A Demand-Supply Cooperative Responding Strategy in Power System with High Renewable Energy Penetration

The specific power system considered in this paper is depicted in Fig. 1. It comprises wind turbines, industrial loads, conventional thermal plants, and the power transmission network.

In light of energy and environmental considerations, there is a need to enhance wind power utilization on the supply side. However, it will potentially introduce overload risks in the transmission network [25, 26]. To address this concern, it is imperative to comprehensively promote the flexibility of industrial power demand and corresponding thermal power dispatch.

As shown in Fig. 3, we incorporate both price-based and incentive-based IDR programs into PDSCR and analyze components of IPE electricity fares. The price-based IDR program relies on the contracted electricity prices. Specifically, IPEs can reduce production cost by shifting their loads to low-price periods and avoiding power usage during high-price periods, as illustrated in Fig. 3(a). Consequently, the price variations among different time periods encourage industrial demand shifting, that is, absorbing excess wind power or preventing power shortages. This program serves as a foundational approach for basically determining intraday electricity demand. However, unforeseen fluctuations in wind power supply would lead to power imbalances. In practice, unilateral price adjustments within the contract are not accepted by IPEs on the intraday basis, because such adjustments may disrupt production and incur additional cost.

On the other hand, driven by additional financial incentives, industrial customers further adjust their demand to address intraday power imbalances, known as incentive-based IDR. Specifically, financial compensations or penalties drive industrial power demand to increase during peak periods or decrease during valley periods of wind power supply, as depicted in Fig. 3(b). For electricity transactions, the predictable supply and demand on the market need to be cleared ahead of the day. For uncertain events, the real-time response outside of the contract serves as a supplement. Based on this frame, the IPE production costs can be calculated as the difference between the total fare and the compensation fare, as shown in Fig. 3(c).

In Fig. 2, the PDSCR strategy is divided into two steps, corresponding to the day-ahead and intraday time scales, respectively. In Step 1, the utility center holds decision-making authority over electricity pricing and the scheme for thermal unit operations in day-ahead PDSCR. This can be viewed as a centralized pre-dispatch strategy. In this process, the utility center firstly formulates the price-based IDR program and the day-ahead thermal unit commitment. Subsequently, the IPE gives a production plan to determine industrial power demand. The relationship among these entities is described based on a Stackelberg game framework, formulated as a bi-level multi-objective optimization. Moreover, the day-ahead scheme encompasses thermal generators’ ON/OFF states, reserves, electricity prices, etc.

However, issues like branch overloads or power shortages may arise if actual wind power significantly deviates from its predictions. To address these challenges, Step 2 focuses on enhancing the response to accommodate power deviations, i.e., the actual thermal power output and IDR program would be updated. In terms of the utility center’s scheme, electricity pricing and thermal unit operation schemes remain unchanged. On this basis, the intraday PDSCR strategy emphasizes autonomous cooperation and offers corresponding financial incentives, including the thermal power compensation and incentive-based IDR. Specifically, the intraday PDSCR strategy determines thermal power, wind power integration, industrial production behavior, and reserve usage.

As flexible industrial loads, the parallel manufacturing process (PMP) system is introduced in this subsection. Within the PMP, IPE can achieve the optimal price cost by optimizing the PMP schedule. In this paper, the optimization model of the PMP system is established, and its process with ${\rm R}$ procedures is shown in Fig. 4.

The $i^{th}$ production procedure is denoted as ${\rm P}_{i}$ ($i=$ $1,2,...,{\rm R}$), in which there exist ${\rm C}$ machines, ${\rm M}_{i,1},{\rm M}_{i,2},…,{\rm M}_{i,{\rm C}}$, and there are buffers (${\rm B}_{1},{\rm B}_{2},…,{\rm B}_{{\rm R}-1}$) between two adjacent procedures, which are used to store projects and provide flexibility for the PMP system to conduct adjusting operations. The process constraint of the PMP system is explained in (A.1)-(A.7) of Appendix A. Accordingly, in the production cycle consisting of ${\rm T}$ time intervals, the dispatch model of IPE is formulated as follows.

where ${N_{{\rm{P}},t,i}}$ is the quantity of processed projects of procedure ${{\rm{P}}_{i}}$ and time slot ${t}$. Similarly, the storage quantity of buffer ${{\rm{B}}_{i}}$ is denoted as ${N_{{\rm{B}},t,i}}$. ${c_{P,i}}$ and ${c_{{\rm{B}},i}}$ are cost coefficients of ${P_{i}}$ and ${B_{i}}$, ${\alpha_{t}}$ is the electricity price in $t$ slot, and ${{\rm{C}}_{{\rm{FIXED}}}}$ is the fixed cost. The optimal electricity cost is consider as the interest on the demand side.

In day-ahead PDSCR, the utility center coordinates thermal power supply and industrial demand to respond to the peak-valley trend of wind power supply. On this basis, it is also necessary to guarantee cost optimality on the demand-supply side. However, this cooperative response is not a simple multi-objective optimization problem, and is with a bi-level optimization existing in the order of priority among multiple individuals.

Specifically, on the supply side, the utility center devotes to minimizing outputs of operational thermal power units and enhancing wind power utilization. This amounts to optimizing the overall cost associated with thermal power generations. On the demand side, the utility center employs price-based IDR to impact industrial load shifting. As shown in Fig. 5, the utility center establishes the long-term electricity pricing scheme based on the potential for IDR adoption. Then, industrial customers determine the optimal production schedule considering electricity prices and their specific industrial requirements. This process represents a sequential decision-making approach. The Stackelberg theory introduces the concept of priority among game participants. This priority dictates the relative importance of each participant’s objectives. Therefore, this game model is proved effective in characterizing the hierarchical game relationships [27]. On this basis, we assume a two-individual game distinguished between leader 1 and follower 2, and their utility function and behavior are ($F_{1},b_{1}$) and ($F_{2},b_{2}$). That is, equilibrium $b_{1}^{*}$ in Starkelberg game is as (2).

where $B_{1}$ represents the leader behavior set. $O$ stands for the following response function for given $b_{1}^{*}$. If $O$ is the optimal response function ${b_{2}^{*}}\in{O_{2}}(b_{1}^{*})$, the two behavior $({b_{1}^{*}},{b_{2}^{*}})$ will be Starkelberg equilibrium. Just as in the mathematical programming with equilibrium constraints (MPEC), the optimal response function can be transformed as constraints, according to the convex nature of follower optimization. This transformation is achieved by deducing it through the Karush-Kuhn-Tucker (KKT) condition [28, 29]. Therefore, relevant constraints in the bi-level optimization model are established as (3). The specific constraints is formulated as (A.8)-(A.14) in Appendix A.

where ${{\cal L}_{{\rm{PMP}}}}$ is the Lagrange function, ${f_{neq}}(x,y)$ is inequality constraint vector, the corresponding dual variable vector $\lambda$ is non-negative, and $\circ$ is the Hadamard product [30]. Then, the following is the proposed bi-level and multi-objectives optimization model in day-ahead PDSCR, where the decision variable of utility center is $x=[{s_{t,p}},{P_{{\rm{T}},t,p}},{R_{{\rm{T}},t,p}},{P_{{\rm{W}},t,l}},{{\alpha_{t}}}]$ and the IPE variable is $y=[{N_{{\rm{P}},t,i}},{N_{{\rm{B}},t,i}}]$. $x$ includes thermal On/Off status ${s_{t,p}}$, power outputs ${P_{{\rm{T}},t,p}}$, reserves ${R_{{\rm{T}},t,p}}$, wind power integration ${P_{{\rm{W}},t,l}}$, electricity price ${{\alpha_{t}}}$ for time $t$, thermal generator $p$, wind unit $l$, and number of thermal power units $N_{\rm G}$. $y$ includes numbers of processed projects ${N_{{\rm P},t,i}}$ and buffer ${N_{{\rm{B}},t,i}}$ for time $t$ and process $i$. In addition, a security coefficient (ASC) is introduced to quantify the risk based on the available transfer margin of the branch. This margin of a branch is described by the difference between the transmitted power and its rated value during the dispatch periods. In (4), ${C_{{\rm{ASC}}}}$ is defined as the summation of minimum margins of all the branches and periods. It is noteworthy that the margin should be maximized.

where $T{P_{t,j}}$ and $T{P_{j,\max}}$ denote transmitted power and the maximum margin of branch $j$ $(j=1,2,…,b)$, respectively. Generally, the more wind power utilization means the less thermal cost. On this basis, we formulate the thermal cost and the $C_{\rm{ASC}}$ as the upper optimization objectives, because the utility center is in dominance. Here, the bi-objectives optimization would obtain the optional optimal solutions, i.e., Pareto solutions. The reason is that different preferences for the bi-objectives will cause different optimal solutions. Pareto optimality means there is no feasible solutions $x_{1}$ and $x_{2}$ where objectives $J_{i}(x_{1})\geq J_{i}(x_{2}),\forall i\in\{1,2\}$ with $J_{i}(x_{1})>J_{i}(x_{2}),$ for some $i$, i.e., there is no complete advantage among all the Pareto solutions. Therefore, the decision-making involves the trade-off for Pareto solutions.

On the other hand, the industrial electricity cost is concerned by IPE. In this paper, this electricity cost is formulated as the lower optimization objectives, the performance of which depends on the upper decision. The specific bi-level optimization model is established as (5), the supplemental constraints of the thermal power and grid are shown in (B.2)-(B.14) of Appendix B. This is an MPEC model, where $\min\;[{J_{1}},{J_{2}}]$ represents the optimal leader’s utility, and $\arg\min\;{J_{{\rm{PMP}}}}$ represents the optimal follower’s utility. The optimal response function is defined as the constraint (3), specific forms of which are provided in equations (A.8)-(A.14).

where $F({P_{{\rm T},t,i}})$ and $S_{{\rm T},t,p}$ represent thermal generation cost function and starting cost. In this multi-objective problem, the Pareto solutions can be obtained by $\varepsilon$-constraint optimization method [31].

The intraday PDSCR strategy is based on the combination of thermal power dispatch and incentive-based IDR, and is autonomous and profit-driven. Considering the thermal power commendation and the incentive strategy within the IDR scheme, the updated electricity cost function of the PMP system could be reconstructed as (6), which introduces the incentive ${S_{t}^{\rm{P,R}}}$. The updated cost function of the thermal plants is formulated as (7), where the electricity sales revenue $J_{3}$ and thermal power commendation ${S_{{\rm T},t,k}^{\rm R}}$ are introduced.

where $N_{\rm T}$ represents the number of thermal plants, and ${C_{\rm{outres}}^{t}}$ stands for the purchasing cost of reserve electricity. UP denotes the updated decision, where ${s_{t,p}}$, ${R_{{\rm{T}},t,p}}$ and ${{\alpha_{t}}}$ follow the day-ahead decision. The sum of ${S_{{\rm T},t,k}^{\rm R}}$ and $S_{t}^{\rm{P,R}}$ should be proportional to the increase of wind power utilization $\Delta{P_{\rm{W},t}}$.

where ${s_{t}}$ is the commendation constant. It means that multi-individual cooperation on the demand-supply side could produce more financial commendation. Evidently, for each independent period $t$, the total profit ${\psi_{t}}(x,y)$ is constructed by the thermal cutbacks, the commendation of wind power utilization, and the reserve power cost.

where $[\cdot]_{t}$ represents the $t$ item. In addition, wind power utilization and $C_{\rm{ASC}}^{{\rm{UP}}}$ are also the main indicators on power dispatch. Then, the maximization of cooperative profit ${\psi_{t}}(x,y)$ is established as the optimal profit constraint to characterize profit-driven response. Thus, the model of intraday PDSCR is formulated as the multi-objectives optimization in (10). The supplemental constraints of the thermal power and grid are shown in (B.16)-(B.19) of Appendix B.

Regarding wind power fluctuations, the intraday PDSCR is devoted to balancing power supply and demand, while also alleviating the issue of branch overload. Correspondingly, the cooperative mechanism allows the demand-supply side to receive financial incentives. However, the implementation of PDSCR introduces cost variations in thermal power generation and industrial production. If the financial compensation fails to benefit individuals, or if the profit distributions are unreasonable, it would lead to the conflict of interests. In such cases, achieving effective autonomous cooperation becomes challenging. In the case study of Section V, we will test the performance of non-cooperation scenarios, in order to quantify the extent of negative impacts caused by conflicts.

Note that the Nash bargaining theory is an effective framework for cooperation and conflicts of interest [32]. As shown in Fig. 6, it assumes there exists a bargaining process where multiple individuals discuss profit distributions. Each individual subsequently implements their response based on the outcome of these negotiations. However, it is worth noting that this theory primarily focuses on bargaining balanced solutions within the context of two-party linear utilities. In this subsection, we extend an optimal bargaining model to address the issue of multi-individual nonlinear utilities, that is, the Multi-Individual Optimal Marginal Model (MIOMM). Initially, we demonstrate the convexity and feasibility of MIOMM. Subsequently, we propose and derive Multi-Individual Profit Distribution Marginal Solutions (MIPDMSs) based on different distribution criteria, which in turn serve as multi-individual response constraints.

Note that the problem has the following characteristics: (i) The potential conflicts of interests exist among individuals, i.e., the profit distribution of IPE and thermal plants. (ii) As shown in (8), the distribution of total profit is directly proportional to the increase of wind power utilization. Then, we propose the equal and contribution-based profit distributions. In this case, the more distribution means the more profit for each individual. For demand and supply sides, this incentive is the profit motivation for improving wind power utilization. (iii) The behavior is not mandatory, it means each individual can refuse to respond. To this end, the profit utility $u_{k}$ could be established in the scenario of multi-individual negotiation [33], which is presented as follows.

where $S$ and $D$ denote the cooperative and divergent sets. ${s_{k}}(b)$ is the behavioral profit function for individual $k$, $b$ is the cooperative behavior and ${d_{n}}$ represents the disagreeable behavior. Here, the benefits pair $(S,D)$ provides individuals with the options of cooperation and non-cooperation. In the cooperative process, Nash has provided a linear equilibrium model called Nash bargaining solution (NBS) [34]. In PDSCR, we expend this equilibrium to multi-interest and nonlinear profit utility, and this multi-individual model is formulated as (12).

where $U_{k}$ and $V$ represent individual costs of thermal plant $k$ and IPE, respectively. Their original costs are $U_{k}^{0}$, $V_{0}$. Then, profit utility functions of are defined as ${u_{k}}={{U_{k}}-U_{k}^{0}}$, $v=V-{V_{0}}$, $({u_{k}},v)\in S$. In (12), the profit non-inferiority ensures the motivation of individual participation. In intraday PDSCR, the specific profit utility function are redefined as (13), where $U_{k}$, $V$, $U_{k}^{0}$ and $V_{0}$ are established in independent period ${U_{t,k}}$, $U_{t,k}^{0}$, ${V_{t}}$ and $V_{t}^{0}$. There is a quadratic relationship between the multi-individual profit function and the related decision variable.

where $S_{k}^{V}$ denotes the income coefficient, which indicates the increased electricity sales revenue in thermal plant $k$. The total profit ${\psi_{t}}(x,y)$ can be further reformulated as follows.

Note that the set $({u_{k,t}},v_{t})$ of profit utility functions is multidimensional and quadratic. In the PDSCR strategy, the convexity of $({u_{k,t}},v_{t})$ is the necessary and sufficient condition of MIOMM optimality. Therefore, we derive the convexity of the profit utility, and the lemma and proof are shown as follows.

Lemma 1: For $(u,v)$, $u\in{S_{1}}:\{u={x^{\top}}Ax+{c_{1}}^{\top}x|a_{1}^{\top}x\leq{b_{1}},a_{2}^{\top}x={b_{2}},u\in{\mathbb{R}^{n}}\},v\in{S_{2}}:\{v={c_{2}}^{\top}y|a_{3}^{\top}y\leq{b_{3}},a_{3}^{\top}y={b_{3}},y\in\mathbb{R}{{}^{m}}\}$, set $S:\{(u,v)|{u_{i}},{v_{j}}>0,u\in\mathbb{R}{{}^{n}},v\in\mathbb{R}{{}^{m}}\}$ is convexity, where $A=diag\{{a_{1}},{a_{2}},...,{a_{n}}\}$.

Proof: The detailed proof could be referred to Appendix C.

$\hfill\blacksquare$

In MIOMM, we further study the optimality of cooperative total profit ${\psi_{t}}(x,y)$, which is to prove the equivalent optimality in both models. On this basis, we could replace $\max({\psi_{t}}(x,y))$ with MIOMM.

Proposition: The maximization of multiplication $\prod\limits_{{\chi_{m}}\in\chi}{{\chi_{m}}}$ achieves the maximization of accumulation $\sum\limits_{{\chi_{m}}\in\chi}{{\chi_{m}}}$ for ${\chi_{m}}$.

Proof: The multiplication and accumulation are defined as ${f_{\Pi}}$ and ${f_{\Sigma}}$ ,

In the linear function ${f_{\Sigma}}(\chi)$, $d$ denotes the direction vector and $\forall{d_{m}}\in d$, ${d_{m}}>0$. Obviously, ${f_{\Sigma}}(\chi+d)>{f_{\Sigma}}(\chi)$, i.e., $d$ is able to represent the optimal direction of ${f_{\Sigma}}$. For multiply ${f_{\Pi}}(\chi+d)$, the first-order Taylor unfolding is as (16) when ${\chi_{m}}\geq 0$.

where $o(d)>0$. When ${d_{m}}>0$, the partial derivative $\nabla{f_{\Pi}}{(\chi)^{\top}}d$ can be expressed as (17).

Therefore, ${f_{\Pi}}(\chi+d)>{f_{\Pi}}(\chi)$, i.e., the maximization of accumulation ${f_{\Sigma}}$ is ensured by maximizing the multiplication ${f_{\Pi}}$.

$\hfill\blacksquare$

On the basis of MIOMM, the optimal marginal solution is derived in the convex feasible region called MIPDMS. We assume that multi-individuals would require to divide the cooperative profits, according to the equal or contribution-based criteria. It could respectively represent the fairness and rationality in cooperation, which are the responding motivations for multi-individuals. In the equal MIPDMS, this criteria provide the same profit to participants. In contribution-based MIPDMS, profit utility is formulated according to their actual contributions in cooperation. Accordingly, both the above solutions are feasible, which are demonstrated as follows.

Firstly, the equal MIPDMS in the fair distribution condition is developed within the above multi-individual model, i.e., maximizing the multiplication of multi-individual utility.

Lemma 2: For thermal plants and PMP, the profit is distributed by each individual ${u_{m,t}}$ or ${v_{t}}$ in equal MIPDMS as (18).

Proof: For the NBS objective as (12), the function could be shown as follows.

where the composite function (19) is convex, the minimum will correspond to the primeval optimal solution. The partial derivative to $u_{m,t}$ is as (20).

In the optimal point, ${\nabla_{{u_{m,t}}}}=0$, i.e. $\psi(x,y)=\sum\limits_{k=1}^{N_{\rm{T}}}{{u_{k,t}}}+{u_{m,t}}$. If the profit is distributed equally, we have the following formulations.

$\hfill\blacksquare$

On the other hand, we consider the rational distribution condition to derive the contribution-based MIPDMS. In this solution, the more responding contribution of individuals means the more proportion of total profit.

Lemma 3: For thermal plants and PMP, the profit is divided by each individual in contribution-based MIPDMS as (22-23).

where ${\sigma_{k,t}}=\sum\limits_{p\in k}^{{N_{T}}}{({P_{T,p,t}}-P_{T,p,t}^{UP})}$ represents the ranger of adjusted power of the $k^{th}$ thermal plant.

Proof: Based on (20), we obtain the following formulation.

When the profit distribution is based on the adjusted weights, the total profit function is reformulated as (25).

where ${{{{\sigma_{k,t}}}}/{{{\sigma_{m,t}}}}}$ is defined as corresponding weights between ${{\sigma_{m,t}}}$ and ${{\sigma_{k,t}}}$.

$\hfill\blacksquare$

On the basis of the contribution-based MIPDMS, we explore the specific individual profit boundary as follows.

Lemma 4: For IPE profit $v_{t}$ as (23), upper boundary: $v_{t}\leq{\rm max}[u_{k,t}]$, $i=1,2,...,n$, distribution proportion boundary: $v_{t}\in[\psi(x,y)/(n+1),\psi(x,y)/2]$.

Proof: Firstly, $u_{m,t}={\rm max}[u_{k,t}]$, $\sigma_{m,t}={\rm max}[\sigma_{k,t}]$, and the relationship of $u_{k,t}$ and $v_{t}$ can be represented as follows.

Therefore, ${\rm max}[u_{k,t}]$ is the upper boundary of IPE profit $v_{t}$. For the IPE profit $v_{t}$, ${v_{t}}=\{1-\sum\limits_{j=1}^{N_{\rm{T}}}{[{\sigma_{j,t}}/(\sum\limits_{k=1}^{N_{\rm{T}}}{{\sigma_{k,t}}}+{\sigma_{j,t}})]}\}\cdot\psi(x,y)$, its 1st and 2nd partial derivatives are shown as follows.

There are two types of extreme points in feasible domain $\sigma_{m,t}\in[0,\sum\limits_{k=1}^{N_{\rm{T}}}{{\sigma_{k,t}}}]$. (i) For ${\sigma_{m,t}}\in\{{\sigma_{k,t}}|{\sigma_{1,t}}={\sigma_{2,t}}=...={\sigma_{{N_{\rm{T}}},t}},k\in 1,2,...,{N_{\rm{T}}}\}$, $\nabla_{{\sigma_{m,t}}}^{2}{v_{m,t}}=\frac{{2(n-1)\cdot{\sigma_{m,t}}}}{{{{[({N_{\rm{T}}}+1)\cdot{\sigma_{m,t}}]}^{3}}}}>0$, and $\sigma_{m,t}$ corresponds the local lower boundary. For $\forall{\sigma_{m,t}}\in[0,\sum\limits_{k=1}^{N_{\rm{T}}}{{\sigma_{k,t}}}/{N_{\rm{T}}}]$ and $\forall{\sigma_{m,t}}\in(\sum\limits_{k=1}^{N_{\rm{T}}}{{\sigma_{k,t}}}/{N_{\rm{T}}},\sum\limits_{k=1}^{N_{\rm{T}}}{{\sigma_{k,t}}}]$, which constitutes the complete feasible domain, ${\nabla_{{\sigma_{m,t}}}}{v_{m,t}}\leq 0$ and ${\nabla_{{\sigma_{m,t}}}}{v_{m,t}}\geq 0$ separately, and $\sigma_{m,t}$ corresponds the global lower boundary, where $v_{t}=\frac{\psi(x,y)}{({N_{\rm{T}}}+1)}$. (ii) For ${\sigma_{m,t}}\in\{{\sigma_{k,t}}|{\sigma_{m,t}}=\sum\limits_{k=1}^{N_{\rm{T}}}{{\sigma_{k,t}}},{\sigma_{k,t}}=0,i\neq m,k=1,2,...,{N_{\rm{T}}}\}$, $\nabla_{{\sigma_{m,t}}}^{2}{v_{m,t}}=0$, $\sigma_{m,t}$ corresponds the global upper boundary, where $v_{t}=\frac{\psi(x,y)}{2}$.

$\hfill\blacksquare$

Accordingly, the total profit optimality constraint could be replaced by MIPDMS, and the intraday PDSCR model is reformulated as (30).

It is worth mentioning that the MIPDMS in different criteria stands for the optimal profit utility sets. In the above two forms of MIPDMS, the demand-supply cooperative response is satisfactory in the profit distribution and is without the conflict of interest. The reason for this is that multi-individual profits are maximized, and the corresponding distribution is fair and rational. Furthermore, specific numerical relationships are analyzed in Section V.

A modified IEEE 24-bus power system is used to conduct case studies, in which two wind farms are located on bus 12 and bus 9, and a PMP system representing an IPE is connected to bus 10. Moreover, we set 6 procedures and 5 buffers in the PMP system, where the maximum projects of $M_{i}$ and $B_{i}$ are 2 and 4. The powers of the unit projects in procedure and buffer are as follows: $\{96,64,24,72,64,42\}$ and $\{8,8,8,8,8,8\}$. The minimum production quantity of the PMP system in a dispatch cycle is set as 25, and information on DR programs is shown in TABLE II, where $\Delta P_{\rm W}$ represents the increase in wind power utilization.

In this paper, three cases are conducted to verify the effectiveness of the PDSCR strategy, while revealing impacts of the equal and the contribution-based MIPDMSs on the performance of demand-supply response. As shown in TABLE III, Case 1 is based on the day-ahead PDSCR strategy, regarded as a benchmark (without cooperation). By comparing Case 1 with Cases 2$\sim$3, we could verify the optimization effectiveness of the intraday PDSCR strategy for wind power utilization and transmission margin enhancement. The difference between Cases 2 and 3 is that the cooperation model respectively implements the equal and the contribution-based MIDPMSs. It means that the cooperative profit is divided equally or not. Therefore, by comparing Case 2 with Case 3, we could analyze the impacts of these two MIPDMSs.

In order to characterize wind power uncertainty in the intraday scheme, the intraday wind power error is assumed to follow the Gaussian distribution. In this case, the forecast value is considered as the mean and the standard deviation is set to 0.08 of the mean value. Based on this distribution model, the Latin hypercube approach [35], a traditional method, is used to sample the wind power values. Then, the sampling values constitute the multiple random scenarios to characterize wind power fluctuation, and the three cases are simulated with 100 scenarios.

In this subsection, we obtain the scheme of Case 1 by the day-ahead PDSCR strategy. The bi-objective Pareto front (PF) is shown as Fig. 7, which is the curve of objective values regarding obtained Pareto solutions. It includes the two upper-level objectives of $C_{\rm{ASC}}$ (optimal boundary: 0.7728) and total thermal cost (optimal boundary: 120652.49$\$$), as well as corresponding wind power curtailments. PF well characterize the trade-off relationship between the above two objectives. In Section III, we illustrate the consistent interests between the objectives of wind power curtailment and thermal cost. It is validated by the wind power curtailment curve in Fig. 7, where the wind power curtailment decreases with the reduction of thermal cost.

Also, it is necessary to verify the rationality of the determined day-ahead PDSCR scheme. Herein, we analyze the PF compromise solution in Fig. 7, which is the center point in the $C_{\rm{ASC}}$ boundary. Specifically, the responding curve of demand and thermal power are shown in Fig. 8(a). During the peak periods of wind power supply, such as 8:00-11:00 and 17:00-19:00, the total power demand shows an upward trend, when thermal power is relatively reduced. In contrast, the total power demand decreases during the valley periods of wind power supply, such as 1:00-6:00 and 12:00-16:00. This tracking trend minimizes the thermal power and the wind power curtailment, and reveals that the shifting of industrial power demand and the thermal operation scheme are matching the operation demand of power system in day-ahead dispatch.

Specifically, the corresponding power demand trend is influenced by price-based IDR, and the specific industrial demand curve is illustrated in Fig. 8(b). When the utility center determines low electricity prices (green areas) during certain periods, such as 3:00-6:00 or 12:00-17:00, industrial demand increases. During these periods, IPEs initiate projects in the $P$ process, as depicted by P1-P6 in Fig. 9, to maximize production. The color depth and size of these projects are directly proportional to the number of ongoing projects. Conversely, when the utility center determines high electricity prices (gray areas) at 7:00 and 10:00, IPE initiates projects in the $B$ process, denoted as B1-B5 in Fig. 9, resulting in a reduction in industrial demand during these periods.

The day-ahead strategy is designed based on the standard wind power scenario without considering power fluctuations. In Case 1, we evaluate the performance of this day-ahead load scheme in uncertain scenarios. Cases 2 and 3, on the other hand, adapt the response scheme to accommodate uncertain scenarios using the intraday PDSCR strategy, where the equal and contribution-based MIPDMSs represent two different profit distribution approaches among cooperative participants. The analysis comprises two primary components: i) Impact of different cases, and ii) Cooperative profit distribution and detailed responding.

i) Impact of different cases

In TABLE IV, we show the simulation results of 10 scenarios, which are selected in 100 random scenarios by stratified sampling based on the wind power fluctuation value. These results include the wind power curtailment, transmission margin $\rm C_{ASC}$, the total cost on the demand-supply side, and PF quality. Herein, hypervolume (HV) is an effective indicator to evaluate the PF quality [36]. Specifically, compared with Case 1, intraday PDSCR strategies in Cases 2 and 3 achieve less wind power curtailment in all the representative scenarios. They also exhibit a more stable $C_{\rm{ASC}}$, with a higher lower boundary (30.88) than that of Case 1 (0.2541). Furthermore, we observe that intraday PDSCR leads to a reduction in total costs, which results from optimizing total profits on the demand and supply side. Therefore, intraday strategies outperform day-ahead schemes in the conditions of wind power fluctuations, which ultimately benefits participating stakeholders on the demand-supply side.

Furthermore, by comparing Case 2 with Case 3, we can analyze which cooperative distribution criteria provide advantages. In this context, HV serves as an objective indicator, which is used to comprehensively assess the performance of multi-objective optimization. In TABLE IV, an examination of the HV values in Cases 2 and 3 reveals that the PF in the equal MIPDMS case outperforms the contribution-based one. Therefore, the PDSCR strategy that is constrained by equal MIPDMS can effectively enhance cooperative responses.

ii) Cooperative profit distribution and detailed responding

The simulation results discussed above indicate that MIPDMSs are advantageous for increasing total profit and facilitating effective cooperation. In this part, we will study the specific profit distribution and the corresponding demand-supply responses within the two MIPDMS scenarios. There exists a positive linear relationship between the increment in wind power utilization $\Delta P_{\rm W}$ and the total profit function ${\psi_{t}}(x,y)$. When the positive fluctuation is small, profits tend to be relatively limited, potentially posing challenges in promoting cooperation. Hence, this paper selects Scenario 5 (fluctuation: 1100MW), where wind power is close to the forecast value, to explore the characteristics of MIPDMS.

In Cases 2 and 3, profit distributions among the participating individuals are illustrated in Fig. 10 and Fig. 11. In these figures, the inner circle represents the final production cost after the response in each time period, while the outer circle represents the responding profit, which is the cost reduction. Note that some individuals are not in operation and do not receive a distribution of cooperative profit, e.g., for the thermal plant 3 during 12:00-24:00.

In Case 2, we set the issue that the equal distribution of responding profits is required by multiple individuals on the demand-supply side. Consequently, in Fig. 10, the multi-individual responding profit is equal during the cooperative periods, such as 11:00-19:00. When no total profit is obtained, individual responding profit is also 0, as observed during 1:00-4:00. This aligns with the equal MIPDMS constraint. In Case 3, we set that the contribution-based distribution of responding profit is required by multiple individuals. Consequently, Fig. 11 shows that cooperative profits vary among individuals. Taking 18:00-19:00 as an example, thermal plant 2 achieves higher profits, compared with thermal plant 1. It indicates that plant 2 has made a more contribution during these periods. These observations reveal that MIPDMS accurately distributes profits, and the fair profit distribution helps mitigate the conflict of interests among multiple individuals.

Moreover, to analyze the specific response scenarios in both cases, we present the load and power curves for demand-supply response in Fig. 12 and Fig. 13. These figures also include the wind power consumption curve and the total profit. It can be seen that the reduction in thermal power supply and the increase in industrial load are advantageous for utilizing wind power. Conversely, in Fig. 13, instances of non-cooperation during 6:00-14:00, indicate that IPE profits are limited to the upper boundary $max(u_{m,t})$ (according to Lemma 4). Note that cooperative strategies include not only financial incentives but also profit limits. Additionally, non-cooperation is significantly less frequent in Case 3. It reveals that the contribution-based MIPDMS is more beneficial to facilitating cooperation.

The modified Yantai 26-bus power system, a real system in China, is used for further study. In this system, the higher proportion of wind power is connected to buses 8 and 20, and the industrial load connects to bus 22. Furthermore, model parameters of the PMP system also follow the data in Subsection A, and the minimum production quantity is set as 20. The specific network topology and system parameters are provided in Appendix D.

For the day-ahead PDSCR strategy, the Pareto solutions are shown in Fig. 14, which regards the two upper-level objectives of $\rm C_{ASC}$ (optimal boundary: 0.8053) and total thermal cost (optimal boundary: 81003.65$\$$), as well as wind power curtailment corresponding each Pareto solution. The responding results for the compromise solution are shown in Fig. 15. Note that in the peak periods of wind power supply, such as 3:00-14:00 in Fig. 15(a), industrial power demand is in a high level in Fig. 15(b). Also, the thermal power is correspondingly minimized. Moreover, in the peak periods of wind power supply, the demand curve is close to the wind power supply, which shows that the day-ahead PDSCR is reasonable and satisfactory.

Moreover, we also simulate the three cases in TABLE III, to further investigate the impact of the equal and contribution on responding performance. Here, a random scenario close to the standard wind power value is selected to make analysis, wind power fluctuation of which is about 301.23 MW. The corresponding simulation results are shown in TABLE V. Similar to the results in TABLE IV, intraday PDSCR for Cases 2 and 3 are conducive to reduce wind power curtailment and demand-supply total cost. It verifies the effectiveness of intraday PDSCR strategy. Furthermore, the total cost on the demand-supply side in Case 3 is lower than that of Case 2, which reveals that contribution-based cooperation would generate more benefits. Furthermore, the contribution-based MIPDMS in Case 3 has a remarkable HV performance, which reveals that the response is more effective, compared with the equal MIPDMS in Case 2. The reason is that superior profit distribution will lead to more effective responding results in profit-driven cooperation. In the numerical perspective, wind power utilization is positive to cooperative profits, and thus the higher wind power proportion will expand the feasible region of profit utility set $({u_{k,t}},v_{t})$. Finally, simulation results verify that the regional expansion of the profit utility set $({u_{k,t}},v_{t})$ is more advantageous to the contribution-based cooperation, which will lead to the superior response effect.