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| Non-Iterative Optimal Power Flow Calculation Based on Holomorphic Embedding Method |
| Liu Chengxi, Xu Shenkai, Lai Qiupin |
| School of Electrical Engineering and Automation Wuhan University Wuhan 430072 China |
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Abstract It is very important to obtain the active power output of each generator by solving the optimal power flow (OPF) for the economic operation of power system. The traditional methods mainly include simplified model, convex relaxation and intelligent algorithm, but these methods also have some shortcomings such as imprecision and dependence on the initial value. Holomorphic embedding method (HEM) is a new method used in power flow calculation and it ensures convergence through proper holomorphic embedding forms. HEM has the advantages of non-iteration, recursion, analyzability, no dependence on the initial value, and the obtained solution is a runnable point. Based on these above advantages, HEM is used to solve the OPF flow, and a non-iterative OPF calculation method based on HEM is proposed.
Firstly, taking a 3-bus power system as an example, the objective function is the lowest generating cost, and the active power output of the generator is optimized. Secondly, constraints and KKT conditions of the optimization model are listed. Based on the basic construction rules of HEM, the holomorphic embedding forms of constraints and KKT conditions are constructed. Thirdly, according to the basic properties of holomorphic functions, the recursive equations of power series coefficients of analytic expressions of bus voltage and generator active power output are derived. Finally, coefficients can be obtained by the cross recursion of the above two sets of equations, and the generating cost is obtained.
Simulation results on IEEE 3-bus, 4-bus, 30-bus and 39-bus New England power systems show that the HEM is convergent when solving the OPF. The calculation result of the active power output of generators and the bus voltage are close to the reference value, and the error is less than 1% in the first two power systems. In the four systems, the convergence accuracy is achieved at the 6th, 8th, 10th and 12th order respectively. For the result of the objective function, the error of the results calculated by HEM reaches to 10-6 in the first two systems. The source of error is the order of recursion, and the process of each recursion is to make the result gradually approximate to the exact solution. However, the error increases with the increase of the scale of the system. In New England 39-bus system, the error is larger because inequality constraints are not considered in the system, which makes the active power output more ideal, so that the generating cost and active power loss are smaller. In the case of a small system scale, the holomorphic embedding method proposed in this paper has a shorter computing time than the interior point method (IPM). However, the computing time in IEEE 30-bus and New England 39-bus systems is longer than the IPM. Finally, the feasibility of the HEM in IEEE 118 bus power system is verified.
The following conclusions can be drawn through the simulation analysis: (1) HEM can be used to obtain the analytical expressions of the bus voltage and the active power output of the generator. The coefficients of the power series are obtained through the cross recursion of the KKT condition and constraints in holomorphic embedding form. (2) Due to the constraint of power flow included in the model, the optimal result obtained is operable in the real system, which has the advantage of being independent of the initial value. (3) The calculation results of HEM meet the accuracy requirements, and it is proved that HEM is feasible in the calculation of OPF.
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Received: 10 March 2022
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2023, Vol. 38 
