM [35] MDE [26] IBF only the best result of each method is presented, since only this result is reported for the 7 benchmark methods in their respective references. As seen, the proposed IBF outperforms all 7 other benchmark methods of Table IV in all 3 subcases. Test Case 4: IEEE 118-bus test system (references of data: [31] and [36]). Obtained results for this test case by the proposed IBF are reported in Table V and compared with the results of 3 other methods. The objective function and constraints of the OPF-SC model used for all methods of Table V are as described for the previous subcase 3.1. However, this test case has NP= =130 decision variables. Observe from Table V that the proposed IBF outperforms all other methods. Even the worst result of the IBF is better than the best result of the two first methods of Table V (i.e., GA [36] and EPGA [37]). Moreover, the best result and even the average result of the IBF are better than the best result of the RDEA [33] and the worst result of the IBF is better than the worst result of the RDEA [33]. Significant differences are observed between the best result of the IBF and the best results of the GA [36] and EPGA [37]. Number of decision variables of this test case is much more than the other test cases of the paper, leading to a much larger solution space. Thus, greater differences between the results of different solution methods may be observed for the IEEE 118-bus test case compared to the other test cases. A solution method may be trapped in a poor local minimum of the complex large solution space of this test case and so great difference is observed between its result and the result obtained by another solution method that has higher exploration capability and can escape from such local minima. All reported results for the IBF in this paper are feasible solutions satisfying the OPF-SC constraints. In other words, all penalty terms penalizing deviations from the OPF-SC constraints become zero along the evolution process of the IBF resulting in final feasible solution. To better illustrate this matter, OF (the fuel cost function), AOF (OF + penalty terms) and sum of penalty terms for the best bacterium of the IBF in a trial run for test case 4 (the largest test case of the paper) are shown in Fig. 7. Observe that in iteration 202, the value of the whole penalty terms becomes zero such that the values of the OF and AOF become the same (4778 $/hr). Moreover, the value of the whole penalty terms remains zero in all remaining iterations from 202 up to 1200 (feasible solutions are obtained in all of these iterations). Test Case 5: IEEE 30-bus test system (references of data: [24] and [31]). In this test case, the efficiency of the proposed IBF to solve OPF-SC problem with another objective function is also evaluated. Obtained results from the IBF with fuel cost (F) and fuel emission (E), owning the quadratic function model, as the objective are shown in Table VI and compared with the results of 4 other methods. Moreover, based on [24], two subcases are considered Table V) Obtained results for test case 4 Generation cost ($/hr) Best Average Worst GA [36] NA NA EPGA [37] NA NA RDEA [33] IBF with NP= =11 decision variables (only including active power generation and voltage magnitude setting of units) as subcase 5.1 and NP= =17 decision variables (similar to test case 2) as subcase 5.2 for each objective function in Table VI. In this numerical experiment, the proposed IBF is compared with two deterministic optimization methods of sequential quadratic programming or SQP (MATPOWER software [31]) and interior point method or IPM (PSAT software [38]) and two stochastic search techniques of PSO and HPSO proposed in [24]. Observe from Table VI that the proposed IBF has better result than all other methods with both the objective functions in both the subcases indicating the superiority of the IBF. Test Case 6: IEEE 118-bus test system (references of data: [31] and [36]). In this numerical experiment a structural analysis is performed in which the proposed improvements for the IBF are applied step by step to BF and the obtained results in each stage are separately reported. IEEE 118-bus test system with the same OPF- SC model and data of the test case 4 is considered here. The obtained results are shown in Table VII. In this Table, the first improvement I1 indicates replacing the tumble of (10) in the step 4 of BF with the proposed mutation operation of (16) and replacingc( i). in (12) of the step 6-2 of BF by. DV ( i1, j, k, l ) DV ( i 2, j, k, l ). However, i1, i 2NS βmax is a constant value in this case. In the BF+I1+I2, the adaptation procedure is also added and βmax is adaptively changed along the iterations of the reproduction loop based on (17). In table VII, I3 includes the improvements applied to steps 11-1 and Also, I4 and I5 are those applied to the steps 15-1 and 19 of the IBF. Observe from Table VII that by applying each improvement to the BF, its performance enhances and better results are obtained step by step. Also, it is seen that the improvements I1 and I2 have higher shares in the enhanced performance of the proposed IBF. By adding all improvements, we reach the good solutions of the IBF. This numerical experiment better illustrates the effectiveness of the proposed improvements for the IBF. Computation times of the proposed IBF for test cases 1-5 are shown in Table VIII. All reported computation times of this paper are average CPU time of 20 trial runs measured on a simple hardware set of Pentium IV 3.6 GHz with 4 GB RAM. Since the computation times of the other methods represented in Tables I, II, IV, V and VI have been measured on different hardware sets Table VI) Obtained results for test case 5 with fuel cost (F) and fuel emission (E) as the objective function (only the best result of the PSO, HPSO and IBF is reported) Fuel Cost F ($/hr) Fuel Emission E (ton/hr) Subcase 5.1 Subcase 5.2 Subcase 5.1 Subcase 5.2 SQP (MATPOWER) NA NA IPM (PSAT) NA NA PSO [24] NA NA HPSO [24] IBF Table VII) Results of the structural analysis on the proposed IBF Generation cost ($/hr) Best Average Worst BF BF+I BF+I1+I BF+I1+I2+I BF+I1+I2+I3+I BF+I1+I2+I3+I4+I5=IBF

8 Fig. 7) Obtained results from the IBF with N ed N re N ch= =1200 iterations for test case 4: AOF and OF (a) and sum of penalty terms (b) employed in their respective references, the computation times are not compared here. Only the BF method is also implemented in this research work and so its computation times, measured on the same hardware set, are also reported in Table VIII. Low computation times of the IBF in this Table indicate its low computation burden to solve the OPF-SC problem. The highest computation time is related to IEEE 118-bus test system, which is about 2 minutes. Moreover, the computation times of the IBF are close to those of the BF. Even in some cases, the computation times of the BF and IBF are approximately the same. The largest difference is related to test case 4 (IEEE 118-bus test system), which is about 5s or 4% more computation time of the IBF, while the results of the IBF for this test case are much better than the results of the BF as shown in Table VII. The most time consuming part of solving OPF-SC problem by the IBF algorithm is related to AC power flow execution, required for computing AOF for each bacterium in each iteration. To reduce the computation time of the IBF for large practical power systems, the AC power flow may be replaced by simpler versions, such as the DC power flow. For instance, with DC network modeling for IEEE 118-bus test system, the computation time of the IBF for solving the OPF-SC problem reduces to less than one minute on the same hardware set. Higher reductions in the computation time are expected for larger power systems. Besides, today s power systems dispatching centers are usually equipped with PAS (power application software) servers, owning high floating point and integer processing rates. Those centers regularly run OPF-SC operational function to update system s control variables. Running the IBF on such servers will lead to lower computation times compared to the Table VIII) Computation times in terms of second for the BF and proposed IBF in the test cases 1-5 Test case BF IBF Subcase Subcase Subcase Subcase Subcase simple hardware setup used in the present research work. The proposed IBF is a robust algorithm and has low sensitivity with respect to the initial points. For instance, difference between the best and worst solutions of the IBF among its 20 trial runs for the largest test case of the paper (IEEE 118-bus test case) is only about 0.2%. As another instance, for test case 1, difference between the best and worst solutions of the IBF among its 20 trial runs is only about 0.07%. The negligible differences between the consecutive executions of the IBF indicate that its results can reasonably be replicated in real world applications. Contingencies have not been taken into account in the previous numerical experiments, since the benchmark methods of these test cases do not consider contingencies. In [30], for the 26-bus test system (test case 1), line 2-7 outage is selected as one of the most critical contingencies from the results of contingency analysis. OPF- SC results for the 26-bus test system considering this contingency obtained by two solution methods of MIGA (mixed-integer genetic algorithm) and evolutionary programming (EP) have been presented in [30]. Obtained results from the proposed IBF for this OPF-SC test case with contingency (line 2-7 outage) are reported in Table IX and compared with the results of MIGA and EP quoted from [30]. It can be seen that the best, average and worst generation costs of the IBF with considering the contingency are slightly more than its best, average and worst generation costs without considering the contingency (reported in Table I), since by considering the contingency, some additional constraints (bus voltage magnitude and branch flow limits in the post-contingent state) are also included in the OPF-SC problem. From Table IX, it can be observed that the proposed IBF outperforms the MIGA and EP. Even the worst result of the IBF is better than the best results of both the MIGA and EP. To further illustrate the capability of the proposed method to solve OPF-SC problem with contingency, IEEE 118 bus test system with 177 contingencies is considered in the numerical experiment of table X. This test system has 186 branches. The 177 contingencies include all single-branch outages of IEEE 118 bus test system, except its 9 islanding contingencies. To consider the effect of the contingencies in the OPF-SC model, the method of sensitivity factors is used here [39]. Table X shows that the generation cost of the final solution obtained by the proposed IBF is considerably lower than the generation cost of the final solution of BF, while the