Open capacity measurement and evaluation of n-1 security criteria considering distribution network dispatching security

In order to ensure the safety of distribution network dispatching, the open capacity evaluation, one of the distribution network operation and maintenance tools, is analyzed. Aiming at the shortcomings of traditional reactive power capacity evaluation that is too simple and cannot effectively handle multiple interconnected feeders, a new method is proposed to consider N-1 to establish security and network reconfiguration. Considering the safety factor and the minimum network loss, the open capacity is evaluated, and the interconnection switch is regenerated, which provides the best scheme for the maintenance of the distribution network. This model not only considers the power flow constraints during normal operation but also considers the switching operation constraints under fault conditions. It has the advantages of high solution quality and feasible calculation results. In order to solve the model, a two-level hybrid algorithm based on generalized Benders decomposition, a dichotomous approximation is proposed. The algorithm adopts the idea of upper approximation, lower test, and two-layer interaction. It continuously reduces the solution space through alternate iterations until the optimal solution is obtained. The test results of 22-node and 70-node systems verify the correctness and effectiveness of the model and algorithm in this paper.


Introduction
The distribution network is an indispensable link in the process of power system energy transmission.It directly faces the end users of electric energy, which is of major purport to guarantee the safety of urban power supply.In recent years, with the continuous growth of load, with each passing day complex network structure and the continuous improvement of users' requirements for power supply reliability and power quality, the safe and economic operation of the distribution network has been subjected to unprecedented tests [1] .In order to cope with these tests, the research on this subject can provide technical support for the long-term follow-up planning of the distribution network.
The open capacity evaluation of a distribution network (OCE) is the planning of load access points under the presupposes that taking into account network security [2][3] .The linchpin is to consider the optimization of the N-1 equipment thermal stability quota.The conventional evaluation of distribution networks is dominated by manual participation and experience, which is limited to load growth on the specified topology until the equipment quota [4] .This method is not only of poor accuracy but also limited to a single topology, which is difficult to apply to the situation of multiple interconnection switches.In this regard, researchers have made many attempts and improvements to fundamentally solve the above problems [5][6][7] .Although these efforts are fruitful, there are still some defects and deficiencies, which are specifically shown in the following aspects: 1) For a long time, the bottleneck of mathematical optimization theory has restricted the solution of integer programming, especially mixed integer nonlinear programming.Thus, the existing open capacity 3) As an important supplement to traditional algorithms, intelligent optimization algorithms have been applied in network reconfiguration, open capacity evaluation, and other fields [17][18][19] .However, the inherent problems of intelligent algorithms, such as high computational complexity and insufficient stability of solutions, are prone to lead to long computational time, poor optimality, and other consequences.Therefore, it is urgent to seek more powerful and efficient algorithms to provide effective assistance for accurate and rapid evaluation of open capacity.
For the sake of the foregoing problems, setting up an evaluation model of distribution network openable capacity takes into account N-1 security and network reconfiguration.The model takes into account both power flow constraints and switch operation constraints, which maximizes the power supply capacity and ensures the feasibility of the results.Then, according to the model, the absolute value operation in the switch constraint is transformed equivalently by introducing auxiliary variables, thus retaining the linear characteristics of the model.Finally, for the sake of the foregoing mixed integer nonlinear programming problem, this paper further proposes a two-level hybrid algorithm of generalized Benders decomposition and dichotomy approximation and uses 22-node and 70-node distribution systems to test its effectiveness.

Objective Function
The assessment of the open capacity of the distribution network is to seek the maximum accessible capacity of the load by optimizing the switch state under the current network conditions.Its goal function can be expressed as: max ( ) where λ is the load growth factor, representing the growth multiple of the system load in a uniform proportion.
The above objective functions shall meet the operation requirements under both normal and fault conditions.Specifically, under normal conditions, network security, and power supply quality shall be guaranteed, such as equipment thermal stability limit, network radiation operation, power flow balance constraint, voltage security constraint, etc.In the fault state, it shall be considered that after feeder N-1 fault, the load transfer of the whole feeder can be realized by only one switching operation without capacity overrun.Other problems, such as voltage and reactive power, are not required, which will be solved after the system is recovered.

Constraint Condition
where N is the number of N-1 faults, and k = 0 A indicates normal state; g i I  is the current of substation g to which node i belongs; S , and L S refer to node, substation and line collection respectively; d i I  is the current of load d; ij I is the current of branch ij, and its reference direction is from i to j.
where B N and G N are the number of nodes and substations, respectively; ij x is the state variable of the branch switch, wherein 0 means open and 1 means closed.
where ,max ij I is the maximum allowable current of branch ij.4) Substation capacity constraints.
, , m a x where ,max g I is the maximum allowable current of the transformer.5) Switch operation constraint after fault.
Formula ( 6) indicates that after the fault at the outlet side of the feeder (at this time, the outlet switch has been disconnected), in order to ensure rapid recovery of the power supply, only one interconnection switch is allowed to be closed for multiple interconnection feeders.Other switch states should be consistent with the normal state.It should be added that the N-1 fault of the main transformer is not taken into account in this paper.This is because the failure rate of the main transformer is far lower than that of the feeder.In addition, the failure of the main transformer has been considered in the N-1 security check of the main network.Therefore, only the feeder N-1 can be considered in the assessment of the open capacity of the distribution network to meet the load margin requirements in most cases.At the same time, the results will not be too conservative, resulting in a waste of resources.
In addition to meeting the above constraints, the evaluation model of the open capacity of the distribution network must also ensure that the network under normal conditions (i.e., k=0) has sufficient security.The specific constraints are as follows: 6) Power flow constraint.B 0, B 0, ( cos sin ) ( ) ( sin cos ) ( ) where g P and g Q are the active and reactive output of substation g, respectively; d P and d Q are the active and reactive demand of load d, respectively; ij P and ij Q are the active and reactive power flow through branch ij, respectively; ij g and ij b are the conductance and susceptance of branch ij, respectively; si g and si b are the conductance and susceptance to ground of branch ij; i U and i θ are the voltage amplitude and phase angle of node I, respectively, ij i j θ θ θ   .
In Formulas ( 7) -( 8), each node load has the same growth factor λ, that is, the load increases in the same proportion.However, in the actual distribution network, the composition and nature of different node loads are different, and there are differences in the growth direction between them.In this regard, according to operating experience, different weights i w can be introduced to guide the node load to increase according to their respective proportions.At this time, the objective formula will be converted . For the convenience of research, this paper adopts a unified load growth factor, which will not be repeated later.
,min ,max G , g g g ,min ,max G , where ,min , where ,min i U and ,max i U are the minimum and maximum value of node voltage amplitude, respectively.9) Line power flow constraint.
,max L 0 , where ij S is the apparent power of branch ij, , and ,max ij S is its maximum value.

Basic Ideas
Formulas (1)-( 14) show an MINLP (mixed-integer nonlinear programming) problem.The calculation difficulty lies in the processing of integer variables and nonlinear constraints.At present, there are mature algorithms (such as cut plane, branch, and bound) and corresponding commercial solvers for mixed integer linear programming.For the MINLP problem, there are no very effective calculation methods and tools.Based on the idea of "decomposition coordination", this paper uses the generalized Benders decomposition algorithm [20] to divide the open capacity evaluation model into the main problem of mixed integer linear programming and the sub-problem of continuous nonlinear programming that are easy to solve.It then coordinates them with the help of feasible cuts to gradually shrink the solution space so as to obtain the optimal solution.

Main Problem
Figure 1 shows the structure of the generalized Benders decomposition algorithm for solving OCE.The main problem in the figure is mainly used to deal with discrete variables ( ij x ) and linear constraints (such as switch operation constraints).The goal of this problem is to maximize the load growth factor.On the basis of satisfying the thermal stability constraints, the switch positions under normal and fault conditions are determined.The main problems are as follows: min s.t.formula(2) ( 6) where α is Benders cut.
The resulting nonconvexity will greatly reduce the convergence of the whole problem.To solve this problem, you can see λ as an outer parameter.Its optimal value is approximated by dichotomy.The specific methods are: 1) The integer linear programming problem is calculated through Formulas ( 1)- (6).λ is obtained, where the maximum value of max   6) Since Formula (16) contains the absolute value operation, i.e., Formula (6), this constraint results in the main problem that cannot be directly calculated.In this regard, a conventional approach is to convert it into a quadratic constraint at the expense of convergence.On the basis of Formula (16), this paper extends the absolute value constraint to the objective function by introducing the auxiliary variable , ij k ε , thus retaining the linear characteristics of the rudimentary problem.The derivation process is as follows: , it can be considered that the switching operation under fault k meets the requirements, and the corresponding capacity cost is zero; When , the switch is forced to meet the requirements of Formula ( 6) by giving a larger unit operating cost.Based on the above ideas, the main problem (16) can be converted into: where ρ is the cost converted from switching operation to capacity.

Calculation Results and Analysis
Figure 3-(a) shows the wiring diagram of a 22-node system.The system has 5 feeders (N=5) and 21 branches.The impedance of each branch is 1.270+j0.390Ω, and the maximum allowable current is 80.0 A. The reference voltage of the system is 10 kV, and the voltage operation limit is 0.93~1.00p. u.The load of nodes 1~3, 7~8, 13~14 is 0.200+j0.150MVA, the load of nodes 4, 10, 15, 18~22 is 0 MVA, and the load of nodes 5~6, 9, 11~12, 17 is 0.5×(0.200+j0.150)MVA, the load of node 16 is 0.8×(0.200+ j0.150)MVA.The following three scenarios are discussed.Scenario 1 is the OCE model without considering N-1 security constraints; that is, Formulas (2)-( 6) only take the normal state k=0. Figure 3-(b) shows the network topology obtained in this scenario.The corresponding calculation time is 1.30 s, and the load growth factor is 2.1642.It can be seen that switches 2-3, 4-16, 14-15, and 9-10 are open, and the reconstruction results meet the radial constraint.In Scenario 1, the effect of N-1 on the branch current is not taken into account.In order to obtain the maximum openable capacity, OCE requires that the load of each feeder under normal conditions should be as small as possible; that is, the load should be balanced as much as possible.At this time, the feeder with the highest load is 20→7→8→9, and the total load is 2.5×(0.200+j0.150)MVA.
Scenario 2 is the OCE model considering feeder N-1 security constraints, where k is taken as 0~5, corresponding to Figures 3-(c~h) respectively.Wherein, Figure 3-(c) is the reconstructed topology under normal operation state, and Figures 3-(d~h) is the topology after 18~22 bus faults in turn.The CPU time of this scenario is 1.90 s, and the load growth factor is 1.3856.It can be seen that compared with Scenario 1, there was a significant reduction.In the figure, the switches opened under normal conditions are 3-4, 4-16, 15-17, and 9-10, and the reconstruction results also meet the radial constraint.By analyzing the five fault states in the figure, it can be seen that in the fault state, except for disconnecting the feeder outlet switch to simulate the N-1 fault, only one interconnection switch is closed (for example, in Figure 3-(f), the bus 20 fault causes the outlet switch 20-7 to open, and the switch 9-10 is closed to ensure the power supply of loads 7~9).The rest of the switch states are consistent with the normal state, which indicates that the switch operation number constraint ( 6) plays a role.The validity of model ( 19) is also verified.Compared with Scenario 1, in order to obtain the maximum openable capacity, considering the OCE of N-1, the load of each feeder in the fault state (the most serious) rather than the normal state should be balanced as far as possible.At this time, the highest load case is feeder 19→6→5 →4→3→2→1 in Figure 3-(d), and the total load is 4.0×(0.200+j0.150)MVA.
Analysis of Figure 3-(c) shows that there are multiple optimal solutions in Scenario 2. In addition to Figure 3-(c), the normal operation state also includes four cases in Figures 3-(i~l), and the corresponding openable capacities of these four cases are equal.However, only one of the five optimal solutions, including Figure 3-(c), has the minimum network loss.To illustrate this problem, Scenario 3 further considers the network loss under normal operation based on Scenario 2. After calculation, the network loss in Figure 3-(c

Conclusion
In order to accurately measure the load-carrying capacity of the distribution network, especially the multi-ring network, so as to effectively formulate the corresponding industrial expansion reporting strategy and upgrading measures, this paper studies the evaluation model and method of the open capacity of the distribution network.The main work is as follows: 1) The switch operation, power flow, and feeder N-1 security constraints are introduced into the calculation of distribution network supply capacity.An evaluation model of distribution network openable capacity considering network reconfiguration is established.The model can convert the switch operation constraints after fault equivalently.It is applicable to various distribution network connection forms, such as single connections and multiple connections.It has the characteristics of strong adaptability, high calculation accuracy, and excellent reliability of results.
2) A two-level hybrid algorithm of generalized Benders-dichotomy approximation for solving OCE is proposed.In the first layer, the idea of dichotomy approximation is used to continuously shrink the range of maximum power supply capacity to drive the solution in the second layer.In the second layer, based on the generalized Benders decomposition strategy, the feasibility test and upper and lower bounds update of the openable capacity are carried out to realize the interaction between the first layer and the second layer.The two layers iterate alternately until the maximum open capacity is obtained.
3) The proposed model and algorithm are tested with 22-bus and 70-bus distribution systems.The test results of the example show that the open capacity evaluation model and algorithm considering network reconfiguration can well solve the problems of oversimplification of traditional models and the inability of solving algorithms to effectively deal with complex constraints.They have a certain reference value for improving the operation and planning level of distribution networks.
It is worth pointing out that with the rapid development of distributed energy technology, it has become normal for a large number of distributed power sources to access the distribution network.After the grid connection of distributed energy, the status quo of unidirectional energy flow in the traditional distribution network has been changed, and the operation and control of the distribution network have become more complex.Therefore, it is necessary to take this impact into account in the subsequent research and, at the same time, promote the assessment of open capacity from load to distributed generation to meet more practical needs.
and maximum output active power of the substation; ,min g Q and ,max g Q are the minimum and maximum output reactive power of the substation.8) Node voltage constraint.,min ,max B

Figure 1 .
Figure 1.Schematic Diagram of Algorithm Structure

λ
= λ + λ at the nth outer cycle is calculated.λ is taken, and the known value is substituted into the improved main problem (16) and sub-problem to solve; 3) When the major sub-problem is feasible, the lower bound is updated ( ) min n λ = λ ; Otherwise, the upper bound is updated ( ) max n λ = λ ; 4) We then judge whether convergence condition 4 max min < 10 λ λ   is true.If yes, the maximum growth factor min λ will be obtained, and the program will exit; Otherwise, we return to step 2).
) is the smallest, about 32.7349 kW, and the network losses in the other four cases are 32.9512kW, 42.4762 kW, 35.5013 kW, and 41.2846 kW, respectively.

Figure 2 .
Figure 2. Schematic Diagram of Algorithm Structure