4.1. Case Descriptions
An actual 20 kV distribution network in China, including two 110-kV substation nodes (i.e., node 1 and node 53, represented by orange solid point), fifty-three 20- kV load nodes, and 53 lines, is used to verify the effectiveness of the proposed model and methods. The total active load of the distribution network is 272.85 MW. The topology of the distribution network is shown in
Figure 3, and the dotted lines in
Figure 3 represent the tie lines in the distribution system. The voltage fluctuation range of node
i at the dispatching period
t is set to be [0.95
, 1.05
].
To analyze the influence of the number of DG connection nodes on the accommodation capacity and flexibility of the distribution network, two scenes with different DG connection nodes are set up, as shown in
Table 1.
Table 1 shows that only node 41 is a WF node, and there are more PV nodes in scene A than in scene B. The maximum accommodation capacity of DG in the distribution network in this area (i.e., the maximum connection capacity of DG) is calculated by the proposed CLPSO algorithm and mixed strategy Nash equilibrium. The basic parameters of the CLPSO algorithm used in this paper are as follows: the number of particles is set to 30; each acceleration factor is 1.494; the upper and lower limit of inertia weight are set to 0.7 and 0.2, respectively; and the maximum number of iterations is set to 100.
Considering the uncertainties of DG, three typical WF output scenarios and four typical PV output scenarios are generated based on measured data. Each scenario is divided into six dispatching periods. The standardized output curve of each dispatching period is shown in
Figure 4.
4.2. Simulation Results
Seven external archival solutions for the maximum accommodation capacity of DG are obtained by the proposed CLPSO algorithm, and the objective function values of each solution are standardized. The proposed method is implemented in CPLEX and solved using the YALMIP (an optimization solution tool which relies on external solvers for the low-level numerical solution of optimization problem) toolbox on a PC with a Core i5 3570 CPU and 4GB of RAM. The running time using the proposed CLPSO algorithm in the basic case is 58504.2 s, which is acceptable in the optimal planning problem. In order to visually display the distribution of three objective functions corresponding to external archives, the Pareto frontier solution set of three-dimensional objective function space is drawn, as shown in
Figure 5. It can be seen from
Figure 5 that the solution set in the external archives is distributed at the Pareto optimal frontier, the distances between the external archives are large, and the congestion degree is low. It can be concluded that the external archive solution set obtained by the proposed CLPSO algorithm contains a variety of decision-making solutions that fully consider the mutual dominance of the three objective functions presented in this paper. The optimal compromise solution could fully weigh the interests of each objective function and achieve the comprehensive optimum.
The weights of the three objective functions calculated by the entropy weight method are 0.3143, 0.3661, and 0.3196, respectively. The difference of the weights of the three objective functions is not significant, which means that the information entropy provided by each objective function in the proposed optimization model is comparable and the economy and flexibility of the distribution network operation after the connection of DG can be considered comprehensively.
The normalized function values of each optimal boundary solution and the optimal solutions with Nash equilibrium decision-making and TOPSIS are presented in
Table 2. In
Table 2, the line capacity margin is the least among the seven Pareto frontier solutions in the optimal solution for the maximizing daily energy supply of DG. The daily energy supply of DG is least in the optimal solution for minimizing nodal voltage amplitude deviation. This means that there is a game relationship among the three objectives. The optimal solution obtained with the mixed strategy Nash equilibrium decision-making, which is a type of decentralized decision-making method, is the same as that obtained with TOPSIS, which is a type of centralized decision-making method. The result shows that the directions along which the particle swarms search to find the optimal solutions for the two methods are consistent. Moreover, because the three objectives are regarded as non-cooperative decision-making participants in a competitive relationship, the proposed mixed strategy Nash equilibrium decision-making model is more suitable for finding the optimal compromise solution and corresponding DG connection solution for the distribution network in this paper.
Table 3 shows the solutions of the mixed strategy Nash equilibrium decision-making with different
ui. It shows that the change in values of parameter
ui does not influence the final optimal solution for the mixed strategy Nash equilibrium decision-making, which demonstrates the robustness and adaptability of the proposed decision-making method.
The total DG accommodation capacities of seven Pareto frontier solutions are shown in
Figure 6. The optimal compromise solution (i.e., solution 3) has a 78.9-MW DG accommodation capacity on the premise of obtaining joint optimal values of multiple objectives. Its DG accommodation capacity is the second largest, following solution 2. Furthermore, the penetration of DG in solution 3 is 28.9%, showing that the distribution network in solution 3 has a better accommodation capacity for DG.
The specific PV and WF accommodation capacities of each node to be built under two different scenes are shown in
Figure 7. In scene A, the number of DG nodes to be built is large and the distribution is scattered. Thus, the total DG accommodation capacity of the distribution network in this area could be maximized on the premise of ensuring the flexibility of the distribution network, in which several nodes do not need to connect large-capacity DG equipment. In scene B, the actual accommodation capacity of the PV nodes to be built is larger and more centralized than for the corresponding nodes in scene A.
The optimal compromise solutions for DG accommodation capacity under two scenes are shown in
Table 4. It can be seen from
Table 4 that when the number of DG connection nodes in the distribution network is reduced, the fitness of each objective function and the total DG accommodation capacity of the optimal compromise solution of the DG configuration are reduced simultaneously. This is because of the limitation of the line capacity; the DG capacity of a single node is limited and the reduction of the connection nodes directly leads to the reduction of the total accommodation capacity. Thus, the dispatchable DG power is also reduced accordingly. At the same time, more decentralized DG connection nodes can provide voltage support and alleviate the pressure of power transmission. The fitness function value of the line capacity margin which represents the average of all line capacity margin values is 58.1% for scene A. It is higher than that for scene B, indicating that the planning strategy of DG, having more connection nodes and more dispersed distribution, can help to improve the flexibility of the distribution network.
Because of the distinct renewable energy consumption requirements at different stages in different regions, it is necessary to carry out sensitivity analysis for abandonment rates of PV and WF in the proposed optimization model.
Table 5 presents the sensitivity analysis results considering abandonment rates of PV and WF, namely 10% (Case 1), 30% (Case 2), and 50% (Case 3), which include the fitness of each objective function and the total DG accommodation capacity of the optimal compromise solution.
It can be seen from
Table 5 that with the increase of permissible abandonment rate, the total accommodation capacity of DG increases after optimization. At peak load stages, the distribution network can use more renewable energy, and the power transmission pressure of the lines in the distribution network is reduced. Eventually, the line capacity margin index increases from 56.5% to 57.1%.
Take scenario 1 and scenario 12 as examples.
Figure 8 shows the voltages of node 41, considering different abandonment rates for all dispatching periods. Scenario 1 is composed of PV scenario 1 and WF scenario 1, while complex scenario 12 is composed of PV scenario 4 and WF scenario 3. When the permissible abandonment rate is 50%, the total accommodation capacity of DG reaches the maximum value of 31.6 MW. Under complex scenario 1, the high output power of WF at dispatching periods 3 and 4, which are load peak stages, can provide adequate voltage support. The bigger accommodation capacity of WF at node 41 will cause higher nodal voltage amplitude, even exceeding the rated voltage. Under scenario 12, the WF output power at load peak stages is lowest and the output level of PV is also low, which makes it difficult to support the nodal voltage of node 41 at a distance. In case 3, although the total accommodation capacity of DG is maximal, the voltages of node 41 for dispatching periods 3 and 4 are 19.9961 kV and 19.9964 kV, respectively, which are still lower than the rated voltage.
Figure 9 represents the voltage of node 5 under all scenarios. It shows that the magnitude of nodal voltage is influenced by the types of scenarios. Under scenario 10, the power of PV, whose output characteristic is consistent with that of load, is the largest, while the power of WF, whose output characteristic is different to that of load, is the smallest. Hence, the value of the nodal voltage at each dispatching period is closest to 20 kV. Under scenario 4, the output of PV is smallest and that of PV is largest, causing the maximum voltage amplitude deviation among all scenarios.