Cooperative Operation Model of Wind Turbine and Carbon Capture Power Plant Considering Benefit Distribution

Abstract

Increasing systematic carbon sinks and clean energy generation proportion are the main ways to reduce the carbon emission of power system. In order to promote wind power accommodation and reduce system carbon emissions, a cooperative operation model of wind turbine and carbon capture power plant (CCPP) is constructed. Then, the model is equivalently transformed into two sub-problems. One is the operation optimization sub-problem of cooperative alliance with the goal of maximizing the alliance benefit. The other is the benefit distribution sub-problem with the goal of fair distributing cooperative benefit. To protect participants’ privacy, the alternating direction method of multipliers (ADMM) is used to realize the distributed solution of the two sub-problems. Finally, the effectiveness of the proposed model is verified by an example, and the sensitivity analysis of the alliance benefit and system carbon emission is carried out with carbon price and carbon capture cost as the sensitivity factors. The example results show that: (1) By providing up and down regulation services to wind turbines, CCPP can obtain ancillary service income and help to reduce the declaration deviation of wind turbines, which can realize multi-win-win situation. (2) Carbon price affects both thermal power units and carbon capture equipment. So, compared with carbon costs, the carbon emissions and the alliance benefit are both more sensitive to carbon price. The model of the paper is constructed under the deviation punishment mechanism, and subsequent research can be expanded in combination with a more detailed imbalance settlement mechanism.

1. Introduction

Since the 1980s, the growth of carbon emissions has led to an about 1.1 degrees Celsius increase in the average temperature of the earth. Global warming means more frequent and severe acute disasters such as heat waves and floods and intensified chronic disasters such as drought and sea-level rise. At present, many countries have made commitments to carbon neutrality. Take China as an example, the goal of carbon neutrality is endogenizing carbon constraints in the process of clean and efficient development of energy. Specifically, by 2060, the power sector will achieve deep decarburization or even negative emission [1]. To achieve this goal, the first is to develop clean energy vigorously. The second is to increase carbon sinks and carbon removal. Carbon sinks mainly refer to natural carbon sinks that fix carbon through natural ecosystems, also including technical carbon sinks. From 2010 to 2016, terrestrial ecosystem carbon sinks averaged 11.1 ± 3.8 billion tons annually in China. In the future, natural carbon sinks will increase slightly but it is difficult to double [2]. It is expected that the energy-related carbon emission in China will fall to 1.4 billion tons in 2060. However, if non-energy carbon emissions are added, it is impossible to completely rely on natural carbon sinks to achieve carbon neutrality, therefore, it is necessary to develop technical carbon sinks [3].

For the rapidly developing renewable energy represented by wind power and photovoltaics, due to the influence of natural factors such as season, terrain and weather, the output of renewable energy units is highly uncertain, resulting in a large deviation between the actual output and the planned output. The late-mover output deviation increases the difficulty and cost of power system scheduling, which will lead to the problem of energy abandonment [4]. The system dispatchers will allocate the scheduling cost to the deviation-responsibility units in the form of punishment, which will affect the income of generating units [5,6]. For carbon capture power plant (CCPP) introducing carbon capture and storage (CCS) technology, there is no process coupling between the power generation system and the carbon capture system. By controlling the energy distribution between the various units of the CCPP, the power generation and the carbon capture level can be independently adjusted. The structural characteristics and operation mechanism of the CCPP make it have good peak shaving performance and standby characteristics [7,8,9]. However, the high cost is still the main reason for restricting the large-scale construction of CCPP. The initial investment cost of carbon capture equipment is still high at present, although there is a large potential for cost reduction, from the current point of view, carbon price is difficult to cover the cost of carbon capture. For the above problems, combined with the operation characteristics of renewable energy units and CCPP, how to improve the benefit of CCPP while reducing the uncertainty of renewable energy output through the cooperative operation of them. Studying this issue is of practical significance for the green development and safe supply of power system, as well as the achievement of the carbon neutrality goals.

Under the background above, this paper constructs a cooperative operation model of wind turbine and CCPP under the deviation punishment mechanism for wind turbines. And the model is equivalently transformed into two sub-problems, which are operation optimization sub-problem and benefit distribution sub-problem. The former corresponds to the global operation optimization problem of cooperative alliance with the goal of maximizing cooperative benefit. The latter corresponds to the fair distribution of cooperative benefit. In order to protect the privacy of each subject, the alternating direction method of multipliers (ADMM) is used to realize the distributed solution to two sub-problems. The effectiveness of the proposed model and method is verified by an example. Finally, taking carbon price and carbon cost as sensitive factors, the sensitivity analysis of the alliance benefit and carbon emission of the system is carried out.

2. Literature Review

In the studies of the internal operation mechanism and flexibility enhancement mechanism of CCPP, Ref. [9] quantitatively evaluated the operational flexibility brought by carbon capture technology to traditional thermal power plants systematically and analyzed the factors affecting the flexibility of power plants. Refs. [10,11] modelled and simulated the CO₂ adsorption process of the capture system after combustion in a carbon capture power plant, but they mainly focused on the internal chemical dynamic process of the adsorber. Refs. [12,13,14] studied the internal energy flow and effective operating range of CCPP and analyzed their peak shaving performance and benefits. The above studies suggest that compared with traditional thermal power plants, the flexibility improvement of CCPP mainly comes from the flexible adjustment of carbon capture equipment capturing energy consumption. Based on the studies of the operation mechanism of CCPP. Ref. [15] established a carbon capture level optimization model considering the loss of electricity sales, and used this model to calculate the carbon emission price sensitive range of CCPP, thereby realizing the optimal operation of CCPP under different carbon emission prices. Ref. [16] defined the relative carbon capture degree, by optimizing the relative carbon capture degree, decision makers can select the operation strategy of carbon capture equipment in a wider range according to actual needs and realize the flexible operation of CCPP. However, most of the above studies focus on the operation analysis and decision optimization from the independent perspective of CCPP, and lack of research on the dynamic characteristics of their participation in system scheduling.

In the studies of CCPP participating in power system scheduling, Refs. [17,18] analyzed the CCPP optimal decision-making models considering the system peak regulation service and reserve service, respectively. Ref. [19] proposed a multi-objective optimal power flow model considering CCPP, by taking the lowest coal consumption costs, carbon emissions and network loss as the research objectives. Based on the flexible operation mechanism of CCPP, the active power output characteristics and carbon emission characteristics of CCPP were analyzed in Refs. [20,21], which were incorporated into the low-carbon dispatching model. The above studies are mostly based on the perspective of independent power system dispatchers for system optimization, furthermore, when CCPP are associated with renewable energy units, it is necessary to consider the multi-agent decision-making perspective for operation optimization combined with the dynamic characteristics of CCPP.

In the studies of multi-agent decision-making in CCPP and renewable energy combined system, Refs. [22,23,24,25] taking CCPP as a flexible adjustment resource, established a combined optimization model of carbon capture unit, heat storage device, cascade hydropower stations, wind turbines, photovoltaic units, and other power resources. The results show that by flexibly adjusting the output of CCPP, the investment capacity of energy storage battery can be reduced, the utilization rate of renewable energy can be improved, and obvious emission reduction benefits and economic benefits can be obtained. Refs. [26,27] combining CO₂ treatment with wind power abandonment, analyzed the direct and indirect wind power accommodation capacity of CCPP. Refs. [28,29] proposed a combined cycle operation system of carbon capture technology and electro-to-gas technology, which can realize the recycling of CO₂. The basic operation mode and flexible operation mode of CO₂ circulation system were also defined. The calculation results in [29] show that the combined operation of carbon capture and P2G (power-to-gas) can reduce 40% of wind curtailment and 20% of carbon emissions. Although the above studies involve the operation optimization, emission reduction effect analysis and economic benefit analysis of the joint operation system with CCPP, they pay less attention to the benefit distribution after the joint operation of CCPP and renewable energy. In fact, renewable energy generators and CCPP usually belong to different stakeholders, it is necessary to study their benefit distribution methods.

3. Operation Optimization Model of Wind Turbine and CCPP Considering Deviation Penalty

Figure 1 shows a typical energy system composed of wind turbine, thermal power unit, and carbon capture equipment, and assuming their belong to different stakeholders. In cooperative operation mode, thermal power unit and carbon capture equipment can provide up-regulation and down-regulation services to reduce wind power deviation.

3.1. Wind Turbines Operation Model

To maintain the real-time balance of the power system, the electric power dispatching center needs to call other callable units for peak regulation and frequency modulation when there is a deviation between the declared and the actual electricity quantity, which increases the dispatching cost of the system. This part of the cost is caused by the power deviation of the high-uncertainty units, and the power system scheduler will penalize the deviation of the responsible unit to compensate for the system dispatch cost. Therefore, the power deviation will affect the power sales profit of the unit. If not combined with other flexible resources, the wind turbine cannot avoid its randomness, the objective function of the wind turbine electricity declaration optimization model is maximizing the total profit expected from on-grid electricity.

$$\begin{array}{cc}\max & U^{\mathrm{w},0,*}\end{array}=\max {\displaystyle \sum _{n=1}^N{\displaystyle \sum _{t=1}^T{\lambda }_n(}}{p}_t^{\mathrm{w}2\mathrm{g}}{g}_t^{\mathrm{w}2\mathrm{g}}-p_t^{\mathrm{w}2\mathrm{g},+}{g}_{t,n}^{\mathrm{w}2\mathrm{g},0,+}-p_t^{\mathrm{w}2\mathrm{g},-}{g}_{t,n}^{\mathrm{w}2\mathrm{g},0,-}),$$

(1)

where, $U^{\mathrm{w},0,*}$ is total electricity sales profit of wind turbine without other flexible resources; ${\lambda }_n$ is the probability that sample n occurs, $n=1,2,\dots ,N$, N is the number of random samples; $p_t^{\mathrm{w}2\mathrm{g}}$ is the unit price of electricity sales to power grid by wind turbine during t period; $g_t^{\mathrm{w}2\mathrm{g}}$ is the electric quantity sold to power grid by wind turbines without other flexible resources during t period; $p_t^{\mathrm{w}2\mathrm{g},+}$ and $p_t^{\mathrm{w}2\mathrm{g},-}$ are the penalty prices of power grid enterprises for the positive and negative electric quantity deviation respectively to wind turbines; $g_{t,n}^{\mathrm{w}2\mathrm{g},0,+}$ and $g_{t,n}^{\mathrm{w}2\mathrm{g},0,-}$ are the positive and negative deviation electric quantity of wind turbines respectively without other flexible resources in t period under the nth sample; $g_t^{\mathrm{w}2\mathrm{c}}$ is electric quantity sold to carbon capture equipment from the wind turbine in t period; $g_{t,n}^{\mathrm{w}}$ is the forecasted power generation of wind turbines in the nth sample during t period.

In the objective function, both positive deviation and negative deviation are positive numbers, and at least one of them is 0, so:

$$g_{t,n}^{\mathrm{w}2\mathrm{g},0,+}=\max \{g_{t,n}^{\mathrm{w}}-g_t^{\mathrm{w}2\mathrm{g}},0\},\forall t,$$

(2)

$$g_{t,n}^{\mathrm{w}2\mathrm{g},0,-}=\max \{g_t^{\mathrm{w}2\mathrm{g}}-g_{t,n}^{\mathrm{w}},0\},\forall t.$$

(3)

Combining with the objective function, Equations (2) and (3) can be transformed into linear constraint conditions:

$$g_{t,n}^{\mathrm{w}2\mathrm{g},+}\ge 0,$$

(4)

$$g_{t,n}^{\mathrm{w}2\mathrm{g},-}\ge 0,$$

(5)

$$g_{t,n}^{\mathrm{w}2\mathrm{g},0,+}\ge g_{t,n}^{\mathrm{w}}-g_t^{\mathrm{w}2\mathrm{g}},$$

(6)

$$g_{t,n}^{\mathrm{w}2\mathrm{g},0,-}\ge g_t^{\mathrm{w}2\mathrm{g}}-g_{t,n}^{\mathrm{w}}.$$

(7)

In addition, constraint conditions include the upper and lower limit constraints to unit output and ramp rate constraints:

$$0\le g_t^{\mathrm{w}2\mathrm{g}}\le g_{}^{\mathrm{w},\max },$$

(8)

$$g_{t+1}^{\mathrm{w}2\mathrm{g}}-g_t^{\mathrm{w}2\mathrm{g}}\le \Delta g_{}^{\mathrm{w},\mathrm{u},\max },\forall t,$$

(9)

$$g_t^{\mathrm{w}2\mathrm{g}}-g_{t+1}^{\mathrm{w}2\mathrm{g}}\le \Delta g_{}^{\mathrm{w},\mathrm{d},\max },\forall t,$$

(10)

where, $g_{}^{\mathrm{w},\max }$ is max power generation of wind turbines in unit time. Equations (9) and (10) are ramp rate constraints. $\Delta g_{}^{\mathrm{w},\mathrm{u},\max }$ and $\Delta g_{}^{\mathrm{w},\mathrm{d},\max }$ are allowable max rise variation and max decline variation respectively of power generation in unit time.

Thus, the original optimization model of wind turbines can be expressed as:

$$\max {\displaystyle \sum _{n=1}^N{\displaystyle \sum _{t=1}^T{\lambda }_n(}}{p}_t^{\mathrm{w}2\mathrm{g}}{g}_t^{\mathrm{w}2\mathrm{g}}-p_t^{\mathrm{w}2\mathrm{g},+}{g}_{t,n}^{\mathrm{w}2\mathrm{g},0,+}-p_t^{\mathrm{w}2\mathrm{g},-}{g}_{t,n}^{\mathrm{w}2\mathrm{g},0,-}),$$

(11)

In addition to this, the model (11) also includes constraints Equations (4)–(10).

When there are negative deviations, the actual power generation of wind turbines is less than the planned. When there are positive deviations, the actual power generation of wind turbines is greater than the planned. To avoid deviation penalty, combined wind turbines obtain the up and down regulation services from thermal power unit and carbon capture equipment, respectively. Wind turbines need to pay for ancillary services. Therefore, the operation model is as follows:

$$\max U^{\mathrm{w}}=U^{\mathrm{w}2\mathrm{g}}-U^{\mathrm{h}2\mathrm{w}}-U^{\mathrm{c}2\mathrm{w}},$$

(12)

$$U^{\mathrm{w}2\mathrm{g}}={\displaystyle \sum _{n=1}^N{\displaystyle \sum _{t=1}^T{\lambda }_n(p_t^{\mathrm{w}2\mathrm{g}}{g}_t^{\mathrm{w}2\mathrm{g}}-p_t^{\mathrm{w}2\mathrm{g},+}{g}_{t,n}^{\mathrm{w}2\mathrm{g},+}-p_t^{\mathrm{w}2\mathrm{g},-}{g}_{t,n}^{\mathrm{w}2\mathrm{g},-})}} ,$$

(13)

$$U^{\mathrm{h}2\mathrm{w}}={\displaystyle \sum _{t=1}^T{g}_t^{\mathrm{w}2\mathrm{h}}{p}_{}^{\mathrm{w}2\mathrm{h}}},$$

(14)

$$U^{\mathrm{c}2\mathrm{w}}={\displaystyle \sum _{t=1}^T{g}_t^{\mathrm{w}2\mathrm{c}}{p}_{}^{\mathrm{w}2\mathrm{c}}},$$

(15)

$$g_{t,n}^{\mathrm{w}2\mathrm{g},+}\ge g_{t,n}^{\mathrm{w}}-g_t^{\mathrm{w}2\mathrm{g}}-g_t^{\mathrm{w}2\mathrm{c}},$$

(16)

$$g_{t,n}^{\mathrm{w}2\mathrm{g},-}\ge g_t^{\mathrm{w}2\mathrm{g}}-g_{t,n}^{\mathrm{w}}-g_t^{\mathrm{w}2\mathrm{h}},$$

(17)

$$g_{t,n}^{\mathrm{w}2\mathrm{g},+}\ge 0,g_{t,n}^{\mathrm{w}2\mathrm{g},-}\ge 0,$$

(18)

$$g_t^{\mathrm{w}2\mathrm{c}}\ge 0,g_t^{\mathrm{w}2\mathrm{h}}\ge 0,$$

(19)

where, $U^{\mathrm{w}}$ is profit of wind turbine after combined with other flexible resources; $U^{\mathrm{w}2\mathrm{g}}$ is the income of wind turbines electricity sold to power grid after combined with other flexible resources; $U^{\mathrm{c}2\mathrm{w}}$ is the income obtained by carbon capture equipment selling down-regulation service to wind turbine; $U^{\mathrm{h}2\mathrm{w}}$ is the income obtained by thermal power units selling up- regulation service to wind turbines; $g_{t,n}^{\mathrm{w}2\mathrm{g},+}$ and $g_{t,n}^{\mathrm{w}2\mathrm{g},-}$ are positive and negative deviation of wind turbine after combined with other flexible resources, respectively, under the sample n in t period; $g_t^{\mathrm{w}2\mathrm{h}}$ is the electric quantity of up- regulation service purchased by wind turbines from thermal power units during t period; $p_{}^{\mathrm{w}2\mathrm{h}}$ is the unit price of up-regulation service purchased by wind turbines from thermal power units; $g_t^{\mathrm{w}2\mathrm{c}}$ is the electric quantity of down-regulation service purchased by wind turbines from carbon capture equipment during t period; $p_{}^{\mathrm{w}2\mathrm{c}}$ is the unit price of down- regulation service purchased by wind turbines from carbon capture equipment.

3.2. Thermal Power Unit Operation Model

The regulation electric quantity sold to wind turbines is essentially incremental electric quantity for thermal power unit and carbon capture equipment. So, they need to maximize the incremental profit of the electric quantity. The profit model of thermal power units is:

$$\max U^{\mathrm{h}}=U_{}^{\mathrm{h}2\mathrm{w}}-{\displaystyle \sum _{t=1}^T{g}_t^{\mathrm{h}2\mathrm{w}}(c^{\mathrm{h}}+{\delta }^{\mathrm{h}}{p}_t^{\mathrm{c}})},$$

(20)

$$0\le g_t^{\mathrm{h}}+g_t^{\mathrm{h}2\mathrm{w}}\le g_{}^{\mathrm{h},\max },$$

(21)

$$g_{t+1}^{\mathrm{h}}+g_{t+1}^{\mathrm{h}2\mathrm{w}}-(g_t^{\mathrm{h}}+g_t^{\mathrm{h}2\mathrm{w}})\le \Delta g_{}^{\mathrm{h},\mathrm{u},\max },$$

(22)

$$g_t^{\mathrm{h}}+g_t^{\mathrm{h}2\mathrm{w}}-(g_{t+1}^{\mathrm{h}}+g_{t+1}^{\mathrm{h}2\mathrm{w}})\le \Delta g_{}^{\mathrm{h},\mathrm{d},\max },$$

(23)

where, $U^{\mathrm{h}}$ is the incremental profit of thermal power unit; $g_t^{\mathrm{h}2\mathrm{w}}$ is the up- regulation electric quantity sold to wind turbines by thermal power unit in t period; ${\delta }^{\mathrm{h}}$ is carbon emission intensity of thermal power unit; $c^{\mathrm{h}}$ is the unit electric cost of thermal power units; $p_t^{\mathrm{c}}$ is carbon price in t period carbon market. $g_t^{\mathrm{h}}$ is planned power generation of thermal power unit in t period; $g_{}^{\mathrm{h},\max }$ is the max power generation in unit time of thermal power unit; $\Delta g_{}^{\mathrm{h},\mathrm{u},\max }$ is the max up-regulation power generation in unit time of thermal power unit; $\Delta g_{}^{\mathrm{h},\mathrm{d},\max }$ the max down-regulation power generation in unit time of thermal power unit.

3.3. Carbon Capture Equipment Operation Model

Carbon capture equipment needs to maximize the incremental profit of selling down- regulation services. Its profit model is:

$$\max U^{\mathrm{c}}=U^{\mathrm{c}2\mathrm{w}}+{\displaystyle \sum _{t=1}^T{g}_t^{\mathrm{c}2\mathrm{w}}{\delta }^{\mathrm{c}}(}{p}_t^{\mathrm{c}}+s^{\mathrm{c}}-c^{\mathrm{c}}),$$

(24)

$$0\le g_t^{\mathrm{c}}+g_t^{\mathrm{c}2\mathrm{w}}\le g_{}^{\mathrm{c},\max },$$

(25)

$$g_{t+1}^{\mathrm{c}}+g_{t+1}^{\mathrm{c}2\mathrm{w}}-(g_t^{\mathrm{c}}+g_t^{\mathrm{c}2\mathrm{w}})\le \Delta g_{}^{\mathrm{c},\mathrm{u},\max },$$

(26)

$$g_t^{\mathrm{c}}+g_t^{\mathrm{c}2\mathrm{w}}-(g_{t+1}^{\mathrm{c}}+g_{t+1}^{\mathrm{c}2\mathrm{w}})\le \Delta g_{}^{\mathrm{c},\mathrm{d},\max },$$

(27)

where, $U^{\mathrm{c}}$ is the incremental profit of carbon capture equipment; $g_t^{\mathrm{c}2\mathrm{w}}$ is the down- regulation electric quantity sold to wind turbine by carbon capture equipment in t period; ${\delta }^{\mathrm{c}}$ is the electricity-CO₂ transformation factor of carbon capture equipment; $s^{\mathrm{c}}$ is the unit subsidy for carbon capture equipment capturing CO₂; $c^{\mathrm{c}}$ is the unit cost of carbon capture equipment capturing CO₂; $g_t^{\mathrm{c}}$ is planned electricity consumption of carbon capture equipment; $g_{}^{\mathrm{c},\max }$ is the max electricity consumption of carbon capture equipment in unit time; $\Delta g_{}^{\mathrm{c},\mathrm{u},\max }$ is the max up-regulation electricity consumption of carbon capture equipment in unit time; $\Delta g_{}^{\mathrm{c},\mathrm{d},\max }$ is the max down-regulation electricity consumption of carbon capture equipment in unit time.

4. Multi-Agent Cooperative Nash Bargaining Model of Wind Turbine and CCPP

As independent rational individuals, all participants hope to reach a consensus through negotiations, and seek a balanced strategy to maximize the benefits of all participants. How to determine the transaction electricity and transaction price reasonably is the focus of all participants.

4.1. The Basic Principle of Nash Bargaining

Nash bargaining theory belongs to the cooperative game, which can take into account both individual and collective interests. The solution that maximizes the Nash product in Equation (28) is the equilibrium solution of the Nash bargaining game problem. The Nash bargaining solution can make participants of cooperative alliance obtain the Pareto optimal benefits.

$$\{\begin{array}{c}\max {\displaystyle \prod _{i=1}^I(U_i-U_i^0)}\\ \mathrm{s}.\mathrm{t}.\begin{array}{cc}& U_i\ge U_i^0\end{array}\end{array},$$

(28)

where, I is the count of participants in negotiations; $U_i$ is benefits of participant i; $U_i^0$ is the disagreement point, namely the benefit before the cooperation negotiation of participants; $U_i\ge U_i^0$ represents the benefit of negotiation participants i is enhanced through cooperation. The goal of Nash bargaining is to maximize the benefit enhancements for all partners.

4.2. Wind Turbine and CCPP Cooperative Operation Nash Bargaining Model

Using Nash bargaining principle to research the cooperative operation among wind turbine, thermal power unit, and carbon capture equipment:

$$\{\begin{array}{c}\max {\displaystyle \prod _{i=1}^I(U^{\mathrm{w}}-U^{\mathrm{w},0,*})U^{\mathrm{h}}{U}^{\mathrm{c}}}\\ \mathrm{s}.\mathrm{t}.\begin{array}{cc}& U^{\mathrm{w}}\ge U^{\mathrm{w},0,*}\end{array}\\ U^{\mathrm{h}}\ge 0\\ U^{\mathrm{c}}\ge 0\end{array},$$

(29)

where, $U^{\mathrm{w},0,*}$ is the optimal operating profit of the wind turbine without cooperation, namely Nash bargaining disagreement point. Thermal power unit and carbon capture equipment consider profit increment in negotiation, so their disagreement point is zero. In addition, the model (29) also includes Equations (12)–(27).

4.3. Equivalent Conversion of Cooperative Operation Nash Bargaining Model

Essentially, the Nash bargaining cooperative game model Equation (29) is a non-convex nonlinear optimization problem, which is difficult to solve directly. In this paper, a series of equivalent transformations of the model was done. Then, the model was converted into the following two easily solved sub-problems, which are operation optimization sub-problem and benefit distribution sub-problem. The optimal solution of the original problem Equation (29) can be obtained by solving two subproblems in turn.

When finding the maximum value of model (29), it needs to satisfy three conditions for the mean inequality: variables in the inequality are positive, the sum of variables in the inequality is constant, and when variables in the inequality equaling, the inequality becomes a equality. It is easy to know that the objective function of model (29) satisfies the first condition. If the value of $(U^{\mathrm{w}}-U^{\mathrm{w},0,*})+U^{\mathrm{h}}+U^{\mathrm{c}}$ is a constant, namely satisfying the second condition, the objective function of model (29) can obtain the maximum. Because $U^{\mathrm{w},0,*}$ is constant, and:

$$(U^{\mathrm{w}}-U^{\mathrm{w},0,*})+U^{\mathrm{h}}+U^{\mathrm{c}}=U^{\mathrm{w}}+U^{\mathrm{h}}+U^{\mathrm{c}}-U^{\mathrm{w},0,*}.$$

(30)

So, if $U^{\mathrm{w}}+U^{\mathrm{h}}+U^{\mathrm{c}}$ is constant, the condition to make the inequality become an equality can be satisfied. Furthermore, the larger the value of $U^{\mathrm{w}}+U^{\mathrm{h}}+U^{\mathrm{c}}$, the larger the objective function value of model (29) is. Introduce process variables ${\omega }^{\mathrm{h}}$, ${\omega }^{\mathrm{w}}$ and ${\omega }^{\mathrm{c}}$:

$$\{\begin{array}{c}{\omega }^{\mathrm{h}}=-{\displaystyle \sum _{t=1}^T{g}_t^{\mathrm{h}2\mathrm{w}}(c^{\mathrm{h}}+{\delta }^{\mathrm{h}}{p}_t^{\mathrm{c}})}\\ {\omega }^{\mathrm{w}}=U^{\mathrm{w}2\mathrm{g}}\\ {\omega }^{\mathrm{c}}={\displaystyle \sum _{t=1}^T{g}_t^{\mathrm{c}2\mathrm{w}}{\delta }^{\mathrm{c}}(}{p}_t^{\mathrm{c}}+s^{\mathrm{c}}-c^{\mathrm{c}})\end{array},$$

(31)

Then:

$$(U^{\mathrm{w}}-U^{\mathrm{w},0})+U^{\mathrm{h}}+U^{\mathrm{c}}={\omega }^{\mathrm{w}}+{\omega }^{\mathrm{h}}+{\omega }^{\mathrm{c}}-U^{\mathrm{w},0}.$$

(32)

Thus, the alliance benefit maximization problem can be transformed into operation optimization sub-problem.

$$\{\begin{array}{c}\max {\omega }^{\mathrm{w}}+{\omega }^{\mathrm{h}}+{\omega }^{\mathrm{c}}\\ \begin{array}{cc}\mathrm{s}.\mathrm{t}.& {\omega }^{\mathrm{h}}=-{\displaystyle \sum _{t=1}^T{g}_t^{\mathrm{h}2\mathrm{w}}(c^{\mathrm{h}}+{\delta }^{\mathrm{h}}{p}_t^{\mathrm{c}})}\end{array}\\ {\omega }^{\mathrm{w}}=U^{\mathrm{w}2\mathrm{g}}\\ {\omega }^{\mathrm{c}}={\displaystyle \sum _{t=1}^T{g}_t^{\mathrm{c}2\mathrm{w}}{\delta }^{\mathrm{c}}(}{p}_t^{\mathrm{c}}+s^{\mathrm{c}}-c^{\mathrm{c}})\\ g_t^{\mathrm{c}2\mathrm{w}}=g_t^{\mathrm{w}2\mathrm{c}}\\ g_t^{\mathrm{h}2\mathrm{w}}=g_t^{\mathrm{w}2\mathrm{h}}\end{array}.$$

(33)

In addition to this, Model (33) also includes constraints Equations (13), (16)–(19), (21)–(23), (25)–(27). Model (33) essentially maximizes the overall benefits of the multi-agent cooperative alliance. It can be seen from Equations (32) and (33) that in the alliance benefit maximization model, regulation service transaction volume $g_t^{\mathrm{h}2\mathrm{w}}$ and $g_t^{\mathrm{c}2\mathrm{w}}$ offset each other in the superposition process. So, if establishing the total benefit maximization model of the alliance directly, regulation service transactions volume among the participants will offset in superposition, which induces the sum of regulated service transactions cannot be determined. It is one of the necessary reasons for introducing the Nash bargaining model in this paper. By solving sub-problem 2, the regulation service transaction price and transaction amount can be determined. Values ${\omega }^{\mathrm{w},*}$, ${\omega }^{\mathrm{h},*}$ and ${\omega }^{\mathrm{c},*}$ can be obtained by solving sub-problem 1. Insert them into Equation (28):

$$\{\begin{array}{c}\max ({\omega }^{\mathrm{w},*}-U^{\mathrm{h}2\mathrm{w}}-U^{\mathrm{c}2\mathrm{w}}-U^{\mathrm{w},0,*})({\omega }^{\mathrm{h},*}+U^{\mathrm{h}2\mathrm{w}})({\omega }^{\mathrm{c},*}+U^{\mathrm{c}2\mathrm{w}})\\ \begin{array}{cc}\mathrm{s}.\mathrm{t}.& {\omega }^{\mathrm{w},*}+U^{\mathrm{w}2\mathrm{c}}-U^{\mathrm{h}2\mathrm{w}}\ge U^{\mathrm{w},0,*}\end{array}\\ {\omega }^{\mathrm{h},*}+U^{\mathrm{h}2\mathrm{w}}\ge 0\\ {\omega }^{\mathrm{c},*}+U^{\mathrm{c}2\mathrm{w}}\ge 0\end{array}.$$

(34)

In addition to this, the model (34) also includes Equations (14) and (15).

The natural logarithmic function is a strictly monotonically increasing convex function. Take the logarithm of Equation (34) and convert the original problem from maximizing to minimizing. Then, model (34) is equivalent to the following benefit distribution sub-problem.

$$\{\begin{array}{c}\min -[\ln ({\omega }^{\mathrm{w},*}-U^{\mathrm{h}2\mathrm{w}}-U^{\mathrm{c}2\mathrm{w}}-U^{\mathrm{w},0,*})+\ln ({\omega }^{\mathrm{h},*}+U^{\mathrm{h}2\mathrm{w}})+\ln ({\omega }^{\mathrm{c},*}+U^{\mathrm{c}2\mathrm{w}})] \\ \begin{array}{cc}\mathrm{s}.\mathrm{t}.& {\omega }^{\mathrm{w},*}+U^{\mathrm{w}2\mathrm{c}}-U^{\mathrm{h}2\mathrm{w}}\ge U^{\mathrm{w},0,*}\end{array}\\ {\omega }^{\mathrm{h},*}+U^{\mathrm{h}2\mathrm{w}}\ge 0\\ {\omega }^{\mathrm{c},*}+U^{\mathrm{c}2\mathrm{w}}\ge 0\end{array}.$$

(35)

In addition to this, the model (35) also includes Equations (14) and (15).

5. Solution of Wind Turbine and CCPP Cooperative Nash Bargaining Model

To protect the privacy of the participants involved in the negotiation, the alternating direction multiplier method (ADMM) is used to solve the sub-problems. Because the ADMM algorithm has good convergence, simple form and strong robustness, it is often used to solve the optimization problem with separable variables. The detailed solution idea is shown in reference [30].

5.1. Solution of Operation Optimization Sub-Problem Based on ADMM

For the alliance benefit maximization model (33), first, ream that:

$$\{\begin{array}{c}{\overline{g}}_t^{\mathrm{h}2\mathrm{w}}=g_t^{\mathrm{h}2\mathrm{w}}\\ \begin{array}{c}{\overline{g}}_t^{\mathrm{c}2\mathrm{w}}=g_t^{\mathrm{c}2\mathrm{w}}\\ {\overline{g}}_t^{\mathrm{w}2\mathrm{h}}=g_t^{\mathrm{w}2\mathrm{h}}\\ {\overline{g}}_t^{\mathrm{w}2\mathrm{c}}=g_t^{\mathrm{w}2\mathrm{c}}\end{array}\end{array}.$$

(36)

where, ${\overline{g}}_t^{\mathrm{h}2\mathrm{w}}$ and ${\overline{g}}_t^{\mathrm{c}2\mathrm{w}}$ are regulation electricity quantity sold to wind turbine by carbon capture equipment and thermal power unit respectively in t period; ${\overline{g}}_t^{\mathrm{w}2\mathrm{h}}$ and ${\overline{g}}_t^{\mathrm{w}2\mathrm{c}}$ are regulation electricity quantity purchased by wind turbine from carbon capture equipment and thermal power unit respectively in t period. When ${\overline{g}}_t^{\mathrm{h}2\mathrm{w}}={\overline{g}}_t^{\mathrm{w}2\mathrm{h}}$ and ${\overline{g}}_t^{\mathrm{c}2\mathrm{w}}={\overline{g}}_t^{\mathrm{w}2\mathrm{c}}$, it shows that they negotiate a consensus on regulation services trade.

Then, take the inverse value of the objective function in (33), and convert the model from maximizing to minimizing. By introducing Lagrange multipliers and penalty factors, the augmented Lagrangian function of the model (33) objective function can be obtained.

$$\begin{array}{l}L=-({\omega }^{\mathrm{w}}+{\omega }^{\mathrm{h}}+{\omega }^{\mathrm{c}})+{\displaystyle \sum _{t=1}^T{\lambda }_t^{\mathrm{wc}}({\overline{g}}_t^{\mathrm{c}2\mathrm{w}}-{\overline{g}}_t^{\mathrm{w}2\mathrm{c}})}+{\displaystyle \sum _{t=1}^T{\lambda }_t^{\mathrm{wh}}({\overline{g}}_t^{\mathrm{h}2\mathrm{w}}-{\overline{g}}_t^{\mathrm{w}2\mathrm{h}})}+\\ \frac{{\rho }^{\mathrm{wc}}}{2}{\displaystyle \sum _{t=1}^T{\Vert {\overline{g}}_t^{\mathrm{c}2\mathrm{w}}-{\overline{g}}_t^{\mathrm{w}2\mathrm{c}}\Vert }_2^2}+\frac{{\rho }^{\mathrm{wh}}}{2}{\displaystyle \sum _{t=1}^T{\Vert {\overline{g}}_t^{\mathrm{h}2\mathrm{w}}-{\overline{g}}_t^{\mathrm{w}2\mathrm{h}}\Vert }_2^2}\end{array}$$

(37)

According to the principle of ADMM algorithm, the Equation (37) can be decomposed to obtain the distributed optimization operation models of wind turbine, thermal power unit, and carbon capture equipment:

(1)

Distributed optimal operation model of wind turbine

$$\begin{array}{l}\min {\displaystyle \sum _{n=1}^N{\displaystyle \sum _{t=1}^T{\lambda }_n(g_{t,n}^{\mathrm{w}}}}{c}^{\mathrm{w}}-p_t^{\mathrm{w}2\mathrm{g}}{g}_t^{\mathrm{w}2\mathrm{g}}+p_t^{\mathrm{w}2\mathrm{g},+}{g}_{t,n}^{\mathrm{w}2\mathrm{g},+}+p_t^{\mathrm{w}2\mathrm{g},-}{g}_{t,n}^{\mathrm{w}2\mathrm{g},-})+\\ {\displaystyle \sum _{t=1}^T{\lambda }_t^{\mathrm{wc}}({\overline{g}}_t^{\mathrm{c}2\mathrm{w}}-{\overline{g}}_t^{\mathrm{w}2\mathrm{c}})}+{\displaystyle \sum _{t=1}^T{\lambda }_t^{\mathrm{wh}}({\overline{g}}_t^{\mathrm{h}2\mathrm{w}}-{\overline{g}}_t^{\mathrm{w}2\mathrm{h}})}+\\ \frac{{\rho }^{\mathrm{wc}}}{2}{\displaystyle \sum _{t=1}^T{\Vert {\overline{g}}_t^{\mathrm{c}2\mathrm{w}}-{\overline{g}}_t^{\mathrm{w}2\mathrm{c}}\Vert }_2^2}+\frac{{\rho }^{\mathrm{wh}}}{2}{\displaystyle \sum _{t=1}^T{\Vert {\overline{g}}_t^{\mathrm{h}2\mathrm{w}}-{\overline{g}}_t^{\mathrm{w}2\mathrm{h}}\Vert }_2^2}\end{array}$$

(38)

In addition to this, the model (38) also includes Equations (16)–(19).

(2)

Distributed optimal operation model of thermal power unit

$$\min {\displaystyle \sum _{t=1}^T{g}_t^{\mathrm{h}2\mathrm{w}}(c^{\mathrm{h}}+{\delta }^{\mathrm{h}}{p}_t^{\mathrm{c}})}+{\displaystyle \sum _{t=1}^T{\lambda }_t^{\mathrm{wh}}({\overline{g}}_t^{\mathrm{h}2\mathrm{w}}-{\overline{g}}_t^{\mathrm{w}2\mathrm{h}})}+\frac{{\rho }^{\mathrm{wh}}}{2}{\displaystyle \sum _{t=1}^T{\Vert {\overline{g}}_t^{\mathrm{h}2\mathrm{w}}-{\overline{g}}_t^{\mathrm{w}2\mathrm{h}}\Vert }_2^2}.$$

(39)

In addition to this, the model (39) also includes Equations (21)–(23).

(3)

Distributed optimal operation model of carbon capture equipment

$$\min {\displaystyle \sum _{t=1}^T{g}_t^{\mathrm{w}2\mathrm{c}}{\delta }^{\mathrm{c}}(}{c}^{\mathrm{c}}-p_t^{\mathrm{c}})+{\displaystyle \sum _{t=1}^T{\lambda }_t^{\mathrm{wc}}({\overline{g}}_t^{\mathrm{c}2\mathrm{w}}-{\overline{g}}_t^{\mathrm{w}2\mathrm{c}})}+\frac{{\rho }^{\mathrm{wc}}}{2}{\displaystyle \sum _{t=1}^T{\Vert {\overline{g}}_t^{\mathrm{c}2\mathrm{w}}-{\overline{g}}_t^{\mathrm{w}2\mathrm{c}}\Vert }_2^2}.$$

(40)

In addition to this, the model (40) also includes Equations (25)–(27).

First, the distributed algorithm 1 is established for the operation optimization sub-problem of cooperative alliance, the specific steps are as follows:

(1) Setting the max iterations times is $k^{\max }=50$. Convergence precision is $\xi ={10}^{-5}$. Penalty factors are ${\rho }^{\mathrm{wc}}={\rho }^{\mathrm{wh}}={10}^{-4}$. Iteration times is initialized to $k=0$. Initial regulation service electricity expected by carbon capture equipment and thermal power unit to wind turbine are ${\overline{g}}_{t,k}^{\mathrm{c}2\mathrm{w}}={\overline{g}}_{t,k}^{\mathrm{h}2\mathrm{w}}=0$. Lagrange multipliers are ${\lambda }_t^{\mathrm{wc}}={\lambda }_t^{\mathrm{wh}}=0$.

(2) For wind turbine, according to regulation service electricity ${\overline{g}}_{t,k}^{\mathrm{c}2\mathrm{w}}$ and ${\overline{g}}_{t,k}^{\mathrm{h}2\mathrm{w}}$ expected by carbon capture equipment and thermal power unit to provide, solve model (38) and obtain regulation service electricity ${\overline{g}}_{t,k+1}^{\mathrm{w}2\mathrm{h}}$ and ${\overline{g}}_{t,k+1}^{\mathrm{w}2\mathrm{c}}$ expected to be purchased.

(3) For thermal power unit, according to up-regulation service electricity ${\overline{g}}_{t,k+1}^{\mathrm{w}2\mathrm{h}}$ expected by wind turbine to purchase, solve model (39) and obtain regulation service electricity ${\overline{g}}_{t,k+1}^{\mathrm{h}2\mathrm{w}}$ expected to be sold.

(4) For carbon capture equipment, according to down- regulation service electricity ${\overline{g}}_{t,k+1}^{\mathrm{w}2\mathrm{c}}$ expected by wind turbine to purchase, solve model (39) and obtain regulation service electricity ${\overline{g}}_{t,k+1}^{\mathrm{c}2\mathrm{w}}$ expected to be sold.

(5) Update Lagrange multipliers according to Equation (41):

$$\{\begin{array}{c}{\lambda }_{t,k+1}^{\mathrm{wh}}={\lambda }_{t,k}^{\mathrm{wh}}+{\rho }^{\mathrm{wh}}({\overline{g}}_{t,k+1}^{\mathrm{h}2\mathrm{w}}-{\overline{g}}_{t,k+1}^{\mathrm{w}2\mathrm{h}})\\ {\lambda }_{t,k+1}^{\mathrm{wc}}={\lambda }_{t,k}^{\mathrm{wc}}+{\rho }^{\mathrm{wc}}({\overline{g}}_{t,k+1}^{\mathrm{c}2\mathrm{w}}-{\overline{g}}_{t,k+1}^{\mathrm{w}2\mathrm{c}})\end{array}.$$

(41)

(6) Update iterations times: k = k + 1;

(7) Judging convergence of the algorithm, if the termination condition (42) of iterative is satisfied, the iteration terminates.

$$\{\begin{array}{c}\max ({\displaystyle \sum _{t=1}^T{\Vert {\overline{g}}_t^{\mathrm{c}2\mathrm{w}}-{\overline{g}}_t^{\mathrm{w}2\mathrm{c}}\Vert }_2^2},{\displaystyle \sum _{t=1}^T{\Vert {\overline{g}}_t^{\mathrm{h}2\mathrm{w}}-{\overline{g}}_t^{\mathrm{w}2\mathrm{h}}\Vert }_2^2)<\xi }\\ \begin{array}{cc}\mathrm{or}& k>k_{\max }\end{array}\end{array},$$

(42)

Otherwise step 2 is repeated, until the convergence condition or the max iterations times is satisfied.

5.2. Solution of Benefit Distribution Sub-Problem Based on ADMM

By solving the operation optimization sub-problem, the optimal expected transaction regulation electricity between wind turbine and carbon capture, wind turbine and thermal can be obtained as ${\overline{g}}_t^{\mathrm{w}2\mathrm{c},*}$ and ${\overline{g}}_t^{\mathrm{w}2\mathrm{h},*}$ simultaneously.

According to Equations (14), (15), and (36), regulation electricity trading volume $U^{\mathrm{h}2\mathrm{w}}$ and $U^{\mathrm{c}2\mathrm{w}}$ can be expressed as:

$$\{\begin{array}{c}{U}^{\mathrm{h}2\mathrm{w}}={\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{h},*}{p}_{}^{\mathrm{w}2\mathrm{h}}}\\ U^{\mathrm{c}2\mathrm{w}}={\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{c},*}{p}_{}^{\mathrm{w}2\mathrm{c}}}\end{array},$$

(43)

Substituting Equation (43) into benefit distribution model (35), it can obtain:

$$\{\begin{array}{c}\min -[\ln ({\omega }^{\mathrm{w},*}-{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{c},*}{p}_{}^{\mathrm{w}2\mathrm{c}}}-{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{h},*}{p}_{}^{\mathrm{w}2\mathrm{h}}}-U^{\mathrm{w},0,*})+\\ \ln ({\omega }^{\mathrm{h},*}+{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{h},*}{p}_{}^{\mathrm{w}2\mathrm{h}}})+\ln ({\omega }^{\mathrm{c},*}+{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{c},*}{p}_{}^{\mathrm{w}2\mathrm{c}}})] \\ \begin{array}{cc}\mathrm{s}.\mathrm{t}.& {\omega }^{\mathrm{w},*}+{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{c},*}{p}_{}^{\mathrm{w}2\mathrm{c}}}-{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{h},*}{p}_{}^{\mathrm{w}2\mathrm{h}}}\ge U^{\mathrm{w},0,*}\end{array}\\ {\omega }^{\mathrm{h},*}+{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{h},*}{p}_{}^{\mathrm{w}2\mathrm{h}}}\ge 0\\ {\omega }^{\mathrm{c},*}+{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{c},*}{p}_{}^{\mathrm{w}2\mathrm{c}}}\ge 0\end{array}$$

(44)

With the principle of ADMM algorithm, introducing ancillary variable ${\overline{p}}_{}^{\mathrm{w}2\mathrm{c}}$, ${\overline{p}}_{}^{\mathrm{w}2\mathrm{c}}$,${\overline{p}}_{}^{\mathrm{c}2\mathrm{w}}$ and ${\overline{p}}_{}^{\mathrm{h}2\mathrm{w}}$ to decouple transaction price. ${\overline{p}}_{}^{\mathrm{w}2\mathrm{c}}$ and ${\overline{p}}_{}^{\mathrm{w}2\mathrm{c}}$ are unit price of regulation service expected by wind turbine to purchase from carbon capture equipment and thermal power unit respectively; ${\overline{p}}_{}^{\mathrm{c}2\mathrm{w}}$ and ${\overline{p}}_{}^{\mathrm{h}2\mathrm{w}}$ are the unit price of regulation service expected by carbon capture equipment and thermal power unit to wind turbine. So, the augmented Lagrangian function of the model (44) objective function can be expressed as:

$$\begin{array}{l}\min -[\ln ({\omega }^{\mathrm{w},*}+{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{c},*}{\overline{p}}_{}^{\mathrm{w}2\mathrm{c}}}-{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{h},*}{\overline{p}}_{}^{\mathrm{w}2\mathrm{h}}}-U^{\mathrm{w},0,*})+\\ \ln ({\omega }^{\mathrm{h},*}+{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{h},*}{\overline{p}}_{}^{\mathrm{h}2\mathrm{w}}})+\ln ({\omega }^{\mathrm{c},*}-{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{c},*}{\overline{p}}_{}^{\mathrm{c}2\mathrm{w}}})]+\\ {\displaystyle \sum _{t=1}^T{\gamma }_t^{\mathrm{wc}}({\overline{p}}_{}^{\mathrm{c}2\mathrm{w}}-{\overline{p}}_{}^{\mathrm{w}2\mathrm{c}})}+{\displaystyle \sum _{t=1}^T{\gamma }_t^{\mathrm{wh}}({\overline{p}}_{}^{\mathrm{h}2\mathrm{w}}-{\overline{p}}_{}^{\mathrm{w}2\mathrm{h}})}+\\ \frac{{\psi }^{\mathrm{wc}}}{2}{\displaystyle \sum _{t=1}^T{\Vert {\overline{p}}_{}^{\mathrm{c}2\mathrm{w}}-{\overline{p}}_{}^{\mathrm{w}2\mathrm{c}}\Vert }_2^2}+\frac{{\psi }^{\mathrm{wh}}}{2}{\displaystyle \sum _{t=1}^T{\Vert {\overline{p}}_{}^{\mathrm{h}2\mathrm{w}}-{\overline{p}}_{}^{\mathrm{w}2\mathrm{h}}\Vert }_2^2}\end{array}$$

(45)

According to the principle of ADMM algorithm decomposition, Equation (45) can be decomposed to obtain the distributed optimization models for the regulation service price of wind turbine, thermal power unit, and carbon capture equipment:

(1)

Distributed optimal operation model of wind turbine

$$\{\begin{array}{c}\begin{array}{l}\min -[\ln ({\omega }^{\mathrm{w},*}+{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{c},*}{\overline{p}}_{}^{\mathrm{w}2\mathrm{c}}}-{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{h},*}{\overline{p}}_{}^{\mathrm{w}2\mathrm{h}}}-U^{\mathrm{w},0,*})+\\ {\displaystyle \sum _{t=1}^T{\gamma }_t^{\mathrm{wc}}({\overline{p}}_{}^{\mathrm{c}2\mathrm{w}}-{\overline{p}}_{}^{\mathrm{w}2\mathrm{c}})}+{\displaystyle \sum _{t=1}^T{\gamma }_t^{\mathrm{wh}}({\overline{p}}_{}^{\mathrm{h}2\mathrm{w}}-{\overline{p}}_{}^{\mathrm{w}2\mathrm{h}})}+\\ \frac{{\psi }^{\mathrm{wc}}}{2}{\displaystyle \sum _{t=1}^T{\Vert {\overline{p}}_{}^{\mathrm{c}2\mathrm{w}}-{\overline{p}}_{}^{\mathrm{w}2\mathrm{c}}\Vert }_2^2}+\frac{{\psi }^{\mathrm{wh}}}{2}{\displaystyle \sum _{t=1}^T{\Vert {\overline{p}}_{}^{\mathrm{h}2\mathrm{w}}-p_{}^{\mathrm{w}2\mathrm{h}}\Vert }_2^2}\end{array}\\ \mathrm{s}.\mathrm{t}.{\omega }^{\mathrm{w},*}+{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{c},*}{\overline{p}}_{}^{\mathrm{w}2\mathrm{c}}}-{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{h},*}{\overline{p}}_{}^{\mathrm{w}2\mathrm{h}}}\ge U^{\mathrm{w},0,*}\end{array}.$$

(46)

(2)

Distributed optimal operation model of thermal power unit

$$\{\begin{array}{c}\min -\ln ({\omega }^{\mathrm{h},*}+{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{h},*}{\overline{p}}_{}^{\mathrm{h}2\mathrm{w}}})+{\displaystyle \sum _{t=1}^T{\gamma }_t^{\mathrm{wh}}({\overline{p}}_{}^{\mathrm{h}2\mathrm{w}}-{\overline{p}}_{}^{\mathrm{w}2\mathrm{h}})}+\frac{{\psi }^{\mathrm{wh}}}{2}{\displaystyle \sum _{t=1}^T{\Vert {\overline{p}}_{}^{\mathrm{h}2\mathrm{w}}-{\overline{p}}_{}^{\mathrm{w}2\mathrm{h}}\Vert }_2^2}\\ \mathrm{s}.\mathrm{t}.{\omega }^{\mathrm{h},*}+{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{h},*}{\overline{p}}_{}^{\mathrm{h}2\mathrm{w}}}\ge 0\end{array}.$$

(47)

(3)

Distributed optimal operation model of carbon capture equipment

$$\{\begin{array}{c}\min -\ln ({\omega }^{\mathrm{c},*}-{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{c},*}{\overline{p}}_{}^{\mathrm{c}2\mathrm{w}}})+{\displaystyle \sum _{t=1}^T{\gamma }_t^{\mathrm{wc}}({\overline{p}}_{}^{\mathrm{c}2\mathrm{w}}-{\overline{p}}_{}^{\mathrm{w}2\mathrm{c}})}+\frac{{\psi }^{\mathrm{wc}}}{2}{\displaystyle \sum _{t=1}^T{\Vert {\overline{p}}_{}^{\mathrm{c}2\mathrm{w}}-{\overline{p}}_{}^{\mathrm{w}2\mathrm{c}}\Vert }_2^2}\\ \mathrm{s}.\mathrm{t}.{\omega }^{\mathrm{c},*}-{\displaystyle \sum _{t=1}^T{\overline{g}}_t^{\mathrm{w}2\mathrm{c},*}{\overline{p}}_{}^{\mathrm{c}2\mathrm{w}}}\ge 0\end{array}.$$

(48)

The distributed algorithm 2 is established for benefit distribution sub-problem. The specific steps are as follows:

(1) Setting the max iterations times is $k^{\max }=50$. Convergence precision is $\xi ={10}^{-8}$. Penalty factors are ${\psi }^{\mathrm{wc}}={\psi }^{\mathrm{wh}}=1$. Iteration times is initialized to $\mathrm{k}=0$. Initial regulation service price expected by carbon capture equipment and thermal power unit to wind turbine are ${\overline{p}}_k^{\mathrm{c}2\mathrm{w}}={\overline{p}}_k^{\mathrm{h}2\mathrm{w}}=0$. Lagrange multipliers are ${\gamma }_t^{\mathrm{wc}}={\gamma }_t^{\mathrm{wh}}=0$.

(2) For wind turbine, according to regulation service price ${\overline{p}}_k^{\mathrm{h}2\mathrm{w}}$ and ${\overline{p}}_k^{\mathrm{c}2\mathrm{w}}$ expected by carbon capture equipment and thermal power unit, solve model (46) and obtain regulation service price ${\overline{p}}_{k+1}^{\mathrm{w}2\mathrm{h}}$ and ${\overline{p}}_{k+1}^{\mathrm{w}2\mathrm{h}}$ expected to pay.

(3) For thermal power unit, according to the regulation service price ${\overline{p}}_{k+1}^{\mathrm{w}2\mathrm{h}}$ expected by wind turbine, solve model (47) and obtain the regulation service price ${\overline{p}}_{k+1}^{\mathrm{h}2\mathrm{w}}$ expected to sell.

(4) For carbon capture equipment, according to the regulation service price ${\overline{p}}_{k+1}^{\mathrm{w}2\mathrm{c}}$ expected by wind turbine, solve model (48) and obtain the regulation service price ${\overline{p}}_{k+1}^{\mathrm{c}2\mathrm{w}}$ expected to sell.

(5) Update Lagrange multipliers according to Equation (49):

$$\{\begin{array}{c}{\gamma }_{t,k+1}^{\mathrm{wh}}={\gamma }_{t,k}^{\mathrm{wh}}+{\psi }^{\mathrm{wh}}({\overline{p}}_{k+1}^{\mathrm{h}2\mathrm{w}}-{\overline{p}}_{k+1}^{\mathrm{w}2\mathrm{h}})\\ {\gamma }_{t,k+1}^{\mathrm{wc}}={\gamma }_{t,k}^{\mathrm{wc}}+{\psi }^{\mathrm{wc}}({\overline{p}}_{k+1}^{\mathrm{c}2\mathrm{w}}-{\overline{p}}_{k+1}^{\mathrm{w}2\mathrm{c}})\end{array}.$$

(49)

(6) Update iterations times: k = k + 1;

(7) Judging convergence of the algorithm, if the termination condition (50) of iterative is satisfied, the iteration terminates.

$$\{\begin{array}{c}\max ({\displaystyle \sum _{t=1}^T{\Vert {\overline{p}}_{}^{\mathrm{c}2\mathrm{w}}-{\overline{p}}_{}^{\mathrm{w}2\mathrm{c}}\Vert }_2^2},{\displaystyle \sum _{t=1}^T{\Vert {\overline{p}}_{}^{\mathrm{h}2\mathrm{w}}-{\overline{p}}_{}^{\mathrm{w}2\mathrm{h}}\Vert }_2^2)<\xi }\\ \begin{array}{cc}\mathrm{or}& k>k_{\max }\end{array}\end{array},$$

(50)

Otherwise step 2 is repeated, until the convergence condition or the max iterations times is satisfied.

6. Case Analysis

6.1. Optimization Results of Independent Operation

In this paper, a 200 MW wind turbine was selected as the analysis object. The average value of typical daily wind speed in 12 months of a city was selected as the mean value of wind speed probability distribution. The standard deviation of wind speed data in each period was set to 0.65~0.95 m/s. The relationship between a wind turbine output and the wind speed shows as Equation (A1). Then, 1000 groups of sample data were generated by Monte Carlo simulation.

Based on the parameters of the wind turbine, the output of each period under 1000 sets of samples was calculated. IBM SPSS Statistics 21 software was used to cluster the wind turbine output data by k-means clustering method. 10 groups of basic sample data were generated by clustering and the corresponding probability values of different samples were calculated. The parameters of wind turbine and 10 groups of basic output sample data are in

Table A1. Basic parameters of the wind turbine.

	Rated Power/(MW)	Cut-in Wind Speed/(m/s)	Cut-out Wind Speed/(m/s)	Rated Wind Speed/(m/s)	Climbing Limit/(MW/h)
Value	200	2.8	22.8	12.5	60

,

Table A2. Clustering results of wind turbine power output.

Period	Power Output of the Wind Turbine/(MW)
	Sample1	Sample2	Sample3	Sample4	Sample5	Sample6	Sample7	Sample8	Sample9	Sample10
1	200	86	200	14	200	200	149	99	43	200
2	200	87	200	13	200	0	146	71	32	200
3	200	73	200	11	200	200	166	29	22	200
4	200	59	200	10	200	200	138	23	19	200
5	200	50	200	9	200	200	137	15	15	200
6	200	34	200	10	200	200	82	13	14	200
7	200	31	200	11	200	200	77	14	17	200
8	200	53	200	11	200	200	96	14	19	200
9	200	58	200	12	200	176	98	16	24	200
10	200	85	200	19	200	91	142	24	37	200
11	200	146	200	26	200	86	121	40	59	200
12	200	200	200	34	200	113	138	66	79	0
13	200	200	200	38	200	156	125	65	91	200
14	200	200	200	38	200	169	119	85	115	200
15	200	200	200	41	200	192	96	101	159	200
16	200	200	161	37	200	200	96	146	200	200
17	200	200	188	35	200	200	109	195	200	200
18	200	200	200	27	200	200	96	200	189	0
19	197	200	200	16	200	173	61	200	182	0
20	155	200	154	14	200	158	51	200	158	0
21	99	200	110	12	200	151	44	200	114	0
22	94	200	94	12	200	77	38	200	97	200
23	104	200	141	13	187	51	40	200	112	200
24	141	200	125	14	200	77	34	200	89	200

and

Table A3. Cluster center probability of wind turbine output.

	Sample1	Sample2	Sample3	Sample4	Sample5	Sample6	Sample7	Sample8	Sample9	Sample10
Probability	4.66%	13.42%	0.82%	35.07%	5.21%	1.64%	10.14%	2.47%	24.66%	1.92%

.

Based on the above clustering samples, Matlab R2020a software and Yalmip optimization toolbox were used to optimize the uncombined wind turbine electricity quantity declaration firstly. The optimization results are shown in

Table 1. Optimization results of wind turbine output declaration in independent operation.

Period	Electricity Quantity Declaration/(MWh)	Period	Electricity Quantity Declaration/(MWh)	Period	Electricity Quantity Declaration/(MWh)
1	149	9	98	17	200
2	146	10	142	18	200
3	166	11	146	19	200
4	138	12	200	20	200
5	137	13	200	21	200
6	82	14	200	22	200
7	77	15	200	23	200
8	96	16	200	24	200

:

In Table 1, based on the generation data and price data of the wind turbine in this paper, when the wind turbine was not combined with other flexible resources, decision makers of wind turbine chose to declare according to the maximum output during 12–24 periods. Further, based on the optimal declaration of electricity quantity, the positive and negative deviation electricity quantity in each period (based on the weighted results of sample occurrence probability) is as follows:

From Figure 2 and Figure 3, the on-grid electricity benefit of the uncombined wind turbine is 302,320 yuan. However, when the wind turbine pursuits profit maximized, there would be a large deviation between the declared electricity quantity and the actual electricity quantity, which would increase the cost of power system rescheduling and be disadvantageous to the safe and stable operation of power system. In addition, under the same penalty price for positive and negative deviation, the declared electricity quantity of the wind turbine was virtually high, that was, the negative deviation was significantly higher than the positive deviation.

6.2. Optimization Results of Cooperative Operation

Assuming that the planned power generation/consumption is half of the maximum power generation/consumption, the ADMM algorithm was used to realize the distributed solution of the two sub-problems. Relevant parameters for thermal power unit and carbon capture equipment were set out in

Table A4. Other basic parameters.

Parameters	Value	Parameters	Value
$p_t^{\mathrm{w}2\mathrm{g}}$/(¥/MWh)	400	$g_{}^{\mathrm{h},\max }$/(MWh)	300
$p_t^{\mathrm{w}2\mathrm{g},+}$/(¥/MWh)	600	$\Delta g_{}^{\mathrm{h},\mathrm{u},\max }$/(MWh)	125
$p_t^{\mathrm{w}2\mathrm{g},-}$/(¥MWh)	600	$\Delta g_{}^{\mathrm{h},\mathrm{d},\max }$/(MWh)	−125
$g_{}^{\mathrm{w},\max }$/(MWh)	200	${\delta }^{\mathrm{c}}$/(ton/MWh)	0.76
$\Delta g_{}^{\mathrm{w},\mathrm{u},\max }$/(MWh)	60	$s^{\mathrm{c}}$/(¥/ton)	105
$\Delta g_{}^{\mathrm{w},\mathrm{d},\max }$/(MWh)	−60	$c^{\mathrm{h}}$/(¥/ton)	270
${\delta }^{\mathrm{h}}$/(ton/MWh)	0.841	$g_{}^{\mathrm{c},\max }$/(MWh)	30
$c^{\mathrm{h}}$/(¥/MWh)	302.4	$\Delta g_{}^{\mathrm{c},\mathrm{u},\max }$/(MWh)	30
$p_t^{\mathrm{c}}$/(¥/ton)	58.5	$\Delta g_{}^{\mathrm{c},\mathrm{d},\max }$/(MWh)	−30

. Figure 4 shows convergence results of various participants’ benefit and alliance benefit. The algorithm converges after 12 iterations and the solving is highly efficient.

The total benefit of the alliance is 454,241 yuan in cooperative operation. Because thermal power unit and carbon capture equipment had incremental benefits, the total benefit of the combined system increased by 151,921 yuan. For the combined system, its increased benefit was from the decreased deviation penalty to the wind turbine, that was, the declaration deviation of the wind turbine was decreased in the case of thermal power unit and carbon capture equipment providing regulation services. The electricity declaration positive and negative deviation of the wind turbine are shown in Figure 5 and Figure 6.

When the wind turbine was not combined with other resources, the weighted sum of the wind turbine declaration positive deviation and declaration negative deviation were 114.31 MWh and 2032.70 MWh, respectively, with a total of 2147.01 MWh. After the wind turbine combined with thermal power unit and carbon capture equipment, the weighted sum of the wind turbine declaration positive deviation and declaration negative deviation were 599.49 MWh and 746.42 MWh, respectively, with a total of 1345.91 MWh. Compared with the uncombined case, the declaration deviation of the wind turbine was reduced by 801.10 MWh after combination.

The up-regulation electricity provided by thermal power unit and the down-regulation electricity provided by carbon capture equipment (expressed by negative number) are shown in Figure 7. To obtain more on-grid electricity income, the wind turbine had more negative deviation of electricity declaration before the league. So, thermal power unit provided a lot of up-regulation service to the wind turbine in period with more negative deviation, the trading volume was greater. Similarly, the wind turbine chose to trade with carbon capture equipment in 1, 3–9 periods with large positive deviation. By providing up and down regulation services, thermal power unit and carbon capture equipment obtained ancillary service income and reduced the declaration deviation of the wind turbine, which achieved tripartite win-win. In addition, the randomness of the wind turbine was reduced, which would also reduce the difficulty of power system rescheduling and improve the safety and reliability of power system.

Furthermore, Figure 8 shows the convergence results of regulation service transaction price in benefit distribution sub-problem. It can be seen that the algorithm converges after 11 and 14 iterations, respectively, which shows that the distributed algorithm, based on the ADMM, has good convergence characteristics for solving both operation optimization problem and benefit distribution problem. By the algorithm, the distributed and efficient solution of the two sub-problems can be realized while protecting the privacy information of each participant.

According to the convergence results, the transaction price of up-regulation service between the wind turbine and thermal power unit was 415.455 ¥/MWh. The transaction price of down-regulation service between the wind turbine and carbon capture equipment was 191.562 ¥/MWh. On the transaction electricity quantity of each participant, it could be calculated that the increased income of the wind turbine, thermal power unit and carbon capture equipment, which were 17,875 yuan, 120,770 yuan and 13,275 yuan, respectively. All participants had increased income.

6.3. Sensitivity Analysis

As an emerging technology, carbon capture has high technical iteration speed and great potential for cost reduction. According to the Annual Report of China Carbon Dioxide Capture Utilization and Storage (2021), the post-combustion capture cost in China will fall from 230–310 ¥/ton in 2025 to 190–280 ¥/ton and 70–120 ¥/ton in 2030 and 2060, respectively. In addition, for carbon capture projects, policy subsidy is an important part of cost recovery and an important way to reduce costs, especially in the early stages of technological development. In May 2021, China National Development and Reform Commission issued Pollution Control and Energy Conservation and Carbon Reduction within the Central Budget Investment Special Management Approach, which pointed out that investment in energy saving and carbon reduction projects did not exceed 15 per cent of total investment, and the central and national authorities provide full subsidies of 15 per cent for related projects. That is, China will provide subsidies for energy saving and carbon reduction projects, including carbon capture projects, up to 15% of the investment. In carbon capture revenue, the main revenue source is carbon market transactions. At present, carbon price level is low in China. According to the 2020 China Carbon Price Survey jointly issued by China Carbon Forum and ICF International Consulting Company, the average price of carbon market in China is expected to rise from 49 ¥/ton in 2020 to 71 ¥/ton in 2025, and to 93 ¥/ton in 2030.The average price expected in 2050 is 167 ¥/ton.

Unit cost and benefits of thermal power unit and wind turbine are relatively fixed, and their future changes will not be large. So, taking carbon cost and carbon price as sensitive factors, the corresponding changes of the system total benefit and carbon emissions under the sensitive factors changing were analyzed respectively. Carbon cost is essentially the difference between carbon capture costs and subsidy. The calculation results are shown in Figure 9.

Figure 9 shows corresponding changes of the system carbon emissions with carbon cost and carbon price changing. The carbon emissions show a downtrend. At the carbon price of 150–200 ¥/ton, the system carbon emission was negative, that is, the carbon captured quantity of carbon capture system was greater than the emission of thermal power unit at this point. In this time, the system was in increasing carbon sink state, and the system carbon emission was relatively insensitive to carbon cost. This was because compared with carbon cost, carbon price affects both thermal power unit and carbon capture equipment.

Figure 10 shows corresponding changes of the alliance benefit with carbon cost and carbon price changing. Same as system carbon emissions, the alliance benefit was also more sensitive to carbon price. With the carbon price increasing, the alliance benefit decreased significantly. When the carbon price and carbon cost were 200 ¥/ton, the alliance benefit reached the lowest level, which was 316,627 yuan. However, due to the premise of Nash equilibrium, the lowest alliance benefit was more than the independent operation benefit of wind turbine (302,320 yuan). Alliance participants were still profitable.

In summary, it can be known that compared with carbon cost decreasing, carbon price is the key factor affecting carbon emission and benefit of the alliance. Since thermal power unit are also affected by carbon prices, it can also be extended to the following conclusion. Under the goal to minimize carbon emissions of the alliance of wind turbines and CCPP, the policy-based emission-reducing effect by simply subsidizing to increase carbon sinks is worse than the market-based emission-reducing effect by improving carbon price. Further, the primary task of promoting the development of carbon capture technology is to promote the development of the carbon market and attract more entities to participate in carbon trading, so that carbon capture equipment can obtain reasonable benefits to support its sustainable development.

7. Conclusions

(1)

In this paper, the cooperative operation model of wind turbine and CCPP is converted into the operation optimization sub-problem and the benefit distribution sub-problem. The proposed distributed algorithm, based on the ADMM, has good convergence characteristics for solving above problems and can make the distributed and efficient solution of the two sub-problems realized while protecting the privacy information of each participant.

(2)

Under the deviation punishment mechanism, there are a lot of positive and negative deviations between the declared electricity and the actual electricity generation of wind turbine. By providing up and down regulation services to wind turbine, thermal power unit and carbon capture equipment can obtain ancillary service income and reduce the declaration deviation of wind turbine, which realizes multi-win-win situation.

(3)

Carbon price affects both thermal power unit and carbon capture equipment. So, compared with carbon cost, the carbon emission and the alliance benefit are both more sensitive to carbon price. Thus, for alliance of wind turbine and CCPP, the policy-based emission-reducing effect by simply subsidizing to increase carbon sinks is worse than the market-based emission-reducing effect by improving carbon price.