Nash Bargaining-Based Coordinated Frequency-Constrained Dispatch for Distribution Networks and Microgrids

Abstract

As the penetration of distributed renewable energy continues to increase in distribution networks, the traditional scheduling model that the inertia and primary frequency support of distribution networks are completely dependent on the transmission grid will place enormous regulatory pressure on the transmission grid and hinder the active regulation capabilities of distribution networks. To address this issue, this paper proposes a coordinated optimization method for distribution networks and microgrid clusters considering frequency constraints. First, the confidence interval of disturbances was determined based on historical forecast deviation data. On this basis, a second-order cone programming model for distribution networks with embedded frequency security constraints was established. Then, microgrids performed economic dispatch considering the reserves requirement to provide inertia and primary frequency support, and a stochastic optimization model with conditional value-at-risk was built to address uncertainties. Finally, a cooperative game between the distribution network and microgrid clusters was established, determining the reserve capacity provided by each microgrid and the corresponding prices through Nash bargaining. The model was further transformed into two sub-problems, which were solved in a distributed manner using the ADMM algorithm. The effectiveness of the proposed method in enhancing the operational security and economic efficiency of the distribution networks is validated through simulation analysis.

1. Introduction

With the transition, the installed capacity of wind and solar power is continuously growing, gradually replacing the generation capacity of traditional coal-fired power plants [1,2]. Due to the limited ability of wind turbines and solar PV to provide inertia and primary frequency response (PFR), the inertia and frequency regulation resources of the power system are becoming increasingly constrained, posing severe challenges to the frequency security of the power system [3,4,5,6].

In response to the insufficiency of inertia and PFR capabilities in the power system, several studies have introduced frequency security constraints into the optimal dispatch models [7,8,9,10,11,12,13,14]. Reference [11] introduced the concept of frequency security margin into the dispatch model, modeled the maximum frequency deviation after a disturbance using a piecewise linear method, and incorporated it into the optimal dispatch model to optimize the base power and active reserves. Reference [12] used an analytical method to model the frequency security constraints for energy storage participation in frequency response, explored the value of inertia and PFR, and incorporated frequency response resources into the integrated market for clearing. Reference [13] proposed a convex hull approximation method to linearize the maximum frequency deviation constraint, obtained the optimal combination of inertia and droop coefficients, and provided inertia support and PFR through reserving backup capacity in energy storage. However, these studies have only focused on the optimal dispatch of transmission system, neglecting the frequency response capabilities of resources in the distribution side [15,16]. Reference [17] introduced frequency security constraints based on transmission and distribution coordination, where both the distribution and transmission networks provided bidirectional frequency support. Nonetheless, the coordination between the distribution networks and microgrids was not included.

With the development of the electricity market, numerous studies have focused on the coordination between distribution networks and microgrids to enhance the overall stability and efficiency of the power system. Reference [18] established a cooperative game model for multi-microgrids on the distribution side, enhancing energy complementarity among microgrids. This approach improved the overall energy efficiency by leveraging the diverse generation resources of different microgrids, thereby achieving optimized power sharing and reducing operational costs. Similarly, reference [19] proposed a coordination model between the distribution network and microgrids, where the distribution network guided the operational strategies of microgrids through electricity price signals. In this model, the distribution network set the price to incentivize microgrid participation in the electricity market and optimize power flow. However, these game-theoretic models mainly focus on power exchange optimization, neglecting frequency security and the microgrids’ potential to provide inertia support and PFR. This oversight limits the applicability of these models, especially in scenarios where maintaining frequency stability is crucial, such as during grid disturbances or a high penetration of renewable energy sources.

The ability of microgrids to provide inertia support and PFR has become a significant research focus in recent years, as these capabilities are vital for maintaining frequency stability in low-inertia power systems. Reference [20] integrated a dynamic frequency security model into the microgrid scheduling framework, incorporating the use of storage units within microgrids to provide inertia and PFR. This approach helped mitigate frequency fluctuations caused by variable renewable energy sources like PV and sudden load changes. By leveraging the storage systems’ fast response, microgrids could actively support the grid’s frequency, thus enhancing overall stability. On the other hand, Reference [21] addressed frequency disturbances in microgrid operations by implementing load shedding strategies during islanded operation. However, this approach primarily focused on ensuring frequency stability within isolated microgrids rather than considering the support to distribution networks during interconnected operations.

In summary, current studies have not fully considered frequency security constraints in the optimal dispatch of distribution networks. Although some studies have explored the frequency response capabilities of microgrids, they have not leveraged microgrids to provide inertia support and PFR to the distribution networks. Additionally, how to incentivize microgrids to actively provide inertia and PFR services remains an issue that requires further investigation. To excavate the capability of microgrids to provide inertia and PFR support, thereby reducing the reserve costs of distribution networks, this paper constructs a coordinated frequency-constrained dispatch model for distribution networks and microgrids, exploring a novel interactive mode between microgrids and distribution networks towards future demand for stronger frequency regulation capability of distribution networks. The main contributions are as follows:

(1) A second-order cone programming model for distribution networks with embedded frequency security constraints was established. Based on an equivalent frequency response model, constraints such as maximum frequency rate of change, maximum frequency deviation, and quasi-steady-state frequency were established. From this, the optimal combination of inertia and droop coefficients that met the safe operation requirements of the distribution network was determined, thereby defining the reserve requirements of the distribution network.

(2) A bi-level coordinated optimization model for the distribution network and microgrids was established, where the distribution network and microgrids negotiated reserve capacity and reserve prices through cooperative bargaining, thereby reducing the reserve costs for inertia and PFR resources of the distribution network and improving the revenue of microgrids. The alternating direction method of multipliers (ADMM) [22] was used to solve the problem in a distributed manner, achieving a collective optimum for the distribution network and microgrids.

2. The Interaction Framework Between Distribution Networks and Microgrids

The interaction framework between distribution networks and microgrids is shown in Figure 1, which is essentially a bi-level optimization dispatch model. Unlike the traditional model that only involves power exchange [23], this paper incorporated frequency security constraints into the distribution network model to address disturbances in the distribution network. On this basis, to explore the inertia support and PFR capabilities of microgrids, the distribution network and microgrids engage in cooperative bargaining to negotiate the active reserve capacity required for providing frequency response services and its pricing. The negotiation process is as follows:

(1)

The distribution network calculates the required inertia, droop coefficients, and corresponding reserve capacity for each period based on the optimization model considering frequency security constraints.

(2)

The distribution network interacts with microgrids for reserve energy exchange through Nash bargaining. In each round of the game, the distribution network adjusts and communicate their reserve demand based on the current supply–demand situation. The microgrids then adjust their reserve supply and electricity trading volumes for each period according to the distribution network’s demand and provide feedback. Through multiple iterations, the distribution network and microgrids gradually optimize their strategies until the reserve demand matches the reserve supply.

(3)

Following the same interaction process as in step 2, the distribution network and microgrids engage in reserve price negotiations until the price offered by the distribution network matches the price required by the microgrids.

In the model constructed in this paper, the microgrid includes distributed PV systems, wind turbines, battery energy storage systems, as well as electric boilers and absorption chillers, allowing for integrated energy utilization. By optimizing its energy interaction scheme with the distribution network, the microgrid meets various energy demands and enhances its operational economics by providing reserve services.

3. Mathematical Model

3.1. Optimization Model for Distribution Networks

3.1.1. Frequency Security Modeling

The transmission network can be modeled as an equivalent generator model [24]. The microgrid provides inertia and PFR to distribution networks using virtual synchronous generator technology, resulting in the dynamic frequency response model of distribution networks, shown in (1) [14].

$$2(H_g+{\displaystyle \sum _{i\in \mathrm{m}}{H}_i}){\displaystyle \frac{d\mathsf{\Delta }f(t)}{dt}}+D_0\mathsf{\Delta }f(t)=\mathsf{\Delta }{P}_g+{\displaystyle \sum _{i\in \mathrm{m}}\mathsf{\Delta }{P}_i}-\mathsf{\Delta }P$$

(1)

where $H_g$ represents the inertia provided by the transmission network, $H_i$ represents the virtual inertia provided by microgrid $i$, and $D_0$ represents the system damping. $\mathsf{\Delta }{P}_g$ and $\mathsf{\Delta }{P}_i$ are the response power of the transmission network and the microgrid $i$, $\mathsf{\Delta }P$ is the disturbance in distribution networks, and $\mathsf{\Delta }f$ is the frequency deviation.

The frequency response model of the distribution network is constructed using an equivalent generator model, and its control block diagram is shown in Figure 2 [11].

In Figure 2, $F_{\mathrm{H}}$ and $T_{\mathrm{R}}$ represent the high-pressure turbine power generation proportion and the generator reheating time, respectively. $H_{\mathrm{total}}$ is the system equivalent inertia, and $D_{\mathrm{total}}$ is the system equivalent droop coefficient, which are calculated as [25]:

$$H_{\mathrm{total}}=H_g{\displaystyle \frac{P_g}{P_d}}+{\displaystyle \sum _{i\in \mathrm{m}}{H}_i}{\displaystyle \frac{P_i}{P_d}}$$

(2)

$$D_{\mathrm{total}}=D_g{\displaystyle \frac{P_g}{P_d}}+{\displaystyle \sum _{i\in \mathrm{m}}{D}_i}{\displaystyle \frac{P_i}{P_d}}$$

(3)

where $P_g$ is the maximum output of the transmission network, and $D_g$ is the droop coefficient it provides. $P_d$ is the maximum load demand of distribution networks. $P_i$ is the maximum output of the microgrid, and $D_i$ is the droop coefficient provided by microgrid i.

Using the equivalent system frequency response model, the time-domain model of the frequency response can be obtained [13]:

$$\mathsf{\Delta }f={\displaystyle \frac{\mathsf{\Delta }P}{D_0+D_{\mathrm{total}}}}[1+\alpha e^{-\xi {\omega }_{n}t}\sin ({\omega }_{r}t+\varphi )]$$

(4)

where

$$\left\{\begin{array}{l}{\omega }_n^2={\displaystyle \frac{D_{\mathrm{total}}+D_0}{2H_{\mathrm{total}}{T}_{\mathrm{R}}}}\\ \xi ={\displaystyle \frac{D_0{T}_{\mathrm{R}}+2H_{\mathrm{total}}+D_{\mathrm{total}}{F}_{\mathrm{H}}{T}_{\mathrm{R}}}{2(D_0+D_{\mathrm{total}})}}\\ {\omega }_r={\omega }_n\sqrt{1-{\xi }^2}\\ \alpha =\sqrt{{\displaystyle \frac{1-2T_{\mathrm{R}}\xi {\omega }_n+{T_{\mathrm{R}}}^2{{\omega }_n}^2}{1-{\xi }^2}}}\\ \varphi =\arctan ({\displaystyle \frac{{\omega }_r{T}_{\mathrm{R}}}{1-\xi {\omega }_n{T}_{\mathrm{R}}}})-\arctan ({\displaystyle \frac{\sqrt{1-{\xi }^2}}{-\xi }})\end{array}\right.$$

(5)

The natural frequency ${\omega }_n$, damping frequency ${\omega }_r$, damping ratio $\xi$, and coefficients $\alpha$ and $\varphi$ are functions of $H_{\mathrm{total}}$ and $D_{\mathrm{total}}$, as defined in (5). The physical meanings of these coefficients can be found in reference [26].

Through the time-domain model, the dynamic frequency indicators of the frequency response can be transformed into the following constraints [11]:

$$2H_{\mathrm{total}}\overline{f_{\mathrm{rate}}}\ge \left|\mathsf{\Delta }P\right|$$

(6)

$$(D_0+D_{\mathrm{total}})\overline{f_{\mathrm{ss}}}\ge \left|\mathsf{\Delta }P\right|$$

(7)

$$\left({\displaystyle \frac{D_{\mathrm{total}}+D_0}{1+\sqrt{1-{\xi }^2}\alpha e^{-\xi {\omega }_n{t}_{\mathrm{nadir}}}}}\right)\overline{\mathsf{\Delta }{f}_{\mathrm{nadir}}}\ge \left|\mathsf{\Delta }P\right|$$

(8)

Formula (6) represents the constraint of the maximum rate of frequency change, (7) represents the quasi-steady frequency constraint, and (8) represents the maximum frequency deviation constraint. $\overline{f_{\mathrm{rate}}}$, $\overline{f_{\mathrm{ss}}}$ and $\overline{\mathsf{\Delta }{f}_{\mathrm{nadir}}}$ represent the maximum allowable rate of change of frequency (RoCoF), quasi-steady-state frequency and frequency deviation of the system, respectively. $t_{\mathrm{nadir}}$ denotes the time to reach the maximum frequency deviation, which can be calculated by:

$$t_{\mathrm{nadir}}={\displaystyle \frac{1}{{\omega }_r}}\arctan ({\displaystyle \frac{{\omega }_r{T}_{\mathrm{R}}}{\xi {\omega }_r{T}_{\mathrm{R}}-1}})$$

(9)

Let $F(H_{\mathrm{total}},D_{\mathrm{total}})={\displaystyle \frac{D_{\mathrm{total}}+D_0}{1+\sqrt{1-{\xi }^2}\alpha e^{-\xi {\omega }_n{t}_{\mathrm{nadir}}}}}$. Through piecewise linearization [27], a set of linear constraints on maximum frequency deviation can be obtained.

$$({\beta }_h^C+{\beta }_h^H{H}_{\mathrm{total}}+{\beta }_h^D{D}_{\mathrm{total}})\times \overline{\mathsf{\Delta }{f}_{\mathrm{nadir}}}\ge \left|\mathsf{\Delta }P\right|,\forall h$$

(10)

where ${\beta }_h^C$, ${\beta }_h^H$, and ${\beta }_h^D$ represent a set of hyperplanes approximating $F(H_{\mathrm{total}},D_{\mathrm{total}})$. For the specific derivation process, please refer to reference [11].

3.1.2. Optimization Model of Distribution Networks

(1)

Objective Function

The cost of distribution networks is divided into two parts:

$$C^{\mathrm{DN}}=\min C^{\mathrm{P},\mathrm{DN}}+C^{\mathrm{R},\mathrm{DN}}$$

(11)

where $C^{\mathrm{P},\mathrm{DN}}$ and $C^{\mathrm{R},\mathrm{DN}}$ are the cost of electricity interaction and reserve service from the transmission network and microgrids, respectively. The electricity interaction cost can be expressed by (12):

$$C^{\mathrm{P},\mathrm{DN}}={\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}{P}_{g,t}}{c}_t^{\mathrm{buy}}\mathsf{\Delta }t+{\displaystyle \sum _{i\in \mathrm{m}}({\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}{P}_{i,t}^{\mathrm{buy}}}{c}_t^{\mathrm{buy}}-{\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}{P}_{i.t}^{\mathrm{sell}}}{c}_t^{\mathrm{sell}})}\mathsf{\Delta }t$$

(12)

$$C^{\mathrm{R},\mathrm{DN}}={\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}(R_{g,t}\times c_t^{\mathrm{R}}+{\displaystyle \sum _{i\in \mathrm{m}}{R}_{i,t}\times c_{i,t}^{\mathrm{R}}})}\mathsf{\Delta }t$$

(13)

where $P_{g,t}$ is the electricity purchased from the transmission network at period $t$, $c_t^{\mathrm{buy}}$ is the electricity purchase price; $P_{i,t}^{\mathrm{buy}}$ and $P_{i.t}^{\mathrm{sell}}$ are the amounts of electricity purchased from and sold to microgrid $i$, respectively, and $c_t^{\mathrm{sell}}$ denotes the electricity sale price. $R_{g,t}$ and $R_{i,t}$ are the reserve capacity provided by the transmission network and microgrid $i$, respectively, while $c_t^{\mathrm{R}}$ and $c_{i,t}^{\mathrm{R}}$ are the corresponding price for providing reserve services. $N_{\mathrm{T}}$ is the scheduling cycle, and $\mathsf{\Delta }t$ is the time step.

The relationship between the provided reserve and inertia and droop coefficients is given by [28]:

$$R_{g,t}\ge 2H_{g,t}{P}_g\overline{f_{\mathrm{rate}}}+D_{g,t}{P}_g\overline{\mathsf{\Delta }{f}_{\mathrm{nadir}}}$$

(14)

$$R_{i,t}\ge 2H_{i,t}{P}_i\overline{f_{\mathrm{rate}}}+D_{i,t}{P}_i\overline{\mathsf{\Delta }{f}_{\mathrm{nadir}}},\forall i$$

(15)

where $H_{g,t}$/$D_{g,t}$ and $H_{i,t}$/$D_{i,t}$ are the inertia and droop coefficient from the transmission network and microgrid $i$ at period $t$, respectively.

(2)

Operational Constraints

In the established distribution network model, PVs and Static Var Compensators (SVCs) are considered. Using second-order cone relaxation of the distflow equations, the steady-state operational constraints are expressed as [29]:

$$p_{j,t}={\displaystyle \sum _{k:j\to k}{P}_{jk,t}}-{\displaystyle \sum _{u:u\to j}(P_{uj,t}-r_{uj}{l}_{uj,t})},\forall j\in N_{\mathrm{bus}}$$

(16)

$$q_{j,t}={\displaystyle \sum _{k:j\to k}{Q}_{jk,t}}-{\displaystyle \sum _{u:u\to j}(Q_{uj,t}-x_{uj}{l}_{uj,t})},\forall j\in N_{\mathrm{bus}}$$

(17)

$$v_{j,t}=v_{u,t}-2(r_{uj}{P}_{uj,t}+x_{uj}{Q}_{uj,t})+(r_{uj}^2+x_{uj}^2)l_{uj,t},\forall (u,j)\in N_{\mathrm{line}}$$

(18)

$$\underset{¯}{p_u}\le p_{u,t}\le \overline{p_u},\forall u\in N_{\mathrm{bus}}$$

(19)

$$\underset{¯}{q_u}\le q_{u,t}\le \overline{q_u},\forall u\in N_{\mathrm{bus}}$$

(20)

$$\underset{¯}{v_u}\le v_{u,t}\le \overline{v_u},\forall u\in N_{\mathrm{bus}}$$

(21)

$$l_{uj}\le \overline{l_{uj}},\forall (u,j)\in N_{\mathrm{line}}$$

(22)

$${‖\begin{array}{l}2P_{uj,t}\\ 2Q_{uj,t}\\ l_{uj}-v_{u,t}\end{array}‖}_2\le l_{uj}+v_{u,t},\forall u\in N_{bus},\forall (u,j)\in N_{line}$$

(23)

$$p_{j,t}=P_{j,g,t}-P_{j,i,t}^{\mathrm{sell}}+P_{j,i,t}^{\mathrm{buy}}+P_{j,t}^{\mathrm{PV}}-P_{j,t}^{\mathrm{load}}$$

(24)

$$q_{j,t}=Q_{j,g,t}+Q_{j,t}^{\mathrm{SVC}}-Q_{j,t}^{\mathrm{load}}$$

(25)

where $p_{j,t}$ and $q_{j,t}$ represent the active and reactive power flowing out of node $j$ at period $t$. $P_{jk,t}$ and $Q_{jk,t}$ represent the active and reactive power flowing from node $j$ to node $k$ at period $t$, respectively, with $j\to k$ indicating the direction of power flow. $r_{uj}$ and $x_{uj}$ represent the impedance and reactance parameters between nodes $u$ and $j$. $v_{j,t}$ denotes the square of the voltage at node $j$ at period $t$, and $l_{uj,t}$ denotes the square of the line current between nodes $u$ and $j$. $P_{j,t}^{\mathrm{load}}$ and $Q_{j,t}^{\mathrm{load}}$ represent the active and reactive load power at node $j$ at period $t$. $P_{j,g,t}$ and $Q_{j,g,t}$ represent the injected active and reactive power from the transmission network connected to node $j$ at period $t$; if node $j$ is not the connection node between the distribution network and the transmission network, they are set to 0. $P_{j,i,t}^{\mathrm{sell}}$ and $P_{j,i,t}^{\mathrm{buy}}$ represent the selling and purchasing power from the microgrid $i$ connected to node $j$ at period $t$. If there is no microgrid connection, their values are set to 0. $P_{j,t}^{\mathrm{PV}}$ represents the PV generation connected to node $j$ at period $t$. $Q_{j,t}^{\mathrm{SVC}}$ represents the reactive power from the SVC connected to node $j$ at period $t$. $N_{\mathrm{bus}}$ represents the set of nodes in the distribution network, and $N_{\mathrm{line}}$ represents the set of lines in the distribution network.

The operational constraints for PVs in the distribution network at period $t$ are:

$$0\le P_{j,t}^{\mathrm{PV}}\le P_{j,t,\max }^{\mathrm{PV}},\forall j\in N_{\mathrm{PV}},\forall t$$

(26)

where $P_{j,t,\max }^{\mathrm{PV}}$ represents the maximum predicted active power output of the PV.

In the frequency security constraints, it is necessary to consider the uncertainty of disturbances. Disturbances in distribution networks come from PVs power fluctuations ${\xi }^{\mathrm{PV}}$ and load power fluctuations ${\xi }_{\mathrm{d}}$. Assuming that the disturbance sources ${\xi }^{\mathrm{PV}}$ and ${\xi }_{\mathrm{d}}$ follow a normal distribution [30], the cumulative distribution function (CDF) of the disturbances is obtained based on historical data [10]:

$$F_{\left|\mathsf{\Delta }P\right|}(x)=\mathrm{Pr}(\left|\mathsf{\Delta }P\right|\le x)$$

(27)

where

$$\left|\mathsf{\Delta }P\right|=\left|{\xi }^{\mathrm{PV}}+{\xi }_{\mathrm{d}}\right|$$

(28)

Let $\mathrm{Pr}$ represent the probability, and set the risk coefficient as $\alpha$. Then, the disturbance that needs to be satisfied within the confidence interval for each period is:

$$F_{\left|\mathsf{\Delta }{P}_t\right|}(K_{t,\alpha })=1-\alpha$$

(29)

where $K_{t,\alpha }$ is the upper quantile of $\left|\mathsf{\Delta }{P}_t\right|$. The dynamic frequency constraints of the distribution network for each period are transformed into:

$$2H_{t,\mathrm{total}}\overline{f_{\mathrm{rate}}}\ge K_{t,\alpha }$$

(30)

$$(D_0+D_{t,\mathrm{total}}\overline{)f_{\mathrm{ss}}}\ge K_{t,\alpha }$$

(31)

$$({\beta }_h^C+{\beta }_h^H{H}_{t,\mathrm{total}}+{\beta }_h^D{D}_{t,\mathrm{total}})\times \overline{\mathsf{\Delta }{f}_{\mathrm{nadir}}}\ge K_{t,\alpha },\forall h$$

(32)

where $H_{t,\mathrm{total}}$ and $D_{t,\mathrm{total}}$ are the system equivalent inertia and droop coefficient. ${\beta }_h^C$, ${\beta }_h^H$ and ${\beta }_h^D$ are the hyperplane coefficients mentioned in Section 3.1.1.

3.2. Microgrid Operation Optimization Model

(1)

Objective Function

The optimization objective of microgrid $i$ is to minimize scheduling costs. To account for uncertainty, a stochastic optimization approach was employed, and the impact of uncertainty risk is quantified using conditional value-at-risk (CVaR) [31,32]. The objective function is shown as follows:

$$C_i=\min C_i^{\mathrm{E}}+{\sigma }_i{C}_i^{\mathrm{CVaR}}-C_i^{\mathrm{R}}$$

(33)

where $C_i^{\mathrm{E}}$ represents the expected dispatch cost for all scenarios, $C_i^{\mathrm{CVaR}}$ represents the risk cost, and ${\sigma }_i$ is the risk coefficient used to balance the relationship between dispatch cost and risk cost. $C_i^{\mathrm{R}}$ represents the reserve benefits. It can be calculated as follows:

$$C_i^{\mathrm{R}}={\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}{\mathrm{c}}_{i,t}^{\mathrm{R}}\times R_{i,t}\mathsf{\Delta }t}$$

(34)

The dispatch cost includes the degradation cost of electrical storage $C_i^{\mathrm{ESS}}$, heat storage systems $C_i^{\mathrm{HS}}$, and the cost of electricity interaction $C_i^{\mathrm{EX}}$, calculated as follows:

$$C_i^{\mathrm{E}}=C_i^{\mathrm{ESS}}+C_i^{\mathrm{EX}}+C_i^{\mathrm{HS}}$$

(35)

Each cost is calculated according to the following:

$$C_i^{\mathrm{ESS}}={\displaystyle \sum _{\omega =1}^S{\rho }_{i,\omega }}{\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}{K}^{\mathrm{ESS}}\left(P_{i,t,\omega }^{\mathrm{ESS},\mathrm{ch}}+P_{i,t,\omega }^{\mathrm{ESS},\mathrm{dis}}\right)\mathsf{\Delta }t}$$

(36)

$$C_i^{\mathrm{EX}}={\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}{c}_t^{\mathrm{sell}}{P}_{i,t}^{\mathrm{sell}}\mathsf{\Delta }t-}{c}_t^{\mathrm{buy}}{P}_{i,t}^{\mathrm{buy}}\mathsf{\Delta }t$$

(37)

$$C_i^{\mathrm{HSS}}={\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}{K}^{\mathrm{HSS}}\left(P_{i,t}^{\mathrm{HSS},\mathrm{ch}}+P_{i,t}^{\mathrm{HSS},\mathrm{dis}}\right)\mathsf{\Delta }t}$$

(38)

In (36), $K^{\mathrm{ESS}}$ is the degradation coefficient of the energy storage system, and $P_{i,t,\omega }^{\mathrm{ESS},\mathrm{ch}}$ and $P_{i,t,\omega }^{\mathrm{ESS},\mathrm{dis}}$ represent the charging and discharging power of the energy storage system, respectively. ${\rho }_{i,\omega }$ is the probability of scenario $\omega$. $K^{HSS}$ represents the degradation coefficient of the heat storage electric boiler, while $P_{i,t}^{\mathrm{HSS},\mathrm{ch}}$ and $P_{i,t}^{\mathrm{HSS},\mathrm{dis}}$ denote the heat storage power and heat release power of the electric boiler at period $t$, respectively.

The scheduling risk cost $C_i^{\mathrm{CVaR}}$ is calculated according to the following:

$$C_i^{\mathrm{CVaR}}=\zeta +{\displaystyle \frac{1}{1-\alpha }}{\displaystyle \sum _{\omega =1}^S{\rho }_{i,\omega }}{\left[C_{i,\omega }-\zeta \right]}^{+}$$

(39)

$$C_{i,\omega }=C_{i,t}^{\mathrm{EX}}+C_{i,t}^{\mathrm{HS}}+{\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}{K}^{\mathrm{ESS}}}\left(P_{i,t,\omega }^{\mathrm{ESS},\mathrm{ch}}+P_{i,t,\omega }^{\mathrm{ESS},\mathrm{dis}}\right)\mathsf{\Delta }t$$

(40)

where $\zeta$ is an auxiliary variable, whose optimal value is the Value at Risk (VaR), $\alpha$ is the confidence level of CVaR, $C_{i,\omega }$ represents the scheduling cost under scenario $\omega$. ${\left[x\right]}^{+}=\max \left\{x,0\right\}$.

(2)

Operational Constraints

(a)

Electric power Balance Constraint

$$\begin{array}{l}{\stackrel{⌢}{P}}_{i,t}^{\mathrm{PV}}+\mathsf{\Delta }{\stackrel{⌢}{P}}_{i,t,\omega }^{\mathrm{PV}}-P_{i,t,\omega }^{\mathrm{PV},\mathrm{loss}}+{\stackrel{⌢}{P}}_{i,t}^{\mathrm{Wind}}+\mathsf{\Delta }{\stackrel{⌢}{P}}_{i,t,\omega }^{\mathrm{Wind}}-P_{i,t,\omega }^{\mathrm{Wind},\mathrm{loss}}+\\ P_{i,t}^{\mathrm{sell}}-P_{i,t}^{\mathrm{buy}}=P_{i,t,\omega }^{\mathrm{ESS},\mathrm{ch}}-P_{i,t,\omega }^{\mathrm{ESS},\mathrm{diss}}+{\stackrel{⌢}{P}}_{i,t}^{\mathrm{e},\mathrm{load}}+\mathsf{\Delta }{\stackrel{⌢}{P}}_{i,t,\omega }^{\mathrm{e},\mathrm{load}}+P_{i,t}^{\mathrm{EB}}\end{array}$$

(41)

where ${\stackrel{⌢}{P}}_{i,t}^{\mathrm{PV}}$, ${\stackrel{⌢}{P}}_{i,t}^{\mathrm{Wind}}$, and ${\stackrel{⌢}{P}}_{i,t}^{\mathrm{e},\mathrm{load}}$ represent the predicted power of PV, wind, and electrical loads at period $t$, respectively; $\mathsf{\Delta }{\stackrel{⌢}{P}}_{i,t,\omega }^{\mathrm{PV}}$, $\mathsf{\Delta }{\stackrel{⌢}{P}}_{i,t,\omega }^{\mathrm{Wind}}$, and $\mathsf{\Delta }{\stackrel{⌢}{P}}_{i,t,\omega }^{\mathrm{e},\mathrm{load}}$ represent the power prediction errors of PV, wind, and loads in scenario $\omega$; $P_{i,t,\omega }^{\mathrm{PV},\mathrm{loss}}$ and $P_{i,t,\omega }^{\mathrm{Wind},\mathrm{loss}}$ represent the curtailed power of PV and wind; $P_{i,t,\omega }^{\mathrm{ESS},\mathrm{ch}}$ and $P_{i,t,\omega }^{\mathrm{ESS},\mathrm{diss}}$ represent the charging and discharging power of battery energy storage in scenario $\omega$; $P_{i,t}^{\mathrm{EB}}$ represents the electrical power of the heat storage electric boiler.

(b)

Thermal Power Balance Constraint

$${\widehat{P}}_{i,t}^{\mathrm{he},\mathrm{load}}+P_{i,t}^{\mathrm{AC},\mathrm{in}}+{\eta }^{\mathrm{HSS}}{P}_{i,t}^{\mathrm{HSS},\mathrm{ch}}-P_{i,t}^{\mathrm{HSS},\mathrm{diss}}/{\eta }^{\mathrm{HSS}}=P_{i,t}^{\mathrm{EB}}\cdot {\eta }^{\mathrm{EB}}$$

(42)

where ${\widehat{P}}_{i,t}^{\mathrm{he},\mathrm{load}}$ represents the predicted thermal load power, $P_{i,t}^{\mathrm{AC},\mathrm{in}}$ represents the input power of the absorption chiller, ${\eta }^{\mathrm{HSS}}$ represents the efficiency of heat storage and release, and $P_{i,t}^{\mathrm{HSS},\mathrm{ch}}$ and $P_{i,t}^{\mathrm{HSS},\mathrm{diss}}$, respectively, represent the heat storage and release power of the heat storage electric boiler.

(c)

Cooling Power Balance Constraint

$${\widehat{P}}_{i,t}^{\mathrm{c},\mathrm{load}}=P_{i,t}^{\mathrm{AC},\mathrm{in}}\cdot {\eta }^{\mathrm{AC}}$$

(43)

where ${\widehat{P}}_{i,t}^{\mathrm{c},\mathrm{load}}$ represents the predicted cooling load power, and ${\eta }^{\mathrm{AC}}$ represents the efficiency of the absorption chiller.

(d)

Equipment Constraint

The following formulas represent the upper and lower bounds constraints of the electrical power of the absorption chiller and the heat storage electric boiler.

$$0\le P_{i,t}^{\mathrm{AC},\mathrm{in}}\le P_{i,\max }^{\mathrm{AC},\mathrm{in}}$$

(44)

$$0\le P_t^{\mathrm{EB}}\le P_{i,\max }^{\mathrm{EB}}$$

(45)

(e)

Battery Energy Storage System Operation Constraints

The State of Charge (SOC) in adjacent periods must satisfy the following relationship:

$$SO{C}_{i,t,\omega }-SO{C}_{i,t-1,\omega }={\displaystyle \frac{\left(P_{i,t,\omega }^{\mathrm{ESS},\mathrm{ch}}\eta -P_{i,t,\omega }^{\mathrm{ESS},\mathrm{diss}}/\eta \right)\mathsf{\Delta }t}{E_i^{\mathrm{ESS},\mathrm{cap}}}}$$

(46)

In (46), $SO{C}_{i,t,\omega }$ represents the State of Charge (SOC) of the battery energy storage system, and $\eta$ represents the charging and discharging efficiency. $E_i^{\mathrm{ESS},\mathrm{cap}}$ represents the maximum capacity of the energy storage system.

The remaining capacity of the battery energy storage system at each period during operation must satisfy the upper and lower bounds constraints as shown in the following:

$$SO{C}_{\min }\le SO{C}_{i,t,\omega }\le SO{C}_{\max }$$

(47)

Additionally, the charging and discharging power of the battery energy storage system must be limited within the following range:

$$0\le P_{i,t,\omega }^{\mathrm{ESS},\mathrm{ch}}\le {\delta }_{i,t,\omega }^{\mathrm{ESS}}{\lambda }^{\mathrm{ESS}}{E}_i^{\mathrm{ESS},\mathrm{cap}}$$

(48)

$$0\le P_{i,t,\omega }^{\mathrm{ESS},\mathrm{dis}}\le (1-{\delta }_{i,t,\omega }^{\mathrm{ESS}}){\lambda }^{\mathrm{ESS}}{E}_i^{\mathrm{ESS},\mathrm{cap}}$$

(49)

where ${\delta }_{i,t,\omega }^{\mathrm{ESS}}$ is a binary variable representing the charging and discharging state of the battery energy storage system, where ${\delta }_{i,t,\omega }^{\mathrm{ESS}}=1$ indicates that the system is in charging state, and ${\delta }_{i,t,\omega }^{\mathrm{ESS}}=0$ indicates discharging state; ${\lambda }^{\mathrm{ESS}}$ is the maximum charging and discharging rate of the battery energy storage system.

Energy storage systems participate in frequency response. To ensure sufficient power and energy during frequency response, reserve constraints should be established [13]:

$$P_{i,t,\omega }^{\mathrm{ESS},\mathrm{ch}}+P_{i,t,\omega }^{\mathrm{ESS},\mathrm{dis}}+R_{i,t}^{\mathrm{ESS}}\le {\lambda }^{\mathrm{ESS}}{E}_i^{\mathrm{ESS},\mathrm{cap}}$$

(50)

$$SO{C}_{\min }+R_{i,t}^{\mathrm{ESS}}\mathsf{\Delta }{t}_{\mathrm{f}r}\le SO{C}_{i,t,\omega }\le SO{C}_{\max }-R_{i,t}^{\mathrm{ESS}}\mathsf{\Delta }{t}_{\mathrm{f}r}$$

(51)

where $R_{i,t}^{\mathrm{ESS}}$ represents the reserve capacity provided by the energy storage at period $t$, which is equal to the reserve capacity that the microgrid provides to the distribution network. $\mathsf{\Delta }{t}_{\mathrm{f}r}$ denotes the primary frequency response time.

(f)

Operation Constraints of the Heat Storage System in the Heat Storage Electric Boiler

The HOC in adjacent time periods must satisfy the following relationship:

$$HO{C}_{i,t}-HO{C}_{i,t-1}={\displaystyle \frac{\left(P_{i,t}^{\mathrm{HSS},\mathrm{ch}}{\eta }^{\mathrm{HSS}}-P_{i,t}^{\mathrm{HSS},\mathrm{diss}}/{\eta }^{\mathrm{HSS}}\right)\mathsf{\Delta }t}{E_i^{\mathrm{HSS},\mathrm{cap}}}}$$

(52)

In (52), $HO{C}_{i,t}$ represents the thermal storage state of the thermal storage system at period $t$. $E_i^{\mathrm{HSS},\mathrm{cap}}$ represents the maximum capacity of the thermal storage system.

$$HO{C}_{\min }\le HO{C}_{i,t}\le HO{C}_{\max }$$

(53)

The remaining capacity of the heat storage system at each time period during operation must satisfy the upper and lower bounds constraints as shown in (53).

$$0\le P_{i,t}^{\mathrm{HSS},\mathrm{ch}}\le {\delta }_{i,t}^{\mathrm{HSS}}{E}_i^{\mathrm{HSS},\mathrm{cap}}$$

(54)

$$0\le P_{i,t}^{\mathrm{HSS},\mathrm{diss}}\le (1-{\delta }_{i,t}^{\mathrm{HSS}})E_{i,t}^{\mathrm{HSS},\mathrm{cap}}$$

(55)

In (54), ${\delta }_{i,t}^{\mathrm{HSS}}$ is a binary variable representing the heat storage and release state of the heat storage system, where ${\delta }_{i,t}^{\mathrm{HSS}}=1$ indicates that the system is in the heat storage state, and ${\delta }_{i,t}^{\mathrm{HSS}}=0$ indicates the heat release state.

(g)

Electricity Interaction Constraints

The constraints on purchasing and selling electricity are as follows:

$$0\le P_{i,t}^{s\mathrm{e}\mathrm{l}\mathrm{l}}\le {\delta }_{i,t}^{\mathrm{EX}}{P}_{\max }^{\mathrm{sell}}$$

(56)

$$0\le P_{i,t}^{\mathrm{buy}}\le \left(1-{\delta }_{i,t}^{\mathrm{EX}}\right)P_{\max }^{\mathrm{buy}}$$

(57)

where ${\delta }_{i,t}^{\mathrm{EX}}$ is a binary variable representing the purchase and sale status of electricity, where ${\delta }_{i,t}^{\mathrm{EX}}=1$ indicates that the microgrid purchases electricity from the distribution network, and ${\delta }_{i,t}^{\mathrm{EX}}=0$ indicates that the microgrid sells electricity to the distribution network; $P_{\max }^{\mathrm{sell}}$ and $P_{\max }^{\mathrm{buy}}$ represent the maximum purchase and sale power, respectively.

3.3. Coordination Optimization Model of Distribution Networks and Microgrids

The distribution network and microgrids belong to different interest groups and engage in a game over reserve capacity and pricing. The following diagram represents the overall mathematical model:

This paper introduces frequency security constraints (30)–(32) into the traditional distribution network optimization model, which were derived from the frequency response model in Section 3.1.1. For the microgrid optimization model, we considered reserve capacity constraints (50) and (51) to realize inertia and PFR support for distribution networks. The coordinated model as described in Figure 3 was solved to obtain the reserve capacity interaction and reserve price interaction between distribution networks and the microgrids.

4. Cooperative Bargaining Model for Distribution Networks and Microgrids

In the cooperative model, each microgrid and the distribution network are rational entities, prioritizing their own interests when cooperating. Therefore, the cooperative model should fully consider the interests of all parties involved. The distribution network aims to reduce its reserve costs by obtaining inertia and PFR support from microgrids. Meanwhile, microgrids aim to gain reserve benefits, thereby reducing their operational costs. Hence, the cooperative model can be described as follows:

$$\left\{\begin{array}{l}\max (C^{\mathrm{DN},0}-C^{\mathrm{DN},\mathrm{co}}){\displaystyle \prod _{i\in \mathrm{m}}(C_i^0-C_i^{\mathrm{co}})}\\ s.t.\\ C^{\mathrm{DN},0}-C^{\mathrm{DN},\mathrm{co}}\ge 0,\\ C_i^0-C_i^{\mathrm{co}}\ge 0,\forall i\in \mathrm{m}\end{array}\right.$$

(58)

where $C^{\mathrm{DN},\mathrm{co}}$ and $C_i^{\mathrm{co}}$ represent the costs for the distribution network and microgrid $i$ after participating in cooperation, and $C^{\mathrm{DN},0}$ and $C_i^0$ represent the costs when not participating in cooperation, which is the negotiation breakdown point. When the microgrid $i$ does not participate in cooperation, the reserve capacity $R_i$ is 0.

By solving the equilibrium solution of problem (58), the distribution network and microgrid can derive optimal bargaining strategies $\left\{R_i,c_i^{\mathrm{R}}\right\}$.

4.1. Reformatting the Problem

The optimization problem (58) inherently forms a non-convex, nonlinear optimization problem considering reserve capacity and reserve price as optimization variables. To facilitate its solution, this paper transforms it into two sub-problems: minimizing the total system costs and benefit allocation. By sequentially optimizing sub-problems, the optimal solution to the original problem (58) was obtained.

(1)

Sub-problem 1: minimizing the total system costs.

The total system cost is the sum of the objective functions of all entities within the system. Let $C_g^{\mathrm{R}}={\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}(R_{g,t}\times c_{g,t})\mathsf{\Delta }t}$, it represents the reserve costs from the transmission network. Therefore, the overall cost is:

$$\min C^{\mathrm{P},\mathrm{DN}}+C_g^{\mathrm{R}}+{\displaystyle \sum _{i\in \mathrm{m}}(C_i^{\mathrm{E}}+{\sigma }_i{C}_i^{\mathrm{CVaR}})}$$

(59)

In cooperation, the distribution network does not dispatch the purchase and sale power of microgrids; thus, reducing the total system cost is transformed into a problem of reducing reserve costs from the transmission network by utilizing reserves from the microgrid.

(2)

Sub-problem 2: benefit allocation.

By solving sub-problem 1, the reserve capacity after interaction between the distribution network and each microgrid can be obtained. Substituting the reserve into (58), the reserve price can be calculated, and the objective function becomes:

$$\max \left\{(C^{\mathrm{DN},0}-C^{\mathrm{P},\mathrm{DN},*}-C_g^{\mathrm{R},*}-{\displaystyle \sum _{i\in m}{\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}{R}_{i,t}^{*}{c}_{i,t}^{\mathrm{R}}}}){\displaystyle \prod _{i\in m}(C_i^0-C_i^{\mathrm{E},*}-{\sigma }_i{C}_i^{\mathrm{CVaR},*}+{\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}{R}_{i,t}^{*}{c}_{i,t}^{\mathrm{R}}})}\right\}$$

(60)

where the superscript $*$ denotes the solution obtained from solving problem 1. Then, by taking the logarithm, the product problem is transformed into a sum problem, as follows:

$$\min \left\{\begin{array}{l}-\ln \left[(C^{\mathrm{DN},0}-C^{\mathrm{P},\mathrm{DN},*}-C_g^{\mathrm{R},*}-{\displaystyle \sum _{i\in m}{\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}{R}_{i,t}^{*}{c}_{i,t}^{\mathrm{R}}}})\right]-\\ {\displaystyle \sum _{i\in \mathrm{m}}\ln \left[(C_i^0-C_i^{\mathrm{E},*}-{\sigma }_i{C}_i^{\mathrm{CVaR},*}+{\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}{R}_{i,t}^{*}{c}_{i,t}^{\mathrm{R}}})\right]}\end{array}\right\}$$

(61)

The derivation process is detailed in reference [33]. Problem 2 is likewise a convex problem, and it was solved using a distributed solving algorithm.

4.2. Distributed Solution Algorithm

Based on the derivation in Section 4.1, the problem was transformed into two sub-problems (59) and (61). The ADMM algorithm was used to solve subproblem 1 and subproblem 2 separately, obtaining the optimal solution to the original problem.

4.2.1. Problem 1: System Cost Minimization

In problem 1, without considering the interaction price of reserve capacity, each microgrid and the distribution network are coupled through the reserve capacity $R_{i,t}$ they provide. Consistency constraints are introduced to achieve decoupling.

$$R_{i,t}^{\mathrm{DN}}=R_{i,t}^{\mathrm{m}}$$

(62)

In (62), $R_{i,t}^{\mathrm{DN}}$ is the reserve capacity that the distribution network expects the microgrid $i$ to provide, and $R_{i,t}^{\mathrm{m}}$ represents the reserve capacity that microgrid $i$ is willing to provide to the distribution network. When (62) holds, problem 1 converges. The ADMM algorithm is used for decomposition and computation, resulting in the optimization model for problem 1.

The optimization model for the distribution network is:

$$\left\{\begin{array}{l}\min L_{\mathrm{pr}1}^{\mathrm{DN}}=C^{\mathrm{P},\mathrm{DN}}+C_g^{\mathrm{R}}+\\ {\displaystyle \sum _{i\in \mathrm{m}}{\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}\left[{\lambda }_{i,t,\mathrm{pr}1}\left(R_{i,t}^{\mathrm{DN}}-R_{i,t}^{\mathrm{m}}\right)+\frac{{\rho }_{\mathrm{pr}1}}{2}{‖R_{i,t}^{\mathrm{DN}}-R_{i,t}^{\mathrm{m}}‖}_2^2\right]}}\\ s.t.(16)–(26),(30)–(32)\end{array}\right.$$

(63)

where $L_{\mathrm{pr}1}^{\mathrm{DN}}$ is the augmented Lagrangian function for the distribution network; ${\rho }_{\mathrm{pr}1}$ is the penalty factor; ${\lambda }_{i,t,\mathrm{pr}1}$ is the Lagrange multiplier for the interactive power between the distribution network and microgrid $i$ at period $t$.

The optimization model of microgrid $i$ regarding subproblem 1 is:

$$\left\{\begin{array}{l}\min L_{i,\mathrm{pr}1}=C_i^{\mathrm{E}}+{\sigma }_i{C}_i^{\mathrm{CVaR}}\\ +{\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}\left[{\lambda }_{i,t,\mathrm{pr}1}\left(R_{i,t}^{\mathrm{DN}}-R_{i,t}^{\mathrm{m}}\right)+\frac{{\rho }_{\mathrm{pr}1}}{2}{‖R_{i,t}^{\mathrm{DN}}-R_{i,t}^{\mathrm{m}}‖}_2^2\right]}\\ s.t.(41)–(57)\end{array}\right.$$

(64)

where $L_{i,\mathrm{pr}1}$ is the augmented Lagrangian function for microgrid $i$.

The update equations for variables during the iterative process are as follows:

$$\left\{\begin{array}{l}{R}_i^{\mathrm{DN}}(w+1)=\underset{R_i^{\mathrm{m}}(w)}{\arg \min }{L}_{\mathrm{pr}1}^{\mathrm{DN}}(R_i^{\mathrm{m}}(w),{\lambda }_{i,\mathrm{pr}1}(w))\\ R_i^{\mathrm{m}}(w+1)=\underset{R_i^{\mathrm{DN}}(w+1)}{\arg \min }{L}_i(R_i^{\mathrm{DN}}(w+1),{\lambda }_{i,\mathrm{pr}1}(w))\\ {\lambda }_{i,\mathrm{pr}1}(w+1)={\lambda }_{i,\mathrm{pr}1}(w)+{\rho }_{\mathrm{pr}1}\left(R_i^{\mathrm{DN}}(w+1)-R_i^{\mathrm{m}}(w+1)\right)\end{array}\right.$$

(65)

where $w$ represents the number of iterative steps. The convergence criterion is given by:

$$\left\{\begin{array}{l}{r}_{\mathrm{pr}1}(w)={‖R_i^{\mathrm{DN}}(w)-R_i^{\mathrm{m}}(w)‖}_2\le {\epsilon }_{\mathrm{pr}1,\mathrm{pri}}\\ s_{\mathrm{pr}1}(w)={‖R_i^{\mathrm{DN}}(w)-R_i^{\mathrm{DN}}(w-1)‖}_2\le {\epsilon }_{\mathrm{pr}1,\mathrm{dual}}\end{array}\right.$$

(66)

where $r_{\mathrm{pr}1}(w)$ and $s_{\mathrm{pr}1}(w)$ represent the primal and dual residuals of subproblem 1 at the $w$-th iteration, respectively. ${\epsilon }_{\mathrm{pr}1,\mathrm{pri}}$ and ${\epsilon }_{\mathrm{pr}1,\mathrm{dual}}$ denote the convergence accuracy of the primal and dual residuals of subproblem 1 at the $w$-th iteration.

4.2.2. Problem 2: Benefit Allocation

In problem 2, which is the benefit allocation problem, the reserve capacities of each microgrid have already converged. The distribution network and each microgrid use the reserve price $c_{i,t}^{\mathrm{R}}$ at each period as the coupling variable. Consistency constraints were introduced to achieve decoupling.

$$c_{i,t}^{\mathrm{D}}=c_{i,t}^{\mathrm{m}}$$

(67)

In (67) $c_{i,t}^{\mathrm{D}}$ and $c_{i,t}^{\mathrm{m}}$ represent the bidding prices of the distribution network and the microgrid, respectively. The optimization model for Problem 2 of the distribution network is as follows:

$$\left\{\begin{array}{l}\min L_{\mathrm{pr}2}^{\mathrm{DN}}=-\ln (C^{\mathrm{DN},0}-C^{\mathrm{P},\mathrm{DN},*}-C_g^{\mathrm{R},*}-{\displaystyle \sum _{i\in \mathrm{m}}{\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}{R}_{i,t}^{*}{c}_{i,t}^{\mathrm{D}}}})\\ +{\displaystyle \sum _{i\in \mathrm{m}}{\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}\left[{\lambda }_{i,t,\mathrm{pr}2}\left(c_{i,t}^{\mathrm{DN}}-c_{i,t}^{\mathrm{m}}\right)+\frac{{\rho }_{\mathrm{pr}2}}{2}{‖c_{i,t}^{\mathrm{DN}}-c_{i,t}^{\mathrm{m}}‖}_2^2\right]}}\\ s.t.\\ C^{\mathrm{DN},0}-C^{\mathrm{P},\mathrm{DN},*}-C_g^{\mathrm{R},*}-{\displaystyle \sum _{i\in \mathrm{m}}{\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}{R}_{i,t}^{*}{c}_{i,t}^{\mathrm{D}}}}\ge 0\end{array}\right.$$

(68)

In (68) $L_{\mathrm{pr}2}^{\mathrm{DN}}$ represents the augmented Lagrangian function for the distribution network in problem 2; ${\lambda }_{i,t,\mathrm{pr}2}$ represents the Lagrange multiplier, and ${\rho }_{\mathrm{pr}2}$ is the penalty factor.

The optimization model for the microgrid is as follows:

$$\left\{\begin{array}{l}\min L_{i,\mathrm{pr}2}=-\ln (C_i^0-C_i^{\mathrm{E},*}-{\sigma }_i{C}_i^{\mathrm{CVaR}}+{\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}{R}_{i,t}^{*}{c}_{i,t}^{\mathrm{m}}})\\ +{\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}\left[{\lambda }_{i,t,\mathrm{pr}2}\left(c_{i,t}^{\mathrm{DN}}-c_{i,t}^{\mathrm{m}}\right)+\frac{{\rho }_{\mathrm{pr}2}}{2}{‖c_{i,t}^{\mathrm{DN}}-c_{i,t}^{\mathrm{m}}‖}_2^2\right]}\\ s.t.\\ C_i^0-C_i^{\mathrm{E},*}-{\sigma }_i{C}_i^{\mathrm{CVaR}}+{\displaystyle \sum _{t=1}^{N_{\mathrm{T}}}{R}_{i,t}^{*}{c}_{i,t}^{\mathrm{m}}}\ge 0\end{array}\right.$$

(69)

Variable updates during the iteration process are similar to those in Problem 1 and are not reiterated here.

5. Case Study

5.1. Parameter Settings

This paper assumes that the transmission network is equivalent to a generator. Its parameters are in

Table 1. Equivalent generator parameters for the transmission network.

Parameter	Value Range
Inertia $H_g(s)$	2–20
Droop Coefficient $D_g(p.u.)$	15–60
Reheat Time Constant $T_{\mathrm{R}}(s)$	8
Aggregated fraction $F_{\mathrm{H}}$	0.25

[13]:

The distribution network adopted the IEEE 33-bus case. Three microgrids were distributed at nodes 24, 30, and 12. SVCs were installed at nodes 6, 17, 30, and 33. PV sources with capacities of 300 kW, 600 kW, and 400 kW were installed at nodes 2, 7, and 11, respectively. The structure is shown in Figure 4.

Microgrid 1 had 1200 kW of PV, 500 kW of wind turbines, a 1000 kW heat storage boiler, and a 500 kW absorption chiller. Its cooling, heating, and electrical loads were 120 kW, 120 kW, and 650 kW. Microgrid 2 was identical to Microgrid 1. Microgrid 3 had a 400 kW wind turbine and loads at 90% of Microgrid 1, with the rest of the same setup. Microgrids 1 and 2 had 500 kWh of energy storage, and Microgrid 3 had 400 kWh, all with a 0.5 charge/discharge rate.

The selling price of electricity from the distribution network to the microgrid is shown in Figure 5. The price remained the same throughout each hour. The distribution network purchased electricity from the microgrids and the transmission network at 80% of the selling price, while reserve was priced at 20% of the selling price [14].

5.2. Model and Algorithm Verification

This section first presents the convergence results of the interaction model and then demonstrates the feasibility of the constructed model through a sequential analysis of internal scheduling results of the microgrid, distribution networks, and the interaction results between them.

(1)

Algorithm Convergence Verification

Figure 6 shows the convergence results of the primal and dual residuals of the reserve capacity. It can be observed that the algorithm converged after 63 iterations, which met the requirement for fast convergence.

Figure 7 shows the convergence results of the primal and dual residuals for Problem 2, denoting the benefit allocation process. It can be seen that Problem 2 also converged quickly, meeting the convergence criteria after 18 iterations.

(2)

Verification of Microgrid Optimization Model

To verify the accuracy of the microgrid model, this part presents the internal scheduling results after interaction, using Microgrid 1 as an example.

Figure 8 shows the power generation of the PV and wind turbines in Microgrid 1, displaying the expected values for each scenario on the next day. The power balance results for the electric, thermal, and cooling loads are presented in Figure 9.

Figure 9 shows the power balance of heating, cooling, and electricity in the microgrid. It can be observed that the heating, cooling, and electricity demands were all met. For the electricity balance, we can see that during periods 1–32 and 66–96, the PV and wind power generation were insufficient to meet the microgrid’s electricity demand, resulting in power purchases from the distribution network. During periods 33–66, the PV generation in the microgrid was relatively high, allowing the microgrid to sell excess electricity back to the distribution network. In the heat balance chart, the microgrid showed a significant heat generation from the electric boiler at period 26, driven by a lower electricity price at this time. This allowed the microgrid to store heat initially and release it during periods 72–80, thereby reducing electricity usage in these later periods and lowering power purchase costs.

In the internal scheduling of the microgrid, it responded to time-of-use electricity pricing, which was primarily reflected in the results of the charging and discharging of energy storage systems.

From Figure 10, it can be seen that the energy storage in the microgrid charged during periods 1–32 (off-peak) and discharged during periods 71–80 (peak), thereby increasing the microgrid’s revenue. During periods 33–70, the PV generation was relatively high; due to the maximum selling limit, the charging and discharging behavior of the energy storage was closely related to the PV generation. Charging occurred during peak PV generation periods (e.g., periods 41 and 42), while discharging took place during periods of lower PV output (e.g., periods 37 and 56), aiming to sell more electricity to the distribution network and maximize revenue.

(3)

Verification of Distribution Networks Optimization Model

The frequency response results during the maximum disturbance scenario are depicted in Figure 11 to validate the effectiveness of frequency safety constraints. A disturbance risk coefficient of 0.05 was used, focusing on the period with the highest potential disturbance (0.2715 of the maximum load). Parameters $\overline{f_{\mathrm{rate}}}$, $\overline{\mathsf{\Delta }{f}_{\mathrm{ss}}}$, and $\overline{\mathsf{\Delta }{f}_{\mathrm{nadir}}}$ were set to 0.5 Hz/s, 0.25 Hz, and 0.5 Hz respectively [34].

From Figure 11, it can be observed that the maximum frequency deviation of the distribution network is 0.48 Hz, which was less than the allowable limit of 0.5 Hz for the system. The quasi-steady-state frequency deviation was 0.2 Hz, which was also below the permissible limit of 0.25 Hz. The maximum RoCoF will be provided in the next chapter, and all values remained within the allowable range for the system.

In Figure 12, the proportions of each component participating in frequency response after optimization are given.

From Figure 12, it can be seen that all three microgrids participated in the frequency response, providing part of the inertia support and PFR to the distribution network. Due to the larger energy storage capacity of Microgrid 1 and Microgrid 2, they had a greater ability to participate in the frequency response (active reserves were both 12% for Microgrid 1 and 2, while Microgrid 3 had only 10%).

Figure 13 shows the disturbance magnitude and the required reserve capacity for each period. During periods 1–26 and 77–96 (with 15 min time steps), the output from the PVs in the distribution network was low, resulting in smaller disturbances and lower reserve requirements. In periods 28–76, the distribution network experienced high load levels, and PVs contributed to significant active power fluctuations. The additional reserves provided by the microgrids were insufficient, requiring the transmission network’s participation in providing additional frequency reserves.

The interaction between the distribution network and the microgrid also involves electric energy exchange, during which the voltage should be maintained within a reasonable range. Figure 14 shows the voltage distribution of all nodes in the distribution network throughout the day. It can be observed that during the interaction between the distribution network and the microgrid cluster, the voltage remained within a normal range, ensuring the stable operation of the distribution network.

5.3. Comparison Results

To validate the effectiveness of frequency safety constraints and the economic viability of cooperative game interactions between the distribution network and the microgrid, this study compares the interaction model proposed in reference [18]. In the model presented in reference [18], the distribution network interacts solely with the microgrid in terms of electric energy. In contrast, this study includes not only electric energy interactions with the microgrid but also interactions regarding reserve capacity.

We first compared the frequency response results of the model proposed in this paper with [18].

From Figure 15, it can be seen that with a disturbance of 0.2715 times the maximum load, the frequency of the distribution network could be maintained within the specified range (with a maximum frequency deviation of 0.48 Hz and a quasi-steady-state frequency deviation of 0.2 Hz) after considering frequency safety constraints and reserve capacity. In contrast, when only electric energy interactions were considered, ensuring frequency safety in the distribution network became challenging (with a maximum frequency deviation of 1.77 Hz and a quasi-steady-state frequency deviation of 0.44 Hz).

The frequency response results of the distribution network for each time period are shown in Figure 16, with disturbances selected from the maximum disturbance within the confidence interval.

Figure 16 presents the maximum frequency deviation, maximum rate of change of frequency, and quasi-steady-state frequency deviation encountered during maximum disturbances for each time period. When frequency safety constraints and reserves were not considered, the distribution network experienced significant frequency violations during periods 27–74, primarily due to fluctuations in PV generation. However, after incorporating frequency safety constraints, the distribution network optimized its inertia and droop coefficients, and reserves were provided by the microgrid and transmission network, allowing the system frequency to remain consistently within the safe range.

In the model proposed in this study, the energy storage in the microgrid was required to provide frequency regulation reserves for the distribution network; therefore, its operational state differed from that in the model presented in reference [18]. Figure 17 provides the SOC variation curve of Microgrid 1 for analysis.

Figure 17 shows the SOC curve of Microgrid 1’s battery storage. As shown in Figure 17, in the case of electricity interaction only, the microgrid bought electricity during off-peak hours (1–32, 89–96) and sold electricity during peak hours (69–80) to obtain profits. This resulted in rapid SOC increases during charging in periods 1–32 and significant SOC drops during discharging in periods 69–80. In other periods (33–68), the storage discharged when microgrid generation was insufficient and charged when generation was ample. When considering the interaction between energy and reserve capacity, the storage system similarly charged during off-peak periods, leading to an increase in SOC, thereby providing sufficient upward frequency regulation (energy discharge capability). However, to ensure adequate downward regulation capability (energy absorption), the SOC should not be too high. Therefore, from periods 31 to 69, the SOC remained lower than it would have been in the model proposed in [18]. Conversely, from periods 81 to 86, the SOC was higher to ensure sufficient downward regulation capability (energy absorption). It is also worth noting that the significant discharging of the storage system during peak periods (65–80) was delayed until after period 74. This was because the storage system still needed to provide frequency regulation reserves during periods 65–74, requiring some power reserve and preventing large discharges.

In the mode proposed in this paper, reserve capacity and reserve pricing were determined through a cooperative game between the distribution network and the microgrid. In this cooperative game, both the distribution network and the microgrid can benefit. We also provide a cost comparison to analyze the economic differences between the model in [18] and the model presented in this paper.

Figure 18 displays the negotiated prices between each microgrid and the distribution network. During periods when additional reserves were not required, the negotiated prices were 0. The prices negotiated between the distribution network and microgrids were lower than the reserve prices from the transmission network, enabling the distribution network to achieve cost savings through microgrid reserves.

Table 2. Comparison of cost for each component.

Entity	Model Without Considering Reserve (RMB)			The Proposed Model (RMB)
	Electricity Trading	Reserve Cost	Total Cost	Electricity Trading	Reserve Cost	Total Cost
Distribution network	26,839	1358	28,197	26,756	1233	27,989
Microgrid 1	−1263	0	−1396	−1235	−181	−1546
Microgrid 2	−816	0	−857	−791	−171	−1002
Microgrid 3	−726	0	−763	−704	−138	−880

presents the cost breakdown for each entity under two models, where negative values signify profits. In the absence of frequency regulation participation from microgrids, the distribution network’s frequency regulation demand was entirely fulfilled by the transmission network. The result indicates that each microgrid had strategically reduced its electricity trading to enhance its reserve income under the proposed model. Through this collaborative approach, the overall revenue of all three microgrids had increased. Specifically, the reserve costs incurred by the distribution network decreased by RMB 125, and the total cost was reduced by RMB 208, which underscores a successful reduction in operational expenses for distribution networks. Consequently, the collaboration between distribution networks and microgrids not only aids in diminishing the total costs for both the distribution system and the microgrid cluster but also enables all parties to derive benefits from this cooperative game.

6. Conclusions

This paper constructed a Nash bargaining-based coordinated frequency-constrained dispatch model for distribution networks and microgrids, where microgrids gained revenue through providing the inertia and PFR service, and the distribution network reduced its reserve requirement and costs from the transmission grid. The experimental results indicate that the proposed method can ensure frequency safety in the distribution network under disturbances. During the maximum disturbance, the maximum frequency deviation was reduced by 72.6%, the quasi-steady-state frequency deviation was reduced by 71.1%, and the maximum frequency change rate was reduced by 80.2%. Additionally, by utilizing reserve from the microgrid, the adjustment pressure on the transmission network was effectively reduced, with reserve capacity decreasing by 35%. Through Nash bargaining, the reserve costs for the distribution network were reduced by 9.2%, leading to a total cost reduction of 0.73%, and the total revenues for the three microgrids increased by 10.74%, 16.92%, and 15.33%, respectively.

In the future, we will further analyze the interactions and frequency support mechanisms among multiple entities, including the transmission network, distribution network, and microgrids, to fully tap into the flexible resources on the distribution side.