Robust Planning of Distributed Generators in Active Distribution Network Considering Network Reconfiguration

Abstract

The energy crisis and environmental concerns have accelerated the development of the active distribution network (ADN) with a high proportion of renewable energy, which poses a challenge to the operation of the power system. Moreover, using active management means to promote the consumption of renewable energy is an important task of ADN. Therefore, as an important operation means, the network reconfiguration is used to enhance the adjustable capacity of the power system at the planning stage. Firstly, a “wind–light–load” uncertain scenario set is constructed to address the uncertainty of wind speed, lighting, and load. On this basis, a robust optimization model for distributed power generation taking into account network reconstruction and in ADN is proposed. In addition, the distributed generator (DG) permeability indicator is introduced in the planning model to improve the ADN ability of absorbing renewable energy. A linearized AC power flow model is utilized to calculate the power flow. Finally, via simulation in an IEEE 33-bus system and IEEE 69-bus system, the influence of network reconfiguration and robustness on distributed generator planning, economy and reliability of ADN is analyzed, and the validity of the model is verified.

1. Introduction

With the continuous shortage of fossil energy and the increasing demand for environmental protection, the proportion of clean energy in the energy system is increasing. Especially in recent years, with the rapid development of wind power generation and photovoltaic power generation technology, wind turbine power generation and photovoltaic power generation have been widely used. In order to make reasonable use of various resources and improve the controllability of the power system, active distribution networks have emerged.

On the one hand, DG devices improve the utilization of clean energy and increases the flexibility of the network. On the other hand, the uncertainty of DG also has a certain impact on the stability and safe operation of the network. Therefore, it is necessary to plan DG reasonably to make ADN operate more safely and reliably, and make full use of renewable energy for power generation.

ADN emphasizes that various controllable sources in modern distribution networks, especially renewable energy generation, should be passively absorbed to actively managed and utilized [1]. Therefore, it is necessary to consider the operation characteristics of ADN while planning distribution network. An optimal allocation model of renewable energy considering the special supply capacity of ADN under the active management mode is proposed [2]. A double-layer programming model of ADN is established to promote the efficient utilization of intermittent DG [3]. The literature [4,5] establishes the multi-time-scale optimization model and multi-objective DG location and capacity planning model considering the investment level and operation simulation level, respectively. In the ADN planning stage, a joint planning operation optimization model considering the economic dispatching strategy of DG and the energy storage system (ESS) is developed in [6]. Although some operation characteristics of ADN are considered in the DG planning stage, few scholars consider network reconfiguration in the DG planning stage, although network reconfiguration is among the important operation means of distribution network.

At the same time, the high proportion of renewable energy integration into the power grid is another major feature of ADN. The large-scale penetration of renewable energy increases the uncertainty of distribution network, posing challenges to distribution network planning. Therefore, references [7,8] establish the corresponding multi-objective programming model for the uncertainty of intermittent DG output and load behavior. Considering the uncertainty of DG output and load growth, an ADN planning model is established with DG and grid as planning objects [9,10]. A probability model considering the uncertainty of DG is proposed in [11], and used the theory of multi-state system to transform the stochastic problem into the deterministic problem. DG planning model is developed based on multi-scenario analysis solution [12,13,14]. Considering the intermittency of DG and the correlation between wind speed, light intensity and load, the chance constrained programming method is proposed [15]. Taking into account the random fluctuation of wind turbine output power, a probability analysis model of wind power generation is established [16]. The uncertainty of DG is considered in the planning stage in the above documents, but the robust optimization model is rarely used. According to [17,18], robust optimization model has significant adaptability to the uncertainty of DG and load, and should be widely used. However, the above literature only deducts the uncertainty of DG and does not consider improving the consumption of renewable energy. In the context of a high proportion of renewable energy connected to the power grid, it is crucial to consider the consumption of renewable energy during the planning phase.

In the planning stage, some scholars gradually consider the characteristics of active management of active distribution networks. The authors in [19,20] considered resource active management in ADN planning. Additionally, ESSs offer new opportunities in ADN and can be utilized in active network management in order to improve the decision making on the network expansion planning [21,22]. However, as an important active management method, network reconfiguration is ignored in the existing research. To solve the above research gaps, a robust planning model considering network reconfiguration and DG permeability is proposed in this paper. Significantly, distribution network reconfiguration means to change the network topology by changing the section switch and tie switch of the network, so that the distribution network can achieve the best power flow distribution under the constraints of the system, so as to achieve a balanced load, meet the requirements of the new load, and increase the voltage stability, the purpose of minimizing network loss [23]. Under the active management mode, the distribution network has multiple active management measures. As an important active management measure to optimize power flow and reduce network loss, network reconfiguration should be considered in the distributed generation planning stage. The essence of distribution network reconfiguration is to change the topology of the distribution system according to the load change, and to solve the optimal combination of the opening and closing states of the tie switch and the section switch under certain constraints. Therefore, the distribution network reconfiguration is a “0–1” combination of tie switch and section switch states. Therefore, the specific structure of the switchgear is not be introduced.

Therefore, the main contributions of this paper are as follows:

(1)

The “wind–light–load” uncertain sets are developed to describe the uncertainty of wind speed, light intensity and loads. Moreover, the DG permeability indicator is introduced in the planning model.

(2)

Network reconfiguration, as an important operation means of active distribution, is considered in the planning model to optimize the location and capacity of DG. The planning scheme dispatches the existing resources in the active distribution network to meet the load demand, thus avoiding the redundancy of planning resources as much as possible.

(3)

The two-level optimization is utilized to decompose the planning model. At the same time, a linearized AC power flow model is utilized to calculate the power flow. Therefore, the planning level and operating level are mixed integer optimization models.

The remainder of this paper is organized as follows. The framework for DG planning is proposed in Section 2. The wind–light–load uncertain sets are developed in Section 3. Then, the robust programming model for DG planning is represented in Section 4. Section 5 verifies the effectiveness of the proposed model and demonstrates the simulation results. Conclusions are given in Section 6.

2. The Framework for DG Planning

2.1. The Structure of ADN

Active distribution network is a public distribution network with flexible topology, which adopts active management of a distributed power supply. The structure of ADN is shown in Figure 1. As shown in Figure 1, various DGs (such as wind power, photovoltaic, etc.) are converted into corresponding AC or DC modes through power electronic components, and then incorporated into the system through step-up transformers. Communication, automation and other relevant electrical equipment can achieve close integration with the power grid by proper connection. In addition, the user side is equipped with advanced metering infrastructure (AMI) represented by smart meters, which is used to achieve real-time collection of electricity information and bidirectional interoperability between the power grid and users [24].

The active feature of active distribution network technology is mainly reflected in the system operation control mode. Under the active distribution network, through advanced automation technology, the regional supply side and demand side resources can be actively managed to achieve the optimization of specific operation objectives of the system. Therefore, as an important means of active management, network reconfiguration needs to be considered in the planning stage. Moreover, it is crucial to maximize the consumption of green energy through active management. Based on this, a robust planning model considering network reconfiguration and DG permeability is proposed in this paper. The framework for DG planning is described in Section 2.2.

2.2. The Framework for DG Planning

The framework for DG planning is illustrated in Figure 2. The impacts of wind speeds on failures of components in IPGS are molded. Firstly, we determine the fluctuation proportion range. Based on it, the scene of “wind speed–light intensity–load” is generated. And then, the electrical power generated by wind turbines and photovoltaics is calculated, which is described in Section 3. Based on the above uncertain scenario set, a robust optimization model for distributed generators in the active distribution network considering network reconfiguration and DG permeability is built. According to the model characteristics, a two-level optimization method is used to decompose the model. The specific description of the planning model can be found in Section 4.

3. The Wind–Light–Load Uncertain Sets

Since the hierarchical robust planning model aims to adapt to the uncertainty development of DG and load [25], the key lies in the construction of the uncertainty set.

In this paper, the DGs to be selected are a wind turbine generator (WTG) and photovoltaic generator (PVG). According to the characteristics of uncertainty of wind speed, light intensity and load, statistical prediction values are made by collecting historical data of wind speed, light intensity and load in typical areas. This paper takes 24 h data as the research object of simulation operation, so as to ensure that different timing characteristics of wind power, photovoltaic power generation and load are fully reflected. The process of constructing the uncertain sets is shown in Figure 3 and described as follows:

First, determine the fluctuation proportion range of wind speed, light intensity and load as: $\left[{{\displaystyle -\delta }}_w,{{\displaystyle \delta }}_w\right]$, $\left[{{\displaystyle -\delta }}_p,{{\displaystyle \delta }}_p\right]$, $\left[{{\displaystyle -\delta }}_l,{{\displaystyle \delta }}_l\right]$.

Then, Latin hypercube sampling technology is used to randomly generate the percentage of fluctuation at 24 h, and according to the historical data of wind speed, light intensity and load, a multi-scene set of “wind speed–light intensity–load” is generated.

Finally, according to Formulas (1) and (2), wind speed and light intensity are converted into active power of wind power ${{\displaystyle P}}_{WTG}$ and active power of photovoltaic power generation ${{\displaystyle P}}_{PVG}$ [26].

$${{\displaystyle P}}_{WTG}=\left\{\begin{array}{l}0,0\le {{\displaystyle V}}_h\le {{\displaystyle V}}_{ci}or{{\displaystyle V}}_{co}\le {{\displaystyle V}}_h\\ {{\displaystyle P}}_{WTG}^r\frac{{{\displaystyle V}}_h-{{\displaystyle V}}_{ci}}{{{\displaystyle V}}_r-{{\displaystyle V}}_{ci}},{{\displaystyle V}}_{ci}\le {{\displaystyle V}}_h\le {{\displaystyle V}}_r\\ {{\displaystyle P}}_{WTG}^r,{{\displaystyle V}}_r\le {{\displaystyle V}}_h\le {{\displaystyle V}}_{co}\end{array}\right.$$

(1)

$${{\displaystyle P}}_{PVG}=\left\{\begin{array}{l}{{\displaystyle P}}_{PVG}^r\frac{I}{{{\displaystyle I}}_r},I\le {{\displaystyle I}}_r\\ {{\displaystyle P}}_{PVG}^r,I>{{\displaystyle I}}_r\end{array}\right.$$

(2)

where ${{\displaystyle P}}_{WTG}^r$ is denoted as the rated capacity of WTG. ${{\displaystyle V}}_h$ is denoted as the wind speed at the hub of the WTG impeller. ${{\displaystyle V}}_{ci}$, ${{\displaystyle V}}_r$, and ${{\displaystyle V}}_{co}$ are, respectively, denoted as the cut in wind speed, rated wind speed and cut out wind speed of the WTG. ${{\displaystyle P}}_{PVG}^r$ is denoted as the rated capacity of PVG. I is the light intensity. ${{\displaystyle I}}_r$ is denoted as the rated light intensity of PVG.

4. Robust Programming Model for DG Planning

Aiming at the uncertainty of wind speed, light and load, this paper develops a DG robust planning model considering network reconfiguration on the basis of the uncertain sets.

4.1. The Objective Function

The objective function of this paper is to minimize the annual comprehensive cost. It includes the fixed annual investment cost of DG ${{\displaystyle C}}_I$, the annual operation and maintenance cost of DG ${{\displaystyle C}}_{OM}$, the network loss cost ${{\displaystyle C}}_{loss}$, and the power purchase cost from the superior grid ${{\displaystyle C}}_{up}$. In particular, the restructuring costs ${{\displaystyle C}}_C$ is considered. At the same time, according to the characteristics of ADN, the operation process of ADN is considered in the planning stage, so the active management cost of DG ${{\displaystyle C}}_{AM}$ should also be included in the objective function.

The objective function expression is

$$\min F={{\displaystyle C}}_I+{{\displaystyle C}}_{OM}+{{\displaystyle C}}_{loss}+{{\displaystyle C}}_{up}+{{\displaystyle C}}_{AM}$$

(3)

The specific expression of each part of the cost in Equation (3) is as follows.

(1)

DG investment cost ${{\displaystyle C}}_I$:

$${{\displaystyle C}}_I={{\displaystyle R}}_{WTG}{\displaystyle \sum _{i=1}^{{{\displaystyle N}}_{bus}}({{\displaystyle c}}_{WTG,i}^I\times {{\displaystyle p}}_{WTG,i}^r)+{{\displaystyle R}}_{PVG}}{\displaystyle \sum _{i=1}^{{{\displaystyle N}}_{bus}}({{\displaystyle c}}_{PVG,i}^I\times {{\displaystyle p}}_{PVG,i}^r)}$$

(4)

$${{\displaystyle R}}_{WTG}=\frac{d{{\displaystyle \left(1+d\right)}}^{{{\displaystyle y}}_{WTG}}}{{{\displaystyle \left(1+d\right)}}^{{{\displaystyle y}}_{WTG}}-1}$$

(5)

$${{\displaystyle R}}_{PVG}=\frac{d{{\displaystyle \left(1+d\right)}}^{{{\displaystyle y}}_{PVG}}}{{{\displaystyle \left(1+d\right)}}^{{{\displaystyle y}}_{PVG}}-1}$$

(6)

where ${{\displaystyle R}}_{WYG}$ and ${{\displaystyle R}}_{PVG}$ are denoted as the present value to equivalent annual coefficients of WTG and PVG, respectively. d is the discount rate. ${{\displaystyle y}}_{WYG}$ and ${{\displaystyle y}}_{PVG}$ are denoted as the economic service life of WTG and PVG, respectively. ${{\displaystyle N}}_{bus}$ is denoted as the number of distribution network nodes. ${{\displaystyle c}}_{WTG,i}^I$ and ${{\displaystyle c}}_{PVG,i}^I$ are denoted as the fixed investment costs per unit capacity of WTG and PVG installed at node i, respectively. ${{\displaystyle p}}_{WTG,i}^r$ and ${{\displaystyle p}}_{PVG,i}^r$ are denoted as the rated capacities of the WTGs and PVGs installed at node i, respectively.

(2)

Operation and maintenance cost of DG ${{\displaystyle C}}_{OM}$:

$${{\displaystyle C}}_{OM}={\displaystyle \sum _{s=1}^{{{\displaystyle N}}_s}{\displaystyle \sum _{i=1}^{{{\displaystyle N}}_{bus}}\left({{\displaystyle c}}_{WTG,i}^{OM}{{\displaystyle p}}_{WTG,i,s}+{{\displaystyle c}}_{PVG,i}^{OM}{{\displaystyle p}}_{PVG,i,s}\right)}}{{\displaystyle t}}_s$$

(7)

where ${{\displaystyle N}}_S$ is denoted as the number of ADN operation scenarios. ${{\displaystyle t}}_s=8760\times {{\displaystyle p}}_s$ is denoted as the duration of scenario s in a year, and 8760 is the number of hours in a year; ${{\displaystyle p}}_s$ is the probability of occurrence of scenario s; ${{\displaystyle c}}_{WTG,i}^{OM}$ and ${{\displaystyle c}}_{PVG,i}^{OM}$ are denoted as the operation and maintenance costs of the unit electricity generated by the WTG and PVG installed at node i, respectively. ${{\displaystyle p}}_{WTG,i,s}$ and ${{\displaystyle p}}_{PVG,i,s}$ are denoted as the active outputs of WTGs and PVGs installed at node i under scenario s, respectively; ${{\displaystyle N}}_s$ is denoted as the number of scenarios.

(3)

Network loss cost ${{\displaystyle C}}_{loss}$:

$${{\displaystyle C}}_{loss}={\displaystyle \sum _{s=1}^{{{\displaystyle N}}_s}{\displaystyle \sum _{h=1}^H{{\displaystyle c}}_{t,h}^{loss}{{\displaystyle p}}_{t,h,s}^{loss}\cdot {{\displaystyle t}}_s}}$$

(8)

where h is denoted as the number of daily operation period; H is denoted as the total number of daily periods; ${{\displaystyle c}}_{t,h}^{loss}$ is denoted as the network loss price in t period; ${{\displaystyle p}}_{t,h,s}^{loss}$ is denoted as the network loss in period t.

(4)

Power purchase cost from the power system ${{\displaystyle C}}_{up}$:

$${{\displaystyle C}}_{up}={\displaystyle \sum _{s=1}^{{{\displaystyle N}}_s}{{\displaystyle \rho }}_s}{{\displaystyle p}}_{sub,s}{{\displaystyle t}}_s$$

(9)

where ${{\displaystyle p}}_{sub,s}$ is denoted as the active power of purchasing electricity from the superior power grid in scenario s. ${{\displaystyle \rho }}_s$ is denoted as power purchase cost for ADN to purchase power from the power grid in scenario s.

(5)

Reconstruction cost ${{\displaystyle C}}_C$:

$${{\displaystyle C}}_C={\displaystyle \sum _{j=1}^{{{\displaystyle N}}_s}{\displaystyle \sum _{i=1}^N{{\displaystyle E}}_{{{\displaystyle B}}_j}\left|{{\displaystyle s}}_{j,i}-{{\displaystyle s}}_{j,i-1}\right|}}$$

(10)

where ${{\displaystyle E}}_{{{\displaystyle B}}_j}$ refers to the cost caused by changing the primary opening and closing state of the switch numbered j; ${{\displaystyle S}}_{j,i}$ refers to the on-off state of the switch numbered j in the i th period. S_j,i = 1 indicates that the switch is closed, and S_j,i = 0 indicates that the switch is open.

(6)

The active management costs of DG ${{\displaystyle C}}_{AM}$:

$${{\displaystyle C}}_{AM}={\displaystyle \sum _{s=1}^{{{\displaystyle N}}_s}{\displaystyle \sum _{i=1}^{{{\displaystyle N}}_{bus}}\left({{\displaystyle c}}_{WTG,i}^{AM}{{\displaystyle p}}_{WTG,i,s}+{{\displaystyle c}}_{PVG,i}^{AM}{{\displaystyle P}}_{PVG,i,s}\right)}}{{\displaystyle t}}_s$$

(11)

where ${{\displaystyle c}}_{WTG,i}^{AM}$ and ${{\displaystyle c}}_{PVG,i}^{AM}$ are denoted as the active management costs of the WTGs and PVGs installed at node i, respectively.

4.2. Constraints

The constraints in this paper are system power flow constraints, node voltage constraints, branch power constraints, DG operation constraints, switching times constraints, and network topology constraints. In particular, according to [27], one of the characteristics of ADN different from traditional distribution networks is that it has a certain proportion of distributed controllable resources. Therefore, in the planning stage of ADN, the proportion of DG input should be guaranteed to meet the needs of ADN, so as to ensure more stable operation of the power grid. Therefore, DG permeability constraints should be considered. The specific formulas for each constraint are shown as following.

(1)

Power flow constraint:

A linearized AC power flow model proposed is utilized here. The linearization method has been proved applicable in [28].

The linearized AC power flow model can be expressed as

$${{\displaystyle k}}_{ij\_1}^{}=\frac{{{\displaystyle x}}_{ij}}{{{\displaystyle r}}_{ij}^2+{{\displaystyle x}}_{ij}^2},{{\displaystyle k}}_{ij\_2}^{}=\frac{{{\displaystyle r}}_{ij}}{{{\displaystyle r}}_{ij}^2+{{\displaystyle x}}_{ij}^2}$$

(12)

$${{\displaystyle P}}_{ij}^{s,t}={{\displaystyle k}}_{ij\_1}\cdot \left({{\displaystyle \delta }}_i^{s,t}-{{\displaystyle \delta }}_j^{s,t}\right)+{{\displaystyle k}}_{ij\_2}\cdot \left({{\displaystyle V}}_i^{s,t}-{{\displaystyle V}}_j^{s,t}\right)$$

(13)

$${{\displaystyle Q}}_{ij}^{s,t}=-{{\displaystyle k}}_{ij\_2}\cdot \left({{\displaystyle \delta }}_i^{s,t}-{{\displaystyle \delta }}_j^{s,t}\right)+{{\displaystyle k}}_{ij\_1}\cdot \left({{\displaystyle V}}_i^{s,t}-{{\displaystyle V}}_j^{s,t}\right)$$

(14)

$${{\displaystyle P}}_i^{s,t}={\displaystyle \sum _{i=1,i\ne j}^M{{\displaystyle k}}_{ij\_1}}\cdot \left({{\displaystyle \delta }}_i^{s,t}-{{\displaystyle \delta }}_j^{s,t}\right)+{{\displaystyle k}}_{ij\_2}\cdot \left({{\displaystyle V}}_i^{s,t}-{{\displaystyle V}}_j^{s,t}\right)$$

(15)

$${{\displaystyle Q}}_i^{s,t}={\displaystyle \sum _{i=1,i\ne j}^M-{{\displaystyle k}}_{ij\_2}}\cdot \left({{\displaystyle \delta }}_i^{s,t}-{{\displaystyle \delta }}_j^{s,t}\right)+{{\displaystyle k}}_{ij\_1}\cdot \left({{\displaystyle V}}_i^{s,t}-{{\displaystyle V}}_j^{s,t}\right)$$

(16)

$$\left|{{\displaystyle P}}_{ij}^{s,t}\right|\le {{\displaystyle P}}_{ij}^{\max ,t}$$

(17)

$$\left|{{\displaystyle Q}}_{ij}^{s,t}\right|\le {{\displaystyle Q}}_{ij}^{\max ,t}$$

(18)

where ${{\displaystyle x}}_{ij}$ and ${{\displaystyle r}}_{ij}$ are denoted as reactance and resistance of the branch ij. ${{\displaystyle \delta }}_i^{s,t}$ and ${{\displaystyle V}}_i^{s,t}$ are voltage angle and voltage magnitudes at bus m. ${{\displaystyle P}}_i^{\max ,t}$ and ${{\displaystyle Q}}_i^{\max ,t}$ are denoted as maximum active power and maximum reactive power between node i and j.

(2)

WTG operation constraints:

$$\left\{\begin{array}{l}(1-{{\displaystyle W}}_{WTG,i}^t){{\displaystyle P}}_{WTG,i,s}^{\max }\le {{\displaystyle P}}_{WTG,i,s}^t\le {{\displaystyle P}}_{WTG,i,s}^{\max }\\ 0\le {{\displaystyle W}}_{WTG,i}^t\le {{\displaystyle W}}_{WTG,i}^{\max }\end{array}\right.$$

(19)

where ${{\displaystyle W}}_{WTG,i}^t$ is the removal ratio of the active output of the WTG installed at node i.

(3)

PVG operation constraints:

$$\left\{\begin{array}{l}(1-{{\displaystyle W}}_{PVG,i}^t){{\displaystyle P}}_{PVG,i,s}^{\max }\le {{\displaystyle P}}_{PVG,i,s}^t\le {{\displaystyle P}}_{PVG,i,s}^{\max }\\ 0\le {{\displaystyle W}}_{PVG,i}^t\le {{\displaystyle W}}_{PVG,i}^{\max }\end{array}\right.$$

(20)

where ${{\displaystyle W}}_{PVG,i}^t$ is denoted as the removal ratio of the active output of the PVG installed at node i.

(4)

Power flow balance on each node:

$${{\displaystyle P}}_{WTG,i,s}^t+{{\displaystyle P}}_{PVG,i,s}^t+{{\displaystyle P}}_i^{s,t}-{{\displaystyle L}}_{i,load}^t=0$$

(21)

where ${{\displaystyle L}}_{i,load}^t$ is denoted as power load at node i and time t.

(5)

Switching times constraint:

$$\left\{\begin{array}{l}{\displaystyle \sum _{n=1}^N\left|{{\displaystyle \omega }}_{n+1,l}-{{\displaystyle \omega }}_{n,l}\right|\le {{\displaystyle T}}_s}\\ {\displaystyle \sum _{l=1}^{{{\displaystyle n}}_b}{\displaystyle \sum _{n=1}^N\left|{{\displaystyle \omega }}_{n+1,l}-{{\displaystyle \omega }}_{n,l}\right|\le {{\displaystyle T}}_t}}\end{array}\right.$$

(22)

where ${{\displaystyle T}}_s$ is denoted as the upper limit value of switching action times of the first branch; ${{\displaystyle T}}_t$ is the upper limit value of all switch operation times in the whole period of time.

(6)

DG permeability constraint:

DG permeability limit at the planning stage can enable DG to be absorbed locally. Therefore, the penetration rate of clean energy increases and network loss reduces. The DG permeability is limited by

$${{\displaystyle \zeta }}^{\min }\le \frac{{\displaystyle \sum _{i\in {{\displaystyle B}}_{DG}}{\displaystyle \sum _{s\in S}{\displaystyle \sum _{t\in T}{{\displaystyle P}}_{DG,s,t,i}}}}}{{{\displaystyle P}}_{s,t}^{\max }}\le {{\displaystyle \zeta }}^{\max }$$

(23)

where ${{\displaystyle \zeta }}^{\max }$ and ${{\displaystyle \zeta }}^{\min }$ are denoted as the upper and lower limits of DG permeability, respectively; ${{\displaystyle B}}_{DG}$ is denoted as the installation of DG node set; ${{\displaystyle P}}_{DG,s,t,i}$ is denoted as installed capacity for DG; ${{\displaystyle P}}_{s,t}^{\max }$ is denoted as the maximum load of planning stage t under scenario s.

(7)

Topology constraints of networks:

ADN is a radial connectivity network in each operation scenario [29].

5. Two Level Solution Method

The robust programming model built in this paper mainly includes network reconfiguration and DG programming. Because the two parts are nonlinear programming and have many dimensions, the overall solution is difficult. Therefore, in this paper, the two-level solution method is first used to decompose the model, so as to reduce the dimension of the model and reduce the difficulty of solution.

The bilevel programming model adopted in this paper is specifically expressed as:

$$\left\{\begin{array}{l}\min F\left({{\displaystyle x}}^Y,{{\displaystyle x}}^{DG}\right)={{\displaystyle C}}^Y+{{\displaystyle C}}^{DG}\\ s.t.G\left({{\displaystyle x}}^Y,{{\displaystyle x}}^{DG}\right)\le 0\\ H\left({{\displaystyle x}}^Y,{{\displaystyle x}}^{DG}\right)=0\\ \min f\left({{\displaystyle x}}^Y\right)={{\displaystyle C}}^Y\\ s.t.g({{\displaystyle x}}^Y)\le 0\\ h\left({{\displaystyle x}}^Y\right)=0\end{array}\right.$$

(24)

where $F\left(\cdot \right)$ and $f\left(\cdot \right)$ are the upper and lower objective functions, respectively; $G\left(\cdot \right)$, $H\left(\cdot \right)$, $g\left(\cdot \right)$ and $h\left(\cdot \right)$ are upper and lower layer constraints.

The upper level model aims to solve the location and capacity of DG, so as to minimize the comprehensive cost. The constraints are related to DG planning. The lower level model aims to solve the optimal reconfiguration combination of the network, so as to minimize the network operation cost. The constraints are the number of network reconfiguration constraints and network operation related constraints.

The upper layer and the lower layer are solved separately and interrelated, that is, the upper layer transfers the DG planning results to the lower layer, and the lower layer transfers the reconstructed topology and operating costs to the upper layer. The overall block diagram of the two-layer model solution method is shown in Figure 4.

6. Case Studies

The IEEE 33-bus distribution system [30] is taken as an example to verify the effectiveness of the planning model and solution method in this paper. The topology of the IEEE 33-bus distribution system is shown in Figure 5.

Switches 1–32 are section switches, and switches 33–37 are interconnection switches. The voltage level is 10 kV, DG can be installed at nodes 7, 14, 24 and 25, and the rest are conventional load nodes.

The maximum allowable DG capacity for each node to be selected is 2000 kW. The DG types to be selected are WTG and PVG, and the rated capacity of a single WTG is 100 kW. Cut in wind speed, rated wind speed and cut out wind speed are 4.3 m/s, 17.9 m/s and 7.7 m/s, respectively. The investment cost per unit capacity of WTG is 7800 USD/kW, the operation and maintenance cost per unit generating capacity is 0.30 USD/(kW • h), and the active management cost per unit power is 0.08 USD/(kW • h). The maximum allowable reduction rate of WTG is 20%. The rated capacity of a single PVG is 200 kW, and the rated light intensity of PVG is 500 W/m². The investment cost per unit capacity of PVG is 8200 USD/kW, the operation and maintenance cost per unit power generation is 0.30 USD/(kW • h), the active management cost per unit electricity is 0.08 USD/(kW • h). The maximum allowable reduction rate of PVG is 20%. The economic service life of WTG and PVG is 20 years. The discount rate is 0.06. The unit cost for ADN to purchase electricity from the superior power grid is 0.48 USD/(kW • h). The network loss cost per unit electricity is 0.7 USD/(kW • h). The cost caused by the switch changing the once on/off state is USD 50. The permeability of DG ranges from 0.3 to 0.7 [18]. The switching times of network reconfiguration are constrained to 5 [23]. The IEEE 33-bus network load and line impedance are presented in [30].

The planning scheme experiments are performed on a computer with Intel(R) Core(TM) i7-8565U CPU and 8.00 GB RAM using MATLAB R2020b and Gurobi 9.1.2.

There are four cases to verify the proposed method.

Case 1: Network reconfiguration is not considered and deterministic planning model is adopted.

Case 2: Robust planning model is adopted without considering network reconfiguration.

Case 3: Network reconfiguration is considered and a deterministic planning model is adopted.

Case 4: Network reconfiguration is considered and a robust planning model is adopted.

It is worth noting that the deterministic model in Cases 1 and 3 refers to the assumption that wind speed, light intensity and load do not fluctuate during the planning process, that is, a basic scene is utilised during the planning process.

The planning scheme and various costs obtained from the simulation are shown in

Table 1. Planning scheme of Case 1 to Case 4.

	Bus	7	14	24	25
Case 1	Element Type	WTG	WTG	PVG	PVG
	Capacity (kW)	400	400	400	400
Case 2	Element Type	WTG	WTG	PVG	PVG
	Capacity (kW)	400	400	600	400
Case 3	Element Type	WTG	WTG	PVG	PVG
	Capacity (kW)	200	300	600	400
Case 4	Element Type	WTG	WTG	PVG	PVG
	Capacity (kW)	300	300	600	600

and Figure 6, respectively.

6.1. Analysis of the Influence of Network Reconfiguration on Planning Results

The impact of network reconfiguration on the planning results of DG in ADN is analyzed as follows.

(1)

Comparative analysis of Case 1 and Case 3

In Case 1, the network reconfiguration is not considered and the deterministic planning model is adopted. Only the ADN planning model with DG is solved. In this planning scheme, WTGs are installed at nodes 7 and 14 with an installed capacity of 400 kW, and PVGs are installed at nodes 24 and 25 with an installed capacity of 400 kW. The maximum load in summer is 4000 kW, and the penetration rate of clean energy is 40.0%, meeting the requirements of ADN for high penetration rate of clean energy. The annual average comprehensive cost is USD 7.4136 million, including USD 250,300 for network loss, USD 1,862,800 for DG, and USD 5,305,000 for power purchase from the superior power grid.

In Case 3, network reconfiguration is considered, but deterministic planning model is used to plan the network. In this planning scheme, considering the network reconfiguration, the distribution network can change the network topology through the action of branch switches and tie switches, so as to optimize the network power flow. Therefore, under the premise of a certain load, the total installed capacity of DG is reduced. As shown in Table 1, the total installed capacity of DG is 1500 kW (penetration rate is 37.5%), which reduces 100 kW. Therefore, the total cost of DG is reduced by USD 89,700. By optimizing the power flow, the system power loss is reduced by 38.35%, which is of great significance to energy conservation. The power purchase cost decreased by USD 299,700. At the same time, because the planning scheme reconstructs the network of the system, the reconstruction cost is USD 38,400. It can be seen that this cost is low, because in the planning year, the load fluctuation is mainly reflected in seasonal changes, so the number of reconstructions is less. To sum up, the annual average comprehensive cost of the system decreased by USD 447,000.

(2)

Comparative analysis of Case 2 and Case 4

Since both Case 2 and Case 4 adopt robust planning models, the system has a good adaptability to the uncertainty of load, wind power and photovoltaic, and the system does not need to be reconstructed frequently. Therefore, the total installed DG capacity of Case 2 and Case 4 is the same, which is 1800 kW. At the same time, the reconstruction cost of Case 4 is lower than that of Case 2, which is USD 19,200. However, the operation and maintenance cost of DG and the power purchase cost from the large power grid in Case 4 are slightly lower than that in Case 2. At the same time, the network loss cost of Case 4 is greatly reduced, and the annual average comprehensive cost is reduced by USD 147,500. Therefore, it is still advantageous to consider network reconfiguration when using robust programming model.

Through the above comparison, it is found that network reconfiguration is considered in ADN planning, which is of great significance to reduce network loss, delay DG investment and make the distribution system more economical.

6.2. Analysis of the Robust Model on Planning Result

Robust programming model can increase the adaptability of the system to uncertainty. The specific analysis is as follows.

(1)

Comparative analysis of Case 1 and Case 2

In Case 1, the total installed capacity of DG is 1600 kW, the penetration rate of clean energy is 40.0%. The annual average comprehensive cost is USD 7.4136 million, of which the network loss cost is USD 250,300, the DG comprehensive cost is USD 1.8628 million, and the power purchase cost from the superior power grid is USD 5.305 million.

In Case 2, the robust planning model is used to plan the network without considering the network reconfiguration, and only the upper model is solved. In this planning scheme, the robust planning model is adopted. The planning scheme is more conservative, and the total installed capacity of DG increases. As shown in Table 1, the total installed capacity of DG is 1800 kW (penetration rate is 45.0%), increasing by 200 kW. DG’s comprehensive expenses increased by USD 184,200. However, the utilization rate of DG in the system has been improved, so the cost of purchasing power from the large power grid has been reduced by USD 770,900. At the same time, due to the improved utilization of DG, the source load distance is greatly shortened, reducing the network loss by 20.18%, which is of great significance for saving electric energy and improving the utilization of clean energy. The annual average comprehensive cost of case 2 is USD 637,200 lower than that of Case 1.

(2)

Comparative analysis of Case 3 and Case 4

The network reconfiguration is considered, but a deterministic planning model is adopted in Case 3, while network reconfiguration is considered and a robust programming model is adopted in Case 4. Table 1 and Figure 6 show that when considering network reconfiguration, the impact of the robust optimization model on the planning results is similar to the comparative analysis of Case 1 and Case 2. Due to the adoption of robust programming model, compared with Case 3, the total installed capacity of DG in Case 4 is increased by 300 kW, thus the comprehensive cost of DG is increased by USD 280,400. However, the power purchase cost and network loss decreased by USD 497,800 and USD 101,100, respectively. The reconstruction cost was reduced by USD 19,200. The comprehensive cost decreased by USD 337,700. In particular, the network loss cost of Scenario 4 is only USD 53,200, which is significantly reduced compared with other scenarios. This is precisely because Case 4 considers both network reconfiguration and a robust optimization model. Thus, the power flow is optimized, the DG is absorbed locally, the transmission power on the line is reduced, and the network loss is reduced.

To sum up, using a DG robust planning model considering network reconfiguration to plan ADN can not only reduce network loss and delay DG investment, but also improve DG utilization, reduce power purchase cost from large power grid, and improve the economy of distribution network.

6.3. Analysis of the Influence of Robust Model on System Reliability

In order to analyze the impact of the robust planning model on system reliability in the planning stage, this paper introduces two evaluation indicators, namely, load shedding cost ${{\displaystyle C}}_{LL}$ and discarding wind (discarding light) cost ${{\displaystyle C}}_{LW}$. Because the influence of the fluctuation of light intensity on the system is similar to that of wind speed, this paper only takes wind speed fluctuation and load fluctuation as examples for analysis. The formulas of load shedding cost ${{\displaystyle C}}_{LL}$ and discarding wind cost ${{\displaystyle C}}_{LW}$ are as follows:

$${{\displaystyle C}}_{LL}={\displaystyle \sum _{s\in S}{\displaystyle \sum _{t\in T}\mathsf{\Delta }t{\displaystyle \sum _{j\in {{\displaystyle B}}^{load}}{{\displaystyle c}}_{ENS}{{\displaystyle p}}_{j,s,t}^{ENS}}}}$$

(25)

where S is the scene set. T is the planning period set. ${{\displaystyle B}}_{load}$ is a collection of load nodes. $\mathsf{\Delta }t$ is the time interval. ${{\displaystyle c}}_{ENS}$ is the unit price of load loss penalty. ${{\displaystyle p}}_{j,s,t}^{ENS}$ is the active power of load loss.

$${{\displaystyle C}}_{LW}={\displaystyle \sum _{s\in S}{\displaystyle \sum _{t\in T}\mathsf{\Delta }t{\displaystyle \sum _{j\in {{\displaystyle B}}^{wind}}{{\displaystyle c}}_{WNS}{{\displaystyle p}}_{j,s,t}^{WNS}}}}$$

(26)

where ${{\displaystyle B}}_{wind}$ is a collection of nodes for installing fans; ${{\displaystyle c}}_{WNS}$ is the penalty unit price for wind abandonment; ${{\displaystyle p}}_{j,s,t}^{WNS}$ is the active power of abandoned wind.

When the load and wind speed increase by 10%, 20% and 30%, respectively. The calculation results of ${{\displaystyle C}}_{LL}$ and ${{\displaystyle C}}_{LW}$ are shown in

Table 2. Cost under different load and wind speed fluctuation.

	Load Shedding Cost (104 USD)		Discarding Wind Cost (104 USD)
Range	Robust Model	Deterministic Model	Robust Model	Deterministic Model
10%	26.00	26.20	6.40	50.90
20%	52.00	52.30	11.10	56.60
30%	77.00	77.90	17.10	62.20

(${{\displaystyle c}}_{ENS}$ is taken as 0.6 USD/(kW • h), ${{\displaystyle c}}_{WNS}$ as 0.4 USD/(kW • h)).

It can be seen from Table 2 that when the load fluctuates by 10%, 20% and 30%, respectively, the load loss costs of the robust planning model built in this paper are, respectively, USD 26,520,000, USD 520,000 and USD 770,000, which are correspondingly reduced by USD 2000, USD 3000, and USD 9000 compared with the deterministic planning model. It can be concluded that the robust programming model built in this paper can ensure that the system has higher security and reliability, can adapt to the impact of load fluctuation, and can ensure the safe use of power by the load. With the increase in load fluctuation, the advantages of the model against disturbance become more prominent.

When the wind speed fluctuates by 10%, 20% and 30%, respectively, the wind abandonment cost of the robust planning model built is USD 64,000, USD 111,000, and USD 171,000, respectively. Compared with the deterministic planning model, the wind abandonment cost is significantly reduced, with a corresponding reduction of USD 445,000, USD 455,000, and USD 451,000. It can be concluded that the robust planning model built in this paper is more suitable for the characteristics of high penetration and large uncertainty of ADN clean energy, can use clean energy more efficiently, and conforms to the development trend of the future distribution network.

In order to further verify the effectiveness of the proposed method, the computation time of a planning model for different cases is represented, which is shown in

Table 3. The computation times of Case 1 to Case 4.

	Case 1	Case 2	Case 3	Case 4
Computation Time	100.23 s	858.06 s	163.58 s	1176.09 s

.

It can be seen from Table 3 that the computation times of Case 1 and Case 2 are relatively short because both cases adopt deterministic models and only complete the planning under a basic scene of “wind speed–light intensity–load”. Moreover, because the planning model of Case 3 considers network reconfiguration, the calculation time is increased by 63.35 s. This is because when Case 2 and Case 4 adopt the robust planning model, multiple uncertain scenes are considered. The computation time of the proposed method is 1176.09 s, which is acceptable for distributed generators planning.

6.4. Analysis of the Influence of DG Permeability Constraint on System Reliability

In order to disscuss the influence of the DG permeability constraint on system reliability, we set Case 5, and the specific description is as follows:

Case 5: Network reconfiguration is considered and a robust planning model is adopted, but the DG permeability constraint is ignored.

The planning scheme and various costs obtained from the simulation are shown in

Table 4. Planning results of Case 4 and Case 5.

		Case 4				Case 5
	Bus	7	14	24	25	7	14	24	25
Investment Decision	Element Type	WTG	WTG	PVG	PVG	WTG	WTG	PVG	PVG
	Capacity (kW)	300	300	600	600	300	400	600	600
Cost (104 USD)	Network loss cost	5.32				10.32
	DG Investment Cost	126.61				132.81
	DG operation cost	43.72				39.84
	DG Proactive Management Cost	35.02				30.78
	Power purchase cost	450.3				495.3
	Reconstruction cost	1.92				2.01
	Total cost	662.89				711.06

.

In order to disscuss the influence of DG permeability constraint on system reliability, we compare the results of Case 4 and Case 5.

As shown in

Table 5. Planning scheme of scenario A ADN scenario B.

	Bus	15	32	42	48	62
Scenario A	Element Type	PVG	WTG	PVG	WTG	WTG
	Capacity (kW)	600	400	600	500	400
Scenario B	Element Type	PVG	WTG	PVG	WTG	WTG
	Capacity (kW)	600	300	600	600	400

, the total installed capacity of DG in Scenario 5 has increased by 100 kW compared to the total installed capacity of DG in Scenario 4, resulting in an increase of USD 62,000 in investment costs. But the DG operation cost and DG proactive management cost decrease by USD 38,800 and USD 42,400, respectively. The reason is that the planning model in Case 5 ignores DG constraints, resulting in lower utilization of DG, resulting in lower operating and management costs for DG. In addition, in order to maintain the balance of the power grid, the purchase of electricity from the grid increased in Case 5 compared with Case 4, so that the power purchase cost increased by USD 50,000 in Case 5. It can be seen that it is crucial to consider DG permeability constraints during the DG planning stage, which can increase the absorption capacity of renewable energy and help build a low-carbon and friendly power system.

6.5. Test on Large-Scale System

In order to further verify the effectiveness of the planning model, the proposed model is performed on a large-scale IPGS with the IEEE 69-bus power system [30], as shown in Figure 7.

Switches 1–68 are section switches, and switches 69–74 are interconnection switches. The voltage level is 10 kV; DG can be installed at nodes 15, 32, 42, 48 and 62; and the rest are conventional load nodes. Other parameters of the power system are the same as in the previous case.

As described in Section 1, a robust planning model is utilized in [17,18], but reconfiguration is not considered. To further verify the contributions of the proposed method, the planning results of two scenarios are compared. It is similar to Case 2 and Case 4 of the IEEE 33-bus system: a robust planning model is adopted without considering network reconfiguration in Scenario A. Network reconfiguration is considered and a robust planning model is adopted in Scenario B.

The planning scheme and various costs obtained from the simulation are shown in Table 5 and Figure 8, respectively.

It is consistent with the conclusion in Section 6.1, and the system does not need to be reconstructed frequently. The reason is that the system has a good adaptability to the uncertainty of load, wind power and photovoltaic due to a robust planning model adopted in both Scenario A and Scenario B. Therefore, the total installed DG capacity of Scenario A and Scenario B is the same, which is 2500 kW. The reconstruction cost of Scenario B is lower than that of Scenario A. At the same time, the network loss cost of Scenario B is greatly reduced, and the annual average comprehensive cost is reduced by USD 360,100. Therefore, it is still advantageous to consider network reconfiguration when using robust programming model.

It can be seen from

Table 6. The computation time of Case 1 to Case 4.

	Scenario A	Scenario B
Computation Time	1312.83 s	2116.96 s

that the computation time of Scenario B is 854.13 s longer than that of Scenario A because network reconfiguration is considered.

7. Discussion and Conclusions

This paper proposes a robust planning model considering network reconfiguration and DG permeability. The planning scheme dispatches the existing resources in the active distribution network to meet the load demand. Moreover, the “wind–light–load” uncertain sets are introduced in the planning model to describe the uncertainty of clean energy. Additionally, the two-level optimization is utilized to decompose the planning model.

The results of the test systems are shown as follows:

When planning DG in ADN, considering network reconfiguration will not only affect the planning results of DG, but also affect the economic costs of the system. When considering network reconfiguration, in some scenarios, it is not necessary to add a large number of DGs to meet the demand of load growth, just adjust the switch state and optimize the power flow to meet the safe operation of ADN. Although a small amount of reconstruction cost will be generated, the network loss cost will be greatly reduced, and the comprehensive cost of DG will also be reduced, thus reducing the annual comprehensive cost of the system and improving its economy. In addition, considering the uncertainty of wind speed, light intensity and load, the planning system has better adaptability and stability, and also greatly improves the reliability of the system. Moreover, when DG permeability is considered at the planning stage, the utilization rate of DG increases, and DG can be consumed locally, reducing network losses, thereby reducing the purchase of electricity from the large power grid and making the system more economical.

In summary, the planning method proposed in this article can effectively guide DG planning under the high proportion of renewable energy access, thereby improving DG consumption and delaying DG investment. However, this paper only plans DG in ADN, but does not plan and analyze grid structure and energy storage devices of distribution network, which is also the next research direction.