Table of contents
2.2 Optimal location and capacity of DG (analytical method)
2.3 Optimal power factor determination and DG allocation using GA
3 Simulation results and discussion
3.1 Simulation of 33-node test power distribution system
3.2 Simulation of 69-node test power distribution system
1 Overview
In order to make the network loss of the system reach the lowest value, various methods have been proposed to determine the optimal location and capacity of distributed generating units.
In this paper, the analytic method and genetic algorithm are combined to optimize the configuration of multiple distributed power sources in the distribution network, so as to minimize the network loss of the system.
This combination guarantees convergence accuracy and speed for multiple distributed genset configurations. In this paper, active power, power factor and location of DGs are simultaneously considered when minimizing distribution network losses. If the DG is installed by the DG owner, the utility will only dictate the maximum generation of the DG. However, if the DG was installed by it, both the size and location of the DG will be determined by the utility. This method is applied to 33-node and 69-node test distribution networks. Simulation results show that this method has lower loss compared with other methods.
This paper proposes a new approach, which is a hybrid approach that uses genetic algorithms to search for a wide range of location combinations and power factors of DGs, and uses analytical methods to calculate the location and capacity of each DG . While this is achieved at the expense of requiring the number of DG units to be pre-specified, this opens up the potential to examine the benefits of strategic placement of different numbers of DGs.
The method is applied to 33-node and 69-node test distribution networks, and the results show that the method is accurate and effective in the optimal configuration of distributed generating units in distribution networks. The innovation points of this paper are as follows:
Analytical and heuristic search methods are combined to achieve both high speed and accurate convergence. The dependence of the active power flow of the slack nodes on the active power generated by the distributed generation is considered as a new constraint to minimize the network loss of the distribution network.
Using the deterministic equation of the optimal output active power of distributed generation, according to the network loss coefficient and network demand, the problem of network loss minimization of distribution network is solved analytically. In the process of minimizing distribution network losses, the active power, power factor, and location of distributed generation are simultaneously considered.
The structure of this paper is as follows:Section 2 Mathematical Model
Section 3 studies and discusses the simulation and results of the placement of multiple DG units.Finally, Section IV concludes the paper.
2 Mathematical model
2.1 Problem statement
The active network loss in the network can be expressed as a function of the power generation of different units. According to the following relationship, it is called the Kron equation:
Equation (1) can be expressed in the following matrix form:
In (2), the matrices B, B0, and B00 are loss coefficient matrices. In general, these coefficients are not constant and depend on the load value and generation. However, they can be calculated in the base case of system operation.
This paper considers the following assumptions: the distribution network is a radial system fed at slack nodes, identified with a number 1 and connected to a sub-distribution network or transmission network, and distributed generation has a constant power factor.
2.2 Optimal location and capacity of DG (analytical method)
Assume that Ng DG units are installed in the buses Kn1, Kn2, . . . using constant power factor , , . . . Assuming that the slack bus is a generating unit, there is a generating unit in the network. Network loss can be calculated according to (1). Assume that DG is installed on busbars 2, 3, ..., +1. If the derivative of (1) with respect to is zero, the network loss will be minimal. It should be noted that ... in (1) indicates that the power produced by the different DGs is independent, and that the power produced by the slack bus depends on these variables as follows:
It should be noted that PD is assumed to be constant in a certain state of the network.
Differentiating (3), we can get 
Since ∂PL/∂Pi and ∂PD/∂Pi are equal to 0, (4) can be written as follows:
As shown in (5), P1 depends on the power generation of different DGs. On the other hand, in the case of the minimum system loss, the ratio of the active power change produced by the slack bus to the active power change produced by the DG unit is equal to -1. In order to minimize (1) under the constraints of (3), the Lagrangian relaxation method is adopted , as follows:
The partial differential function should be equal to zero, that is:
Equation (8) can be written in matrix form as follows:
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P can be calculated from (9) using the following equation:
where x, E, and F can be calculated according to the following equations, respectively:
Each element of P is determined as follows:
For known x values, the best one can be calculated according to (10) , substituting (10)-(14) into (3), the following formula can be written:
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Expanding (15) yields the following equation:
where parameters a, b and c are calculated based on the following equations:
a is calculated by the following formula:
By substituting E from (12), (17) can be written as:
Since B = BT, (18) can be simplified as follows:
b is calculated according to the following equation:
Considering (13), b can be calculated according to the following equation:
Finally, c can be calculated as follows:
Equation (16) has the following two roots:
Since b is negative [consider (21)], x1 is a large number, and the corresponding Pi also has a large value [according to (14)]. This answer is not acceptable because Pi in units is too large. Therefore, the following answer is the only accepted answer:
Using (10)–(24), determine the best value.
2.3 Optimal power factor determination and DG allocation using GA
In this section, the DG power factor and its location are determined to have the minimum value of the system loss. Genetic algorithm is a general optimization method that has been used for optimization problems in different fields. The genetic algorithm is carried out in several steps, such as:
In this paper, the optimization problem of each DG unit considers three variables. These variables are the active power of the DG, the power factor of the DG, and the location of the DG. The active power of DG is obtained through analytical schemes and mathematical methods (24). The power factor and the position of DG are determined by using continuous and discrete GA, respectively. In GA, the chromosomes are the problem variables, that is, the power factor and the position of the DG. Therefore, assuming that there are ng DG units, the length of the chromosome in GA will be equal to 2ng, including ng genes of power factors (PF1, PF2, ..., PFn) and ng genes of DG connection positions (D1, D2, . ..., Dn).
Figure 1 Chromosome morphology considered in this paper.
In other words, in the first step of the GA process , a set of possible answers is randomly generated, and these answers are called scenarios or chromosomes. In this paper, the form of a chromosome is considered as shown in Figure 1. In a next step , each chromosome is assigned a number as a possible answer based on its fitness. The above numbers are determined by the fitness function, which will be optimized by the GA. Finally , GA selects some chromosomes for crossover, mutation and replacement operations by selecting operators and according to the fitness of chromosomes. These operators generate a new population and the process is repeated until a stopping condition is reached. To calculate the fitting function corresponding to the chromosome, the network loss is calculated according to (2), and (24) is used to determine the optimal power generation of DG. After the power flow runs, the power system loss is determined according to (1) and assigned to a chromosome as its fitness value.
GA should find the minimum value of the fitness function by varying the power factor and the position of different DGs. This paper adopts the combination of analysis method and heuristic search method to solve the optimal distribution problem of distributed power, as shown in Figure 2. The main benefits of using this method are as follows :
Because the power headroom algebra of the distributed power supply is too wide, the GA convergence speed is slow, and an accurate solution may not be obtained. In this paper, the genetic algorithm is used to determine the installation location and power factor of the distributed generator set, and the analytical method is used to determine the optimal power generated by the distributed generator set.
Using only analytical methods leads to complex and nonlinear equations, since the differential of the loss coefficient with respect to the DGs power factor should be calculated and the loss factor is a nonlinear and complex function of the DGs power factor. Furthermore, the DGs position is a discrete parameter, and its derivative with respect to the DG position is meaningless. Therefore, heuristic search algorithms should be used to optimize DG allocation. Considering these two issues, this paper proposes a method combining analysis and heuristic search.
Figure 2 Flowchart of the proposed method
3 Simulation results and discussion
The method is applied to two test distribution networks (33-node and 69-node systems) shown in Fig. 3 and Fig. 4. The algorithm is implemented in the Matlab environment, and the power flow calculation is performed using MATPOWER software.
In this study, DG has two different operating modes : DG can generate only active power (unity power factor mode) and DG can generate both active and reactive power (non-unity power factor mode).
Figure 3 33-node test power distribution system
Figure 4 69-node test power distribution system
3.1 Simulation of 33-node test power distribution system
This section will consider two different DG operation modes in the following two cases.
3.1.1 Scenario 1: Unified power factor mode for DG operation.
In this case, it is assumed that the DG produces active power and does not produce/consume reactive power. Different numbers of DGs are allocated in the network using the proposed method. In Table 1, the proposed method is compared with other methods, namely Loss Sensitivity Factor (LSF), Improved Analytical Method (IA) and Exhaustive Load Flow (ELF) method [33]. From Table 1, it can be seen that the proposed method performs better than other methods in reducing loss. In the case of placing a DG, among the four algorithms, three algorithms suggest installing a DG on bus 6. To show the performance of the proposed method in finding the optimal power generation of the DG installed on bus 6, the network losses and the power generated by the DG installed on bus 6 are calculated, as shown in Fig. 5. As shown in Fig. 5, if the DG installed on bus 6 generates 2.706 MVA, the loss reaches the minimum value (0.09922 MW). As shown in Table 1, the proposed method finds this power value that the DG installed on bus 6 should produce in order to minimize losses.
Table 1 Simulation results of the first scenario (33 nodes test power distribution system)
Figure 5 Network loss of power generation (DG installed on 6 nodes)
3.1.2 Scenario 2: DG operation mode with non-uniform power factor.
In this case, it is assumed that the power factor of DG is not necessarily equal to 1, and the simulation results are shown in Table 2. From Table 2, it can be seen that the proposed method achieves the lowest network loss.
In the case of placing a DG, the suggested algorithm suggests installing at busbar 30. The installed DG has a power of 1844.85 kVA and a lagging power factor of 0.767. Figure 6 shows the loss vs. DG location and its power factor. As shown in the figure, the combination of analytical method and genetic algorithm achieves the minimum network loss by installing a DG on bus 30 (Fig. 7) with a lagging power factor of 0.767. To resolve the problem analytically, the voltage inequality constraint for the bus (ie Vmin < Vbus < Vmax) cannot be included in the optimization problem. Therefore, after the optimization procedure is completed, the bus voltage and line current can and should be checked to ensure the inequality constraints of the bus voltage. Table 3 lists the minimum and maximum voltages of the 33 bus systems after installing DG units.
Table 2 Simulation results of Scenario 2 (33-node test distribution system)
Table 3 The minimum and maximum voltages of the 33 nodes tested power distribution system after installing DG
Fig.6 Relationship between system loss and DG position and its power factor
Figure 7 Installing a DG on bus 30, the minimum value of network loss
3.2 Simulation of 69-node test power distribution system
This section simulates the 69-bus test power distribution system. The following two scenarios consider two different DG operation modes respectively.
3.2.1 Scenario 1: Unity power factor mode of DG operation:
In this case, it is assumed that DG works in unity power factor mode and can only generate active power. The results of the DG assignment are listed in Table 4. As listed in the table, the proposed method results in lower network loss compared to other methods.
Table 4 Simulation results of the first scenario (69-node test distribution system)
3.2.2 Scenario 2: Non-uniform power factor mode of DG operation:
At this time, DG can generate active and reactive power, and its power factor is not necessarily equal to 1. The results of DG optimal configuration in 1969 are shown in Table 5. Table 6 lists the minimum and maximum voltages of the 69 busbar system after DG unit installation.
Table 5 Simulation results of Scenario 2 (69-node test distribution system)
Table 6 Minimum and maximum voltages for 69-bus test power distribution system after DG installation
4 Conclusion
This paper proposes a method combining analytical method and genetic algorithm, which is used for the configuration of multiple distributed generating sets in the distribution network, so that the system network loss is minimized. This method uses genetic algorithm to find the optimal installation location of distributed generation, and uses a new analytical formula to determine the capacity of distributed generation. The method is compared with IA, LSF and ELF methods in terms of loss reduction. The results show that this method achieves the lowest loss compared to other methods.