Tuned reactive power dispatch through modified differential evolution technique

Abstract

This paper explores the capability of modified differential evolution (MDE) technique for solving the reactive power dispatch (RPD) problem. The proposed method is based on the basic differential evolution (DE) technique with a few modifications made into it. DE is one of the strongest optimization techniques though it suffers from the problem of slow convergence while global minima appear. The proposed modifications are tried to resolve the problem. The RPD problem mainly defines loss minimization with stable voltage profile. To solve the RPD problem, the generator bus voltage, transformer tap setting and shunt capacitor placements are controlled by the MDE approach. In this paper, IEEE 14-bus and IEEE 30-bus systems are chosen for MDE implementation. The applied modification show much improved result in comparison to normal DE technique. Comparative study with other soft-computing technique including DE validates the effectiveness of the proposed method.

Keywords

reactive power dispatch (RPD) / modified differential evolution (MDE) / differential evolution algorithm with localizations around the best vector (DELB)

Introduction

In the emerging field of power system, a number of variables are continuously beating the stability of the normal power generation, transmission and distribution environment. In this connection, the first variable to be described is the bus voltage profile followed by active power and the reactive power. In 1960, Carpentier proposed optimal power flow (OPF) study where real power loss minimization with respect to cost effectivity was discussed. Since then several studies have been conducted to solve the same problem. The OPF [1] problem becomes more complex due to inclusion of cost minimization in it. In that scenario, the reactive power dispatch (RPD) [2] problem gains its importance in the emerging field of power system studies due to its balanced voltage profile with real power loss minimisation aspect. The RPD problem can be solved by controlling bus voltages of different generator buses; placing the transformer tap settings to a few load buses. One more control mechanism related to the RPD problem is usually applied, i.e., shunt capacitor placement in a few buses which helps to compensate reactive power by absorbing lagging Var from the system and keeping the bus voltage up to the required level. Apart from these control variables, there are a few state variables which are to be kept in their specific operating zone during the optimization. These variables are bus voltage angle, load bus voltage magnitude, slack bus real power generation and generator reactive power generation which are dependent on objective function. In this paper focus is given to control real power generation for a fixed amount of load. According to the problem criteria, the RPD problem is considered as a single objective combinatorial optimization problem which has several equality and inequality constraints. Besides, two IEEE standard bus systems (14-bus and 30-bus) are considered to solve the RPD problem.

The RPD problem was already solved by various linear and non linear programming method [3,4]. As the RPD problem is a non linear multi-constrained optimization problem, traditional linear programming method are not sufficient enough to fetch the optimum results. In this situation soft-computing techniques were attempted to solve the problem. Presently these techniques are gaining the highest success in solving such kind of nonlinear, single or multi objective optimization problem in the various emerging field of research. Certainly the RPD problem was evaluated with several evolutionary algorithms such as evolutionary programming (EP), tabu search (TS), simulated annealing (SA), genetic algorithm (GA) and particle swarm optimization (PSO), some of which had proven to be very promising. In conjunction with the RPD problem, differential evolution (DE) is also attempted to solve it. Wu and Ma examined the EP based RPD problem in 1995 [5]. TS method was elaborated by Abido in 2002 [6]. SA was applied to solve the RPD problem by Roa-Sepulveda and Paves-Lazo [7], and Raha et al. [8] to solve the RPD problem in the IEEE bus system. GA [9] was approached by several researchers to solve the same problem. Subbaraj and Rajnarayanan showed self adaptive real coded genetic algorithm (SARGA) with through survey [10]. Yoshida et al. applied PSO in detail for solving the same problem [11]. Mahadevan and Kannan elaborated the comprehensive learning particle swarm optimization (CLPSO) technique for RPD [12]. The DE technique in the OPF area was presented by Abou et al. [13] and Liang et al [14]. Varadarajan and Swarup explained the importance of the RPD problem and the solution methodology by DE in Ref. [15]. All these techniques were tested while keeping their advantages and limitations in mind. In this paper to solve the RPD problem, the modified differential evolution (MDE) technique is chosen, and two modifications are applied according to the problem requirement.

The DE technique was developed by Storn and Price [16] in 1995. The DE technique was approached to solve several non linear problems, the result of which shows very encouraging performance [17]. DE is found to be more robust as it gives minimum standard deviation among the solutions obtained from random trials. Although DE is one of the strongest solving optimization techniques, it suffers from the problem of slow convergence [18]. To overcome the drawbacks of DE, a few modifications are performed with the DE technique. After modification the new technique was termed as MDE with localizations around the best vector (DELB) [19]. The formulated MDE technique was applied to the objective function for both testing cases. In parallel with modifications in the basic DE technique, parameter tuning was also set aside with the MDE. The novelty of the paper is the proposal of new modifications with the DE technique in solving the RPD problem for the IEEE 14-bus system and the IEEE 30-bus system. The obtained result is compared with previously performed soft-computing technique-based work including normal DE. While comparing, the distinguished result shows the effectiveness of the proposed method. The tuned parameters of the proposed technique show much improvement though some limitations are considered for simplicity during the problem formulation.

Problem formulation

The RPD problem is a very old classical problem which appears for the smooth conduct of power by minimizing the active power losses keeping the system stability uninterrupted. System stability is dependent upon balanced voltage profile which is controlled by several means such as control of generator bus voltage, transformer tap setting control, and shunt capacitor placement. Thus considering the entire constraints including generator bus voltages, the RPD problem is developed.

RPD problem can be mathematically expressed as

(1)\min \mathit{f}\left(\mathit{x}\mathit{,}\mathit{u}\right)\mathrm{.}

Now f(x,u) is the objective function which typically includes minimization of total real power losses in the transmission system (RPD), where x represents the state or dependent variable of the system and u represents the control or independent variable.

The state vector consists of:

1) Generator real power output at slack bus (P_g1),

2) Generator reactive power output (Q_G),

3) Load bus voltage (V_L), and

4) Bus voltage angle (θ).

Hence, BoldItalic can be expressed as

(2)x^{\mathrm{T}}=[P_{\mathrm{G}\mathrm{1}},Q_{\mathrm{G}\mathrm{1}},\mathrm{...},Q_{\mathrm{G}\mathrm{N}\mathrm{G}},V_{\mathrm{L}\mathrm{1}},\mathrm{...},V_{\mathrm{L}\mathrm{N}\mathrm{L}},{\theta }_{\mathrm{1}},\mathrm{...},{\theta }_{n\mathrm{b}\mathrm{u}\mathrm{s}}]\mathrm{.}

where NG, NL are the number of the generator and number of load buses respectively and nbus are the total buses in the network.

The control variables consists of:

1) Generator bus voltages V_G,

2) Shunt Var compensation Q_C, and

3) Transformer taps setting T.

Hence, u can be expressed as

(3){\mathit{u}}^{\mathrm{T}}=[V_{\mathrm{G}\mathrm{1}},\mathrm{...},V_{\mathrm{G}\mathrm{N}\mathrm{G}},Q_{\mathrm{C}\mathrm{1}},\mathrm{...},Q_{\mathrm{C}\mathrm{N}\mathrm{C}},T_{\mathrm{1}},\mathrm{...},T_{\mathrm{N}\mathrm{T}}].

where NC and NT are the number of the Var compensators and regulating transformers respectively.

In connection with this, the RPD problem can be expressed as

(4)\min P_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}={\displaystyle \sum _{i=1}^{\mathrm{N}\mathrm{G}}{P}_{\mathrm{g}\mathrm{e}\mathrm{n}}}-{\displaystyle \sum _{\begin{array}{c}i=1,\\ i\ne \mathrm{s}\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{b}\mathrm{u}\mathrm{s}\end{array}}^{n\mathrm{b}\mathrm{u}\mathrm{s}}{P}_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}},}

where P_gen and P_load is the total generated power and the total load power connected to the given system. P_loss is the power loss in the network. In this paper P_gen and P_load is calculated with load flow study via Newton Raphson method [20]. The minimization of the above objective function is subjected to a number of equality and inequality constraints.

Equality constraint:

(5)g(x,u)=\mathrm{0},

i.e., can be represented by typical load flow equations:

(6)\begin{array}{l}{P}_{\mathrm{G}i}-P_{\mathrm{D}i}-V_i\hfill \\ \cdot {\displaystyle \sum _{j=1}^{n\mathrm{b}\mathrm{u}\mathrm{s}}{V}_j\left[G_{ij}\cos \left({\delta }_i-{\delta }_j\right)+B_{ij}\sin \left({\delta }_i-{\delta }_j\right)\right]=0,}\hfill \end{array}

(7)\begin{array}{l}{Q}_{\mathrm{G}i}-Q_{\mathrm{D}i}-V_i\hfill \\ \cdot {\displaystyle \sum _{j=1}^{n\mathrm{b}\mathrm{u}\mathrm{s}}{V}_j\left[G_{ij}\sin \left({\delta }_i-{\delta }_j\right)-B_{ij}\cos \left({\delta }_i-{\delta }_j\right)\right]=0,}\hfill \end{array}

where P_G is the real power generation, Q_G is the reactive power generation, P_D is the active load demand, Q_D is the reactive load demand, {\delta }_i and {\delta }_j is the voltage angle of bus i and j respectively, G_ij and B_ij are the conductance and the susceptance between bus i and j respectively.

Inequality constraint

(8)\mathit{h}\left(x,u\right)\le 0.

These constraints are

1) Generation constraints: The generator bus voltages, reactive power outputs are restricted by their lower and upper limits as

(9)V_{\mathrm{G}i}^{\min }\le V_{\mathrm{G}i}\le V_{\mathrm{G}i}^{\max }(i=\mathrm{1},\mathrm{...},\mathrm{N}\mathrm{G}),

(10)Q_{\mathrm{G}i}^{\min }\le Q_{\mathrm{G}i}\le Q_{\mathrm{G}i}^{\max }(i=\mathrm{1},\mathrm{...},\mathrm{N}\mathrm{G}).

2) Transformer constraints: The transformer tap settings are restricted by their lower and upper limits as

(11)T_i^{\min }\le T_i\le T_i^{\max }(i=\mathrm{1},\mathrm{...},\mathrm{N}\mathrm{T}).

3) Shunt VAR constraints: The shunt VAR compensators are restricted by their lower and upper limits as

(12)Q_{\mathrm{c}i}^{\min }\le Q_{\mathrm{c}i}\le Q_{\mathrm{c}i}^{\max }(i=\mathrm{1},\mathrm{...},\mathrm{N}\mathrm{C}).

4) Security constraints: This includes the constraints of voltage at load buses as

(13)V_{\mathrm{L}i}^{\min }\le V_{\mathrm{L}i}\le V_{\mathrm{L}i}^{\max }(i=\mathrm{1},\mathrm{...},\mathrm{N}\mathrm{L}).

Once the objective function is fixed, the MDE technique is applied to solve the problem in hand.

Modified differential evolution

The MDE technique evolves from the core concept of the DE technique. DE was proposed by Storn and Price in 1995 [16]. It is a very promising meta-heuristic optimization technique which deals with non-linear, non-differentiable, multi-objective, and multi-constraint objective function. Before going to the details of MDE, the normal DE technique needs to be precisely explained. DE involves four basic steps: initialization, mutation, crossover and selection.

Overview of DE

At the first step, DE initiates a set of target vector x_ij with the help of random number generation by using the uniform probability distribution function. In the mutation phase, DE randomly selects three perturbation points from the population pool. Including the randomly chosen points three sets of target vectors are formed, i.e., x_ij(r₁), x_ij(r₂) and x_ij(r₃) where r₁, r₂, r₃ are the random numbers, and r_{\mathrm{1}}\ne r_{\mathrm{2}}\ne r_{\mathrm{3}}. By combining these three sets of vectors, the mutant vector v_ij is formed which is shown in Eq. (14), where, i\in \{\mathrm{1},\cdots ,N_P\}, j\in \{\mathrm{1},\mathrm{...},D\}, and N_P and D is the number of population and dimension of the required target vector respectively.

Mutant vector is represented as

(14)v_{i,j}(t+\mathrm{1})=x_{r_{\mathrm{1}},j}(t)+f_{\mathrm{m}}(x_{r_{\mathrm{2}},j}(t)-x_{r_{\mathrm{3}},j}(t)),

where f_m is the scaling factor or mutation factor (f_{\mathrm{m}}\in (\mathrm{0},\mathrm{1})) and t is the generation counter.

In the crossover stage, the new offspring or trial vector is generated depending upon the crossover factor (C_{\mathrm{R}}\in \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}). The trial vector (y_ij) is expressed as

(15)

At the selection stage of DE, a comparison is performed to generate better or efficient offspring between the target vector and the trial vector set. The firm one of the two will be chosen for the next generation.

This way the DE process continues till the terminating condition arises. At the final generation the obtained offspring is to be considered as the final solution. The terminating condition may be fixed as setting of optimum value or reaching to the maximum generation number.

MDE technique

Among many meta-heuristic techniques approachable to solving non-linear, non-differentiable, multi-objective, and multi-constraint objective function, DE shows very commendable result though there is a big drawback with the technique. The DE method usually suffers the problem of slow convergence as the region of global minima appears. As remedy to this problem a new idea [19] is approached, according to which, modification in the DE technique may be performed via two ways of which the new DE algorithm with localizations around the best vector (DELB) is chosen as a solving tool.

DELB　Similar to the DE algorithm, the DELB method is preceded with four basic steps like initialization, mutation, crossover and selection. The highlighting point of the DELB technique is that this concept tries to search an even better solution compared to the generated trial vector or target vector solution. At the selection stage of the DELB technique, a fine tuning is introduced by forming two more hybrid vector sets and a comparative study of their corresponding fitness value. The hybrid vectors are formed by combining the trial vector and the target vector set. This new modification may slow down the processor speed though the probability of getting good result increases. In the DELB method, at the selection stage, the searching process gets emphasized, and the technique is termed as localization around the best vector or the DELB algorithm.

The DELB method can be demonstrated as.

After initialization the mutant vector is generated using Eq. (14). In this case f_m is selected by random number generation (f_{\mathrm{m}}\in \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}(\mathrm{0},\mathrm{1})).

In the third step i.e. crossover, the trial vector (y_ij) will be selected as presented in Eq. (15). In the selection step, the tuned selection is implemented where, instead of choosing the final solution between the trial vector and the target vector, two more hybrid vector sets are developed. The fitness values of the new vectors are compared according to the conditions as explained in the steps given below (Steps a to d). For every generation, the better fitness value generating vector will be considered as the fit solution. As a terminating condition reaching to the final generation (Gen_max) is considered here and the corresponding value will be said as the final solution vector. The introduced tuned vector may decelerate the process but the probability of getting good value increases. The tuned selection process is implemented using the following steps:

Step a: The fitness value is calculated with the help of both the trial and mutant vector. If the mutant vector gives a better result it will be considered as the final solution and Step a will again be started for the next generation. Otherwise if Rⁱ<w and if the fitness value of the target vector (x) is greater than the fitness value of the trial vector (y), a new set of vector r_i is framed which is expressed as

(16)r_i=x_{\mathrm{b}}-(y_i-x_{\mathrm{b}}),

where Rⁱ is a random number between 0 to 1, w is a constant which equals 0.5, x_b is the current best target vector and y_i is the trial vector set.

Step b: In this step, by putting both the r_i set and the y_i set, the fitness value is calculated. If the r_i set gives a better result, v_i will be replaced by the r_i hybrid vector set and the process will be again started from Step a else Step c will be continued where a new set of vector c_i will be considered which is expressed as

(17)c_i=v_{\mathrm{b}}+\mathrm{0.5}(y_i-v_{\mathrm{b}}),

where v_b is the mutant vector set.

Step c: In this step, the fitness function is calculated by putting the c_i set and the result is compared with the fitness value by the trial vector set. If the c_i set gives a better result, c_i will replace v_i and the process will be again started from Step a else Step d will be initiated.

Step d: Here v_i will be replaced by y_i and the process will again be started from Step a for the next generation. Thus the final v_i will be declared as the fitness vector for the said problem. In this paper the searching process will be continued till the final generation comes. The DELB algorithm is explained by the flow chart illustrated in Fig. 1.

MDE implementation in RPD problem

A very old classical problem (RPD) being solved with the help of MDE technique is discussed in this paper. Certainly there is innovativeness in the proposed DE-based modification. The proposed modification not only brings the improved result but also reduces the processing time. Therefore the same problem with a better solution is the novelty of the paper via the MDE technique.

The proposed algorithm is developed by MATLAB-7.1 software programming tool. The MDE-based results are evolved and the algorithm is compared with other soft computing methodology-based results including the normal DE technique. The RPD problem i.e. the minimization of active power loss is controlled by varying V_G, T and Q_c and keeping all the equality and inequality constraints under its operating domain. The limiting value of the P_g, V_L, θ, Q_G1 is checked throughout the program execution such that no limit violation occurs. The Newton Raphson iteration technique [20] is applied to solve the load flow study used during the MDE formulation. The optimization process stops whenever the preset maximum generation is reached. While IEEE 14-bus system is considered, the total ten control variables are determined. In the case of IEEE 30-bus system, three test cases with 11, 13 and 19 number of variables are controlled.

MDE implementation with the controlled variables

Initially a set of target vector (x) is generated within the maximum and minimum boundary limit using Eq. (18). The boundary values are listed in Table 1.

(18)x_{ij}^{(\mathrm{0})}=x_j^{\min }+{\sigma }_j(x_j^{\max }-x_j^{\min }),

where i=1, 2,…, N_p (the number of population) and j=1,2,…,D. Equation (18) is considered here for the initial generation. The IEEE 14-bus network<FootNote>

Power system test case archive, December 2006 [Online]. Available: http://www.ee.washington.edu/research/pstca/

</FootNote> consists of 14 buses, among which bus 1 is considered as the slack bus having 1.07 pu as magnitude. In the same network, buses 2, 3, 6 and 8 are the PV or generator buses whose voltages are controlled (V_g2, V_g3, V_g6 and V_g8) and the remaining 9 are load buses. In the network, three tap changing transformers are placed at the three lines (T_4-7, T_4-9 and T_5-6). In addition, buses 9 and 14 are chosen for the shunt compensation (Q_C9, Q_C14). The target vector is framed as

(19)x^{\left(0\right)}=\left[\begin{array}{ccccc}{x}_{\mathrm{V}\mathrm{g}{1}_{i=1}}& x_{\mathrm{V}\mathrm{g}{1}_{i=2}}& x_{\mathrm{V}\mathrm{g}{1}_{i=3}}& \mathrm{...}& x_{\mathrm{V}\mathrm{g}{1}_{i=N_{\mathrm{p}}}}\\ x_{\mathrm{V}\mathrm{g}{2}_{i=1}}& x_{\mathrm{V}\mathrm{g}{2}_{i=2}}& x_{\mathrm{V}\mathrm{g}{2}_{i=3}}& \mathrm{...}& x_{\mathrm{V}\mathrm{g}{2}_{i=N_{\mathrm{p}}}}\\ x_{\mathrm{V}\mathrm{g}{3}_{i=1}}& x_{\mathrm{V}\mathrm{g}{3}_{i=2}}& x_{\mathrm{V}\mathrm{g}{3}_{i=3}}& \mathrm{...}& x_{\mathrm{V}\mathrm{g}{3}_{i=N_{\mathrm{p}}}}\\ x_{\mathrm{V}\mathrm{g}{6}_{i=1}}& x_{\mathrm{V}\mathrm{g}{6}_{i=2}}& x_{\mathrm{V}\mathrm{g}{6}_{i=3}}& \mathrm{...}& x_{\mathrm{V}\mathrm{g}{6}_{i=N_{\mathrm{p}}}}\\ x_{\mathrm{V}\mathrm{g}{8}_{i=1}}& x_{\mathrm{V}\mathrm{g}{8}_{i=2}}& x_{\mathrm{V}\mathrm{g}{8}_{i=3}}& \mathrm{...}& x_{\mathrm{V}\mathrm{g}{8}_{i=N_{\mathrm{p}}}}\\ x_{\mathrm{T}{8}_{i=1}}& x_{\mathrm{T}{8}_{i=2}}& x_{\mathrm{T}{8}_{i=3}}& \mathrm{...}& x_{\mathrm{T}{8}_{i=N_{\mathrm{p}}}}\\ x_{\mathrm{T}{9}_{i=1}}& x_{\mathrm{T}{9}_{i=2}}& x_{\mathrm{T}{9}_{i=3}}& \mathrm{...}& x_{\mathrm{T}{9}_{i=N_{\mathrm{p}}}}\\ x_{\mathrm{T}{10}_{i=1}}& x_{\mathrm{T}{10}_{i=2}}& x_{\mathrm{T}{10}_{i=3}}& \mathrm{...}& x_{\mathrm{T}{10}_{i=N_{\mathrm{p}}}}\\ x_{\mathrm{Q}\mathrm{c}{9}_{i=1}}& x_{\mathrm{Q}\mathrm{c}{9}_{i=2}}& x_{\mathrm{Q}\mathrm{c}{9}_{i=3}}& \mathrm{...}& x_{\mathrm{Q}\mathrm{c}{9}_{i=N_{\mathrm{p}}}}\\ x_{\mathrm{Q}\mathrm{c}{14}_{i=1}}& x_{\mathrm{Q}\mathrm{c}{14}_{i=2}}& x_{\mathrm{Q}\mathrm{c}{14}_{i=3}}& \mathrm{...}& x_{\mathrm{Q}\mathrm{c}{14}_{i=N_{\mathrm{p}}}}\end{array}\right].

In the case of the DELB technique, after the formation of the target vector, the mutant vector and trial vector will be developed with the help of Eqs. (19), (14) and (15) respectively. To get the best vector around the local position, the tuned-selection is introduced here. The final solution will be obtained by following the previously explained steps of the tuned-selection via Eqs. (16) and (17).

Similar to the previous case, the target vector matrix for the IEEE 30-bus system can be framed with the different controlled variables. The IEEE 30-bus network<FootNote>

Power system test case archive, December 2006 [Online]. Available: http://www.ee.washington.edu/research/pstca/

</FootNote> consists of 30 buses, among which bus 1 is considered as the slack bus having 1.10 pu as magnitude. In the same network, buses 2, 5, 8, 11 and 13 are the PV or generator buses whose voltages can be controlled (V_g2, V_g5, V_g8, V_g11 and V_g13) and the remaining 24 are load buses. In the network, four tap changing transformers are placed at the lines (T_6-9, T_6-10, T_11-12 and T_28-27). In addition, the shunt compensation in a few buses are considered to compensate for the loss. Keeping generator bus voltage control and transformer tap settings position fixed, three cases are tested by varying shunt capacitor placement. In the first case, buses 10 and 24 are chosen for the shunt compensation (Q_c10, Q_c24) [15]. In the second case, buses 6, 17, 18 and 27 are chosen for the shunt compensation (Q_c6, Q_c17, Q_c17, and Q_c27) [10,12]. In the third case, nine buses i.e. (Q_c10, Q_c12, Q_c15, Q_c17, Q_c20, Q_c21, Q_c23, Q_c24, Q_c29) [12,10] are chosen for the shunt compensation. Now for the three cases, three different target vectors will be generated via Eq. (18). Thereafter the mutant vector and trial vector will be developed using Eqs. (14) and (15). Finally the tuned selection step will provide the optimum solution via Eqs. (16) and (17). The obtained results for the three cases are compared with previously experimented evolutionary algorithms-based result.

Terminating condition

In this paper the terminating condition for the proposed MDE technique is set as reaching at the maximum generation number (Gen_max). When the generation counter reaches the maximum generation number set by the programmer, the execution stops. The solution reaching that point is declared as the final optimum value.

Simulation result and analysis

The result of the DE optimization technique depends on the proper choice of the number of population (N_p), mutation factor (f_m), and crossover factor (C_R), in the test case of MDE, the mutant factor is selected via random number generation (f_m\in (0,1)). Hence the choice of the rest parameters is very crucial for the result. The terminating condition is fixed here as reaching the maximum generation number (Gen_max=50).

IEEE 14-Bus system for RPD problem

In this paper parameter determination is implemented by solving the RPD problem in the case of IEEE 14-bus system. During the execution, N_p and C_R both are varied to get the results. Initially N_p is kept constant and C_R is varied as 0.2, 0.5, 0.6, 0.7, 0.8 and 0.9, and six sets of result are observed. By further increasing N_p, six sets of results are taken. This way the total 24 sets (4×6) of result (N_p=20, 30, 40 and 60) are obtained, among which the best, the average and the worst value for each population with the respective C_R is kept aside. Of the four best results for different N_p (20, 30, 40 and 60), the finest one is considered as the final value for the proposed method. The corresponding N_p and C_R is chosen as the best parameter set for the proposed method. The line parameters and load flow data for both the systems are taken from footnote 1. At light load condition the base value of transmission line loss was 13.49 MW [15] for the IEEE 14-bus system. Regarding observation one point may be stated that all the results are within system limit which is given in Table 1. Therefore stability of the system remains uninterrupted.

DELB method based result Initially the DELB method is applied to solve the RPD problem. The programme execution is started with N_p=20 with different C_R as mentioned earlier. For each case, thirty sets of trial run are considered. Of all the cases the best value providing parameters (N_p, C_R) are selected as the final parameters for higher test cases (IEEE 30-bus system). The variation of N_p and C_R for the DELB method is given in Table 2. One point to be mentioned is that Table 2 shows only the best value (P_loss in MW) for all the 24 cases considered. For N_p = 20 with C_R = 0.7, the minimum P_loss value was 13.0928 MW which subsequently reduces to 13.0829 MW, 13.0696 MW and 13.0532 MW with N_p=30, 40 and 60 and C_R=0.9, 0.6 and 0.7 respectively. From Table 2 it can be implied that the medium C_R and a higher N_p gives better result. In this case, the percentage loss reduces up to 3.23795% which is shown in Table 3 where detailed values for the controlled variables, i.e., V_G, T and Q_c in pu value are also given. The power loss value with respect to the number of generations are graphically presented in Figs. 2 to 5 for the given populations (N_p=20. 30, 40 and 60). And Figures 2 to 5 show the best, the average and the worst responses for the corresponding N_p and C_R.

From the graphical view point (Figs. 2 and 3) it is cleared that the power loss curve has got very early convergence (reaching the optimum value quickly) when N_p=20 and 30 is attempted. In Figs. 4 and 5, the convergence occurs at the 24th and 48th generation respectively. According to the working principle of the DELB technique, it is a comprehensively screening-based optimization tool. From that point of view, Figs. 4 and 5 show more accuracy compared to Figs. 2 and 3. One more point to be mentioned is that as N_p increases the accuracy and the result improves.

The proposed methodology, the DELB method, is compared with the previously applied techniques to solve the same problem. The comparison proves the effectiveness of the proposed method (Table 4).

IEEE 30-Bus system for RPD problem

In this paper the IEEE 30-bus system is also considered for solving the RPD problem with the MDE approach. As the parameters (f_m and C_R) for a particular N_p has already been obtained from the previous case for the IEEE 30-bus system, the results are taken only by varying N_p (20, 30, 40 and 60). Here attempt is made to solve the RPD problem by varying the position of the shunt capacitor via three ways. For the three cases, the target vector formation is considered in the same manner as shown in Section 4.1. The line parameters and load flow data for both the systems are considered from footnote 1. Initially the base value of transmission line loss was 5.66 MW [15] for Case 1, 5.988 MW for Case 2 [21] and 5.8120 MW for Case 3 [22] of the considered IEEE 30-bus system. According to the result obtained, one thing to be mentioned is that all the controlled variables comes within range as mentioned in Table 1. Thus the stability of the system remains incessant.

DELB method based result Initially the DELB method is approached to the above mentioned cases using different N_p (20, 30, 40 and 60) with C_R = 0.7, 0.9, 0.6 and 0.7 respectively. Due to the implementation of the said parameters, 16 set (4×4) of results are obtained for the specified three cases, of which (one set contains the best, the average and the worst value) only the best values are shown in Table 5. Like the previous results, the higher N_p (60) gives the finest results for all the three cases. In Table 6 controlled variables are given in detail for the three cases. It is worth mentioning that the power loss reduction value is quite significant here. Case 3 shows minimum power loss and Case 2 shows maximum percentage of power loss reduction while solving it. For Case 1 the percentage of power loss comes at 13.78621%, where it is 23.07348% for Case 2 and 22.55867% for Case 3.

From Fig. 6, it can be concluded that the optimum value comes nearly at the 40th generation. The best value for Case 1 comes at 4.8797 MW where the average and the worst value are above 5 MW as given in Fig. 6. In the cases of Figs. 7 and 8 the convergence occurs even after the 40th and 30th generation respectively. The best, the average and the worst value are 4.60636, 4.70571 and 4.73765 MW respectively for Case 2. For Case 3 the best value reaches 4.50089 MW with the average value of 4.70571 MW and the worst value of 4.73383 MW. Eventually for both Cases 2 and 3, the best, the average and the worst values are very close. According to the optimization principle of the DELB method, the data searching process is much emphasized. Therefore, it takes little time to get a better result. All the figures obtained show that the optimum value comes at the later generation.

Finally these three cases are compared with the previously performed evolutionary technique for validation. Table 7 gives a comparative study for Case 1 where the results obtained are compared with the normal DE and PSO technique. The result shows improvement compared to the previously applied evolutionary technique mentioned here.

Table 8 shows the comparison between the proposed techniques with the recently developed two evolutionary techniques. In this case the proposed method gives better result compared to the CLPSO and PSO method. While considering the SARGA method, it can be concluded that the DELB-based result is partially weak for a higher population number.

Lastly the results of Case 3 are also compared with a few evolutionary techniques which are given in Table 9, in which it can be observed that the proposed method gives very good response compared to other techniques.

From the three sets of comparative study, it can be stated that the proposed methods not only produce good results, but also reduce the CPU operating time as well as the number of generations. Hence the utilization of modification brings a value in it.

Conclusions

In this paper, the MDE has been successfully implemented to solve the RPD problem. During the trial runs some result appear poor but the average runs generate quite stable result. The distinguished results are compared with other soft-computing technique based results including the normal DE. The comparison validates the effectiveness of the proposed method in this paper. Previously lots of work in the RPD domain has been conducted using many optimization techniques such as DE, PSO, GA etc. Application of this type of modification with the DE technique in the RPD domain for the IEEE 14-bus and 30-bus system is rare. Therefore the application of the DELB technique with the problem mentioned has its unique novelty. Meanwhile, for the sake of simplicity, voltage index and sensitivity analysis, the RPD analysis in deregulated power environment, implementation of hybrid DE with the PSO or SA technique remains untouched in this paper which will be included the study in the future. In this paper, only a few points have been focused: firstly higher population shows improved result, medium and high crossover factor provides better result, and convergence comes according to the criteria set by the proposed methods. Moreover due to proper choice of terminating condition the required operating time for the CPU is comparatively reduced. Overall, it can be summarized that the proposed method shows improved result with the same cost of arrangement.

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