Demand Response of Integrated Zero-Carbon Power Plant: Model and Method

Abstract

An integrated zero-carbon power plant aggregates uncontrollable green energy, adjustable load, and storage energy resources into an entity in a grid-friendly manner. Integrated zero-carbon power plants have a strong demand response potential that needs further study. However, existing studies ignore the green value of renewable energy in power plants when participating in demand response programs. This paper proposed a mathematical model to optimize the operation of an integrated zero-carbon power plant considering the green value. A demand response mechanism is proposed for the independent system operator and the integrated zero-carbon power plants. The Stackelberg gaming process among these entities and an algorithm based on dichotomy are studied to find the demand response equilibrium. Case studies verify that the mechanism activates the potential of the integrated zero-carbon power plant to realize the load reduction target.

1. Introduction

An integrated zero-carbon power plant (IZP) effectively consolidates uncontrollable green electricity, adjustable loads, and energy storage resources into a grid-friendly entity. Green electricity primarily refers to various types of green energy, such as residential photovoltaic (PV), micro-wind turbines, biomass power generation, heat pumps, green hydrogen, etc. Adjustable loads mainly refer to charging and discharging piles, swapping and discharging stations, electric vehicles, cold storage, and large air-conditioning users [1] (p. 32). Energy storage mainly involves adopting electrical, thermal, cold, and hydrogen technologies. Integrated zero-carbon power plants exhibit significant potential for further exploration in the field of demand response. They can provide great potential for load reduction if the power demand exceeds supply in the peak load period.

As an approximate representation, the interactive operational mode of a virtual power plant (VPP) with the power grid holds valuable reference significance [2] (pp. 230–231). Reference [3] proposed a new framework for the optimal virtual power plant energy management problem considering correlated demand response. Reference [4] proposed a demand response real-time pricing method in VPP considering the smooth renewable energy consumption requirement. Reference [5] focused on the optimal demand response strategy of a commercial building-based virtual power plant with real-world implementation in a heavily urbanized area. Reference [6] developed a data mining-driven incentive-based demand response scheme to model electricity trading between a VPP and its participants, which induces load curtailment of consumers by offering them incentives and also makes maximum utilization of distributed energy resources. Reference [7] developed a data-driven approach for VPP resource planning, in which BES sizing and demand response customer selection are optimized synergistically to maximize VPPs’ profit in the electricity market. Reference [8] investigated the impact of using a demand response program and battery energy storage system on the VPP’s internal electricity market and also cost-minimization analysis from a utility viewpoint. However, they failed to account for the green value associated with renewable energy in the IZP. Consequently, the current studies cannot fully harness the IZPs’ potential when participating in demand response programs.

Regarding demand response strategies, price-based demand response is the primary category. In the price-based demand response model [9] (p. 2870), load aggregators employ fluctuating retail electricity prices to influence adaptable loads, such as air conditioning loads, to optimize their electricity consumption schedules. This includes three primary methods: time-of-use, real-time, and critical peak pricing. Reference [10] proposed robust optimization methods to analyze optimization strategies for the electricity consumption schedules of thermostatically controlled loads during demand response periods when retail electricity prices exhibit a high correlation. Reference [11] further employed stochastic robust optimization methods to investigate real-time energy management strategies for load aggregators in price-based demand response, focusing on thermostatically controlled loads. Reference [12] developed a comprehensive single-family residential air conditioning model based on OpenStudio and EnergyPlus and formulated dynamic demand response control strategies based on real-time published retail electricity prices. Incentive-based demand response is another vital category of demand response, where demand response implementers devise plans to reduce user loads and provide economic compensation to users. Reference [13] proposed a distributed direct load control scheme based on a dual-layer communication control architecture, enhancing the accuracy of load reduction for residential users participating in demand response. Reference [14] developed a direct load control execution framework based on smart sensors, optimizing execution through smart sensors to minimize the discomfort of electricity users. Reference [15] introduced an interruptible load management method based on dynamic optimal power flow, optimizing interruptible load schemes while considering factors such as advance notification time and short-term and long-term price discounts. Reference [16] aimed to minimize generation costs and load shedding costs and proposed an optimized combination scheme where generators and interruptible loads jointly share the system’s reserve capacity.

In the above-mentioned demand response programs, the gaming process between the independent system operator (ISO) and an IZP is usually a Stackelberg game. The equilibrium of the Stackelberg game can be calculated to obtain the optimal demand response strategies. An iterative algorithm is the most adopted method to calculate this equilibrium since the IZP information is usually unavailable for the ISO, which is responsible for the power dispatch to maintain the supply and demand balance of the entire power system. In a limited information environment, leaders cannot access the reaction functions of followers. Typically, numerical iterative methods are used to find the equilibrium solution of Stackelberg games. Reference [17] proposed a demand response model for load aggregators and electricity users based on coupon incentives. They also introduced an iterative algorithm based on traversal methods to find the optimal incentive settings in the equilibrium state. Reference [18] considered uncertainties in load demand, spot prices, electricity user privacy, and line flow and proposed an optimization algorithm for residential user demand response using deep reinforcement learning. Reference [19] proposed an optimization solution that combines long-term memory networks and reinforcement learning algorithms to design demand response incentive strategies for electricity retailers when user response behavior is unknown. Reference [20] introduced an incentive-based real-time compensation setting strategy using deep learning and reinforcement learning algorithms to improve the operational security and economics of the power system. The aforementioned studies help market participants gradually learn the optimal equilibrium solution of Stackelberg games in a limited information environment through iterations. However, trial and error interactions with the environment usually require huge attempts to achieve a precise expensive solution.

The contributions of the paper are shown in Figure 1.

(1)

A mathematical model that considers the variable green value of renewable energy output is introduced for IZP operations.

(2)

A demand response mechanism is explored for IZPs through a Stackelberg game. IZPs make a great contribution to reducing the peak load in the power grid by interacting with the ISO.

(3)

An iterative algorithm is put forward based on the idea of dichotomy aiming at identifying the optimal demand response strategy in the incomplete information environment of the gaming process efficiently and precisely.

The paper is organized as follows: the model of integrated smart zero-carbon power plants and the ISO is in Section 2, the proposed demand response mechanism and the algorithm to calculate the optimal strategy are designed in Section 3, Section 4 demonstrates case studies, and Section 5 concludes the entire paper.

2. Model of Integrated Smart Zero-Carbon Power Plants and the ISO

The role of the ISO is to perform a power dispatch algorithm to maintain the power supply and demand balance. The ISO will forecast the load profile and set the load reduction target of IZPs in the peak load period. The ISO will transmit the load reduction target to the IZPs participating in the demand response programs. Upon receiving the load reduction signal, the IZPs will formulate the optimal operational plan considering the green value of renewable energy. The IZPs will receive financial compensation in the load reduction program. Note that the compensation rate is usually fixed beforehand in bilateral contracts [21] (p. 557) (not discussed in our article). The data interaction between the ISO and IZPs is illustrated in Figure 2.

2.1. Integrated Zero-Carbon Power Plant Model

Integrated zero-carbon power plants typically incorporate a control center responsible for overseeing a multitude of power generators, power consumers, and energy storage devices, ensuring the efficient operation of the entire power plant while minimizing the operation costs.

(1)

Energy storage device model

The relationship between charging (discharging) power and the energy level in the energy storage device at each period is given as:

$$S_{k,t}^{\mathrm{es}}=S_{k,t-1}^{\mathrm{es}}+\left(P_{k,t}^{\mathrm{cha}}-P_{k,t}^{\mathrm{dis}}\right)\Delta t$$

(1)

The charging and discharging power inequality constraints and energy storage limits constraints of energy storage device k during period t can be expressed as:

$$\{\begin{array}{l}0⩽S_{k,t}^{\mathrm{es}}⩽S_k^{\mathrm{es},\max }\\ 0⩽P_{k,t}^{\mathrm{cha}}⩽U_{k,t}^{\mathrm{cha}}{P}_k^{\mathrm{es},\max }\\ 0⩽P_{k,t}^{\mathrm{dis}}⩽(1-U_{k,t}^{\mathrm{cha}})P_k^{\mathrm{es},\max }\\ U_{k,t}^{\mathrm{cha}}\in \{0,1\}\end{array}$$

(2)

Note that a value of 1 indicates that the energy storage device is in the charging state during period t, and a value of 0 indicates that the energy storage device is discharging during period t.

(2)

Green energy producer model

The green energy production cost of renewable energy power generator i is written in a linear form:

$$C_i^{\mathrm{gG}}={\displaystyle \sum _{t=1}^{24}{c}_i^{\mathrm{gG}}{P}_{i,t}^{\mathrm{gG}}\Delta t}$$

(3)

The profit of renewable energy power generator i to sell renewable energy to the grid is given as:

$$E_{i,\mathrm{grid}}^{\mathrm{gG}}={\displaystyle \sum _{t=1}^{24}\left(e_{t,\mathrm{grid}}^{\mathrm{gG}}+e_{t,\mathrm{grid}}^{\mathrm{gGC}}\right)P_{i,t,\mathrm{grid}}^{\mathrm{gG}}\Delta t}$$

(4)

The green value of renewable energy at period t can be written as an elastic form based on the Cournot model [22] (p. 694):

$$e_{t,\mathrm{grid}}^{\mathrm{gGC}}=e_{\max ,\mathrm{grid}}^{\mathrm{gGC}}-{\chi }_t{\displaystyle \sum _{i=1}^I{P}_{i,t,\mathrm{grid}}^{\mathrm{gG}}}$$

(5)

The balance constraint of power generation in the IZP is:

$${\displaystyle \sum _{i\in \Xi }{P}_{i,t}^{\mathrm{gG}}}={\displaystyle \sum _{i\in \Xi }\left(P_{i,t,\mathrm{self}}^{\mathrm{gG}}+P_{i,t,\mathrm{grid}}^{\mathrm{gG}}\right)}+{\displaystyle \sum _{k\in \mathsf{\Omega }}{P}_{k,t}^{\mathrm{cha}}}$$

(6)

(3)

Energy user model

The energy consumption utility function of energy user j is given in a quadratic form:

$$U_j^{\mathrm{gL}}={\displaystyle \sum _{t=1}^{24}\left(d_j^{\mathrm{gL}}{P}_{j,t}^{\mathrm{gL}}-l_j^{\mathrm{gL}}{\left(P_{j,t}^{\mathrm{gL}}\right)}^2\right)\Delta t}$$

(7)

The cost of purchasing electricity from the power grid is written in a linear form:

$$C_{j,\mathrm{grid}}^{\mathrm{gL}}={\displaystyle \sum _{t=1}^{24}{e}_{t,\mathrm{grid}}^{\mathrm{gL}}{P}_{j,t,\mathrm{grid}}^{\mathrm{gL}}\Delta t}$$

(8)

The upper and lower limits of green electricity consumption for power user j are:

$$P_{j,\min }^{\mathrm{gL}}<P_{j,t}^{\mathrm{gL}}<P_{j,\max }^{\mathrm{gL}}$$

(9)

The balance constraint of power consumption in the IZP is:

$${\displaystyle \sum _{j\in \mathsf{\Psi }}{P}_{j,t}^{\mathrm{gL}}}={\displaystyle \sum _{j\in \mathsf{\Psi }}\left(P_{j,t,\mathrm{self}}^{\mathrm{gL}}+P_{j,t,\mathrm{grid}}^{\mathrm{gL}}\right)}+{\displaystyle \sum _{k\in \mathsf{\Omega }}{P}_{k,t}^{\mathrm{dis}}}$$

(10)

(4)

Dispatching model of the IZP

Typically, the primary objective of the IZP operator is to maximize the overall social welfare or minimize the social cost associated with the entire IZP system:

$$\min C_{\mathrm{total}}={\displaystyle \sum _{i\in \Xi }{C}_i^{\mathrm{gG}}}+{\displaystyle \sum _{j\in \mathsf{\Psi }}{C}_{j,\mathrm{grid}}^{\mathrm{gL}}}-{\displaystyle \sum _{j\in \mathsf{\Psi }}{U}_j^{\mathrm{gL}}}-{\displaystyle \sum _{i\in \Xi }{E}_{i,\mathrm{grid}}^{\mathrm{gG}}}$$

(11)

The equality constraint is as follows:

$${\displaystyle \sum _{i\in \Xi }{P}_{i,t,\mathrm{self}}^{\mathrm{gG}}}={\displaystyle \sum _{j\in \mathsf{\Psi }}{P}_{j,t,\mathrm{self}}^{\mathrm{gL}}}$$

(12)

2.2. Load Reduction Target of the ISO

The ISO will curtail the load of IZPs during peak load forecasting periods in the power system, thus enhancing the power grid’s operational efficiency.

Assume that the forecasting peak load of the power system on a day before the demand response program is $P_{\mathrm{peak},\mathrm{before}}$; the peak load of the power system on a day after the demand response program is $P_{\mathrm{peak},\mathrm{after}}$; the load reduction amount in the peak load period is $L_{\mathrm{reduce}}$.

The relationship of the above three variables is:

$$L_{\mathrm{reduce}}=P_{\mathrm{peak},\mathrm{before}}-P_{\mathrm{peak},\mathrm{after}}$$

(13)

The relationship between the load reduction of each IZP and the load reduction target of the entire system is:

$${\displaystyle \sum _{m=1}^M{L}_{m,\mathrm{reduce}}}=L_{\mathrm{reduce}}$$

(14)

The utility of the ISO to reduce peak load can be written in a quadratic form.

$$U_{\mathrm{reduce}}={\eta }_1{L}_{\mathrm{reduce}}-{\eta }_2{L^2}_{\mathrm{reduce}}$$

(15)

3. Demand Response Mechanism and the Algorithm to Calculate the Optimal Strategy

The interaction between the ISO and the IZP can be described as a typical Stackelberg game [23] (p. 121). In this gaming process, the ISO initiates by issuing a load reduction signal, after which the IZP determines its load reduction quantity.

3.1. Stackelberg Game between ISO and the IZP in the Demand Response Programs

The Stackelberg game between the ISO and the IZPs is given in Figure 3.

(1)

Follower problem of IZP

The ISO determines the load reduction target of the IZP and sends this load reduction plan to each IZP. The constraints in the optimization problem of the IZP m should include the following equality constraints:

$$\left({\displaystyle \sum _{j\in \mathsf{\Psi }}{P}_{j,t_{\mathrm{peak}},\mathrm{grid},\mathrm{after}}^{\mathrm{gL}}}-{\displaystyle \sum _{i\in \Xi }{P}_{i,t_{\mathrm{peak}},\mathrm{grid},\mathrm{after}}^{\mathrm{gG}}}\right)-\left({\displaystyle \sum _{j\in \mathsf{\Psi }}{P}_{j,t_{\mathrm{peak}},\mathrm{grid},\mathrm{before}}^{\mathrm{gL}}}-{\displaystyle \sum _{i\in \Xi }{P}_{i,t_{\mathrm{peak}},\mathrm{grid},\mathrm{before}}^{\mathrm{gG}}}\right)=L_{m,\mathrm{reduce}}$$

(16)

Note that the additional constraints will narrow the feasible region of the optimization problem of the IZP and will cause an increase in the cost. The increase in the cost of the IZP before and after the demand response programs (or the decrease in the profit of the IZP) is:

$$C_{m,\mathrm{IZP},\mathrm{increase}}=C_{m,\mathrm{IZP},\mathrm{after}}-C_{m,\mathrm{IZP},\mathrm{before}}$$

(17)

(2)

Leader problem of ISO

The objective function of ISO is to determine the optimal load reduction target to balance the load reduction utility and the increase of IZP’s cost:

$$\max U_{\mathrm{ISO},\mathrm{total}}=\omega U_{\mathrm{reduce}}-\left(1-\omega \right){\displaystyle \sum _{m=1}^M{C}_{m,\mathrm{IZP},\mathrm{increase}}}$$

(18)

Note that a larger $\omega$ indicates that the load reduction utility is more important than the loss of IZP’s profit, and a smaller $\omega$ indicates that the loss of IZP’s profit is more important than the load reduction utility.

3.2. Algorithm Based on the Principle of Dichotomy to Calculate the Optimal Demand Response Strategy

The iterative algorithm used to compute the equilibrium in the Stackelberg game relies on the principles of dichotomy. Dichotomy is a method that systematically divides the search zone into smaller steps, a technique commonly employed in computer science to search for items within large datasets.

In Figure 4, the blue and yellow parts denote the whole search area and the target, respectively. The green triangles and black lines denote the positions of dichotomy. The main idea is continuously dichotomizing the search area to locate the target. Generally, the algorithm’s time complexity is O(logn) when the searching area is linear.

The proposed algorithm, which the ISO implements for the IZPs, consists of seven steps, which are shown in Figure 5.

Step 1:

ISO sets the range [L_m,reduce,min, L_m,reduce,max] of the load reduction target of IZP m in the demand response program.

Step 2:

ISO initializes the iteration time n = 1 at the beginning of the iteration algorithm.

Step 3:

ISO bisects the action selection range, in other words, the range of the load reduction target.

Sub-range 1 of the load reduction target.

[L_m,reduce,min, (L_m,reduce,max + L_m,reduce,min)/2]

Sub-range 2 of the load reduction target.

[(L_m,reduce,max + L_m,reduce,min)/2, L_m,reduce,max]

Step 4:

ISO selects the mid-point in the two intervals of the load reduction target as follows:

Mid-point 1

T1 = (L_m,reduce,max + 3 × L_m,reduce,min)/2

Mid-point 2

T2 = (3 × L_m,reduce,max + L_m,reduce,min)/2

Step 5:

ISO compares the reward value of these two points U_ISO,total(T1) and U_ISO,total(T2):

If U_ISO,total(T1) > U_ISO,total(T2).

Since the probability of obtaining the optimal value in the left part is larger than that in the right part, the searching region is narrowed to the left part. Thus, the upper limit of the action selection range will be renewed as (L_m,reduce,max + L_m,reduce,min)/2, while the lower limit of the action selection range remains.

If U_ISO,total(T1) < U_ISO,total(T2).

Since the probability of obtaining the optimal value in the right part is larger than that in the left part, the searching region is narrowed to the right part. Thus, the lower limit of the action selection range will be renewed as (L_m,reduce,max + L_m,reduce,min)/2, while the upper limit of the action selection range remains.

Step 6:

ISO renews the iteration time n = n + 1.

Step 7:

ISO compares the value of n and N.

If n = N, end the algorithm.

If n < N, turn to step 3.

The above iterative algorithm employed for computing the Stackelberg game equilibrium is based on the principles of dichotomy to reduce the computational complexity exponentially.

4. Case Study

4.1. Parameter Settings

This demand response program consists of one ISO and three IZPs. Each IZP has six power generators, six power users, and six energy storage devices, and the parameter settings for these components are identical within each IZP.

The coefficient to produce per unit of renewable energy $c_i^{\mathrm{gG}}$ is set to 0.01 USD/kWh. The unit price to buy electricity from the power grid is set to 0.06 USD/kWh. The unit value to sell electricity to the power grid is set to USD 0.005/kWh. The elastic coefficient of green value is set to 0.001 USD/kWh. In an IZP, the maximum amount of power generation is around 390 kW, and the minimum amount of power generation is around 150 kW. The mean value of the upper limits of the six power generators in the IZP is given in Figure 6.

The two coefficients in the utility function of power users are 12 and 0.04, respectively. The mean value of the upper limits of the six power consumers in the IZP is given in Figure 7. In an IZP, the maximum value of the upper load-consuming limit is around 1140 kW, and the minimum value of the upper load-consuming limit is around 180 kW.

The upper energy storage limit for each energy storage device within the IZP is configured at 20 kWh, and the upper charging and discharging power limits are set to the same value of 5 kW.

Figure 8 provides the specific numerical value of green energy. The IZP’s unit price for selling renewable energy to the power grid also fluctuates, ranging from 0.035 USD/kWh to 0.044 USD/kWh.

4.2. Simulation Results

Assuming that ISO’s forecasting peak load period for the next day is at the 16th hour, Figure 9 and Figure 10 illustrate the relationship between the load reduction target set by ISO for each IZP and ISO’s overall utility.

Figure 9 shows the changing process of the ISO’s total utility as each iteration progresses. Specifically, the ISO’s utility is increasing by optimizing the load reduction target step by step. After the third iteration, the total utility is approaching stable at USD 2974.

Figure 10 shows the changing process of the ISO’s load reduction target as each iteration progresses. Specifically, the load reduction target selected by the ISO falls first, then rises, and finally stabilizes. After the sixth iteration, the load reduction target for each IZP is fixed at 110 kW, and the optimal total load reduction amount of the three IZPs is 330 kW.

The Stackelberg game equilibrium is obtained within 10 iterations, which is much faster than the traversal search algorithm (around 10²–10³ iterations). In other words, this algorithm significantly reduces computational complexity and enhances computational accuracy.

Figure 11 shows the total loads of the power system before and after demand response programs. In the 16th hour, the load of the power system is reduced from the peak value of 2809 kW to a lower value of 2479 kW, which is 11.75% less than the peak load. Furthermore, the load in the adjacent periods also changes to guarantee global optimality. Consequently, the load is reduced by 30 kWh in the 15th hour while increasing by 30 kWh in the 17th hour because of load shedding in the peak load period.

The power bought and sold to the power grid by each IZP is given in

Table 1. Detailed value of power bought from the power grid and power sold to the power grid by each IZP.

Time (h)	Sold Amount (kW)	Bought Amount (kW)
1	63.1	132.7
2	78.9	141.6
3	154.7	258.6
4	82.1	155.8
5	151.9	262.1
6	140.9	250.8
7	100.8	0
8	102.7	0
9	150.8	88.5
10	100.8	0
11	100.8	0
12	131.9	90.7
13	95.2	604.6
14	55.3	535.7
15	93.1	583.0
16	77.7	454.9
17	100.8	647.3
18	70.5	544.5
19	107.9	445.6
20	93.3	328.0
21	119.7	374.4
22	119.7	404.2
23	106.9	348.1
24	66.3	308.5

. The amount of power sold to the power grid by each IZP reached the peak value of 154.7kW at the 3rd hour and reached the valley value of 55.3kW at the 14th hour. The power bought from the power grid by each IZP is given in Table 1. The amount of power bought from the power grid by each IZP reached the peak value of 647.2kW in the 17th hour.

The total energy stored in the energy storage devices of each IZP is given in Figure 12. The maximum energy level in the energy storage devices of each IZP reaches the peak value at 77 kW in the 13th hour and the 14th hour. During the operation of the IZP, the energy storage devices act as a buffer to connect the power generation and consumption more smoothly.

To sum up, the ISO’s role is to mitigate the peak load within the power grid to ensure the secure operation of the entire power system, while the IZP devises the optimal internal operational plan in response to the load reduction signal.

5. Conclusions

This paper presents a comprehensive approach to executing demand response programs between IZPs and ISO. Firstly, a mathematical model is proposed to optimize the operation of IZPs, considering the green value of renewable energy resources. Next, a demand response mechanism is introduced for both the ISO and the IZPs to reduce peak loads precisely, and the Stackelberg gaming process between the ISO and multiple IZPs is modeled. Finally, an algorithm based on dichotomy is proposed to calculate the equilibrium. Through demand response, the load level of the power system is reduced by 330 kW in the peak load period, and the total utility of the ISO is improved to USD 2974. The proposed iterative algorithm based on dichotomy converges to the equilibrium within 10 times, much faster than the traversal search methods. Through these efforts, the power system’s load is efficiently reduced, fully harnessing the potential of IZPs in demand response.

Future work will focus on the following two main aspects: developing a more complicated model for IZPs’ internal components, and establishing an underlying architecture for implementing demand response to enhance communication and computational efficiency.