3. Opf Solution Using Gravitational Search Algorithm

In this section, a novel GSA-based approach is proposed to solve the OPF problem in distribution network. The following objective functions are considered: (Case 1) fuel cost minimization, (Case 2) power loss minimization and (Case 3) simultaneous minimization of the fuel cost and power loss. The proposed approach is tested on 11-node distribution network in which three DG units are integrated.

Gravitational search algorithm (GSA) is a newly stochastic search algorithm developed by Rashedi et al. [30]. This algorithm has a great potential to become a break-though optimization method [31]. GSA has been proved as high quality performance method in solving different optimization problems in literature [31], [32-36].

In GSA, the search agents are collection of mutually interacting masses based on the Newtonian gravity and the laws of motion. In this algorithm, agents are considered as objects and their performances are measured by their masses. All these objects attract each other by the gravity force, and this force causes a global movement of all objects towards the objects with heavier masses [30]. The position of the mass corresponds to the solution of the problem, and its gravitational and inertial masses are determined using a fitness function. In other words, each mass presents a solution. The algorithm is navigated by properly adjusting the gravitational and inertial masses. By lapse of time, the masses will be attracted by the heaviest mass representштп an optimum solution in the search space.

The GSA could be considered as an isolated system of masses. It is like a small artificial world of masses obeying the Newtonian laws of gravitation and motion. In a system with N agents (masses), the position of the ith agent is defined by:

for (35)

where n is the search space dimension and defines the position of the ith agent in the dth dimension.

After evaluating the current population fitness, the mass of each agent is calculated as follows [30,36]:

(36)

where:

(37)

where fit_i(t) represents the fitness value of the agent i at time t. best(t) and worst(t) are the best and worst fitness of all agents, respectively and defined as follows (for a minimization problem):

(38)

(39)

According to Newton gravitation theory, the total force acting on the ith agent in the dth dimension at t time is specified as follows:

(40)

where rand_j is a random number in the interval [0, 1]. G(t) is gravitational constant at time t, M_i(t) and M_j(t) are masses of agents i and j, ɛ is a small constant and R_ij(t) is the Euclidian distance between the two agents i and j given by the following equation:

(41)

Kbest is the set of first K agents with the best fitness value and biggest mass, which is a function of time, initialized to K₀ at the beginning and decreased with time. In such a way, at the beginning, all agents apply the force, and as time passes, Kbest is decreased linearly and at the end there will be just one agent applying force to the others.

By the law of motion, the acceleration of the ith agent, at t time in the dth dimension is given by following equation:

(42)

Searching strategy on this notion can be defined to find the next velocity and next position of an agent. Next velocity of an agent is defined as a function of its current velocity added to its current acceleration. Hence, the next position and next velocity of an agent can be computed as follows:

(43)

(44)

where rand_i is a uniform random variable in the interval [0, 1]. This random number is utilized to give a randomized characteristic to the search. represents the position of agent i in dimension d, is the velocity and is the acceleration.

It must be pointed out that the gravitational constant G(t) is important in determining the performance of GSA. It is initialized at the beginning and will be reduced with time to control the search accuracy. In other words, the gravitational constant is a function of the initial value G₀ and time t:

(45)

where α is a user specified constant, t the current iteration and T is the maximum iteration number. Parameters of maximum iteration T, population size N, initial gravitational constant G₀ and constant α control the performance of GSA.

1. GSA implementation

Proposed GSA has been applied to solve the OPF problem for distribution network. The steps of the GSA are listed in Table 6.

Table 6. Steps of GSA.

Step 1 Step 2	Search space identification. Initialization: generate random population between minimum and maximum values of the control variables of the OPF problem.
Step 3	Fitness evaluation of each agent for the OPF problem.
Step 4	Update the G(t), best(t) and worst(t) and Mi(t) for i=1,2,...,N.
Step 5 Step 6 Step 7	Calculation of the total force in different directions. Calculation of acceleration. Updating agents' velocity and position.
Step 8 Step 9	Repeat Steps 3-7 until the stop criteria is reached. Stop.

Program was developed in MATLAB 2008b computing environment and run on a 2.20 Ghz, PC with 3.0 GB RAM. The control parameters of GSA are shown in Table 7.

Table 7. GSA control parameters.

Population size N	50
Maximum iteration number T	100
Initial gravitational constant G0	100
Constant α	20

2. Simulation results

Proposed GSA-based approach has been implemented to solve the optimal distribution power flow on real MV distribution network consisting of three feeders shown in Fig. 12 [37]. Line data and load data of the distribution network are listed in Table 8. It is supposed that three DG units are integrated in the distribution network. DG units at node 8 consist of several fuel cell stacks and DG units at nodes 9 and 10 consist of microturbine units with characteristics given in [4]. We assume that DG8 and DG10 operate in PQ mode with 0.9 power factors (produce reactive power) in all operating conditions, and DG9 is capable to control active power and voltage magnitude independently, and therefore operates in PV mode. DG units data, including cost coefficients, are given in Table 9. Limits for control variables are given in Table 10. Maximum and minimum values for voltages at load buses, including DG units which are modeled as PQ nodes, are 1.05 and 0.95 p.u., respectively.

Fig. 12. Single line diagram of MV distribution network.

Table 8. Line data and Load data.

Send. node	Rec. node		Line parameters				Load at Rec. node
			R (Ω)	X (Ω)	B (S)		PL (kW)	QL (kW)
0	1		0.20900	0.11700	0.000116		1262.4	256.2
0	2		0.20900	0.11700	0.000116		2416.6	490.5
0	3		0.20900	0.11700	0.000116		612.7	124.4
1	4		0.41800	0.23400	0.000232		435.1	88.3
2	5		0.52250	0.29250	0.000290		1148.0	233.0
3	6		0.41800	0.23400	0.000232		0	0
3	7		0.41800	0.23400	0.000232		575.4	116.8
4	8		0.83600	0.46800	0.000464		0	0
5	9		0.41800	0.23400	0.000232		1188.2	241.2
7	10		0.41800	0.23400	0.000232		0	0

Table 9. DG units data.

DG unit	Type	Mode	PDGnom (kW)	QDG (kVAr)		Cost coefficients
						a ($/h)	b ($/kWh)	c ($/kW2h)
DG8	MT	PQ	1000	cosφ=0.9		1	15·10-3	14·10-6
DG9	FC	PV	1800	-1350 ÷ 1350		3	11·10-3	7.9·10-6
DG10	FC	PQ	600	cosφ=0.9		3	1.2·10-3	35·10-6
Market	HV network	Slack node	-	-		0	25·10-3	0

Table 10. Limits of control variables.

Control variables		Min		Max
PDG8 (kW)		0		1000
PDG9 (kW)		0		1800
PDG10 (kW)		0		600
V0 (p.u.)		0.97		1.03
VDG9 (p.u.)		0.98		1.03
QC1 (kVAr)		0		500
QC2 (kVAr)		0		500
QC3 (kVAr)		0		500

Proposed GSA approach has been applied to solve OPF problem for three different objective functions. Table 11 shows optimal settings of control variables and objective function values for base case (control variables are set to nominal values) and optimization cases.

Table 11. Optimal settings of control variables.

Control variable and obj. fun. values	Base case	Case 1: Fuel cost minimization	Case 2: Loss minimization	Case 3: Fuel cost and loss minimization
PDG8 (kW)	1000	342.7	381.9	346.5
PDG9 (kW)	1800	892.2	1800.0	1795.6
PDG10 (kW)	600	317.9	476.3	372.2
V0 (p.u.)	1	1.0039	1.0016	1.0015
VDG9 (p.u.)	1	0.9991	1.0005	1.0009
QC1 (kVAr)	500	287.8	72.1	209.9
QC2 (kVAr)	500	250.0	2.8	10.0
QC3 (kVAr)	500	369.0	42.5	35.7
Fuel cost ($/h)	201.0167	186.2800	193.3815	192.7196
Ploss (kW)	13.63	13.36	7.94	8.84

In order to verify and compare results, the same OPF problem is solved using a genetic algorithm, which is the most popular and most widely used heuristic optimization method. MATLAB realization of the GA is applied, where population size is set to 50, maximum number of generation is set to 100, and other parameters as the default parameters are used. In each case, 10 consecutive test runs were performed for solving the OPF problem using the proposed GSA approach as well as the reference GA approach.

Case 1: Fuel cost minimization. As could be seen in Table 11, the total fuel cost obtained by the proposed GSA is reduced to 186.28 $/h compared to the base case fuel cost of 201.02 $/h. It is clear that control variables settings corresponding to fuel cost based OPF results in reduction of 7.33 % in fuel cost, but insignificant reduction in power losses.

Comparison results are shown in Table 12. In Table 12 it could be seen that results obtained from the GSA are better than those obtained from the reference GA based approach. Fig. 13 shows comparative convergence profiles of GSA and GA for Case 1. It may be observed from this figure that the GSA tends to find the optimal solution faster than the GA.

Table 12. Comparison of simulation results for Case 1.

Methods	Fuel cost ($/h)			The average value of simulation time (s)
	Min	Average	Max
GSA	186.2800	186.2901	186.3489	13.5958
GA	186.2823	186.3289	186.3601	17.8755

Fig. 13. Comparative convergence profiles of GSA and GA for Case 1.

Case 2: Power loss minimization. Minimization of power loss as objective in OPF results in reduction of 41.75% in losses, and 3.8% in fuel cost over the base case. Compared with Case 1, the power loss is reduced by 40.56% but the fuel cost increased by 3.8%.

Comparison results are shown in Table 13. Results presented in Table 13 clearly show that in GSA approach, Ploss is 7.94 kW, representing reduction of 12.46 % in comparison to GA algorithm. Also, comparative convergence profiles of GSA and GA for Case 2 are shown in Fig. 14. According to this figure, GSA tends to find the optimal solution faster than GA and hence has a higher convergence rate.

Fig. 14. Comparative convergence profiles of GSA and GA for Case 2.

Table 13. Comparison of simulation results for Case 2.

Methods	Power loss (kW)			The average value of simulation time (s)
	Min	Average	Max
GSA	7.940	8.915	9.620	11.3548
GA	9.070	10.00	12.17	17.5018

Case 3: Fuel cost and power loss minimization. Simultaneous minimization of the fuel cost and power loss is a multi-objective OPF problem. The best compromise solution obtained by the proposed GSA is 192.7196 $/h and 8.84 kW, which shows 4.13% reduction in fuel cost and 35.14% reduction in power losses of base case values. Results obtained from the proposed GSA technique are compared with results obtained from the reference GA method. Table 14 shows comparison results. On the basis of these results it is clear that the GSA has better performance in comparison with the GA approach.

Table 14. Comparison of simulation results for Case 3.

Methods	Fuel cost ($/h)		Power loss (kW)	The average value of simulation time (s)
	Average		Average
GSA	192.8601		10.0370	11.6222
GA	192.8844		10.1910	18.2849

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Chapter 2: Energy Saving Technologies in transmission, distribution of electrical energy