Active distribution network planning considering linearized system loss

In this paper, various distribution network planning techniques with DGs are reviewed, and a new distribution network planning method is proposed. It assumes that the location of DGs and the topology of the network are fixed. The proposed model optimizes the capacities of DG and the optimal distribution line capacity simultaneously by a cost/benefit analysis and the benefit is quantified by the reduction of the expected interruption cost. Besides, the network loss is explicitly analyzed in the paper. For simplicity, the network loss is appropriately simplified as a quadratic function of difference of voltage phase angle. Then it is further piecewise linearized. In this paper, a piecewise linearization technique with different segment lengths is proposed. To validate its effectiveness and superiority, the proposed distribution network planning model with elaborate linearization technique is tested on the IEEE 33-bus distribution network system.

strategies is given in [1,2,10]. DNP problem considering DGs can be divided into single planning and comprehensive coordinated planning according to the type of decision variables [11]. Single planning is to determine optimal placement and capacity of DGs without considering the planning of substation configuration and feeders [12], while comprehensive coordinated planning is an overall planning of DGs, substations and distribution feeders. And this paper will focus on the latter one, i.e., optimizing the capacities of DG and the optimal distribution line simultaneously. The rest of this paper is organized as follows: Section 2 describes the planning model of ADN. A piecewise linearized method is introduced in Section 3 in order to transfer the traditional planning model into the stand mixed integer linear programming (MILP) problem. Then a piecewise linearization technique with different segment length is proposed. The above method is applied to the IEEE 33-bus distribution network and the results are presented in Section 4. Finally, some conclusions are drawn in Section 5.

Planning model of ADN
In this section, a formulation of the active distribution network planning problem is presented. The planning model, which assumes that the location of DGs and the topology of the network are fixed, optimizes the capacities of DG and distribution line by a cost/benefit analysis and the benefit is quantified by the reduction of the expected interruption cost. The objective function is to maximize the social welfare, or it can be equally described as minimizing the total cost. The optimization model is set up as follows:

Objective function
The objective function for ADN planning: (1) where TC is the abbreviation of total cost, which means the total cost of the ADN planning. 1) DG construction, operation and maintenance costs (2) where N DG is the number of total DG nodes containing DGs; is DG fixed investment annual average cost factor: where r is annual percentage rate, t is planning period. And C equ is the equipment investment cost of DG on node i; C ope is the operate cost of DG on node i; C rep is annual maintenance cost of DG on node i; i  is power factor of ith DG unit; S DGi is the rated capacity of DG on node i; C ins is the fixed installation cost of DG unit on node i [11].

2) feeders investment costs
where k b is line annual investment cost of per unit length; l b is line planning length; T b is line planning capacity, and N l is the number of total distribution lines. where C e is the electricity price; P L is the total load capacity of the distribution system; and T max is the maximum load equivalent hours. 4 (6) The details of calculating P loss will be given in part 3.
where LOL i,t is the power expected interrupt loss on node i in time t, and VOLL is the value of lost load.

Constraints
The constraints (8) enforce the total power balance, where N i is the number of total distribution lines. The constraints (9) enforce line flow limits at every distribution line. The constraints (10) enforce the nodal voltage limits. In this model, the fluctuation range is within 7% of the normal operation voltage. The constraints (11) and (12) are output limits of units, where means the reactive power absorbed by a reactive compensator. And the constraints (13) is co-ordinated voltage regulation (area voltage control) using OLTC, T k [13]. The constraints (14) and (15) represent the N-1 security criterion, where P l,ll means the active power flow in the line "l" when the other line "ll" in the network fails, and H l,ll is a transfer factor.

Traditional linearizition method
The network loss is explicitly analyzed in this section. For normal operation, under the flat voltage assumption, the network loss is appropriately simplified as a quadratic function of difference of voltage phase angle [14]. That is, the power injection in the line (i, j) computed at bus i, , y ij is the admittance of the line (i, j). Here, we assume that 1  ij U in the normal operation of distribution network, then the power loss of the line (i, j) can be obtained as follows: Then it is further piecewise linearized by using 2L piecewise linear blocks as shown in Figure.1.
However, only L piecewise linear blocks are sufficient by using the positive orthant only. In order to achieve this purpose, we need introduce the linearization of absolute sign: where k ij (l) and ) (l ij  means, the slope and value of the lth block of angle, respectively. The quadratic formulation of (18) is piecewise linearized to the above expression with the introduction of absolute sign. While the absolute value is still not a linear expression, a linear expression of the absolute value in (19) is needed, which is obtained by the following math substitution [15]: Then the power flow can be expressed as follows by using the above piecewise linearization methods: Then the entire model can be transferred into a mixed integer linear model and it can be solved by state-of-art mixed integer linear programming (MILP) commercial solver.

An advanced piecewise linearization technique
The piecewise linearization strategy has great effect on the approximation accuracy of the network loss. If less linearization segment is used, the approximation error will be considerable although the computation speed will be fast. If more linearization segment is applied, the approximation accuracy will be better at the expense of a heavier computation burden.
In our paper, it is found that with the variation of number of linearization segment, the value of network loss would change significantly. After careful analysis, it is found that the difference of voltage phase angle is usually very small in reality, which means that the operation point of the simplified quadratic loss formulation is usually around the zero point. Under this condition, even a large number of linearization segment would cause a poor approximation error of network loss. Based on this observation, a piecewise linearization technique with different segment length is proposed. When do the piecewise linearization, more segments is introduced when the operating point is near zero. With the deviation from the zero point, less piecewise linearization segments is introduced. The Figure 1. Piecewise linearization of system loss in a branch. proposed linearization technique is tested on the IEEE 33-bus distribution network in the following case study.

Case study
The proposed model has been applied to IEEE-33 bus distribution network system. The proposed methodology has been developed in GAMS. The IEEE 33-bus system used in this paper has 33 nodes, 37 existing branches, and 32 loads, it is presented in Figure. 2. The optimal planning results of the capacities of DG and distribution line is showed in table 1and  table 2. It should be noted that the node 1 here is a substation, not DGs is connected, this means the capacity showed on node 1 is the power purchased from upstream grid. And it assumes that the installation location of the DG is given in the model, here for the nodes 5,10,20,30, and then we only need to determine their capacities. The distribution line capacity planning results when the number of linearization segments of calculating line loss is 180 is as follows:  Table 3 gives the itemized cost with and without DG units. It is shown that DGs can help reduce the power loss cost, and the total cost is decreased after DG installed. Table 4 shows the variation of run time and total cost versus the number of linearization segment, and the corresponding figure is shown in Figure.3. It can be seen that if more linearization segment is applied, the approximation accuracy will be better at the expense of a heavier computation burden, and finally the approximation result tends to a stable value with the linearization segment arises.

Conclusions
The problem of optimal capacity of DGs and distribution lines in ADN planning has been considered in this paper. The proposed model is based on premises that the location of DGs and the topology of the network are fixed. And a piecewise linearization technique to calculate network loss is explicitly analyzed, and then the model can be transformed into a MILP problem so that it can be solved by  state-of-art MILP commercial solver. An advanced piecewise linearization technique with different segment length is proposed. The effectiveness of the proposed ADN planning model with elaborate linearization technique is verified on the IEEE 33-bus distribution network. The results show that the total cost and line loss cost are decreased after DG installed, and the proposed linearization technique can get a good balance between the approximation accuracy and the computation efficiency.