A novel bi level optimization model for load supply capability issue in active distribution network

Received: 4 November 2016 Revised: 7 October 2017 Accepted: 11 October 2017 DOI: 10.1002/etep.2492 RESEARCH ARTICLE A novel bi level optimization model for load supply capability issue in active distribution network Wei Zhou 1 Kai Sun 1 Hui Sun 1 Zeng Shou 2 Jinsong Liu 2 Wei Huang 3 Zhonghui Wang 2 1 Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, Liaoning Province, China 2 State Grid Liaoning Electric Power Co., Ltd., Shenyang 110006, Liaoning Province, China 3 Anshan Power Supply Company, Anshan 114006, Liaoning Province, China Correspondence Wei Zhou and Hui Sun, Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, Liaoning Province, China. Email: zhouwei@dlut.edu.cn; dutshui@dlut.edu.cn Funding information Science and Technology Project of State Grid, Grant/Award Number: 2017YF 27 Summary This paper develops a novel bi level optimization model for load supply capability (LSC) based on the economic dispatch of distributed generation in active distribution network. The upper subproblem describes the LSC. The lower subproblem denotes the economic dispatch, which is set up as one of the constraint conditions in LSC problem. Thus, the maximum LSC can be obtained on the premise that each operation point achieves the optimal scheduling. Moreover, load static voltage characteristics are considered in this model, which can avoid the inaccuracy of power flow distribution caused by constant power model. To describe the random property of nondispatchable distributed generation, chance constrained programming is adopted in the bi level optimization model. Hierarchical iteration algorithm is applied to solve the bi level optimization model. The upper and lower subproblems are calculated by particle swarm optimization based on stochastic simulation. A modified IEEE 33 bus distribution network is tested to demonstrate the effectiveness and superiority of the proposed model. The analysis results demonstrate that the model conforms to the practical operation of active distribution network. KEYWORDS active distribution network, bi level optimization, economic dispatch, load supply capability Nomenclature: Parameters: a i, b i, c i, Ratio of constant impedance load, constant current load and constant power load in bus i; A,B,C, Fuel cost function constants of diesel generator; C ng, Gas price; G ij + jb ij, The ijth element of the bus admittance matrix; K Li, Load growing pattern parameter; LHV ng, Low calorific value of gas; N, Total number of nodes; N DG, Total number of dispatchable distributed generators; P DGi, max, P DGi, min, Maximum and minimum real power output of distributed generation in bus i; S grid, max, Allowed transformer capacity connected to the main grid; S ij, max, Allowed line capacity from bus i to bus j; Q DGi, max, Q DGi, min, Maximum and minimum reactive power output of distributed generation in bus i; U i, max, U i, min, Maximum and minimum voltage magnitude in bus i; β, Confidence level; η(p FC ), Operating efficiency of fuel cell; η(p MT ), Operating efficiency of micro turbine; ρ grid, Price of exchanged power between ADN and main grid Variables: K DDGi, Distributed generation growing pattern; P DE, Power output of diesel generator; P DDGi, Q DDGi, Real and reactive power of the dispatchable distributed generation in bus i; P 0 DDGi, Base case real power of the dispatchable distributed generation in bus i; P FC, Power output of fuel cell; P grid, Q grid, Exchanged real and reactive power between ADN and main grid; P ij, Q ij, Line real and reactive power flow from bus i to bus j; P Li, Q Li, Real and reactive power demand at actual voltage in bus i; P 0 Li ; Q0 Li, Real and reactive power demand in bus i for base case; P LNi, Q LNi, Real and reactive power demand at rated voltage in bus i; P MT, Power output of micro turbine; P NDDGi, Q NDDGi, Real and reactive power of the nondispatchable distributed generation in bus i; U i, Voltage magnitude in bus i; λ, Load variation factor Int Trans Electr Energ Syst. 2017;e2492. https://doi.org/10.1002/etep.2492 wileyonlinelibrary.com/journal/etep Copyright 2017 John Wiley & Sons, Ltd. 1of13

2of13 ZHOU ET AL. 1 INTRODUCTION Load supply capability (LSC) is an important index for evaluating safety and reliability of distribution network. With the development of distributed energy resources (DERs), active distribution network (ADN) realizes the energy efficiency improvement by involving active generation control and management for network operation. Taking the generation cost as the objective function, the optimal allocation of distributed generation will be obtained on the premise of satisfying the security constraints. 1-4 Because the active behavior and the uncertainty of DERs will lead to obvious change of load flow distribution, the LSC of ADN will be effected to maximize economic and energy use efficiency. Therefore, it is very essential to develop the research about LSC of ADN for assessing power supply level. The present methods for supply capability can be divided into analytical methods and optimization methods. Luo et al 5 proposes a straightforward method to evaluate the power supply capability of distribution system, which is based on N 1 contingency analysis of interconnected main transformers. A linear programming model is set up, and the extended indices are given in Xiao et al. 6 Further, Xiao et al 7 proposes a model for calculating the total supply capability for distribution system considering both feeder and substation transformer contingencies, and load balancing problem is also taken into account. In Chen et al, 8 the total supply capability is acquired with the conditions that all load outages can be restored via network reconfiguration, and the daily load curves are also considered in the proposed model. With the increasing penetration of DERs, particularly from renewable sources, researchers begin to study the impact of DERs on LSC. Current research mainly includes two types: one type is the LSC only considering nondispatchable DERs; another type is the LSC considering both dispatchable DERs and nondispatchable DERs. For the former, most of the studies focus on the treatment method for distributed generations or the effect of DERs uncertainties on LSC. In Wu et al, 9 back forward sweep method in the form of branch current and variable step size repeated power flow algorithm are applied to calculate LSC. The different node types of distributed generations are handled by different methods, and the proposed algorithm has strong processing ability for different types of distributed generations. Based on the above works, Zhang et al 10 define a series of probabilistic indices to evaluate available LSC; the uncertainty of DERs is also involved in the proposed model. For the latter, most of the related LSC models mainly consider the generation characteristics of nondispatchable DERs, such as wind power and solar power generation. The power outputs of dispatchable DERs are always keeping constant. 11 The controllability of energy storage system provides a new solution to improve power supply capability with random fluctuation power source. 12 Complementary scheduling discipline is used to calculate the outputs of controllable distributed generation, 13 and thus, it reflects active generation control and management characteristics for ADN operation. However, distribution networks are evolving towards active networks to support distribution network operation in an efficient way. An ADN operator can determine its operation decision on scheduling of Distributed Generation (DG) units and purchasing from the grid operator. Short term scheduling issue of ADN has been studied to obtain an optimum allocation of dispatchable DERs. 14-22 Therefore, how to evaluate the LSC introducing economic dispatch is an urgent problem. Very few studies are currently concerned with this problem in ADN. There is an interaction between LSC and economic dispatch. The dispatchable distributed generation growing pattern can be regarded as the decision variable, and it has an adverse interest in each of the problem. Therefore, a bi level optimization model can be adopted to solve the LSC issue in ADN based on economic dispatch. There are already several studies about the bi level program and distribution network. A two stage robust model is set up to determine the optimal selling price for a distributed generation owning retailer in Khojasteh and Jadid. 23 Sadeghi Mobarakeh et al 24 propose a bi level approach to formulate the competition, in which each DG problem is the upper level problem and the distribution companies problem is considered as the lower level one. Using the Karush Kuhn Tucker conditions of the lower level, a nonlinear programming formulation is obtained and solved. However, the equivalent equations become very complex, and the dimension of the equation increases dramatically. Lei et al 25 present a bi level optimization model that aims to maximize the benefits of microgrid agents and a distribution company. An iterative process of upper and lower levels is involved to obtain an optimal solution. This paper presents a novel bi level optimization model for solving LSC issue in ADN. Economic dispatch problem is expressed as the lower subproblem, which is regarded as a constraint condition of upper subproblem. It can been realized that distributed generations are dispatched economically in every operating point along with the load increasing. Load static voltage characteristics is also considered in this model, which can avoid the inaccuracy of power flow distribution caused by constant power model. A modified IEEE 33 bus distribution network is tested to demonstrate the effectiveness and superiority of the proposed model. The paper is arranged as follows: Section 2 describes the problem formulation and structure.

ZHOU ET AL. 3of13 The bi level optimization model and solution algorithm of LSC issue are presented for a given distribution network in Section 3. Numerical results and analysis are discussed in Section 4, and the concluding remarks are provided in Section 5. 2 PROBLEM FORMULATION Load supply capability is defined to be the maximum load before an electrical or operating constraint violation is encountered, and the load should increase under an arbitrary load variation. 26 However, loads are supplied by the main grid and DERs because of integration of distributed generation in ADN. The LSC evaluation should take active management and control of DERs into account when load increases. 2.1 Variation pattern of load and generation output In the most of LSC researches, the load is regarded as a constant power load. However, with the growth of the load, the distribution network will be operating in the limit state, the bus voltage will deviate from the rated voltage, and thus the practical load will be changed. There are 3 load types, including constant impedance load, constant current load, and constant power load in ADN. The constant impedance load and the constant current load will be effected by bus voltage. It can be described by load static voltage characteristics. Therefore, it is necessary to introduce load static voltage characteristics to load model. For a given bus voltage, the real power demand and reactive power demand can be expressed as follows: P Li ¼ P LNi a i U 2 i þ b i U i þ c i ; (1) Q Li ¼ Q LNi a i U 2 i þ b i U i þ c i : (2) And the ratio of constant impedance load, constant current load, and constant power load satisfies a i þ b i þ c i ¼ 1: (3) Suppose the load increases continuously according to the given load growing pattern, the load variation factor λ is a scalar quantity that represents the percentage change. P Li ¼ P 0 Lið1 þ λk Li Þ; (4) Q Li ¼ Q 0 Lið1 þ λk Li Þ; (5) To balance the increasing load demand, main grid and DERs will share the power increment. Distributed renewable energy power generations such as photovoltaic power generation and wind power generation, usually work at the maximum power point according the principle of maximum power acquisition of renewable energy, and they are not dispatched by ADN. The power outputs of renewable energy such as wind energy and solar energy are described using Beta distribution. The micro turbine, diesel generator, and fuel cell are mainly considered as dispatchable distributed power supplies. Suppose dispatchable DERs is also increasing in the certain growing pattern, the distributed generation growing pattern will be determined by active management and control principle in ADN. P DDGi ¼ P 0 DDGið1 þ λk DDGi Þ: (6) 2.2 Overall architecture of LSC issue in ADN In ADN, load demand is supplied by main grid and DERs, and the ADN operator determines its operation decision on scheduling of distributed generation units and purchasing from grid operator. Therefore, the economic operation of ADN should be considered when calculating LSC. The main framework for load supplying capability in ADN is given in Figure 1. In the proposed framework of Figure 1, the conventional LSC problem is firstly solved under a given distribution generation growing pattern, and the maximum quantity of total load will be determined. Then the economic dispatch

4of13 ZHOU ET AL. FIGURE 1 Framework for load supplying capability in active distribution network optimization problem under the above load level is addressed. The economic dispatch issue in ADN involves the allocation of generation among distributed generators and main grid so as to achieve a minimum generation cost. Moreover, the randomness of nondispatchable distribution generation will be introduced in the LSC problem. At the same time, the dispatchable distribution generation growing pattern will be obtained. It will have some impact on the LSC problem. So the practical LSC will be evaluated after considering economic dispatch issue. 3 BI LEVEL OPTIMIZATION MODEL AND SOLUTION ALGORITHM OF LSC ISSUE IN ADN 3.1 LSC model in ADN based on economic dispatch The LSC problem can be formulated as a bi level model. It considers two decision makers referred to as leader and follower that will try to optimize their individual objective functions. Upper subproblem denotes the LSC problem, and it is used to calculate the load variation factor λ when distributed generation growing pattern is given. The economic dispatch problem is described as lower subproblem, which can be regarded as a constraint condition of LSC model. It means that the economic dispatch will be maintained at each operating point as the load increases, when we calculate LSC of distribution network. Lower subproblem is applied to calculate the distributed generation growing pattern. Moreover, the stochastic programming theory is used to introduce the uncertainty of renewable energy. Therefore, the mathematical formulation of LSC problem in ADN based on economic dispatch is as follows: 1. Objective function of upper subproblem: The maximum load demand is to be determined as an objective function of LSC problem. maximize E N P 0 Lið1 þ λk Li Þ a i U 2 i þ b i U i þ c i : (7a) λ;u i ;Q DDGi 2. Objective function of lower subproblem: The objective function of economic dispatch problem is defined as the total generation cost of ADN. N DG minimize E C i P 0 K DDGið1 þ λk DDGi Þ þ ρgrid P grid : (7b) DDGi 3. Technical constraints: The LSC problem and economic dispatch problem will consider the same technical constraints expressed in formula (7c) to (7k).

ZHOU ET AL. 5of13 Power balance equations: The total real and reactive power from dispatchable DERs, nondispatchable DERs, exchanged real and reactive power between ADN, and main grid must satisfy the power balance constraint. P NDDGi is regarded as a random variable in the model. Suppose that all the nondispatchable distributed generators operate at constant power factor pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi control mode, that means that Q NDDGi ¼ P NDDGi 1 cos 2 φ=cos φ. Thus,Q NDDGi is also a random variable. P grid þ P NDDGi þ P 0 DDGið1 þ λk DDGi Þ P 0 Lið1 þ λk Li Þ a i U 2 i þ b i U i þ c i Ui N j¼1 U j G ij cos θ ij þ B ij sin θ ij ¼ 0; (7c) Q grid þ Q NDDGi þ Q DDGi Q 0 Lið1 þ λk Li Þ a i U 2 i þ b i U i þ c i Ui N j¼1 U j G ij sin θ ij B ij cos θ ij ¼ 0: (7d) Bus voltage magnitude constraints: The bus voltages U i should be within a permissible range, and U i, min U i U i, max are satisfied within a given confidence level. Pr U i; min U i U i; max β: (7e) Real and reactive power constraints of the dispatchable DERs: The constraints ensure that the DERs are dispatched within their rated capacities. P DDGi; min P DDGi P DDGi; max ; (7f) P DDGi ¼ P 0 DDGið1 þ λk DDGi Þ; (7g) Q DDGi; min Q DDGi Q DDGi; max : (7h) Maximum capacity power constraints of distribution line: The distribution line power is bounded within the allowed qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi line capacity from bus i to bus j, and P 2 2 ij þ Q ij S ij; max should be satisfied within a given confidence level. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pr P 2 2 ij þ Q ij S ij; max β: (7i) Maximum capacity power constraint of the transformer connected to the main grid: The total exchanged power between ADN and main grid is bounded within the capacity of the substation transformer, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 2 grid þ Q grid S grid; max should be satisfied within a given confidence level. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pr P 2 2 grid þ Q grid S grid; max β: (7j) Dispatchable distributed generation growing pattern limits: K DDGi 0: (7k) The first term of Formula 7b denotes the generation cost of all the dispatchable DERs supplies including micro turbine, diesel generator, and fuel cell. The generation cost of micro turbine is equal to The generation cost of diesel generator is expressed as a quadratic function P MT C MT ¼ C ng =LHV ng ηðp MT Þ : (8) C DE ¼ AP 2 DE þ BP DE þ C: (9)

6of13 ZHOU ET AL. The generation cost of fuel cell is similar as that of micro turbine P FC C FC ¼ C ng =LHV ng ηðp FC Þ : (10) The second term is exchanged power cost between ADN and main grid. The optimal power outputs of all the dispatchable DERs supplies can be obtained by solving the economic dispatch model in ADN, and then the distributed generation growing pattern can be calculated using the following formula. P 0 DDGið1 þ λk DDGi Þ ¼ P DDGi : (11) 3.2 Solution algorithm The hierarchical iteration algorithm based on particle swarm optimization of stochastic simulation is adopted to solve the proposed bi level optimization model. The specific steps are as follows: 1. At the beginning of the hierarchical iteration, an initial point is indispensable including load variation factor and distributed generation growing patterns, etc. 2. Upper LSC optimization model is solved using particle swarm optimization algorithm of stochastic simulation when the distributed generation growing patterns keep constant, and then the maximum load variation factor λ can be obtained. It will be transmitted to the lower subproblem. 3. Lower subproblem is solved using particle swarm optimization algorithm of stochastic simulation when the current load variation factor λ keeps constant, and then real power outputs of the dispatchable DERs supplies and distributed generation growing patterns can be obtained. The distributed generation growing patterns will be updated in the upper subproblem. 4. The hierarchical iterations are considered terminated whenever termination conditions are satisfied. The iterative termination conditions are described as λ k λ k 1 ε1 ; K k DG i Kk 1 ε2 ; (12) where ε 1 =10 3, ε 2 =10 3. If not converged, this process is returned to step 2, and then distributed generation growing patterns will be used in the solution process of upper subproblem. The purpose is to continue to repeat the iteration. The flow chart of hierarchical iteration algorithm based on particle swarm optimization of stochastic simulation is given in Figure 2. The detailed formulation of chance constrained bi level optimization model and solution algorithm are given in Appendix A.1. However, for the proposed bi level model, convergence cannot always be guaranteed by the hierarchical iteration algorithm and the dead cycle phenomena possibly appears. 27 The stub table method is adopted to avoid the above problem. The method puts the distributed generation growing pattern into stub table. If the current distributed generation growing pattern KDG kþ1 is the same as the certain vector in stub table, Kkþ1 DG and Kk DG are sent to upper subproblem. And then the objective function of upper subproblem will be changed as follows: maximize min DGi N P 0 Li 1 þ λ k K Li ai U 2 k;i þ b iu k;i þ c i ; N P 0 Li 1 þ λ kþ1 K Li ai U 2 kþ1;i þ b iu kþ1;i þ c i : (13) 4 NUMERICAL RESULTS AND ANALYSIS In this section, the model for determining LSC is tested on a modified IEEE 33 bus distribution network. 8 The related fuel cost function parameters of diesel generator and efficiency function parameters of micro turbine and fuel cell can be found in Chen and Zhu. 28 The probability distribution parameters of wind energy and solar energy are given in Liu 29 and Kaplani and Kaplanis. 30 The IEEE 33 bus distribution network figure is shown in Figure 3. The installed

ZHOU ET AL. 7of13 FIGURE 2 Flow chart of hierarchical iteration algorithm FIGURE 3 IEEE 33 bus distribution network figure capacities and interconnection nodes of distributed generations are shown in Table 1. All the load growing pattern parameters are set to 1. The confidence level is set to 0.95. In the following, firstly, 3 cases are considered to discuss the effects of economic dispatch on LSC in detail. In cases 1 and 2, all the real power outputs of dispatchable DERs supplies are regarded as constants; they are set to the rated power output and half of the rated power output, respectively. Case 3 denotes that economic dispatch is considered under the proposed model.

8of13 ZHOU ET AL. TABLE 1 Installed capacities of distributed generations Number Node Type Installed Capacity, kw Dispatchable Mode 1 A7 WT 300 Nondispatchable 2 A14 DE 600 Dispatchable 3 A24 WT 300 Nondispatchable 4 A25 MT 650 Dispatchable 5 A30 PV 200 Nondispatchable 6 A32 FC 600 Dispatchable From the comparison results in Table 2, it can be seen that both the total generation cost and the value of LSC under case 3 is lower than that of case 1, but higher than that of case 2. At the same time, the LSC increases as the power output of dispatchable DERs rises. This is because the generation costs of dispatchable DERs are significantly higher than that of power supplied by main grid. When the power outputs of dispatchable DERs are set to rated value, the most total generation costs will be obtained. A compromising solution is obtained by the proposed bi level model. From the simulation results in Figures 4 and 5, we observe that all the bus voltage magnitudes are in normal range and have a certain margin whether economic dispatch is involved or not. At the same time, the real power flow of line 14 reaches the rated capacity when all the real power outputs of the dispatchable distributed generations are regarded as constant rated power outputs (case 1); it becomes bottleneck to restrict the growth of LSC. If all the real power outputs of the dispatchable distributed generations are regarded as the half of rated power outputs (case 2) or economic dispatch is applied to optimize the power outputs of the dispatchable distributed generations (case 3), there is no line power flow reaching the limits. The comparison results of cases 1 and 2 illustrate the effects of the real power output of dispatchable distributed generation on the LSC. The LSC begins to fall as real power output of dispatchable distributed generation decreases. The results of case 3 indicate that the increasing load demand cannot meet the requirements of system operation economy and lead to limit the increment in real power system load although there is a certain margin in the physical sense. To evaluate the accuracy and computational performance of the proposed model, the results of proposed model and the traditional equivalent model of the bi level model through KKTs conditions have been compared and analyzed. Here, the uncertainty of renewable energy are ignored. The traditional equivalent model of the bi level model through KKTs conditions is given in Appendix A.2. It can be seen in Table 3. The comparison results demonstrate that the optimization TABLE 2 Comparison results of LSC and total generation cost under different cases Case Total Generation Cost, Yuan LSC, kw Case 1 3309.5 4770 Case 2 3012.1 4577 Case 3 3256.5 4753 Abbreviation: LSC, load supply capability. FIGURE 4 Comparison results of voltage magnitude under different cases

ZHOU ET AL. 9of13 FIGURE 5 Comparison results of line loading rate under different cases solutions of these two models are almost the same. However, the solution process of equivalent model through KKTs conditions is time consuming compared to that of the proposed model. In addition, 3 different load types including constant impedance load, constant current load, and constant power load are adopted to analyze the impact of load model on LSC. The comparison results of LSC can be seen in Figure 6. Not surprisingly, the LSC rises by 24 kw when load type varies from constant power load to constant current load. Because of static voltage characteristics of load demand, the LSC will be underestimated when using constant power load model. Therefore, static voltage characteristics of load demand cannot be ignored in the actual load model. Further study is used to illustrate the influence of various confidence level β on the solutions of LSC problem. As shown in Figure 7, the value of LSC decreases when β increases. That means the reduction of system security will bring a better load supply capability. TABLE 3 Comparison results of LSC under different models Model LSC, kw Computation Time, s The equivalent model through KKTs conditions 4753.5 429 The proposed model 4752.9 10 Abbreviation: LSC, load supply capability. FIGURE 6 Load supply capability of active distribution network under different load types. LSC, load supply capability FIGURE 7 Load supply capability of active distribution network under different confidence levels.

10 of 13 ZHOU ET AL. 5 CONCLUSIONS In this paper, a novel bi level optimization model of LSC issue is developed, which incorporates both economic dispatch and load static voltage characteristics, and thus, the optimal allocation of distributed generation along with the growth of system load level is realized. The proposed model is implemented on a modified IEEE 33 bus distribution network, and the bi level optimization model is solved by the hierarchical iteration algorithm. The results demonstrate that the proposed model can obtain a compromising LSC compared with conventional constant power of distributed generation, but it can meet the practical operation of ADN. Furthermore, by studying the effect of load model on LSC, we find that the LSC will be underestimated when using constant power load model. Therefore, the load static voltage characteristics must be considered when we investigate the LSC in ADN. ACKNOWLEDGEMENTS This work has been supported by the Science and Technology Project of State Grid (2017YF 27). ORCID Wei Zhou http://orcid.org/0000-0001-8794-0705 REFERENCES 1. Lopes JAP, Hatziargyriou N, Mutale J, Djapic P, Jenkins N. Integrating distributed generation into electric power systems: a review of drivers, challenges and opportunities. Electr Pow Syst Res. 2007;77:1189 1203. 2. Choi JH, Kim JC. Advanced voltage regulation method of power distribution systems interconnected with dispersed storage and generation systems. IEEE Trans Power Delivery. 2001;16:329 334. 3. Dimeas AL, Hatziargyriou ND. Operation of a multiagent system for microgrid control. IEEE Trans Power Syst. 2005;20:1447 1455. 4. Wasiak I, Thoma MC, Foote CET, et al. A power quality management algorithm for low voltage grids with distributed resources. IEEE Trans Power Delivery. 2008;23:1055 1062. 5. Luo FZ, Wang CS, Xiao J, Ge SY. Rapid evaluation method for power supply capability of urban distribution system based on N 1 contingency analysis of main transformers. Int J Electr Power Energy Syst. 2010;1063 1068. 6. Xiao J, Li F, Gu WZ, Wang CS, Zhang P. Total supply capability and its extended indices for distribution system: definition, model calculation and applications. IET Gener Transm Distrub. 2011;5:869 876. 7. Xiao J, Li X, Gu WZ, Li FX, Wang CS. Model of distribution system total supply capability considering feeder and substation transformer contingencies. Int J Electr Power Energy Syst. 2015;65:419 424. 8. Chen K, Wu W, Zhang BM, Djokic S, Harrison GP. A method to evaluate total supply capability of distribution systems considering network reconfiguration and daily load curves. IEEE Trans Power Syst. 2015;31:2096 2104. 9. Wu DW, Li GY, Wu JS, Zhan ZF. Calculation of load supply capability for distribution networks with distributed generations. IEEE Pes Asia pacific Power & Energy Engineering Conference, Kowloon, Hong Kong, 2013. 10. Zhang SX, Cheng HZ, Zhang LB, Bazargan M, Yao LZ. Probabilistic evaluation of available load supply capability for distribution system. IEEE Trans Power Syst. 2013;28:3215 3225. 11. Zhang LM, Qi XJ. Load supplying capability for distribution network considering distributed generation randomness. Electric Power Construction. 2015;36:38 44. 12. Guo Z, Wei G, Guo YC, Fan J. Dynamic optimization of energy storage power in distribution network based on power supply capacity. Power Syst Protect Contr. 2015;43:1 8. 13. Liu JP, Gao YJ, Wu YD, Li J, Sun JW. Real time evaluation for power supply capacity of distribution network with distributed generation. International Conference on Power System Technology, Chengdu, China, 2014. 14. Pilo F, Pisano G, Soma GG. Optimal coordination of energy resources with a two stage online active management. IEEE Trans Ind Electron. 2011;58:4526 4537. 15. Golestani S, Tadayon M. Distributed generation dispatch optimization by artificial neural network trained by particle swarm optimization algorithm. International Conference on the European Energy Market, Zagreb, Croatia, 2011. 16. Peik herfeh M, Seifi H, Sheikh El Eslami MK. Two stage approach for optimal dispatch of distributed energy resources in distribution networks considering virtual power plant concept. Int T Electr Energy. 2014;24:43 63.

ZHOU ET AL. 11 of 13 17. Su CL, Chuang HM. Energy resources scheduling in distribution systems. IEEE International Energy Conference, Dubrovnik, Croatia, 2014. 18. Borghetti A, Bosetti M, Grillo S, et al. Short term scheduling and control of active distribution systems with high penetration of renewable resources. IEEE Syst J. 2010;4:313 322. 19. Doostizadeh M, Ghasemi H. Day ahead scheduling of an active distribution network considering energy and reserve markets. Int T Electr Energy. 2012;23:930 945. 20. Nick M, Cherkaoui R, Paolone M. Stochastic day ahead optimal scheduling of active distribution networks with dispersed energy storage and renewable resources. IEEE Conference on Technologies for Sustainability, Portland, Oregon, USA, 2014. 21. Peikherfeh M, Seifi H, Sheikh El Eslami MK. Active management of distribution networks in presence of distributed generations. International Conference on Clean Electrical Power, Ischia, Italy, 2011. 22. Chen QC, Zhao XY, Gan DH. Active reactive scheduling of active distribution system considering interactive load and battery storage. Protect Contr Modern Power Syst. 2017;2:29 23. Khojasteh M, Jadid S. A two stage robust model to determine the optimal selling price for a distributed generation owning retailer. Int T Electr Energy. 2015;25:3753 3771. 24. Sadeghi Mobarakeh A, Rajabi Ghahnavieh A, Haghighat H. A bi level approach for optimal contract pricing of independent dispatchable DG units in distribution networks. Int T Electr Energy. 2016;26:1685 1704. 25. Lei X, Yan WJ, Peng P. An optimal purchase and sale power model considering microgrids. Int T Electr Energy. 2015;25:246 261. 26. Miu KN, Chiang HD. Electric distribution system load capability: problem formulation, solution algorithm, and numerical results. IEEE Trans Power Delivery. 2000;15:436 442. 27. Dai WH, Xiong N, Li M. Generation rescheduling under uncertain load growth pattern. Power Syst Technol. 2012;36:136 140. 28. Chen DW, Zhu GP. An investigation on optimal load distribution of microgrids. Autom Elect Power Syst. 2010;34:45 49. 29. Liu X. Impact of beta distributed wind power on economic load dispatch. Electr Power Compon Syst. 2011;39:768 779. 30. Kaplani E, Kaplanis S. A stochastic simulation model for reliable PV system sizing providing for solar radiation fluctuations. Appl Energy. 2011;97:970 981. How to cite this article: Zhou W, Sun K, Sun H, et al. A novel bi level optimization model for load supply capability issue in active distribution network. Int Trans Electr Energ Syst. 2017;e2492. https://doi.org/10.1002/ etep.2492 APPENDIX A A.1 DETAILED FORMULATION OF CHANCE CONSTRAINED BI LEVEL OPTIMIZATION MODEL AND SOLUTION ALGORITHM The mathematical formulation of LSC problem in ADN based on economic dispatch is a chance constrained bi level optimization model. The hierarchical iteration algorithm is applied to deal with bi level issue. The upper and lower subproblems can be solved directly by the particle swarm optimization algorithm based on stochastic simulation. Different from the conventional particle swarm optimization method, the key issues of particle swarm optimization method in this paper are the treatments of equality equations including random variables and inequality constraint conditions in probabilistic form. Equation 7c and 7d represent the AC power balance constraints for all the possible P NDDGi within the uncertainty set. Take the upper subproblem as an example, P NDDGi will be generated adopting randomizer. For the given decision variables, all the state variables can be obtained using power flow calculation and Monte Carlo simulation technology. They are also regarded as random variables. Based on the particle swarm optimization algorithm, stochastic simulation technique is used to validate the constraint conditions in probabilistic form. The implementation process is as follows: n o Pr g j ðx; ξþ 0; j ¼ 1; 2; k β; (A1) where ξis random variable.

12 of 13 ZHOU ET AL. For any given decision variable x,setn 1 to 0, N r random numbers are generated by the probability density function of ξ. Bring all the random numbers of ξ and decision variable x into formula A1. If the inequality constraint is satisfied, N 1 plus 1. According to law of large numbers, formula A1 is established only when N 1 /N r β. IfN 1 /N r < β, particles produced by particle swarm optimization should be abandoned. New particles should be regenerated until N 1 /N r β. According to the proposed chance constrained bi level optimization model, the realization flow is shown as follows: 1. Import related parameters of the dispatchable DERs and ADN. 2. Import random distribution parameters of nondispatchable DERs. 3. Import particle swarm optimization algorithm parameters: termination conditions and population size. 4. Initialize some particles (decision variables). 5. Calculate state variables using power flow calculation and Monte Carlo simulation technology. 6. Calculate the fitness of each particle. 7. Compare the fitness of the particle with its personal best. If the particle's fitness is better, update the personal best. 8. Compare the fitness of the particle with the global best. If the particle's fitness is better, update the global best. 9. According to formulas A2 and A3, update the position and velocity of the particle. Validate the constraint conditions in probabilistic form. If it is not satisfied, new particles should be regenerated until it is satisfied. v nþ1 i ¼ w v n i þ c 1 r 1 p besti x n i þ c2 r 2 G best x n i w ¼ 1:05 n 1:05 0:4 ; (A2) c x nþ1 i ¼ x n i þ v nþ1 i ; (A3) where n is the current iteration, v n i is the velocity of particle i, x n i is the position of particle i, p besti is the personal best of particle i, G best is the global best, r 1 and r 2 are random numbers between 0 and 1. c is the total iteration, c 1 = c 2 =2. 1. Repeat (6) to (9) until the termination condition is met. A.2 Traditional equivalent model of the bi level model through KKTs conditions Ignoring the uncertainty of renewable energy, the lower subproblem of bi level model can be replaced by a traditional equivalent model through KKTs conditions. The equivalent model can be expressed as follows: maximize N P 0 Lið1 þ λk Li Þ a i U 2 i þ b i U i þ c i ; (A4a) P grid þ P NDDGi þ P 0 DDGið1 þ λk DDGi Þ P 0 Lið1 þ λk Li Þ a i U 2 i þ b i U i þ c i Ui N j¼1 U j G ij cos θ ij þ B ij sin θ ij ¼ 0; (A4b) Q grid þ Q NDDGi þ Q DDGi Q 0 Lið1 þ λk Li Þ a i U 2 i þ b i U i þ c i Ui N j¼1 U j G ij sin θ ij B ij cos θ ij ¼ 0; (A4c) U i; min U i U i; max ; (A4d) P DDGi; min P DDGi P DDGi; max ; (A4e) P DDGi ¼ P 0 DDGið1 þ λk DDGi Þ; (A4f) Q DDGi; min Q DDGi Q DDGi; max ; (A4g)

ZHOU ET AL. 13 of 13 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 2 ij þ Q ij S ij; max ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 2 grid þ Q grid S grid; max ; (A4h) (A4i) K DDGi 0; N DG F ¼ C i P 0 N eq DDGið1 þ λk DDGi Þ þ ρgrid P grid μ i g pq;i N ineq δ i h i þ u i h i; max þ σi h i l i h i; min τ k N ineq ðln u i þ lnl i Þ; F K DDGi ¼ 0; F μ i ¼ 0; F δ i ¼ 0; F σ i ¼ 0; F u i ¼ 0; (A4j) (A4k) (A4l) (A4m) (A4n) (A4o) (A4p) F l i ¼ 0; (A4q) where g pq, i = 0 indicates the balance constraints A4b to A4c; h i, min h i h i, max indicates the inequality constraints A4d to A4j; μ i, δ i, and σ i indicate the Lagrange multipliers, u i and l i indicate the non negative slack variables, τ k > 0 indicates the barrier parameter in kth iteration, which is monotonically decreased to 0 as iteration increases; N eq indicates the number of balance constraints; and N ineq indicates the number of inequality constraints.

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