A Cooperative Operation Optimization Method for Medium- and Low-Voltage Distribution Networks Considering Flexible Interconnected Distribution Substation Areas

Abstract

The high proportion of distributed photovoltaics (PVs) connected to distribution substation areas causes complex operation of the medium- and low-voltage distribution network. New power electronic devices represented by soft open point (SOP) can achieve a flexible interconnection between distribution substation areas. In this paper, a cooperative operation optimization method for medium- and low-voltage distribution networks considering flexible interconnected distribution substation areas is proposed. Firstly, interval affine is used to model the resource aggregation and obtain the flexibility region constraints of distribution substation areas. Then, a multi-scale cooperative operation framework for medium- and low-voltage distribution networks is constructed. The medium-voltage distribution network adopts the centralized method to solve and issue operation strategy instructions. The distribution substation area generates a voltage-reactive power adaptive regulation curve and performs intra-day rolling control. Finally, the case studies show that the proposed method has efficient operation strategy of medium- and low-voltage distribution networks and can alleviate the voltage fluctuation caused by high integration of PVs.

1. Introduction

With the increasing integration of distributed generators (DGs) and rapid development of new types of loads, the time-varying and uncertain operation status of distribution networks is amplified [1]. Distribution networks may face problems such as voltage violations [2] and line overloads [3]. Especially at the level of medium- and low-voltage distribution networks, random fluctuations of DGs and loads make the operation scenarios of the distribution network more complex and varied. Conventional distribution network control methods cannot cope well with these scenarios. Therefore, the operation of the distribution network faces serious challenges [4].

The soft open point (SOP) is a flexible distribution device based on advanced power electronics technology. It enables flexible interconnection and controllable power transmission of feeders, and greatly enhances the operational flexibility of distribution networks [5]. In recent years, the application scenarios of SOP have been continuously expanded, such as multi-terminal interconnection [6], multi-voltage interconnection [7], and AC/DC hybrid interconnection [8]. In low-voltage distribution networks, SOPs facilitate flexible interconnections between distribution substation areas. By balancing power supply and demand of each area, SOPs achieve mutual power support and flexible operation between adjacent station areas, which is an important direction for the flexible transformation of future distribution networks [9]. However, the continuous and rapid power flow regulation capability of SOP significantly differs from traditional discrete regulation methods such as switches and capacitor banks. The coordinated control strategy needs to consider multiple resource types and time scales, thereby significantly increasing system complexity [10]. Due to limitations in computing and communication conditions, conventional operational control of distribution networks focuses more on the high and medium voltage levels, lacking effective regulation capabilities for low-voltage distributed resources [11]. These issues have hindered the further improvement of the distribution network operation performance, and become key problems that urgently need to be solved in the flexible transformation of the distribution networks.

The operation and control of distribution networks can be modeled and solved by optimization methods, which can be mainly divided into two types: centralized and distributed approaches [12]. The centralized method is to collect systemwide data from the distribution network and solve the optimal dispatch problem, then sending it to the terminal units for execution. The centralized method has been applied in scenarios such as regional power system operation optimization [13], network loss minimization [14], and cluster voltage control [15]. The centralized methods can obtain more comprehensive data, and can achieve precise operation strategy solution under multiple objectives. However, centralized methods also face many limitations, including data storage capacity, communication bandwidth, and computational processing performance [16]. These limitations make it difficult to fully consider the massive resources on the low-voltage side in the operation strategy, resulting in increasingly prominent application limitations [17].

The distributed method divides the distribution network into several areas, and decomposes complex large-scale computing problems into parallel small-scale problems. This can reduce data communication and computation, alleviate the pressure of computation and data transmission in the medium-voltage distribution network, and is more suitable for effective management and control of distributed resources [18]. It has been applied in power system optimization scheduling [19], voltage control [20], and state estimation for an integrated energy system [21]. The alternating direction method of multipliers (ADMM) is a widely used distributed optimization method for solving optimization control problems [16]. It solves optimization problems within each area, coordinates between areas through boundary information, and performs alternating iterative solutions to achieve global optimization. Reference [22] proposed a dual time scale voltage control method for unbalanced distribution networks by combining model predictive control with ADMM. Reference [23] developed an ADMM-based distributed power optimization method for wind farms, ensuring turbine control constraints while converging to optimal power outputs. For multi-agent system control, reference [24] introduced a continuous domain real-time ADMM algorithm to address distributed control challenges. Reference [25] proposed a frequency synchronization optimization control method based on ADMM, which solves consensus states for distributed resources within finite horizons.

Distributed methods provide an effective way for solving the optimization scheduling problem of medium- and low-voltage distribution networks. However, existing distributed methods mainly focus on high- and medium-voltage networks, coordinating different distribution networks [26], or transmission and distribution networks [27]. For flexible interconnection station areas in distribution networks, the types of flexible interconnection are diverse, with small capacity of source load units and high uncertainty. Effective operational optimization methods for distribution substation areas still need to be studied.

In response to the above issues, a cooperative operation optimization method for medium- and low-voltage distribution networks considering flexible interconnected distribution substation areas with SOP is proposed in this paper. The main contributions are as follows:

(1) A flexible region resource aggregation method based on interval affine for the distribution substation areas is proposed. The method is aimed at two-port or multi-port series flexible interconnection. It considers flexible operation constraints of devices such as controllable load (CL), distributed photovoltaic (PV), and SOP. The method aggregates the flexibility of the distribution substation areas and uses affine methods to reduce the conservativeness of the intervals. This provides a foundation for resource scheduling of the distribution substation area on the medium-voltage side.

(2) A multi-scale collaborative operation optimization framework for a medium- and low-voltage distribution network is established. A centralized method is used on the medium-voltage side to calculate the day-ahead operation strategy and distribution substation area scheduling instructions. Based on these instructions, the low-voltage distribution substation areas generate voltage-reactive power adaptive adjustment curves for local control. This improves the flexibility and adaptability to fluctuations in the operating status of the distribution network.

(3) A distributed optimization method for flexible interconnected distribution substation areas based on ADMM is proposed. Under the guidance of control instructions from the medium-voltage side, multiple low-voltage distribution substation areas coordinate autonomously to obtain an optimized operation strategy. The method avoids the complex calculations on the medium-voltage side, enhancing the capability of distributed resources to support the operation of a distribution network.

The remainder of this paper is organized as follows. Section 2 introduces typical forms of flexible interconnected distribution substation areas and the collaborative operation architecture of the medium- and low-voltage distribution network. Section 3 establishes the source-load characteristic model in distribution substation areas. Section 4 proposes a distribution substation area resource aggregation method based on interval affine. The centralized operation optimization method for the medium-voltage distribution network is presented in Section 5. Section 6 proposes a distributed operation optimization method for distribution substation areas based on ADMM. The case studies and analysis are presented in Section 7. Finally, the conclusions are discussed in Section 8.

2. Basic Forms and Control Architecture of Flexible Interconnected Distribution Substation Areas

In this section, typical forms of flexible interconnected distribution substation areas are divided into three types. Their respective forms and characteristics are introduced separately. Then, the collaborative operation architecture of the medium- and low-voltage distribution network with flexible interconnected distribution substation areas is studied.

2.1. Typical Forms of Flexible Interconnected Distribution Substation Areas

In the flexible interconnected distribution substation areas, the forms of flexible interconnection mainly include two-port series flexible interconnection, multi-port series flexible interconnection, and flexible interconnection with a common DC bus. The two-port series flexible interconnection model is the simplest form, as shown in Figure 1a. The low-voltage distribution substation areas use a two-port SOP on the secondary side of the transformer to achieve flexible interconnection. This creates a power connection between Area 1 and Area 2.

Figure 1. (a) Two-port series flexible interconnection. (b) Multi-port series flexible interconnection. (c) Flexible interconnection with a common DC bus.

If several distribution substation areas are located close to each other, a multi-port SOP can be used to achieve flexible interconnection among three or more areas. Figure 1b shows the multi-port series flexible interconnection model.

In addition, the DC side of the SOP can be led out to extend the ports of SOP. This forms a flexible interconnection model with a common DC bus, as shown in Figure 1c. The common DC bus can also connect DC devices, including DC loads, DGs, and electric vehicles.

Based on the above low-voltage flexible interconnection models of distribution substation areas, a typical structure of a medium- and low-voltage distribution network with flexible interconnected distribution substation areas is shown in Figure 2. The medium-voltage distribution network contains two flexible interconnected distribution substation areas. The two-area interconnection part uses a two-port series flexible interconnection model, while the three-area interconnection part uses a multi-port series flexible interconnection model.

Figure 2. Structure of distribution network with flexible interconnected distribution substation areas.

2.2. Collaborative Operation Architecture of a Medium- and Low-Voltage Distribution Network

To fully utilize the distributed resources on the low-voltage side and adapt to the fluctuating operating environment of the distribution network, this paper proposes a multi-scale collaborative operation framework for a medium- and low-voltage distribution network with flexible interconnected distribution substation areas, as shown in Figure 3. In this framework, the medium-voltage distribution network considers the flexibility aggregation of low-voltage distribution substation areas and performs a system-level day-ahead centralized solution. This provides the medium-voltage distribution network with an optimized day-ahead operation strategy, which includes control instructions for the flexible interconnected distribution substation areas. These instructions include the voltage reference value $V_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}$, active power reference value $P_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}$, and reactive power reference value $Q_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ for each distribution substation area. Each flexible interconnected distribution substation area generates a voltage-reactive power adaptive adjustment curve based on these reference values. During the day-ahead rolling operation, the distribution substation area calculates the reactive power output target based on the actual voltage at the connection point and the voltage-reactive power adaptive curve. Finally, the flexible interconnected distribution substation area performs distributed optimization to obtain the operation strategy for the current time period.

Figure 3. Collaborative operation framework for a medium- and low-voltage distribution network with flexible interconnected distribution substation areas.

The advantages of the above architecture are as follows. First, the distribution substation areas participate in medium-voltage optimization in an aggregated form, without considering the specific composition of resources within the distribution substation areas. This reduces the scale and difficulty of solving centralized optimization problems. Second, the low-voltage distribution substation areas use a rolling solution method with voltage-reactive power adaptive adjustment, which helps avoid the impact of day-ahead forecast errors and improves the ability to suppress voltage fluctuations. Lastly, the flexible interconnected distribution substation areas use distributed optimization to achieve coordinated and autonomous operation under the guidance of medium-voltage side instructions. This ensures the safe and efficient operation of the distribution substation area, while also supporting the medium-voltage side and dealing with a high proportion of PV access.

3. Distribution Substation Area Modeling

The random fluctuations of load and the large-scale integration of DG lead to significant uncertainty in the distribution substation area. In this section, the uncertainty of uncontrollable load and flexibility regions of three representative flexible devices, CL, PV, and SOP, are modeled.

3.1. Uncertainty Model of Uncontrollable Load

The uncontrollable load in the distribution substation area is the basic load, and its probabilistic characteristics can be based on historical data. The Gaussian Mixture Model (GMM) can be used to obtain its probability density function (PDF) [28]. To establish its interval representation, the mean is taken as the midpoint of the uncertainty region, and the standard deviation is taken as the interval width. This allows the uncertainty region to be expressed using the interval midpoint and width.

The PDF of the active and reactive loads and the uncertainty region of uncontrollable loads are expressed as Equations (1)–(4).

$$\begin{gathered} f\left(L_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\right)={\sum }_{g=1}^{G^{\mathrm{L}}}{\zeta }_g^{\mathrm{L}}{\displaystyle \frac{1}{\sqrt{2\pi {\Sigma }_{g,i,t}^{Ll}}}}\mathrm{e}\mathrm{x}\mathrm{p}(-{\displaystyle \frac{1}{2}}(L_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}-{\mu }_{g,i,t}^{Ll})({\Sigma }_{g,i,t}^{Ll}{)}^{-1}(L_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}- \\ {\mu }_{g,i,t}^{Ll}{)}^{\mathrm{T}},L_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\in \{P_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}},Q_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\} \end{gathered}$$

(1)

$${\mu }_{i,t}^{Ll}={\displaystyle \frac{{\sum }_{g=1}^{G^{\mathrm{L}}}{\zeta }_g^{\mathrm{L}}{\mu }_{g,i,t}^{Ll}}{{\sum }_{g=1}^{G^{\mathrm{L}}}{\zeta }_g^L}}$$

(2)

$${\sigma }_{i,t}^{Ll}=[{\displaystyle \frac{{\sum }_{g=1}^{G^{\mathrm{L}}}{\zeta }_g^{\mathrm{L}}(L_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}-{\mu }_{g,i,t}^{Ll})(L_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}-{\mu }_{g,i,t}^{Ll}{)}^{\mathrm{T}}}{{\sum }_{g=1}^{G^{\mathrm{L}}}{\zeta }_g^{\mathrm{L}}}}{]}^{{\displaystyle \frac{1}{2}}}$$

(3)

$$\begin{gathered} U_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}=\{P_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}},Q_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}|{\mu }_{i,t}^{Pl}-{\sigma }_{i,t}^{Pl}\le P_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\le {\mu }_{i,t}^{Pl}+{\sigma }_{i,t}^{Pl}, {\mu }_{i,t}^{Ql}-{\sigma }_{i,t}^{Ql}\le Q_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\le {\mu }_{i,t}^{Ql}+ \\ {\sigma }_{i,t}^{Ql}\} \end{gathered}$$

(4)

where $L_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\in \{P_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}},Q_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\}$ is the active and reactive load of distribution substation area $i$, $G^{\mathrm{L}}$ is the number of Gaussian components in the active and reactive load, ${\zeta }_g^{\mathrm{L}}$ is the weight coefficient, ${\mu }_{g,i,t}^{Ll}$ is the mean of the gth component, ${\Sigma }_{g,i,t}^{Ll}$ is the variance of the gth component, ${\mu }_{i,t}^{Ll}$ and ${\sigma }_{i,t}^{Ll}$ are the mean and standard deviation for active and reactive loads of distribution substation area $i$, and $U_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}$ is the uncertainty region for uncontrollable loads.

Through the transformation of interval numbers and affine numbers in the following sections, this uncertainty region can be transformed into an affine form (Equation (27)), which is the basis for optimization operation.

3.2. Flexibility Model of CL

The CL can adjust their power based on the operating needs of the distribution network, and have flexible controllability. As shown in Figure 4, when the active power of the CL increases, the reactive power increases linearly based on the power factor. The upper and lower limits of the active power are usually known parameters. Therefore, the flexibility region of CL can be considered deterministic. This characteristic gives the adjustment range of CL clear boundary conditions, and its flexibility region is treated as a deterministic constraint.

Figure 4. Flexibility region of CL.

The flexibility region model of CL is as Equations (5) and (6).

$$P_{i,t}^{\mathrm{C}\mathrm{L},\mathrm{a}\mathrm{c}\mathrm{t}}=P_{i,t}^{\mathrm{C}\mathrm{L}}+{∆P}_{i,t}^{\mathrm{C}\mathrm{L}}$$

(5)

$$F_{i,t}^{\mathrm{C}\mathrm{L}}=\{P_{i,t}^{\mathrm{C}\mathrm{L},\mathrm{m}\mathrm{i}\mathrm{n}}\le P_{i,t}^{\mathrm{C}\mathrm{L},\mathrm{a}\mathrm{c}\mathrm{t}}\le P_{i,t}^{\mathrm{C}\mathrm{L},\mathrm{m}\mathrm{a}\mathrm{x}},Q_{i,t}^{\mathrm{C}\mathrm{L},\mathrm{a}\mathrm{c}\mathrm{t}}=P_{i,t}^{\mathrm{C}\mathrm{L},\mathrm{a}\mathrm{c}\mathrm{t}}\tan {\theta }_i^{\mathrm{C}\mathrm{L}}\}$$

(6)

where ${∆P}_{i,t}^{\mathrm{C}\mathrm{L}}$ is the adjustable active power of the CL, $Q_{i,t}^{\mathrm{C}\mathrm{L},\mathrm{a}\mathrm{c}\mathrm{t}}$ is the actual reactive power of the CL, $P_{i,t}^{\mathrm{C}\mathrm{L},\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum active power of the CL, $P_{i,t}^{\mathrm{C}\mathrm{L},\mathrm{m}\mathrm{i}\mathrm{n}}$ is the minimum active power of the CL, ${\theta }_i^{\mathrm{C}\mathrm{L}}$ is the power factor angle, and $F_{i,t}^{\mathrm{C}\mathrm{L}}$ is the flexibility region of the CL.

3.3. Flexibility Model of PV

The flexibility region of PV is shown in Figure 5. Under constant sunlight intensity, the active power output of PV remains constant, while the reactive power output changes as shown in the A-B segment. When the sunlight intensity decreases, the change in PV output is shown in the B-C segment. Because the capacity of PV power output $S_i^{\mathrm{P}\mathrm{V}}$ is certain, and when the active power output $P_{i,t}^{\mathrm{P}\mathrm{V}}$ decreases, the remaining capacity can be used for reactive power output $Q_{i,t}^{\mathrm{P}\mathrm{V}}$.

Figure 5. Flexibility region of PV.

The flexibility region model of PV can be expressed as Equations (7)–(9).

$$P_{i,t}^{\mathrm{P}\mathrm{V},\mathrm{m}\mathrm{i}\mathrm{n}}\le P_{i,t}^{\mathrm{P}\mathrm{V}}\le P_{i,t}^{\mathrm{P}\mathrm{V},\mathrm{m}\mathrm{a}\mathrm{x}}$$

(7)

$$Q_{i,t}^{\mathrm{P}\mathrm{V},\mathrm{m}\mathrm{i}\mathrm{n}}\le Q_{i,t}^{\mathrm{P}\mathrm{V}}\le Q_{i,t}^{\mathrm{P}\mathrm{V},\mathrm{m}\mathrm{a}\mathrm{x}}$$

(8)

$$(P_{i,t}^{\mathrm{P}\mathrm{V}}{)}^2+{(Q}_{i,t}^{\mathrm{P}\mathrm{V}}{)}^2\le (S_i^{\mathrm{P}\mathrm{V}}{)}^2$$

(9)

where $P_{i,t}^{\mathrm{P}\mathrm{V}}$ and $Q_{i,t}^{\mathrm{P}\mathrm{V}}$ are the active and reactive power outputs of the PV, $P_{i,t}^{\mathrm{P}\mathrm{V},\mathrm{m}\mathrm{i}\mathrm{n}}$ and $P_{i,t}^{\mathrm{P}\mathrm{V},\mathrm{m}\mathrm{a}\mathrm{x}}$ are the minimum and maximum values of the active power output, and $Q_{i,t}^{\mathrm{P}\mathrm{V},\mathrm{m}\mathrm{i}\mathrm{n}}$ and $Q_{i,t}^{\mathrm{P}\mathrm{V},\mathrm{m}\mathrm{a}\mathrm{x}}$ are the minimum and maximum values of the reactive power output.

The active power output of PV is closely related to sunlight intensity, which is affected by weather conditions and shows significant uncertainty [29]. The Beta distribution can effectively describe the random fluctuations of sunlight intensity over a given time period, capturing the probabilistic characteristics of sunlight intensity [30]. The PDF of PV output can be represented by the Beta distribution as Equation (10).

$$f\left(P_{i,t}^{\mathrm{P}\mathrm{V}}\right)={\displaystyle \frac{\Gamma \left({\alpha }^b+{\beta }^b\right)}{\Gamma \left({\alpha }^b\right)\Gamma \left({\beta }^b\right)}}({\displaystyle \frac{P_{i,t}^{\mathrm{P}\mathrm{V}}}{P_{i,t}^{\mathrm{P}\mathrm{V},\mathrm{m}\mathrm{a}\mathrm{x}}}}{)}^{{\alpha }^b-1}(1-{\displaystyle \frac{P_{i,t}^{\mathrm{P}\mathrm{V}}}{P_{i,t}^{\mathrm{P}\mathrm{V},\mathrm{m}\mathrm{a}\mathrm{x}}}}{)}^{{\beta }^b-1}$$

(10)

where $f$ is the PDF of PV output, and ${\alpha }^b$ and ${\beta }^b$ are the shape parameters of the Beta distribution.

Similar to Section 2.1, based on the mean, standard deviation, and flexibility region of the PV output PDF, the interval midpoint and width of the PV output are expressed as Equations (11) and (12).

$${\mu }_{i,t}^{\mathrm{P}\mathrm{V}}={\displaystyle \frac{{\alpha }^b}{{\alpha }^b+{\beta }^b}}{P}_{i,t}^{\mathrm{P}\mathrm{V},\mathrm{m}\mathrm{a}\mathrm{x}}$$

(11)

$${\sigma }_{i,t}^{\mathrm{P}\mathrm{V}}={\displaystyle \frac{{\alpha }^b{\beta }^b}{({\alpha }^b+{\beta }^b{)}^2({\alpha }^b+{\beta }^b+1)}}{P}_{i,t}^{\mathrm{P}\mathrm{V},\mathrm{m}\mathrm{a}\mathrm{x}}$$

(12)

where ${\mu }_{i,t}^{\mathrm{P}\mathrm{V}}$ is the interval midpoint of the PV output, and ${\sigma }_{i,t}^{\mathrm{P}\mathrm{V}}$ is the interval width of the PV output.

The flexibility region of PV is modeled as Equation (13).

$$\begin{gathered} F_{i,t}^{\mathrm{P}\mathrm{V}}=\{P_{i,t}^{\mathrm{P}\mathrm{V}},Q_{i,t}^{\mathrm{P}\mathrm{V}}|{\mu }_{i,t}^{\mathrm{P}\mathrm{V}}-{\sigma }_{i,t}^{\mathrm{P}\mathrm{V}}<P_{i,t}^{\mathrm{P}\mathrm{V}}<{\mu }_{i,t}^{\mathrm{P}\mathrm{V}}+{\sigma }_{i,t}^{\mathrm{P}\mathrm{V}},Q_{i,t}^{\mathrm{P}\mathrm{V},\mathrm{m}\mathrm{i}\mathrm{n}}\le Q_{i,t}^{\mathrm{P}\mathrm{V}}\le \\ \sqrt{(S_i^{\mathrm{P}\mathrm{V}}{)}^2-(P_{i,t}^{\mathrm{P}\mathrm{V}}{)}^2}\} \end{gathered}$$

(13)

In the following sections, the flexibility region of PV will be transformed into affine form, as shown in Equation (28).

3.4. Flexibility Model of SOP

Power electronic devices like SOP can achieve flexible interconnection between distribution substation areas, and exhibit significant flexibility. Generally, the upper and lower power limits of SOP are fixed values, and the flexibility region is defined. The flexibility region of a two-port SOP is shown in Figure 6.

Figure 6. Flexibility region of SOP.

Taking a two-port SOP as an example, assume that the active power transmitted from port 1 is greater than that from port 2. Considering the power loss of SOP, the operating point of port 1 is farther from the origin than that of port 2. For multi-port SOP and SOP with a DC bus, the method for calculating the flexibility region is the same as that for the two-port SOP.

The flexibility regions of two-port, multi-port, and SOPs with a DC bus can be modeled in the following Equations (14)–(17):

$$P_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P},\mathrm{m}\mathrm{i}\mathrm{n}}\le P_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P}}\le P_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P},\mathrm{m}\mathrm{a}\mathrm{x}}$$

(14)

$$Q_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P},\mathrm{m}\mathrm{i}\mathrm{n}}\le Q_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P}}\le Q_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P},\mathrm{m}\mathrm{a}\mathrm{x}}$$

(15)

$$\sqrt{{\left(P_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P}}\right)}^2+{\left(Q_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P}}\right)}^2}\le S_{\beta }^{\mathrm{S}\mathrm{O}\mathrm{P}}$$

(16)

$$F_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P}}=\left\{P_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P},\mathrm{a}\mathrm{c}\mathrm{t}},Q_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P},\mathrm{a}\mathrm{c}\mathrm{t}}\right| P_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P},\mathrm{a}\mathrm{c}\mathrm{t}}=P_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P}},Q_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P},\mathrm{a}\mathrm{c}\mathrm{t}}=Q_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P}}\}$$

(17)

where $P_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P},\mathrm{m}\mathrm{a}\mathrm{x}}$ and $P_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P},\mathrm{m}\mathrm{i}\mathrm{n}}$ are the maximum and minimum values of active power transmitted by the SOP, and $Q_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P},\mathrm{m}\mathrm{a}\mathrm{x}}$ and $Q_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P},\mathrm{m}\mathrm{i}\mathrm{n}}$ are the maximum and minimum values of reactive power injected by the SOP.

4. Distribution Substation Area Resource Aggregation Method Based on Interval Affine

In this section, an interval affine method is used to establish affine models for uncontrollable loads and PV output. Based on these affine models, distribution substation area flexibility aggregation is achieved, and flexibility region constraint for the distribution substation area is obtained. The power transmission relationship between flexible interconnected distribution substation areas is considered, and distribution substation area correlation constraint is established.

4.1. Basic Concept of Interval Affine

The affine analysis method is a mathematical technique used to deal with uncertainties and errors in numerical calculations [31]. It is an extension of the interval analysis method, and affine forms are introduced to represent variables and their uncertainties. Uncertain variables in the distribution network, such as uncontrollable loads and PV outputs, can be described by using affine models.

In affine analysis, uncertain quantities like uncontrollable load values and PV outputs can be represented as the sum of a central value and a finite number of noise components, which in turn represent the uncertain variables in the distribution substation area. The affine form of the uncertain variable $x$ is represented as $\widehat{x}$ in Equation (18).

$$\widehat{x}=x_0+x_1{\epsilon }_1+x_2{\epsilon }_2+\cdots +x_{n_z}{\epsilon }_{n_z}=x_0+{\sum }_{i_z=1}^{n_z}{x}_{i_z}{\epsilon }_{i_z}$$

(18)

where $x_0$ is the central value, representing the value of $x$ under deterministic conditions; $n_z$ is the number of noise components, representing the total number of independent uncertain factors; ${\epsilon }_{i_z}$ is the $i_z$-th noise component, representing the $i_z$-th uncertain factor; and $x_{i_z}$ is the coefficient of the noise component, representing the degree of influence of the $i_z$-th uncertain factor on the uncertain quantity.

Affine numbers and interval numbers can be converted into each other. The process of converting affine numbers to interval numbers is as Equations (19) and (20).

$$x_w={\sum }_{i_z=1}^{n_z}\left|x_{i_z}\right|$$

(19)

$$\left[x\right]=[\underset{\_}{x},\overline{x}]=\left[x_0-x_w,x_0+x_w\right]$$

(20)

where $x_w$ is the affine radius, also called the interval width; $\left[x\right]$ is the interval number; $x_0$ is the interval midpoint, also called the affine center value; and $\underset{\_}{x}$ and $\overline{x}$ are the lower and upper limits of the interval.

The process of converting interval numbers to affine numbers is as Equations (21)–(23).

$$x_0=(\overline{x}+\underset{\_}{x})/2$$

(21)

$$x_w=(\overline{x}-\underset{\_}{x})/2$$

(22)

$$\widehat{x}=x_0+x_w\epsilon$$

(23)

The affine number obtained by converting an interval number contains only one independent noise component, with its noise component coefficient being equal to the interval width, as shown in Figure 7.

Figure 7. Conversion of interval numbers to affine numbers.

4.2. Affine Model for Uncontrollable Load and PV Output

The uncertain quantities in the distribution substation area are the uncontrollable load value and PV output value. Based on the flexibility region model established in Section 3, and using the method described in Section 4.1, affine models for uncontrollable loads and PVs can be constructed as Equations (24)–(26). The affine central value is equal to the interval midpoint, and the noise component coefficient is equal to the interval width.

$${\widehat{P}}_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}={\mu }_{i,t}^{Pl}+{\sigma }_{i,t}^{Pl}{\epsilon }_{P,i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}$$

(24)

$${\widehat{Q}}_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}={\mu }_{i,t}^{Ql}+{\sigma }_{i,t}^{Ql}{\epsilon }_{Q,i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}$$

(25)

$${\widehat{P}}_{i,t}^{\mathrm{P}\mathrm{V}}={\mu }_{i,t}^{\mathrm{P}\mathrm{V}}+{\sigma }_{i,t}^{\mathrm{P}\mathrm{V}}{\epsilon }_{i,t}^{\mathrm{P}\mathrm{V}}$$

(26)

where ${\widehat{P}}_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}$, ${\widehat{Q}}_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}$, and ${\widehat{P}}_{i,t}^{\mathrm{P}\mathrm{V}}$ are the affine values for uncontrollable active load, uncontrollable reactive load, and PV output for substation area $i$, respectively. ${\epsilon }_{P,i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}$, ${\epsilon }_{Q,i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}$, and ${\epsilon }_{i,t}^{\mathrm{P}\mathrm{V}}$ are the noise components for uncontrollable active load, uncontrollable reactive load, and PV output, respectively.

Based on the affine models for uncontrollable load and PV output, the uncertainty region for the uncontrollable load and the flexibility region for the PV output are constructed in affine form in Equations (27) and (28).

$${\widehat{U}}_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}=\{{\widehat{P}}_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}},{\widehat{Q}}_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}|{\widehat{P}}_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}={\mu }_{i,t}^{Pl}+{\sigma }_{i,t}^{Pl}{\epsilon }_{P,i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}},{\widehat{Q}}_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}={\mu }_{i,t}^{Ql}+{\sigma }_{i,t}^{Ql}{\epsilon }_{Q,i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\}$$

(27)

$${\widehat{F}}_{i,t}^{\mathrm{P}\mathrm{V}}=\{{\widehat{P}}_{i,t}^{\mathrm{P}\mathrm{V}},{\widehat{Q}}_{i,t}^{\mathrm{P}\mathrm{V}}|{\widehat{P}}_{i,t}^{\mathrm{P}\mathrm{V}}={\mu }_{i,t}^{\mathrm{P}\mathrm{V}}+{\sigma }_{i,t}^{\mathrm{P}\mathrm{V}}{\epsilon }_{i,t}^{\mathrm{P}\mathrm{V}},Q_{i,t}^{\mathrm{P}\mathrm{V},\mathrm{m}\mathrm{i}\mathrm{n}}\le {\widehat{Q}}_{i,t}^{\mathrm{P}\mathrm{V}}\le \sqrt{(S_i^{\mathrm{P}\mathrm{V}}{)}^2-({\widehat{P}}_{i,t}^{\mathrm{P}\mathrm{V}}{)}^2}\}$$

(28)

where ${\widehat{U}}_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}$ is the affine form of the uncertainty region for uncontrollable load, and ${\widehat{F}}_{i,t}^{\mathrm{P}\mathrm{V}}$ is the affine form of the flexibility region for PV output.

4.3. Distribution Substation Area Flexibility Aggregation Based on the Affine Model

The distribution substation area flexibility region includes the flexibility regions of CL, PV, and SOP. The uncontrollable load and PV account for uncertainty. Therefore, the constraint for the distribution substation area flexibility region is expressed as Equations (29)–(32).

$$F_{i,t}^{\mathrm{T}\mathrm{A}\mathrm{I}}={\widehat{U}}_{i,t}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}+F_{i,t}^{\mathrm{C}\mathrm{L}}-{\widehat{F}}_{i,t}^{\mathrm{P}\mathrm{V}}+F_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P}}$$

(29)

$$F_{i,t}^{\mathrm{T}\mathrm{A}\mathrm{I}}=\left\{P_{i,t}^{\mathrm{T}\mathrm{A}\mathrm{I},\mathrm{a}\mathrm{c}\mathrm{t}},Q_{i,t}^{\mathrm{T}\mathrm{A}\mathrm{I},\mathrm{a}\mathrm{c}\mathrm{t}}\right|(P_{i,t}^{\mathrm{T}\mathrm{A}\mathrm{I},\mathrm{a}\mathrm{c}\mathrm{t}}{)}^2+(Q_{i,t}^{\mathrm{T}\mathrm{A}\mathrm{I},\mathrm{a}\mathrm{c}\mathrm{t}}{)}^2\le (S_T^{\mathrm{T}\mathrm{A}\mathrm{I},\mathrm{m}\mathrm{a}\mathrm{x}}{)}^2\}$$

(30)

$$P_{i,t}^{\mathrm{T}\mathrm{A}\mathrm{I},\mathrm{m}\mathrm{i}\mathrm{n}}\le P_{i,t}^{\mathrm{T}\mathrm{A}\mathrm{I},\mathrm{a}\mathrm{c}\mathrm{t}}\le P_{i,t}^{\mathrm{T}\mathrm{A}\mathrm{I},\mathrm{m}\mathrm{a}\mathrm{x}}$$

(31)

$$Q_{i,t}^{\mathrm{T}\mathrm{A}\mathrm{I},\mathrm{m}\mathrm{i}\mathrm{n}}\le Q_{i,t}^{\mathrm{T}\mathrm{A}\mathrm{I},\mathrm{a}\mathrm{c}\mathrm{t}}\le Q_{i,t}^{\mathrm{T}\mathrm{A}\mathrm{I},\mathrm{m}\mathrm{a}\mathrm{x}}$$

(32)

where, $F_{i,t}^{\mathrm{T}\mathrm{A}\mathrm{I}}$ is the flexibility region of distribution substation area $i$.

4.4. Correlation Model of Flexible Interconnected Distribution Substation Areas

Compared to traditional independent distribution substation areas, flexible interconnected distribution substation areas can transfer power between areas. Their flexibility and regulation capability are correlated. In this section, the correlation constraint of flexible interconnected distribution substation areas is used to describe the power coupling boundaries between distribution substation areas. Specifically, based on the aforementioned typical modes of flexible interconnected distribution substation areas, the correlation of areas in each mode is represented, resulting in the correlation constraint for flexible interconnected distribution substation areas.

For multiple low-voltage distribution substation areas with any number of ports SOP flexible interconnections, assuming there are $m$ areas, the distribution substation area correlation power constraint can be expressed uniformly as Equations (33) and (34).

$$\begin{gathered} {\sum }_{i=1}^m({\widehat{P}}_{i,T}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}+{\sum }_{n=1}^{N_{\mathrm{C}\mathrm{L},i}}{P}_{n,T}^{\mathrm{C}\mathrm{L},\mathrm{m}\mathrm{i}\mathrm{n}}-{\sum }_{n=1}^{N_{\mathrm{P}\mathrm{V},i}}{P}_{n,T}^{\mathrm{P}\mathrm{V},\mathrm{m}\mathrm{a}\mathrm{x}})\le {\sum }_{i=1}^m{P}_{i,T}^{\mathrm{L}\mathrm{V}\mathrm{N},\mathrm{a}\mathrm{c}\mathrm{t}}\le {\sum }_{i=1}^m({\widehat{P}}_{i,T}^{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}+ \\ {\sum }_{n=1}^{N_{\mathrm{C}\mathrm{L},i}}{P}_{n,T}^{\mathrm{C}\mathrm{L},\mathrm{m}\mathrm{a}\mathrm{x}}-{\sum }_{n=1}^{N_{\mathrm{P}\mathrm{V},i}}{P}_{n,T}^{\mathrm{P}\mathrm{V},\mathrm{m}\mathrm{i}\mathrm{n}})+{\sum }_{i=1}^m{P}_{i,T}^{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}+P_{m,T}^{\mathrm{S}\mathrm{O}\mathrm{P},\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}} \end{gathered}$$

(33)

$$F_{m,T}^{\mathrm{L}\mathrm{V}\mathrm{N}}=\{P_{m,T}^{\mathrm{L}\mathrm{V}\mathrm{N},\mathrm{a}\mathrm{c}\mathrm{t}}|P_{m,T}^{\mathrm{L}\mathrm{V}\mathrm{N},\mathrm{a}\mathrm{c}\mathrm{t}}={\sum }_{i=1}^m{P}_{i,T}^{\mathrm{L}\mathrm{V}\mathrm{N},\mathrm{a}\mathrm{c}\mathrm{t}}\}$$

(34)

where $P_{i,T}^{\mathrm{L}\mathrm{V}\mathrm{N},\mathrm{a}\mathrm{c}\mathrm{t}}$ is the actual value of the uncontrollable active load for area $i$, $P_{i,T}^{\mathrm{C}\mathrm{L},\mathrm{m}\mathrm{i}\mathrm{n}}$ and $P_{i,T}^{\mathrm{C}\mathrm{L},\mathrm{m}\mathrm{a}\mathrm{x}}$ are the minimum and maximum active power values of the CL for distribution substation area $i$, $P_{i,T}^{\mathrm{P}\mathrm{V},\mathrm{m}\mathrm{i}\mathrm{n}}$ and $P_{i,T}^{\mathrm{P}\mathrm{V},\mathrm{m}\mathrm{a}\mathrm{x}}$ are the minimum and maximum PV active power output values for distribution substation area $i$, $P_{m,T}^{\mathrm{L}\mathrm{V}\mathrm{N},\mathrm{a}\mathrm{c}\mathrm{t}}$ is the total active load value of the $m$ distribution substation areas; $F_{m,T}^{\mathrm{L}\mathrm{V}\mathrm{N}}$ is the correlation region for the $m$ areas, and $P_{m,T}^{\mathrm{S}\mathrm{O}\mathrm{P},\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}$ is the power loss of the SOP between the $m$ distribution substation areas.

5. Centralized Operation Optimization Method for the Medium-Voltage Distribution Network

In this section, a centralized operation optimization method for the medium-voltage distribution network is presented. The optimization method is formulated by a series of constraints. It focuses on minimizing the distribution network operation costs and enhancing system economic efficiency.

5.1. Operation Optimization Model for the Medium-Voltage Distribution Network

5.1.1. Objective Function of the Medium-Voltage Distribution Network

The centralized optimization model of the medium-voltage distribution network is established. The objective function of the centralized optimization model is described as Equations (35)–(37).

$$\min f^{\mathrm{M}\mathrm{V}\mathrm{N}}=f_1^{\mathrm{M}\mathrm{V}\mathrm{N}}+f_2^{\mathrm{M}\mathrm{V}\mathrm{N}}$$

(35)

$$f_1^{\mathrm{M}\mathrm{V}\mathrm{N}}={\sum }_{T=1}^{N_T}{C}^{\mathrm{L}}({\sum }_{ij}{R}_{ij}^{\mathrm{D}\mathrm{N}}{l}_{T,ij})$$

(36)

$$f_2^{\mathrm{M}\mathrm{V}\mathrm{N}}={\sum }_{T=1}^{N_T}{C}^U{\sum }_{ij}(U_{T,i}^2-{\left(U_i^{\mathrm{t}\mathrm{h}\mathrm{r}}\right)}^2)$$

(37)

where $f^{\mathrm{M}\mathrm{V}\mathrm{N}}$ is the objective function of the medium-voltage distribution network, $f_1^{\mathrm{M}\mathrm{V}\mathrm{N}}$ is the medium-voltage network power loss cost, $f_2^{\mathrm{M}\mathrm{V}\mathrm{N}}$ is the penalty cost for exceeding the voltage limits at medium-voltage nodes, $U_i^{\mathrm{t}\mathrm{h}\mathrm{r}}$ is the upper and lower voltage limits, and $C^{\mathrm{L}}$ and $C^{\mathrm{U}}$ are the weighting coefficients.

5.1.2. Constraints of the Medium-Voltage Distribution Network

The constraints for the medium-voltage distribution network include the power flow constraints, system security constraints, operation constraints of DG units, flexibility region constraints for the distribution substation area, and correlation power constraints for the flexible interconnected distribution substation areas.

The Distflow model is used to describe the network power flow constraints of the medium-voltage side distribution network [4]. The model calculates the active power, reactive power, current magnitude, and node voltage magnitude for each branch. For the many quadratic and product terms in the model, the square of the voltage magnitude $v_{T,i}$ and the square of the current magnitude $l_{T,ij}$ are used instead of the traditional voltage square $V_{T,i}^2$ and current square $I_{T,ij}^2$, thus eliminating the quadratic terms. The power flow constraints for the medium-voltage distribution network are described as Equations (38)–(41).

$$v_{T,j}-v_{T,i}+2\left(R_{ij}^{\mathrm{D}\mathrm{N}}{P}_{T,ij}+X_{ij}^{\mathrm{D}\mathrm{N}}{Q}_{T,ij}\right)-\left(R_{ij}^2+X_{ij}^2\right)l_{T,ij}=0$$

(38)

$${\sum }_{ij}\left(P_{T,ij}-R_{ij}{l}_{T,ij}\right)-{\sum }_{jh}{P}_{T,jh}+P_{T,j}=0$$

(39)

$${\sum }_{ij}\left(Q_{T,ij}-X_{ij}{l}_{T,ij}\right)-{\sum }_{jh}{Q}_{T,jh}+Q_{T,j}=0$$

(40)

$$\left(P_{T,ij}^2+Q_{T,ij}^2\right)-v_{T,i}{l}_{T,ij}=0$$

(41)

The system security constraints are described as Equations (42) and (43).

$${{(V}_{T,i}^{\mathrm{m}\mathrm{i}\mathrm{n}})}^2\le v_{T,i}\le {{(V}_{T,i}^{\mathrm{m}\mathrm{a}\mathrm{x}})}^2$$

(42)

$$l_{T,ij}\le {{(I}_{ij}^{\mathrm{m}\mathrm{a}\mathrm{x}})}^2$$

(43)

where $V_{T,i}^{\mathrm{m}\mathrm{i}\mathrm{n}}$ and $V_{T,i}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ are the minimum and maximum values of the node voltage at node $i$, and $I_{ij}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum allowable current flowing through the line $ij$;

5.2. Solution Method

Convex relaxation is used to transform the linearized Equation (41) into a second-order cone constraint form in Equation (44).

$${‖[2P_{T,ij} 2Q_{T,ij} l_{T,ij}-v_{T,i}{]}^{\mathrm{T}}‖}_2\le l_{T,ij}+v_{T,i}$$

(44)

To evaluate the accuracy of convex relaxation, the cone constraint deviation norm $\mathrm{g}\mathrm{a}\mathrm{p}$ is used as a quantitative indicator in Equation (45).

$$\mathrm{g}\mathrm{a}\mathrm{p}=\Vert l_{T,ij}{v}_{T,i}-P_{T,ij}^2-Q_{T,ij}^2{\Vert }_{\mathrm{\infty }}$$

(45)

When the deviation norm $\mathrm{g}\mathrm{a}\mathrm{p}$ is sufficiently small, it is considered that the second-order cone relaxation satisfies the accuracy requirements. The medium-voltage distribution network then issues the control strategy for the flexible interconnected low-voltage distribution substation areas, including the voltage reference value $V_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}$, active power reference value $P_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}$, and reactive power reference value $Q_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ for each area’s network connection point.

Additionally, the optimization also considers the flexibility region constraint for the distribution substation area, and correlation power constraint for the flexible interconnected distribution substation area, which are derived in Section 4.3 and Section 4.4, as shown in Equations (29)–(34).

5.3. Distribution Substation Area Voltage-Reactive Power Adaptive Regulation Curve Model

During the daily operation of the distribution substation area, due to the fluctuations of DGs and loads, the distribution substation area voltage fluctuates within the range of the reference value. To address this problem, a voltage-reactive power adaptive regulation curve method is employed. This method dynamically collects the voltage at the distribution substation area network connection point, and adjusts the reactive power output of the distribution substation area, enabling real-time response to changes in the distribution substation area. As shown in Figure 8, when the distribution substation area voltage exceeds the reference value, the reactive power output is reduced. Conversely, when the voltage is below the reference, the reactive power output is increased.

Figure 8. Distribution substation area voltage-reactive power adaptive regulation curve.

Each distribution substation area, based on the voltage reference value $V_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ and reactive power reference value $Q_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ from flexible interconnected distribution substation areas, generates the daily voltage-reactive power adaptive regulation constraint for each distribution substation area, as shown in Equation (46).

$$Q_{i,t}=\left\{\begin{array}{c}{Q}_{i,T}^{\mathrm{L}\mathrm{V}\mathrm{N},\mathrm{m}\mathrm{a}\mathrm{x}}, V_{i,t}<0.9V_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}\\ {\displaystyle \frac{Q_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}-Q_{i,T}^{\mathrm{L}\mathrm{V}\mathrm{N},\mathrm{m}\mathrm{a}\mathrm{x}}}{0.1V_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}}}\left(V_{i,t}-V_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}\right)+Q_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}, 0.9V_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}\le V_{i,t}<V_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}\\ {\displaystyle \frac{Q_{i,T}^{\mathrm{L}\mathrm{V}\mathrm{N},\mathrm{m}\mathrm{i}\mathrm{n}}-Q_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}}{0.1V_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}}}\left(V_{i,t}-V_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}\right)+Q_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}, V_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}\le V_{i,t}<1.1V_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}\\ Q_{i,T}^{\mathrm{L}\mathrm{V}\mathrm{N},\mathrm{m}\mathrm{i}\mathrm{n}}, V_{i,t}\ge 1.1V_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}\end{array}\right.$$

(46)

where $Q_{i,T}^{\mathrm{L}\mathrm{V}\mathrm{N},\mathrm{m}\mathrm{a}\mathrm{x}}$ and $Q_{i,T}^{\mathrm{L}\mathrm{V}\mathrm{N},\mathrm{m}\mathrm{i}\mathrm{n}}$ are the maximum and minimum reactive power outputs for the devices in distribution substation area $i$.

During the current daily operation optimization period, each control device measures the voltage at the distribution substation area network connection point. It calculates the target reactive power output based on the voltage-reactive power adaptive regulation constraint.

6. Distributed Operation Optimization Method for Distribution Substation Areas Based on ADMM

In this section, a distributed operation optimization method for distribution substation areas based on ADMM is proposed. The distributed operation optimization method includes low-voltage network loss, node voltage violation penalty, and power deviation penalty in the distribution substation area. Combined with the ADMM algorithm, the process to solve the collaborative optimization problem is described.

6.1. Optimization Model for Flexible Interconnected Distribution Substation Areas

6.1.1. Objective Function of Distribution Substation Areas

For flexible interconnected distribution substation areas, a distributed optimization model for each flexible interconnected distribution substation area is established during the current daily operation optimization period. The objective is to minimize the sum of the low-voltage network loss cost, node voltage violation penalty cost, and power deviation penalty cost for the distribution substation area. The objective function for the distributed optimization model of the distribution substation area is given by Equations (47)–(51).

$$\min f^{\mathrm{L}\mathrm{V}\mathrm{N}}={\sum }_{n=1}^{N_{\mathrm{L}\mathrm{V}\mathrm{N}}}(f_{1,n}^{\mathrm{L}\mathrm{V}\mathrm{N}}+f_{2,n}^{\mathrm{L}\mathrm{V}\mathrm{N}}+f_{3,n}^{\mathrm{L}\mathrm{V}\mathrm{N}}+f_{4,n}^{\mathrm{L}\mathrm{V}\mathrm{N}})$$

(47)

$$f_{1,n}^{\mathrm{L}\mathrm{V}\mathrm{N}}=C^{\mathrm{L}\mathrm{V}}({\sum }_{ij}{r}_{ij}{I}_{t,ij}^2+{\sum }_{i=1}^{N_{\mathrm{S}\mathrm{O}\mathrm{P}}}{P}_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P},\mathrm{L}})$$

(48)

$$f_{2,n}^{\mathrm{L}\mathrm{V}\mathrm{N}}=C^{UL}{\sum }_{ij}(U_{t,i}^2-{\left(U_i^{\prime \mathrm{t}\mathrm{h}\mathrm{r}}\right)}^2)$$

(49)

$$f_{3,n}^{\mathrm{L}\mathrm{V}\mathrm{N}}=C^P{\sum }_{n=1}^{N_{\mathrm{L}\mathrm{V}\mathrm{N}}}\Vert P_{t,n}^{\mathrm{a}\mathrm{c}\mathrm{t}}-P_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}{\Vert }_2$$

(50)

$$f_{4,n}^{\mathrm{L}\mathrm{V}\mathrm{N}}=C^Q{\sum }_{n=1}^{N_{\mathrm{L}\mathrm{V}\mathrm{N}}}\Vert Q_{t,n}^{\mathrm{a}\mathrm{c}\mathrm{t}}-Q_{i,t}{\Vert }_2$$

(51)

where $N_{\mathrm{L}\mathrm{V}\mathrm{N}}$ is the number of flexible interconnected distribution substation areas, $f^{\mathrm{L}\mathrm{V}\mathrm{N}}$ represents the objective function for the flexible interconnected distribution substation area, $f_{1,n}^{\mathrm{L}\mathrm{V}\mathrm{N}}$ represents the power loss cost, including network loss cost and loss cost of the flexible interconnection device in the distribution substation area, $f_{2,n}^{\mathrm{L}\mathrm{V}\mathrm{N}}$ represents the penalty cost for voltage violation at distribution substation area nodes, $I_{t,ij}^2$ is the square of the current flowing through the line $ij$, $U_{t,i}$ is the voltage at node $i$, $U_i^{\prime \mathrm{t}\mathrm{h}\mathrm{r}}$ is the upper and lower voltage limit for the distribution substation area, $f_{3,n}^{\mathrm{L}\mathrm{V}\mathrm{N}}$ represents the active power deviation penalty cost for the flexible interconnected distribution substation area, $P_{t,n}^{\mathrm{a}\mathrm{c}\mathrm{t}}$ is the actual active power value of the distribution substation area, $P_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ is the active power reference value issued by the medium-voltage distribution network, $f_{4,n}^{\mathrm{L}\mathrm{V}\mathrm{N}}$ represents the reactive power deviation penalty cost for the flexible interconnected distribution substation area, $Q_{t,n}^{\mathrm{a}\mathrm{c}\mathrm{t}}$ is the actual reactive power value of the distribution substation area, and $C^{\mathrm{L}\mathrm{V}}$, $C^{UL}$, $C^P$, and $C^Q$ are weight coefficients.

6.1.2. Constraints of Distribution Substation Areas

The constraints considered for the distribution substation area include low-voltage network power flow constraints, system security constraints, operation constraints of DG units, flexible interconnection device SOP constraints, and voltage-reactive power adaptive regulation curve constraints.

The constraint for SOP in the distribution substation areas is expressed as Equations (52) and (53).

$${\sum }_{\beta \in {\mathbf{\Omega }}_{\mathrm{S}\mathrm{O}\mathrm{P}}}{P}_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P}}+{\sum }_{\beta \in {\mathbf{\Omega }}_{\mathrm{S}\mathrm{O}\mathrm{P}}}{P}_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P},\mathrm{L}}=0$$

(52)

$$P_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P},\mathrm{L}}=A_{\beta }^{\mathrm{S}\mathrm{O}\mathrm{P}}\sqrt{{\left(P_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P}}\right)}^2+{\left(Q_{\beta ,t}^{\mathrm{S}\mathrm{O}\mathrm{P}}\right)}^2}$$

(53)

6.2. Solution Method Based on ADMM

Each flexible interconnected distribution substation area performs distributed iterative optimization, interacting with neighboring areas to exchange boundary information. The augmented Lagrangian form for the objective function of the distribution substation area is defined as Equation (54).

$$L_d^{\mathrm{L}\mathrm{V}\mathrm{N}}=f_d^{\mathrm{L}\mathrm{V}\mathrm{N}}+{\lambda }_{j,i} \left(P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_j}^r-P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_i}^{\prime r}\right)+{\displaystyle \frac{\rho }{2}}\Vert P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_j}^r-P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_i}^{\prime r}{\Vert }_2^2$$

(54)

where $f_d^{\mathrm{L}\mathrm{V}\mathrm{N}}$ is the original objective function, $L_d^{\mathrm{L}\mathrm{V}\mathrm{N}}$ is the augmented Lagrangian function of the objective function, ${\lambda }_{j,i}$ is the Lagrange multiplier, $\rho$ is the coefficient for the quadratic penalty term, $\rho >0$, $P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_j}^r$ is the active power transmitted by the flexible interconnection device in distribution substation area $j$ in the $r$-th iteration, and $P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_i}^{\prime r}$ is the active power transmitted by the flexible interconnection device SOP in neighboring distribution substation area $i$ in the $r$-th iteration.

The objective function in augmented Lagrangian form considers the low-voltage network power flow constraints, system security constraints, operation constraints of DG units, flexible interconnection device SOP constraints, and voltage-reactive power adaptive regulation curve constraints. This gives the network power loss cost, flexible interconnection device SOP loss cost, node voltage values, and boundary information exchanged with neighboring areas. The boundary information includes the active power transmitted by the flexible interconnection device in distribution substation area $j$ in the $r$-th iteration, $P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_j}^r$, and the active power transmitted by the flexible interconnection device SOP in neighboring distribution substation area $i$ in the $r$-th iteration $P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_i}^{\prime r}$.

Calculate the primal residual $R_{j,i}^r$ that reflects the feasibility of the original problem, and the dual residual $S_{j,i}^r$ that reflects the feasibility of the dual problem after the $r$-th iteration, as shown in Equations (55) and (56).

$$R_{j,i}^r={∥P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_j}^r-P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_i}^{\prime r}∥}_2^2$$

(55)

$$S_{j,i}^r={∥P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_i}^r-P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_i}^{\prime r-1}∥}_2^2$$

(56)

where $r$ is the iteration number, $P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_i}^r$ is the active power transmitted by the SOP in distribution substation area $i$ in the $r$-th iteration, and $P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_i}^{\prime r-1}$ is the active power transmitted by the SOP in distribution substation area $i$ in the $(r-1)$th iteration.

To determine whether convergence has been achieved, it is necessary to calculate if the values of the primal residual $R_{j,i}^r$ and the dual residual $S_{j,i}^r$ are sufficiently small. The iterative convergence criterion is as Equation (57).

$${‖[R_{j,i}^r S_{j,i}^r{]}^{\mathrm{T}}‖}_{\mathrm{\infty }}\le \epsilon$$

(57)

where $\epsilon$ is the convergence accuracy.

The update for the area interaction parameters is expressed as Equations (58)–(60).

$$P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_j}^{r+1}=\mathrm{a}\mathrm{r}\mathrm{g} \mathrm{m}\mathrm{i}\mathrm{n} (f_d^{\mathrm{L}\mathrm{V}\mathrm{N}}+{\displaystyle \frac{\rho }{2}}{\sum }_{t=1}^{N_t}\Vert P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_j}^r-P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_i}^{\prime ,r}{\Vert }_2^2)$$

(58)

$$P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_i}^{r+1}=\mathrm{a}\mathrm{r}\mathrm{g} \mathrm{m}\mathrm{i}\mathrm{n} (f_d^{\mathrm{L}\mathrm{V}\mathrm{N}}+{\displaystyle \frac{\rho }{2}}{\sum }_{t=1}^{N_t}\Vert P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_j}^{r+1}-P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_i}^{\prime ,r}{\Vert }_2^2)$$

(59)

$${\lambda }_{j,i}^{r+1}={\lambda }_{j,i}^r+\tau \rho (P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_j}^r-P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_i}^{\prime r})$$

(60)

where $P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_j}^{r+1}$ is the active power transmitted by the flexible interconnection device SOP in distribution substation area $j$ in the $(r+1)$th iteration, $P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_i}^{r+1}$ is the active power transmitted by the flexible interconnection device SOP in distribution substation area $i$ in the $(r+1)$th iteration, ${\lambda }_{j,i}^{r+1}$ is the Lagrange multiplier for the active power $P_{t,{\mathrm{L}\mathrm{V}\mathrm{N}}_j}^r$ at distribution substation area $j$ in the $(r+1)$th iteration, and $\tau$ is the step size.

6.3. Solution Procedure

For the coordinated operation of medium- and low-voltage distribution networks with flexible interconnected distribution substation areas, the solution procedure is shown in Figure 9. The specific steps are as follows:

Figure 9. Flowchart of the coordinated operation method of medium- and low-voltage distribution networks with flexible interconnected distribution substation areas.

(1) Input the basic parameter information of the medium- and low-voltage distribution network.

(2) Based on the input parameters, calculate the distribution substation area flexibility region constraint and the distribution substation area correlation power constraint.

(3) The medium-voltage side of the distribution network performs centralized daily solving, while issuing instructions to the distribution substation area side. The instructions include the voltage reference value $V_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}$, active power reference value $P_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}$, and reactive power reference value $Q_{i,T}^{\mathrm{r}\mathrm{e}\mathrm{f}}$ for each distribution substation area.

(4) Each distribution substation area generates the voltage-reactive power adaptive regulation curve.

(5) During the current daily optimization period, the flexible interconnected distribution substation area measures the voltage at its connection point. Based on the generated voltage-reactive power adaptive regulation curve, the distribution substation area calculates the reactive power output target value.

(6) Establish the optimization model for the flexible interconnected distribution substation area, with the objective function set to minimize the total operation costs of the distribution substation area.

(7) The distribution substation area executes the daily distributed optimization and exchanges boundary information with neighboring areas.

(8) If the boundary information error for the $r$-th iteration meets the convergence accuracy $\epsilon$, output the optimization results. Otherwise, update the boundary interaction parameters and return to step 7.

(9) If the daily optimization period has reached the maximum, calculate the results and finish. Otherwise, move to the next daily optimization period and return to step 5.

7. Case Studies and Analysis

In this section, the effectiveness of the cooperative operation optimization method for medium- and low-voltage distribution networks with flexible interconnected distribution substation areas is verified. The proposed method is implemented in MATLAB R2020a, using the CPLEX 12.10 solver in YALMIP [32]. The numerical experiments were carried out on a computer with an Intel Core i7 @ 2.50 GHz processor and 16 GB RAM.

7.1. Case Design

The structure of the modified IEEE 33-node system is shown in Figure 10. Three low-voltage distribution substation areas are added. Areas 1, 2, and 3 are located on different feeders at nodes 25, 29, and 30, respectively. The secondary side of the transformers in the station areas are interconnected through SOP. The rated voltage level of the medium-voltage distribution network is 12.66 kV. PV parameters of the medium-voltage distribution network are shown in Table 1. Load and PV fluctuation curves in the distribution network are shown in Figure 11 [4,12,33]. By multiplying the load reference value by the fluctuation coefficient, the load power value of the time series is obtained.

Figure 10. Structure of the modified IEEE 33-node system.

Table 1. Allocation of PVs in the modified IEEE 33-node system.

Node	Maximum Active Power Output (kW)	Capacity (kWp)
16, 17, 18, 23, 24, 32	120	120
20, 21, 22	240	240

Figure 11. Load and PV fluctuation curves.

The modified practical flexible interconnected distribution substation areas include three areas with the rated voltage level of 0.4 kV. The total active power demand is 175.51 kW, and the total reactive power demand is 164.81 kvar. Area 1 has an active load of 70.30 kW and a reactive load of 71.40 kvar. Area 2 has an active load of 82.70 kW and a reactive load of 81.60 kvar. Area 3 has an active load of 22.51 kW and a reactive load of 11.81 kvar. The area 2 is set as the industrial load, and areas 1 and 3 are resident loads. PV and CL parameters of the flexible interconnected distribution substation areas are shown in Table 2. The capacity of SOP is 100 kVA.

Table 2. Allocation of PVs in the modified practical flexible interconnected distribution substation areas.

	Node	Maximum Active Power Output (kW)	Capacity (kWp)
PV	5, 6, 7, 9, 13, 14, 16, 18, 19, 21, 23, 24, 26, 27, 29	6	6
PV	3, 4, 12	30	30
CL	8, 17	7	/

Set convergence accuracy $\epsilon =1\times {10}^{-3}$, step size $\tau ={\displaystyle \frac{1}{2}}$, secondary penalty coefficient $\rho =0.05$, Lagrange multiplier ${\lambda }_{j,i}=0$, and current iteration $r=0$. The optimization scheduling duration of the medium-voltage distribution network is set to 1 h. The optimization scheduling duration of the distribution substation area is set to 15 min. This paper assumes that the distribution network is in a balanced state.

7.2. Validation of the Flexibility and Adjustability of Distribution Substation Areas

To describe the flexible adjustability of distribution substation areas, the medium-voltage distribution network incorporates flexibility region constraints and the correlation power constraints of the distribution substation area. To validate the impact of considering the flexibility region constraints and correlation power constraints on the operation of medium- and low-voltage distribution networks, the following comparison schemes are set up:

Scheme 1: Without considering the flexible adjustability of distribution substation areas, the centralized optimization on the medium-voltage distribution network is employed.

Scheme 2: Considering the flexible adjustability of distribution substation areas, the centralized optimization on the medium-voltage distribution network is employed.

Comparison of the load fluctuation curves of the distribution substation areas under different schemes is shown in Figure 12. Scheme 1 is closer to the load fluctuation on the medium-voltage side. While in Scheme 2, because the flexibility region constraints and correlation constraints of the distribution substation areas are considered, the adjusted range is wider and the load fluctuations are smoother, which is more beneficial for maintaining the supply–demand balance and economic operation of the distribution network.

Figure 12. Load fluctuation in different schemes for each distribution substation area. (a) Area 1; (b) Area 2; (c) Area 3.

Comparison of operation results of different schemes is in Table 3. Regarding operation costs and voltage fluctuation, as shown in Table 3, the operation cost of the distribution network in Scheme 2 is lower than that in Scheme 1. The voltage range and fluctuation limits are shown in Figure 13 and Figure 14. Scheme 2 has a smaller voltage fluctuation range compared to Scheme 1, and the median voltage is closer to 1.0. This validates that the flexibility region constraints and correlation power constraints of flexible interconnected distribution substation areas lead to a better response to the fluctuations in areas, which is more advantageous for developing an economically optimized operation plan for the distribution network.

Table 3. Comparison of operation results of Scheme 1 and Scheme 2.

	Scheme 1	Scheme 2
Operation cost in the medium-voltage distribution network (USD)	87.8823	87.2683
Maximum voltage (p.u.)	1.0293	1.0287
Minimum voltage (p.u.)	0.9699	0.9780

Figure 13. Voltage fluctuation upper and lower limit box diagram in Scheme 1.

Figure 14. Voltage fluctuation upper and lower limit box diagram in Scheme 2.

7.3. Validation of the Cooperative Operation Optimization Method for Medium- and Low-Voltage Distribution Networks

To validate the effectiveness of the proposed cooperative operation optimization method for medium- and low-voltage distribution networks considering flexible interconnected distribution substation areas, three schemes are set for comparison:

Scheme 3: Without considering the integration of SOP, the initial operation status of the distribution network is obtained.

Scheme 4: Considering the integration of SOP to achieve flexible interconnection, the centralized optimization method adopted in [12] is employed to determine the operation status of the distribution network.

Scheme 5: Considering the integration of SOP to achieve flexible interconnection, the cooperative operation optimization method for medium- and low-voltage distribution networks in this paper is employed to determine the operation status of the distribution network.

The active and reactive power reference values, and the flexibility region upper and lower limits for each distribution substation area in Scheme 3, are shown in Figure 15. After considering the flexibility area of the distribution substation area, the active power of Area 1 and 2 are both positive values. However, due to the PV output exceeding the load demand, there is a situation where the power in Area 3 is negative. Due to the high reactive power output of PVs at night, the reactive power in the areas is lower at night than during the day.

Figure 15. Active and reactive power reference values for each distribution substation area. (a) Area 1; (b) Area 2; (c) Area 3.

The voltage reference values at the connection points of each distribution substation area are shown in Figure 16. Area 1 is closer to the source node and has a higher voltage value. Area 2 and Area 3 are on the same feeder line, so the voltage values are similar.

Figure 16. Voltage reference values of each distribution substation area.

The operation results for the three schemes are shown in Table 4. From the table, it is evident that in Scheme 3, without the integration of SOP, even though the installation and operation costs of the devices are saved, the operation costs remain the highest. However, after the integration of the SOP in the distribution substation areas in Schemes 4 and 5, both the operation costs of the distribution network and the costs of the flexible interconnected distribution substation areas are significantly reduced. Additionally, voltage fluctuations are greatly alleviated, proving the necessity and economic viability of flexible interconnection through SOP in low-voltage distribution substation areas.

Table 4. Comparison of operation results of Schemes 3, 4, and 5.

	Scheme 3	Scheme 4	Scheme 5
Operation cost in the medium-voltage distribution network (USD)	96.3036	83.8272	87.6144
Operation cost in distribution substation areas (USD)	11.3610	6.0026	6.2948
SOP losses (USD)	/	2.8922	3.0285
Maximum voltage (p.u.)	1.0313	1.0279	1.0282
Minimum voltage (p.u.)	0.9659	0.9740	0.9724

Figure 17 compares the active power values transmitted by the SOP between the distribution substation areas in Schemes 4 and 5. Figure 18 compares the reactive power values generated by the SOP in the distribution substation areas in Schemes 4 and 5. It is evident that active power flows from Ports 1 and 3 to Port 2, while Port 2 has the highest reactive power, which aligns with the high demand for power in industrial loads. After the integration of SOP in the distribution substation areas, the loss costs of the distribution network and flexible interconnected low-voltage distribution substation areas in both Scheme 4 and Scheme 5 are significantly reduced compared to Scheme 3. Although the methods in Schemes 4 and 5 are different, the results of the proposed method are consistent with the centralized approach in general.

Figure 17. Comparison of active power values transmitted by SOP in Scheme 4 and Scheme 5.

Figure 18. Comparison of reactive power values generated by SOP in Scheme 4 and Scheme 5.

The maximum and minimum voltage values for Schemes 4 and 5 are compared in Figure 19. The voltage values in both schemes stay within the allowable range, indicating that the proposed method not only reduces the total costs of the distribution network but also alleviates voltage fluctuations to some extent. Furthermore, Scheme 5 exchanges only the transmission power values of SOP between neighboring distribution substation areas, which avoids the need for extensive real-time data transmission between the distribution substation area and the medium-voltage distribution network. This significantly reduces the communication pressure between the medium- and low-voltage distribution networks and the computational pressure on the medium-voltage side, validating the effectiveness of the proposed method.

Figure 19. Comparison of maximum and minimum voltage values in Scheme 4 and Scheme 5.

7.4. Analysis of Handling Source-Load Uncertainty

To verify the impact of source-load uncertainty on the operation of medium- and low-voltage distribution networks, a random error of 20% is added to the baseline source-load fluctuation curves shown in Figure 11, resulting in the actual curve shown in Figure 20. Three schemes are used for comparison and analysis:

Figure 20. Forecasted and actual load and PV fluctuation curves.

Scheme 6: Based on source-load forecast information, the centralized optimization method is employed to determine the operation status of the distribution network.

Scheme 7: Based on source-load forecast information, hourly level strategy commands are issued by the medium-voltage side, and the distribution substation areas perform 15 min rolling distributed optimization based on these strategy commands and actual source-load data to determine the operation status of the network.

Scheme 8: Based on source-load forecast information, hourly level strategy commands are issued by the medium-voltage side, and the distribution substation areas perform calculations based on actual source-load data, adjusting the strategy commands according to the voltage-reactive power adaptive regulation curve, followed by 15 min rolling distributed optimization to determine the operation status of the network.

The operation results of the different schemes are shown in Table 5. Compared with Scheme 6, the operation results obtained are similar to the centralized method in Scheme 8. The difference in operation costs is due to the random fluctuations of the source load. Compared with Scheme 7, Scheme 8 can reduce operation costs and alleviate voltage fluctuations, and provide quicker and more accurate adjustment to source-load fluctuations by adjusting instructions.

Table 5. Comparison of operation results of Schemes 6, 7, and 8.

	Scheme 6	Scheme 7	Scheme 8
Operation cost in the medium-voltage distribution network (USD)	83.8272	86.5748	86.2190
Operation cost in distribution substation areas (USD)	6.0026	6.8451	6.7665
Maximum voltage (p.u.)	1.0279	1.0293	1.0280
Minimum voltage (p.u.)	0.9740	0.9717	0.9734

The comparison of voltage values at the connection point of the distribution substation area in Scheme 7 and Scheme 8 is shown in Figure 21. The trend of voltage before and after adjustment is similar. After revising the strategy, the voltage value in Scheme 8 is closer to the actual fluctuation situation. Therefore, it is better to support the operation of the distribution substation area and enhance the economic efficiency of the distribution network.

Figure 21. The voltage value at the connection point of the distribution substation area in Scheme 7 and Scheme 8.

8. Conclusions

A cooperative optimization method for medium- and low-voltage distribution networks considering flexible interconnected distribution substation areas is proposed in this paper. The impact of flexible interconnection device SOP on low-voltage distribution substation areas is fully considered.

(1) A flexibility region interval affine method for flexible interconnected low-voltage distribution substation areas is proposed. This method enables the aggregation and equivalent representation of the flexibility of distribution substation areas. It supports efficient resource scheduling on the medium-voltage side, and enhances the flexibility of the distribution network in adapting to operational fluctuations.

(2) A multi-scale collaborative operation optimization framework for medium- and low-voltage distribution networks is developed. The medium-voltage side and the distribution substation area side adopt multi-time-scale control. This enhances dynamic response capabilities to fluctuations and improves the economic operation of the distribution network.

(3) A distributed optimization method for flexible interconnected distribution substation areas based on ADMM is proposed. This method alleviates the centralized computational burden on the medium-voltage side, and enables autonomous operation of the flexible interconnected distribution substation areas.

Based on the research in this paper, it is necessary to develop a more detailed operation model considering correlation between individual source and load. The integration of charging loads such as electric vehicles, and other types of distributed generation sources such as small hydroelectric plants and wind farms can be further investigated. Meanwhile, it is significant to consider the unbalanced operation of the system phases in future studies.