Multi-Time Scale Optimal Dispatch of Distribution Network with Pumped Storage Station Based on Model Predictive Control

Abstract

As the penetration of renewable energy increases, the distribution grid faces great challenges in integrating large amounts of distributed energy sources and dealing with their output uncertainty. To address this, a multi-time scale optimal dispatch method based on model predictive control is proposed, including a day-ahead stage and an intra-day rolling stage. In the day-ahead stage, to fully utilize the flexibility of variable speed pumped storage hydropower, the generating/pumping phase modulation condition is considered, not just generating or pumping. Day-ahead optimal dispatch is established with the objective of minimizing the operation economy and node voltage deviation of the distribution network. In the intra-day rolling stage, model predictive control with finite time domain rolling optimal dispatch is used to replace the traditional single-time section optimal dispatch, considering the forecast data of wind, photovoltaic (PV), and load within the finite time domain, so that can respond in advance to smooth the generator output. At the same time, the uncertainty problem of the distribution network is solved effectively by rolling optimization and feedback correction of model predictive control. In order to consider the daily operating capacity balance of energy storage in the intra-day stage, the capacity imbalance penalty is added to the intra-day rolling optimization objective function, so that the energy storage capacity tries to track the results of the day-ahead optimization, achieving the long-term development of energy storage. Simulation analysis proves the feasibility and effectiveness of the proposed method. The proposed method enhances the generation–load–storage coordinated dispatching ability, effectively improving the distribution network’s capability to respond to fluctuations of renewable energy.

1. Introduction

In recent years, with the development of renewable energy, the installed capacity of wind and PV has increased significantly, and its penetration rate in the distribution network has continued to increase [1,2]. However, wind and PV generation has strong volatility and can be intermittent, which leads to the problem of insufficient flexible adjustment ability of the distribution network when accepting wind and PV supply, and the forecast error of wind and PV brings certain challenges to the optimal dispatch of the distribution network [3,4,5].

Pumped storage hydropower (PSH), as the most mature and economically large-scale energy storage technology, has the characteristics of peak shaving and valley filling, which can smooth the fluctuations of wind and PV generation and improve the ability of the power grid to accommodate wind and PV sources [6]. In particular, variable speed pumped storage hydropower (VSPSH) is increasingly favored by the power grid due to its strong flexibility and rapid response characteristics as a flexible adjustment resource [7]. Currently, there are studies on the optimal dispatch of PSH. Reference [8] proposes a day-ahead market structure based on the PSH model, which improves the utilization of PSH resources and the flexibility of the power system. Reference [9] uses the Non-dominated Sorting Genetic Algorithm II (NSGA-II) algorithm to study the economic emission dispatch problem of renewable energy and PSH, considering uncertainties and outages of renewable energy. Reference [10] introduces PSH in wind–fire–water systems to address wind fluctuations. Reference [11] studies an off-grid sustainable power generation system in rural areas composed of wind energy and PSH. The above research only considers the operation conditions of generating and pumping, ignoring the generating/pumping phase modulation condition of VSPSH, and has not fully utilized the function of phase modulation and voltage modulation of VSPSH.

To address the difficulties in wind and PV forecast errors for the optimal dispatch of the distribution network since the wind and PV forecast accuracy increases with a decrease in the time scale, a multi-time scale optimal dispatch method was studied to reduce the impact of forecast error on the optimal dispatch. Reference [12] uses a multi-time scale dispatch model to realize reactive power and voltage control of wind farms. Reference [13] addresses the impact of different time scales load forecast error on optimal dispatch of a microgrid using two stages including day-ahead and intra-day. Reference [14] proposes an optimal dispatch scheme including day-ahead, intra-day, and real-time stages for wind–PV–electrochemical storage–PSH station. However, the above research used traditional multi-time scale optimal dispatch, which is an open-loop optimization that only considers a single time section. When the output of wind and PV fluctuates significantly, it can lead to substantial changes in other controllable power sources and difficulties in meeting generator ramp rates. Unlike the multi-time scale optimal dispatch method, Model Predictive Control (MPC) is widely used in uncertain optimal control problems of power systems due to its rolling optimization and feedback correction features. MPC is an effective approach to address an uncertain optimal dispatch about wind and PV [15]. To minimize the impact of wind forecast error on optimal dispatch, previous studies have combined multi-time scale optimal dispatch method with MPC. Reference [16] uses multi-time scales to achieve economic optimal dispatch of a micro energy grid. Reference [17] for optimal dispatch, uses particle swarm optimization to solve the day-ahead stage and adjust the results of the day-ahead stage using MPC for the intra-day. Compared with the traditional multi-time scale optimal dispatch method, combined with the model predictive control algorithm, the closed-loop optimization of the multi-time section can be realized through the prediction and feedback correction functions of MPC, and the actual state of the system can be fed back, reducing the influence of forecast error and external interference.

Considering the above, the purpose of this study is to coordinate the active power and reactive power resources including VSPSH of the distribution network, to achieve the lowest operation cost and the minimum node voltage deviation of the distribution network, proposing a multi-time scale optimal dispatch method based on model predictive control. The proposed method includes two stages: day-ahead dispatch and intra-day rolling dispatch. The intra-day rolling dispatch adopts model predictive control, rolling updates the forecast information of wind, PV, and load, and feedback on the actual state of the system for optimization.

2. Model of VSPSH

According to topological differences, VSPSH can be divided into doubly-fed types and full-size converter types, both of which have the ability to quickly adjust active power and reactive power. VSPSH has five steady-state operation conditions: generating, pumping, generating phase modulation, pumping phase modulation and shutdown. Considering these five operation conditions, the active power and reactive power constraints of VSPSH can be expressed as follows:

(1)$P_{t,i}^{\mathrm{VSPSH}}=P_{t,i,g}^{\mathrm{VSPSH}}-P_{t,i,p}^{\mathrm{VSPSH}}+P_{t,i,gp}^{\mathrm{VSPSH}}+P_{t,i,pp}^{\mathrm{VSPSH}}$

(2)$Q_{t,i}^{\mathrm{VSPSH}}=Q_{t,i,g}^{\mathrm{VSPSH}}+Q_{t,i,p}^{\mathrm{VSPSH}}+Q_{t,i,gp}^{\mathrm{VSPSH}}+Q_{t,i,pp}^{\mathrm{VSPSH}}$

(3)$\sqrt{{\left(P_{t,i}^{\mathrm{VSPSH}}\right)}^2+{\left(Q_{t,i}^{\mathrm{VSPSH}}\right)}^2}\le S_i^{\mathrm{VSPSH}}$

where t and i represent the tth time and ith VSPSH, $P_{t,i}^{\mathrm{PSH}}$ and $Q_{t,i}^{\mathrm{PSH}}$ are the active and reactive power outputs of the VSPSH, respectively; $P_{t,i,g}^{\mathrm{VSPSH}}$, $P_{t,i,p}^{\mathrm{VSPSH}}$, $P_{t,i,gp}^{\mathrm{VSPSH}}$, $P_{t,i,pp}^{\mathrm{VSPSH}}$ are the active power output of VSPSH under generating, pumping, generating phase modulation, and pumping phase modulation conditions, respectively; $Q_{t,i,g}^{\mathrm{VSPSH}}$, $Q_{t,i,p}^{\mathrm{VSPSH}}$, $Q_{t,i,gp}^{\mathrm{VSPSH}}$, $Q_{t,i,pp}^{\mathrm{VSPSH}}$ are the reactive power outputs of VSPSH under generating, pumping, generating phase modulation, and pumping phase modulation conditions, respectively; and $S_i^{\mathrm{VSPSH}}$ is the rated capacity of the VSPSH converter.

The operation constraints of VSPHS under multi-operation conditions are as follows [18]:

(4)${\mu }_{t,i,p}^{\mathrm{VSPSH}}+{\mu }_{t,i,g}^{\mathrm{VSPSH}}+{\mu }_{t,i,gp}^{\mathrm{VSPSH}}+{\mu }_{t,i,pp}^{\mathrm{VSPSH}}\le 1$

(5)$\begin{array}{c}{\mu }_{t,i,g}^{\mathrm{VSPSH}}{P}_{t,i,\min }^{\mathrm{VSPSH}}\le P_{t,i,g}^{\mathrm{VSPSH}}\le {\mu }_{t,i,g}^{\mathrm{VSPSH}}{P}_{t,i,\max }^{\mathrm{VSPSH}}\\ {\mu }_{t,i,p}^{\mathrm{VSPSH}}{P}_{t,i,\min }^{\mathrm{VSPSH}}\le P_{t,i,p}^{\mathrm{VSPSH}}\le {\mu }_{t,i,p}^{\mathrm{VSPSH}}{P}_{t,i,\max }^{\mathrm{VSPSH}}\\ P_{t,i,gp}^{\mathrm{VSPSH}}={\mu }_{t,i,gp}^{\mathrm{VSPSH}}{P}_{t,i,n}^{\mathrm{VSPSH}}\end{array}$

(6)$\begin{array}{c}-\sqrt{1-{\lambda }^2}{P}_{t,i,g}^{\mathrm{VSPSH}}/\lambda \le Q_{t,i,g}^{\mathrm{VSPSH}}\le \sqrt{1-{\lambda }^2}{P}_{t,i,g}^{\mathrm{VSPSH}}/\lambda \\ -\sqrt{1-{\lambda }^2}{P}_{t,i,p}^{\mathrm{VSPSH}}/\lambda \le Q_{t,i,p}^{\mathrm{VSPSH}}\le \sqrt{1-{\lambda }^2}{P}_{t,i,p}^{\mathrm{VSPSH}}/\lambda \\ {\mu }_{t,i,gp}^{\mathrm{VSPSH}}{Q}_{i,\min }^{\mathrm{VSPSH}}\le Q_{t,i,gp}^{\mathrm{VSPSH}}\le {\mu }_{t,i,gp}^{\mathrm{VSPSH}}{Q}_{i,\max }^{\mathrm{VSPSH}}\\ {\mu }_{t,i,pp}^{\mathrm{VSPSH}}{Q}_{i,\min }^{\mathrm{VSPSH}}\le Q_{t,i,pp}^{\mathrm{VSPSH}}\le {\mu }_{t,i,pp}^{\mathrm{VSPSH}}{Q}_{i,\max }^{\mathrm{VSPSH}}\end{array}$

where ${\mu }_{t,i,p}^{\mathrm{VSPSH}}$, ${\mu }_{t,i,g}^{\mathrm{VSPSH}}$, ${\mu }_{t,i,gp}^{\mathrm{VSPSH}}$, ${\mu }_{t,i,pp}^{\mathrm{VSPSH}}$ are 0–1 variables, and represent whether VSPSH is in generating, pumping, generating phase modulation, and pumping phase modulation conditions, respectively; $P_{t,i,\min }^{\mathrm{VSPSH}}$, $P_{t,i,\max }^{\mathrm{VSPSH}}$ are the minimum and maximum active power output of VSPSH, respectively; $Q_{i,\min }^{\mathrm{VSPSH}}$, $Q_{i,\max }^{\mathrm{VSPSH}}$ are the minimum and maximum reactive power output of VSPSH, respectively; $P_{t,i,n}^{\mathrm{VSPSH}}$ is the minimum active power required to maintain VSPSH idling operation at rated speed; and $\lambda$ is the minimum power factor of the VSPSH converter.

Equation (4) is the constraint that the VSPSH can only be under one operation condition at any time. Equation (5) is the active power constraint of VSPSH under each operation condition. Equation (6) is the reactive power constraint of VSPSH under each operation condition. At the same time, VSPSH needs to meet the daily operating capacity balance constraints and storage boundary constraints, as follows:

(7)$V_{t=0}^{\mathrm{VSPSH}}=V_{t=24}^{\mathrm{VSPSH}}$

(8)$V_{i,\min }^{\mathrm{VSPSH}}\le V_{t,i}^{\mathrm{VSPSH}}\le V_{i,\max }^{\mathrm{VSPSH}}$

where $V_{t,i}^{\mathrm{VSPSH}}$ is the VSPSH storage capacity at time t and $V_{i,\min }^{\mathrm{VSPSH}}$, $V_{i,\max }^{\mathrm{VSPSH}}$ are the minimum and maximum storage capacity of VSPSH.

3. Multi-Time Scale Dispatching Framework of Active Power and Reactive Power Based on MPC

To achieve safe and economical operation of the distribution network, considering that the forecast accuracy of wind, PV, and load improves as the time scale shortens, this paper divides the optimal dispatch of the distribution network into day-ahead and intra-day rolling stages. Additionally, model predictive control is introduced in the intra-day rolling optimization stage. Figure 1 illustrates the multi-time scale active and reactive power coordinated dispatch based on the MPC framework proposed in this paper. The controllable resources include wind, PV, SVG, On-Load Tap Changers (OLTCs), Capacitor Banks (CBs), Gas Turbine (GT), Battery Energy Storage (BES) and VSPSH.

3.1. Day-Ahead Stage

In this paper, the time scale ΔT for the day-ahead optimal dispatch of the distribution network is set to 1 h. In the day-ahead optimization stage, the system inputs the day-ahead forecast data of wind, PV, and load, and minimizes the operation cost and voltage deviation of the distribution network, using mixed-integer second-order cone programming. The purpose of day-ahead optimal dispatch is to determine the operation state of the slow-motion reactive power devices, such as OLTCs and CBs as well as the charging and discharging state of VSPSH and BES. These determinations are used as fixed parameters in the intra-day rolling stage. Due to manufacturing technology and lifespan limitations, the number and speed of OLTC and CB operations are restricted, and the charge and discharge cycles of VSPSH and BES are also limited.

3.2. Intra-Day Rolling Stage

This paper sets the time scale Δt for the intra-day rolling optimal dispatch of the distribution network to 15 min. The intra-day rolling optimization stage inputs ultra-short-term forecast data of wind, PV, and load, OLTC and CB operation states and VSPSH and BES charge and discharge optimization results in the day-ahead stage. A predictive model of MPC is established to predict the system’s output within the prediction horizon of MΔt, it solves the control variables which minimizes the intra-day objective function based on the predictive output of the system, and feeds back the actual state of the system. Since intra-day rolling optimization cannot account for the full 24 h period in one step, it is challenging for VSPSH and BES to ensure the daily operating capacity balance in the intra-day stage. In this regard, this paper adds a capacity imbalance penalty to the objective function in the intra-day rolling stage so that VSPSH and BES track the capacity plan of the day-ahead stage and try to ensure a daily operating capacity balance.

4. Multi-Time Scale Dispatching Model of Active Power and Reactive Power Based on MPC

4.1. Day-Ahead Optimal Dispatching Model

The control variables of day-ahead optimal dispatch include the reactive power of wind and PV, the reactive power of Static Var Generator (SVG), the action states of OLTC and CB, and the active and reactive power of GT, VSPSH, and BES. The day-ahead optimal dispatch aims to minimize the distribution network’s operation cost and node voltage deviation. The operation cost includes the OLTC operation cost, the CB operation cost, the gas turbine generation cost, the VSPSH and BES maintenance cost, and the active power loss cost. Thus, the objective function expression is:

(9)$\min F={\rho }_T\mathsf{∆}{n}_T+{\rho }_c\mathsf{∆}{n}_{CBs}+C_{GT}+C_{VSPSH}+C_{BES}+\mathsf{\Gamma }{P}_{loss}+{\lambda }_v\mathsf{∆}V$

where ${\rho }_T,{\rho }_c$ are OLTC and CBs single-action price; $\mathsf{∆}{n}_T,\mathsf{∆}{n}_{CBs}$ are OLTC and CBs gear changes, respectively; $C_{GT}$ is the generating electricity cost of gas turbines; $C_{VSPSH},C_{BES}$ are the maintenance costs of VSPSH and BES; $\mathsf{\Gamma }$ is the marginal price of power lost; $P_{loss}$ is the line active power loss; ${\lambda }_v$ is the penalty coefficient of voltage deviation; and $\mathsf{∆}V$ is the node voltage deviation.

Day-ahead optimal dispatch constraints include power flow constraints, safe operation constraints and active power and reactive power constraints of controllable resources in the distribution network.

1. Power flow constraints

For radiative networks, the power flow constraint can be expressed as [19]:

(10)$\begin{array}{c}{\displaystyle \sum _{ij\in {\mathsf{\Omega }}_b}\left(P_{ij}-r_{ij}{i}_{ij}\right)=}{P}_j+{\displaystyle \sum _{jk\in {\mathsf{\Omega }}_b}{P}_{jk}}\\ {\displaystyle \sum _{ij\in {\mathsf{\Omega }}_b}\left(Q_{ij}-x_{ij}{i}_{ij}\right)=}{Q}_j+{\displaystyle \sum _{jk\in {\mathsf{\Omega }}_b}{Q}_{jk}}\end{array}$

(11)$\begin{array}{c}{P}_j=P_{j,pre}^{\mathrm{WT}}+P_{j,pre}^{\mathrm{PV}}+P_j^{\mathrm{GT}}+P_j^{\mathrm{VSPSH}}+P_j^{\mathrm{BES}}-P_j^{\mathrm{load}}\\ Q_j=Q_j^{\mathrm{WT}}+Q_j^{\mathrm{PV}}+Q_j^{\mathrm{GT}}+Q_j^{\mathrm{VSPHS}}+Q_j^{\mathrm{BES}}+Q_j^{\mathrm{SVG}}+Q_j^{\mathrm{CBs}}-Q_j^{\mathrm{load}}\end{array}$

(12)$\begin{array}{c}{V}_j^2\ast k_{i j}^2=V_i^2-2\left(r_{ij}{P}_{ij}+x_{ij}{Q}_{ij}\right)+\left(r_{ij}^2+x_{ij}^2\right)I_{ij}^2\\ V_i^2{I}_{ij}^2=P_{ij}^2+Q_{ij}^2\end{array}$

where $P_{ij}$ and $Q_{ij}$ are the active power and reactive power flowing through branch ij, respectively; $P_j$ and $Q_j$ are the active power and reactive power injection values of node j, respectively; $P_{j,pre}^{\mathrm{WT}},P_{j,pre}^{\mathrm{PV}},P_j^{\mathrm{GT}},P_j^{\mathrm{VSPSH}},P_j^{\mathrm{BES}},P_j^{\mathrm{load}}$ are the forecast active power of wind and PV, the active power of GT, VSPSH, BES, and load of node j, respectively; $Q_j^{\mathrm{WT}},Q_j^{\mathrm{PV}},Q_j^{\mathrm{GT}},Q_j^{\mathrm{VSPSH}},Q_j^{\mathrm{BES}},Q_j^{\mathrm{SVG}},Q_j^{\mathrm{CBs}},Q_j^{\mathrm{load}}$ are the reactive power of wind, PV, GT, VSPSH, BES, and load of node j, respectively; $r_{ij}$ and $x_{ij}$ are the resistance and reactance of line ij; $V_i$ is the voltage of node i; k is the gears of OLTC; and $I_{ij}$ is the current of line ij. Formulas (10)–(12) are based on Kirchhoff’s law.

• 2.. Safe operation constraints

(13)$\begin{array}{l}{V}_{i,\min }\le V_i\le V_{i,\max }\\ 0\le I_{ij}\le I_{ij,\max }\end{array}$

where $V_{i,\min }$ and $V_{i,\max }$ are the minimum and maximum voltage of node i, respectively; and $I_{ij,\max }$ is the maximum current of the branch ij.

• 3.. Active power and reactive power constraints of controllable resources

• Active power and reactive power output constraints of wind and PV.

In this model, wind and PV active power at the maximum power point, that is, the active power output of wind and PV in the optimization algorithm is its predicted value.

The reactive power of a Doubly fed Induction Generator (DFIG) includes stator side reactive power and network side reactive power so that the reactive power regulation range of DFIG is:

(14)$\begin{array}{l}{Q}_{\max }^{\mathrm{WT}}=Q_{s,\max }^{\mathrm{WT}}+Q_{c,\max }^{\mathrm{WT}}\\ Q_{\min }^{\mathrm{WT}}=Q_{s,\min }^{\mathrm{WT}}+Q_{c,\min }^{\mathrm{WT}}\\ Q_{\min }^{\mathrm{WT}}\le Q_i^{\mathrm{WT}}\le Q_{\max }^{\mathrm{WT}}\end{array}$

where $Q_{\max }^{\mathrm{WT}}$ and $Q_{\min }^{\mathrm{WT}}$ are the maximum and minimum reactive power output of the WT. $Q_{s,\max }^{\mathrm{WT}}$ and $Q_{s,\min }^{\mathrm{WT}}$ are the maximum and minimum reactive power output of the inverter on the stator side of the WT. $Q_{c,\max }^{\mathrm{WT}}$ and $Q_{c,\min }^{\mathrm{WT}}$ are the maximum and minimum reactive power output of the inverter on the network side of the WT. The specific derivation process is shown in [20].

The reactive power output of PV is limited by the rated power of its converter:

(15)${(P_{i,pre}^{\mathrm{PV}})}^2+{(Q_i^{\mathrm{PV}})}^2\le {(S_{i,N}^{\mathrm{PV}})}^2$

where $S_N^{\mathrm{PV}}$ is the rated capacity of the PV inverter.

• Active power and reactive power output constraints of gas turbines.

As a controllable distributed power source, a gas turbines’s power transmission is through the grid-connected inverter, and its active power and reactive power can be independently controlled and limited by the capacity of the inverter:

(16)${(P_i^{\mathrm{GT}})}^2+{(Q_i^{\mathrm{GT}})}^2\le {(S_{i,N}^{\mathrm{GT}})}^2$

where $S_{i,N}^{\mathrm{GT}}$ is the rated installed capacity of GT.

The active power climb rate constraints of the gas turbine is:

(17)$-R_{i,down}^{\mathrm{GT}}\le P_i^{\mathrm{GT}}(t+1)-P_i^{\mathrm{GT}}(t)\le R_{i,up}^{\mathrm{GT}}$

where $R_{i,up}^{\mathrm{GT}}$ and $R_{i,down}^{\mathrm{GT}}$ are the limit upslope rate and the limit downslope rate of GT, respectively.

• Active power and reactive power constraints of VSPSH.

VSPSH constraints include capacity constraints, multi-operating condition constraints and storage capacity constraints, as shown in Equations (1)–(8).

• Active power and reactive power output constraints of battery energy storage.

BES constraint is similar to VSPSH but it has no phase modulation state. First, BES is constrained by the capacity of the inverter:

(18)${(P_i^{\mathrm{BES}})}^2+{(Q_i^{\mathrm{BES}})}^2\le {(S_{i,N}^{\mathrm{BES}})}^2$

where $S_{i,N}^{\mathrm{BES}}$ is the rated capacity of the BES inverter.

BES cannot be in the state of charging and discharging at the same time, and it needs to meet the daily operating capacity balance, then:

(19)$\begin{array}{c}{u}_{ch,i}+u_{dch,i}\le 1\\ 0\le P_{ch,i}\le u_{ch,i}{P}_{ch,i}^{\max }\\ 0\le P_{dch,i}\le u_{dch,i}{P}_{dch,i}^{\max }\\ P_i^{\mathrm{BES}}=P_{dch,i}-P_{ch,i}\end{array}$

(20)$\begin{array}{c}{E}_{SOC,i}(t+1)=E_{SOC,i}(t)+\eta P_{ch,i}(t)\mathsf{∆}T-P_{dch,i}(t)\mathsf{∆}T/\eta \\ E_{SOC,i}^{\min }\le E_{SOC,i}\le E_{SOC,i}^{\max }\end{array}$

where $u_{ch,i},u_{dch,i}$ are the charging and discharging state of BES, and are 0–1 variables; $P_{ch,i},P_{dch,i}$ are the BES charging and discharging power; $P_{ch,i}^{\max },P_{dch,i}^{\max }$ are the maximum charging and discharging power of BES; $E_{SOC,i}$ is the storage energy of BES; $\eta$ is the charge and discharge efficiency of BES; and $E_{SOC,i}^{\min },E_{SOC,i}^{\max }$ are the minimum and maximum storage energy of BES.

• Reactive power constraints of SVG.

(21)$Q_{\min }^{\mathrm{SVG}}\le Q_i^{\mathrm{SVG}}\le Q_{\max }^{\mathrm{SVG}}$

where $Q_{\min }^{\mathrm{SVG}},Q_{\max }^{\mathrm{SVG}}$ are the minimum and maximum reactive power of SVG, respectively.

• Constraint of capacitor banks.

(22)$\begin{array}{c}{Q}_i^{\mathrm{CBs}}=n_{CBs}{Q}_N^{\mathrm{CBs}}\\ -N_{\max }^{CBs}\le n_{CBs}(t+1)-n_{CBs}(t)\le N_{\max }^{CBs}\\ {\displaystyle \sum _{t=1}^{T-1}{n}_{CBs}(t+1)-n_{CBs}(t)\le }{R}_{\max }^{CBs}\end{array}$

where $n_{CBs}$ is the gear of CBs; $Q_N^{\mathrm{CBs}}$ is the unit reactive power of CBs; $N_{\max }^{CBs}$ is the maximum adjustment gear of CBs; $R_{\max }^{CBs}$ is the maximum daily adjustment number of CBs. Equation (22) constrains the maximum change of CBs in one action and the number of changes in a day.

4.2. Intra-Day Rolling Optimal Dispatching Model

In the day-ahead optimization stage, the optimal dispatch of the distribution network is based on the day-ahead forecast data of wind and solar loads. Due to the long time scale, the forecast error is large in the day-ahead stage, leading to deviations in the optimization results. This paper employs the MPC optimization method to perform rolling optimization of controllable resources in the distribution network every 15 min, enabling intra-day rolling optimal dispatch. The framework for intra-day rolling optimal dispatch is illustrated in Figure 2.

In the intra-day rolling optimization stage, the control variables include the reactive power of wind and PV, the reactive power of SVG, and the active and reactive power of GT, VSPSH and BES. Based on MPC in the intra-day rolling stage, a predictive model is established at time t, the optimization results of OLTC and CBs action states, VSPSH and BES charging/discharging states obtained in the day-ahead stage, as well as the latest ultra-short-term forecast data of wind, PV, and load, and system constraints are input into the model to predict the future outputs in the MΔt period. By minimizing the objective function, the control commands at time $t+\mathsf{∆}t,t+2\mathsf{∆}t,\cdots t+M\mathsf{∆}t$ are obtained, but only the first control command value is applied to the distribution network. The above process is repeated at the next time t + Δt, continuously rolling for optimization.

1. Predictive model of MPC

The control variables are obtained by solving the rolling optimization model, and then the active and reactive power outputs of each distributed power source and energy storage in the finite time domain are predicted. The predictive model is as follows:

(23)$\begin{array}{c}P(k+i|k)=P_0(k)+{\displaystyle \sum _{t=1}^i\mathsf{∆}{u}_1(k+t|k)}\\ Q(k+i|k)=Q_0(k)+{\displaystyle \sum _{t=1}^i\mathsf{∆}{u}_2(k+t|k)}\end{array}$

(24)$\begin{array}{c}P={[P^{\mathrm{GT}}{P}^{\mathrm{VSPSH}}{P}^{\mathrm{BES}}]}^T\\ Q={[Q^{\mathrm{WT}}{Q}^{\mathrm{PV}}{Q}^{\mathrm{GT}}{Q}^{\mathrm{VSPSH}}{Q}^{\mathrm{BES}}{Q}^{\mathrm{SVG}}]}^T\\ \mathsf{∆}{u}_1={[\mathsf{∆}{P}^{\mathrm{GT}}\mathsf{∆}{P}^{\mathrm{VSPSH}}\mathsf{∆}{P}^{\mathrm{BES}}]}^T\\ \mathsf{∆}{u}_2={[\mathsf{∆}{Q}^{\mathrm{WT}}\mathsf{∆}{Q}^{\mathrm{PV}}\mathsf{∆}{Q}^{\mathrm{GT}}\mathsf{∆}{Q}^{\mathrm{VSPSH}}\mathsf{∆}{Q}^{\mathrm{BES}}\mathsf{∆}{Q}^{\mathrm{SVG}}]}^T\end{array}$

where $i=1,2\cdots M$, The initial value of the controlled resource at time k is obtained from the feedback actual measurement. $P(k+i|k)$ is the active output of GT, VSPSH, and BES at k + i time predicted at k time; $Q(k+i|k)$ is the reactive output of WT, PV, GT, VSPSH, BES, and SVG at k + i time predicted at k time; $P_0(k),Q_0(k)$ are the initial values of the controllable resource at the time k, which are obtained from the actual measurement of the feedback; $\mathsf{∆}{u}_1,\mathsf{∆}{u}_2$ are the control variables, that is, the increment of active and reactive power output of the controllable resources.

• 2.. Objective function

To ensure consistency between the day-ahead and intra-day objective functions, the intra-day rolling optimization objective function still includes minimizing the distribution network operation cost and node voltage deviation. Additionally, to maintain the daily operating capacity balance of VSPS and BES as much as possible, the objective function of intra-day incorporates a capacity imbalance penalty. The closer the end of the day is, the greater the penalty for unbalance is, so that the VSPSH and BES capacity can recover to the initial value as much as possible at the end of the day. Therefore, the intra-day rolling optimization objective function is:

(25)$\begin{array}{l}\min F={\rho }_T\mathsf{∆}{n}_T+{\rho }_c\mathsf{∆}{n}_{CBs}+C_{GT}+C_{VSPSH}+C_{BES}+\Gamma P_{loss}+{\lambda }_v\mathsf{∆}V+\gamma \\ \gamma ={\lambda }_{VSPSH}(t)(V_{t,i}^{\mathrm{VSPSH}}-V_{t,i,pre}^{\mathrm{VSPSH}})+{\lambda }_{BES}(t)(E_{SOC,t,i}-E_{SOC,t,i,pre})\end{array}$

where ${\lambda }_{VSPSH}(t),{\lambda }_{BES}(t)$ are the capacity unbalance penalty coefficient of VSPSH and BES, which is related to time t; $V_{t,i,pre}^{\mathrm{VSPSH}},E_{SOC,t,i,pre}$ are the capacity reference value of VSPSH and BES, obtained from the day-ahead optimization results.

• 3.. Constraint condition

The constraints of the intra-day rolling optimization stage are the same as those of the day-ahead optimization stage, such as Equations (1)–(8) and (10)–(22).

• 4.. Feedback correction

Superior to the traditional optimization algorithm and intelligent optimization algorithm, the model predictive control has feedback correction which can update the real-time measurement state of the system at each rolling time, correct the forecast error, and exhibit high anti-interference performance and robustness [21]. The input value of the predictive model in this paper is the ultra-short-term forecast data of the wind, PV, and load. However, due to certain errors between the ultra-short-term forecast value and the actual value or external interference, the reference value of the active power and reactive power output of controllable resources issued in advance is not completely equal to the actual output, and the actual state of the system is not consistent with the predictive state.

In order to eliminate the influence of deviation as possible, this paper uses the actual active power and reactive power output of controllable resources, sampled after previous optimization, as the initial values for the new rolling optimization. Thus, it forms a closed-loop control system and enhances control reliability. Specifically

(26)$$\begin{gathered} P_0(k+1)=P_{real}(k+1) \\ {Q}_0(k+1)=Q_{real}(k+1) \end{gathered}$$

$P_0(k+1),Q_0(k+1)$ are the initial value of active power and reactive power output at k + 1 time; $P_{real}(k+1),Q_{real}(k+1)$ are the actual active power and reactive power output values of controllable resources sampled at k + 1 time.

5. Simulation and Analysis

5.1. The Data of Simulation

The modified IEEE33 node system is used as an arithmetic example in this paper. The IEEE33 node system is accessed to wind turbines, PVs, gas turbines, VSPSH, BES, CBs, and SVG, with the access locations and capacities as shown in Table 1, and the system structural topology as shown in Figure 3.

OLTC is connected to the connecting point of the system and the large power grid, with an adjustable range from 0.95 pu to 1.05 pu, and the adjustment step size is 0.01 pu. Capacitor banks access node 8 with an adjustable capacity of 400 kVar and the adjustment step size is 100 kVar. The active power to keep the VSPSH turbine idling at the phase modulation stage is 20% of the rated power, and the minimum power factor of the VSPSH converter is 0.75. The operating efficiency of VSPSH and BES is 0.8 and 0.9, respectively. The prediction M of Model predictive control is 4, meaning it predicts the system’s output for the next 1 h.

To verify the effectiveness of the proposed method and model, the optimization results were compared with those that did not consider VSPSH phase modulation in the day-ahead stage, and in the intra-day stage, the results of rolling optimization based on MPC are compared with traditional open-loop rolling optimization results, and comparisons are also made with and without considering the daily operating capacity. For multi-time scale simulation needs, Figure 4 and Figure 5 show the day-ahead forecast values and intra-day rolling forecast data for wind, PV, and load. The intra-day rolling forecast data are obtained by adding normally distributed errors to the day-ahead forecast values [22]. In the day-ahead and intra-day dispatch stages, these forecast data of day-ahead and intra-day are brought into Equation (10) and Equations (12) and (13), respectively, to optimize the system.

The optimal model proposed in this paper includes non-convex nonlinear constraints, which need to be transformed into a second-order cone programming problem by convex relaxation, and then solved using a commercial solver [23]. The simulation is conducted in Matlab R2022b, using the Yalmip optimization tool for modeling and Cplex for solving.

5.2. Analysis of Day-Ahead Optimization Results

The day-ahead optimization uses the model proposed in Section 3.1, and the optimization results are shown in Figure 6, Figure 7, Figure 8 and Figure 9, which present the output of each controllable resource in 24 h. Figure 6 shows the changes in OLTC tap positions and CB switching throughout the day. We observed that during peak hours, both OLTC and CBs are at higher positions to meet the voltage regulation needs of the distribution network. Figure 7 illustrates the active power and reactive power outputs and capacity ratio of the two VSPSHS in a day. These data indicate that during the peak load period, VSPSH is mostly working in generating and generating phase modulation, while during the low load period, VSPSH is mostly working in pumping and pumping phase modulation, making full use of the characteristics of VSPSH for peak shaving and valley filling, as well as frequency and voltage regulation, ensuring the safe and economic operation of the distribution network. Both VSPSH’s storage capacities return to their initial states at the end of the day. Figure 8 shows the active power and reactive power output and capacity ratio of two BES in a day. We observed that BES1 and BES2 undergo four charge–discharge cycles within a day, within the allowed number of cycles, and both satisfy the daily operating capacity balance. Figure 9 displays the reactive power output of WT, PV, SVG, and GT and the active power of GT throughout the day. It is evident that the reactive power of WT and SVG fluctuates greatly, mainly because the WT’s fluctuations are largely due to the reactive power being influenced by its active power output, while SVG2 experiences larger fluctuations because it is located at the end of the feeder line, leading the optimization algorithm to use SVG2 for compensating the branch’s end to avoid long-distance reactive power flow. In contrast, the GT’s output is relatively stable due to the higher number of controllable devices in its branch.

After applying the day-ahead optimization model proposed in this paper, the states and outputs of each controllable resource in the distribution network in a day were obtained. From these results, it can be seen that each controllable resource has a reasonable output. The day-ahead optimization results can serve as the basic dispatching command for the distribution network in a day, ensuring the safe and economic operation of the distribution network. Then, during the intra-day phase, these values will be adjusted on a smaller time scale to account for the variability of renewable energy.

Figure 10 shows the optimization results of VSPSH operation conditions considering or not considering generating/pumping phase modulation conditions in the day-ahead stage. We observed that without considering VSPSH phase modulation conditions [8], both VSPSHs perform pumping during low-load night-time and generating during high-load daytime. When phase modulation condition is considered, due to the large reactive load level and long branch length of VSPSH1 and VSPSH2 branches, reactive power compensation effectively reduces system losses. Therefore, VSPSH1 performs phase modulation from 6 to 13 and 18 to 24 h, while VSPSH2 does so from 4 to 15 h to ensure safe and economical operation of the distribution network. From this, it can be inferred that when considering phase modulation conditions, VSPSH can fully utilize its various states and make the most of its flexibility.

5.3. Analysis of Intra-Day Optimization Results

In the intra-day rolling optimization stage, three methods are used for comparative analysis. Method 1: the proposed MPC method considers intra-day operating capacity balance. Method 2: the traditional open-loop rolling optimization method also considers intra-day operating capacity balance [12]. Method 3: the proposed MPC method without considering intra-day operating capacity balance [16].

Figure 11 and Figure 12 show the intra-day variations of active power and reactive power and capacity ratio for VSPSHs and BES units under Method 1 every 15 min. We observed from the figure, VSPSH1, VSPSH2 and BES1 can well ensure the daily operating capacity balance during the intra-day period, while the capacity of BES2 at the end of the day is slightly lower than the initial value, primarily due to significant active power fluctuations from wind and fewer controllable resources on that branch. Figure 13 shows the intra-day reactive power of wind and PV under Method 1. The reactive power of wind exhibits significant fluctuations, while the reactive power of PV is relatively small. This is mainly because wind active power fluctuates throughout the 24 h period, affecting the limits of wind reactive power and the reactive power demand on the branch. In contrast, PV active power fluctuates minimally during the night, almost reaching zero, leading to relatively stable reactive power ability. Figure 14 presents the intra-day reactive power of two SVG under Method 1. We observed that SVG1 experiences frequent fluctuations, while SVG2 exhibits larger amplitude variations.

From Figure 11, Figure 12, Figure 13 and Figure 14, the intra-day optimization stage uses the model predictive control algorithm proposed in this paper, as in Method 1. The distribution network is optimized and dispatched every 15 min, with adjustments made to the day-ahead results, allowing the distribution network operation to adapt to the fluctuations of renewable energy.

Figure 15 illustrates the capacity tracking performance of VSPSH and BES under three different methods compared to the day-ahead optimization results. From Figure 15a–d, we observed that Method 1 provides the best capacity tracking performance, ensuring daily operating capacity balance as much as possible. Method 2, on the other hand, shows poorer capacity tracking compared to Method 1, with noticeable discrepancies between the end of the day and the initial value. This is because Method 1 employs the MPC algorithm, which considers the system’s future state, allowing devices to act in advance and providing feedback to reduce system error. Method 2 only considers current optimization, resulting in inferior tracking performance. Method 3 does not consider daily operating capacity balance, so it does not track the day-ahead optimization results but aims to optimize system economics and safety as much as possible but cannot guarantee the long-term operation of energy storage. Table 2 gives the specific imbalance value of the VSPSH and the BES capacity ratio at the end of the day under different methods. It can be seen that the method proposed in this paper can more effectively ensure the capacity balance of energy storage.

Figure 16 compares the active power variations of GT under Method 1 and Method 2 throughout the day. It is evident from the figure that, since MPC considers the system state across multi-time sections and can respond to system changes in advance, the GT output fluctuations and ramp rates are smaller under Method 1, effectively reducing the mechanical loss of GT.

In summary, the proposed optimal dispatch method for the distribution network in this paper takes full advantage of the flexibility and voltage regulation capability of VSPSH under generating/pumping phase modulation conditions. Additionally, the use of MPC for intra-day rolling optimization of the distribution network effectively achieves daily operating capacity balance and smoothes the active power output of the units.

6. Conclusions

Aiming at the challenges posed by the increasing penetration rate of renewable energy in the distribution network, its forecast accuracy decreases with an increase in the time scale, and the acceptance of large amounts of renewable energy in the distribution network, this paper proposes a multi-time scale active and reactive power coordinated optimal dispatch method of the distribution network with a Pumped Storage Station based on model predictive control. The following are the main contributions of this paper:

• (1). It takes various operation conditions of VSPSH into account, giving full play to the regulation flexibility of VSPSH and coordinates the active and reactive power resources of the distribution network to realize the safe and economic operation of the distribution network.

• (2). It ensures the daily operating capacity balance of energy storage during the intra-day rolling optimization stage, and a capacity imbalance penalty is added to the objective function.

• (3). Compared to traditional rolling optimization algorithms, the proposed method considers optimization in the finite time domain, can respond to the predictable changes of the system in advance, can smooth the output of the units, and effectively tracks the day-ahead optimization results of the energy storage to meet the constraints of the daily operating capacity balance.

The simulation results show that the voltage regulation ability of VSPSH can be brought into full play considering the phase modulation condition. The multi-time scale dispatch method based on model predictive control effectively addresses issues related to the volatility and prediction accuracy of renewable energy.

However, the input data for the model prediction control in this study are based on the forecast values of wind, PV, and load, but the forecast values have errors. Further research should include adding robust optimization based on this study, which can describe the output of wind, PV, and load, and take into account the forecast error of wind, PV, and load, so that the optimization results are more feasible.

Author Contributions

Conceptualization, methodology, software, writing—original draft preparation, and writing—review and editing, Z.W., P.P. and X.Z.; supervision, project administration and funding acquisition, P.P., H.S. and G.C. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Multi-time scale active and reactive power coordinated dispatch based on MPC framework.

Figure 2. Intra-day rolling optimal dispatch framework.

Figure 3. Modified IEEE33 node system.

Figure 4. Day-ahead forecast values for wind, PV and loads.

Figure 5. Intra-day forecast values for wind, PV and loads.

Figure 6. OLTC tap positions and CB switching at the day-ahead stage.

Figure 7. Active power and reactive power outputs and capacity ratio of the two VSPSHS.

Figure 8. Active power and reactive power output and capacity ratio of two BES.

Figure 9. Reactive power of WT, PV, SVG and GT and the active power of GT.

Figure 10. The optimization results of VSPSH operation conditions considering or not considering generating/pumping phase modulation conditions at the day-ahead stage: (a) VSPSH1 operation condition; (b) VSPSH2 operation condition.

Figure 11. Intra-day variations of active power and reactive power and capacity ratio for VSPSHs every 15 min under Method 1.

Figure 12. Intra-day variations of active power and reactive power and capacity ratio for BESs every 15 min under Method 1.

Figure 13. Intra-day reactive power of wind and PV under Method 1.

Figure 14. Intra-day reactive power of two SVGs under Method 1.

Figure 15. The change of VSPSH and BES capacity ratio under different methods at intra-day stage: (a) VSPSH1; (b) VSPSH2; (c) BES1; (d) BES2.

Figure 16. Active power of GT under Method 1 and Method 2 throughout the day.

Table 1

System device configuration parameters.

Device	Position	Capacity (kW)
WT1	24	500
WT2	29	500
PV1	14	300
PV2	20	300
VSPSH1	17	500
VSPSH2	33	300
BES1	18	150
BES2	25	100
SVG1	22	300
SVG2	27	300
GT1	5	250
GT2	11	200

Table 2

The imbalance value of VSPSH and BES capacity ratio at the end of the day under different methods.

Method	VSPSH1	VSPSH2	BES1	BES2
Method 1	0.001	0.011	0.015	0.071
Method 2	0.063	0.082	0.018	0.001
Method 3	0.254	0.286	0.257	0.212

References

1. Han, J.; Lyu, W.; Song, H.; Qu, Y.; Wu, Z.; Zhang, X.; Li, Y.; Chen, Z. Optimization of Communication Network for Distributed Control of Wind Farm Equipped with Energy Storages. IEEE Trans. Sustain. Energy; 2023; 14, pp. 1933-1949. [DOI: https://dx.doi.org/10.1109/TSTE.2023.3264656]

2. Palma-Behnke, R.; Valencia, F.; Vega-Herrera, J.; Núñez-Mata, O. Synthetic Time Series Generation Model for Analysis of Power System Operation and Expansion with High Renewable Energy Penetration. J. Mod. Power Syst. Clean Energy; 2021; 9, pp. 849-858. [DOI: https://dx.doi.org/10.35833/MPCE.2020.000747]

3. Mohandes, B.; Wahbah, M.; El Moursi, M.S.; El-Fouly, T.H. Renewable Energy Management System: Optimum Design and Hourly Dispatch. IEEE Trans. Sustain. Energy; 2021; 12, pp. 1615-1628. [DOI: https://dx.doi.org/10.1109/TSTE.2021.3058252]

4. Tan, H.; Ren, Z.; Yan, W.; Wang, Q.; Mohamed, M.A. A Wind Power Accommodation Capability Assessment Method for Multi-Energy Microgrids. IEEE Trans. Sustain. Energy; 2021; 12, pp. 2482-2492. [DOI: https://dx.doi.org/10.1109/TSTE.2021.3103910]

5. Guo, Z.; Wei, W.; Chen, L.; Dong, Z.Y.; Mei, S. Impact of Energy Storage on Renewable Energy Utilization: A Geometric Description. IEEE Trans. Sustain. Energy; 2021; 12, pp. 874-885. [DOI: https://dx.doi.org/10.1109/TSTE.2020.3023498]

6. Wang, S.; Liu, J.; Chen, H.; Bo, R.; Chen, Y. Modeling State Transition and Head-Dependent Efficiency Curve for Pumped Storage Hydro in Look-Ahead Dispatch. IEEE Trans. Power Syst.; 2021; 36, pp. 5396-5407. [DOI: https://dx.doi.org/10.1109/TPWRS.2021.3084909]

7. Nobile, E.; Sari, G.; Schwery, A. Variable Speed Hydro Pumped Storage as Flexible Enabler of Intermittent Renewable Penetration. Proceedings of the 2018 IEEE Power & Energy Society General Meeting (PESGM); Portland, OR, USA, 5–10 August 2018; pp. 1-5.

8. Huang, B.; Chen, Y.; Baldick, R. A Configuration Based Pumped Storage Hydro Model in the MISO Day-Ahead Market. IEEE Trans. Power Syst.; 2022; 37, pp. 132-141. [DOI: https://dx.doi.org/10.1109/TPWRS.2021.3097270]

9. Jena, C.; Guerrero, J.M.; Abusorrah, A.; Al-Turki, Y.; Khan, B. Multi-Objective Generation Scheduling of Hydro-Thermal System Incorporating Energy Storage with Demand Side Management Considering Renewable Energy Uncertainties. IEEE Access; 2022; 10, pp. 52343-52357. [DOI: https://dx.doi.org/10.1109/ACCESS.2022.3172500]

10. Xia, P.; Deng, C.; Chen, Y.; Yao, W. MILP Based Robust Short-Term Scheduling for Wind–Thermal–Hydro Power System with Pumped Hydro Energy Storage. IEEE Access; 2019; 7, pp. 30261-30275. [DOI: https://dx.doi.org/10.1109/ACCESS.2019.2895090]

11. Pali, B.S.; Vadhera, S. An Innovative Continuous Power Generation System Comprising of Wind Energy Along with Pumped-Hydro Storage and Open Well. IEEE Trans. Sustain. Energy; 2020; 11, pp. 145-153. [DOI: https://dx.doi.org/10.1109/TSTE.2018.2886705]

12. Ouyang, J.; Li, M.; Zhang, Z.; Tang, T. Multi-Timescale Active and Reactive Power-Coordinated Control of Large-Scale Wind Integrated Power System for Severe Wind Speed Fluctuation. IEEE Access; 2019; 7, pp. 51201-51210. [DOI: https://dx.doi.org/10.1109/ACCESS.2019.2911587]

13. Qiu, H.; Gu, W.; Xu, Y.; Zhao, B. Multi-Time-Scale Rolling Optimal Dispatch for AC/DC Hybrid Microgrids with Day-Ahead Distributionally Robust Scheduling. IEEE Trans. Sustain. Energy; 2019; 10, pp. 1653-1663. [DOI: https://dx.doi.org/10.1109/TSTE.2018.2868548]

14. Huang, W.; Li, Y.; Wu, F.; Sun, B.; Jing, R. A Multi-Time Scale Scheduling Method for Wind-PV-Pumped Storage-Electrochemical Energy Storage Integrated Generation System. Proceedings of the 2021 IEEE Sustainable Power and Energy Conference (iSPEC); Nanjing, China, 23–25 December 2021; pp. 442-447.

15. Kouro, S.; Cortés, P.; Vargas, R.; Ammann, U.; Rodríguez, J. Model Predictive Control—A Simple and Powerful Method to Control Power Converters. IEEE Trans. Ind. Electron.; 2009; 56, pp. 1826-1838. [DOI: https://dx.doi.org/10.1109/TIE.2008.2008349]

16. Yang, Y.; Yang, P.; Zhao, Z.; Lai, L.L. A Multi-Timescale Coordinated Optimization Framework for Economic Dispatch of Micro-Energy Grid Considering Prediction Error. IEEE Trans. Power Syst.; 2024; 39, pp. 3211-3226. [DOI: https://dx.doi.org/10.1109/TPWRS.2023.3274751]

17. Zhao, H.; Yang, J.; Li, T.; Mei, X. Coordinated Optimal Dispatching Method for Distributed Energy Storage and Distribution Network Under Multi-time Scale. Proceedings of the 2023 3rd International Conference on Electrical Engineering and Mechatronics Technology (ICEEMT); Nanjing, China, 21–23 July 2023; pp. 403-406.

18. Tan, J.; He, C.; Chen, B. Distributionally Robust Optimal Scheduling Method of Power System Considering Hydropower-photovoltaic-pumped Storage Complementarity and DC Transmission. Proc. CSEE; 2024; 44, pp. 5947-5960.

19. Farivar, M.; Low, S.H. Branch flow model: Relaxations andconvexification—Part I. IEEE Trans. Power Syst.; 2013; 28, pp. 2554-2564. [DOI: https://dx.doi.org/10.1109/TPWRS.2013.2255317]

20. Yan, G.G.; Wang, M.C.; Mu, G.; Cui, Y.; Zhou, Z.Q.; Dai, W.C. Modeling of grid-connected doubly-fed induction generator for reactive power static regulation capacity study. Trans. China Electro-Tech. Soc.; 2008; 23, pp. 98-104.

21. Dong, L.; Chen, H.; Pu, T.; Chen, N.; Wang, X. Multi-time Scale Dynamic Optimal Dispatch in Active Distribution Network Based on Model Predictive Control. Proc. CSEE; 2016; 36, pp. 4609-4617.

22. Xiao, F.; Ai, Q. Multiple time-scale optimal dispatch of demand response resource for microgrid based on model predictive control. Electr. Power Autom. Equip.; 2018; 38, pp. 184-190.

23. Zhu, R.; Wei, H.; Bai, X. Distributionally Robust Optimization of Multi-Energy Dynamic Optimal Power Flow. Proc. CSEE; 2020; 40, pp. 3489-3498.