The Multi-Point Cooperative Control Strategy for Electrode Boilers Supporting Grid Frequency Regulation

Abstract

With the large-scale integration of wind power, photovoltaic, and other renewable energy sources into the power grid, their inherent randomness and variability present significant challenges to the frequency stability of power systems. Conventional thermal power units with limited frequency regulation capabilities face further strain, as frequent power fluctuations accelerate wear and tear, thereby shortening their operational lifespans. This makes it increasingly difficult to meet the demands for frequency regulation. Electrode boilers, as flexible electrical loads, can be retrofitted to enhance their flexibility and participate in grid frequency regulation alongside renewable energy units. This not only improves frequency stability but also reduces wear on generating units. However, the frequency regulation process involves balancing multiple objectives, such as maintaining system frequency stability, ensuring economic efficiency, and optimizing operational effectiveness. Traditional control strategies often struggle to address these competing objectives effectively. To address these challenges, this paper proposes a multi-objective collaborative optimization control decision model for electrode boilers to assist in grid frequency regulation. The model not only meets the frequency regulation requirements but also considers additional constraints, including the operational efficiency of electrode boilers, economic benefits, and equipment degradation. A genetic algorithm is employed to solve the model, and simulation analysis is conducted using the IEEE 14-node system. The results demonstrate that this strategy significantly enhances frequency stability, improves boiler operational efficiency, and boosts economic benefits, offering a viable solution for integrating electrode boilers into grid frequency regulation.

1. Introduction

With the large-scale integration of renewable energy sources, such as wind and photovoltaic power, into the power grid, frequency stability control in power systems is facing increasingly significant challenges. As wind and photovoltaic power, characterized by strong randomness, gradually replace traditional power-generation units, the demand for frequency regulation in power systems has markedly increased. Optimizing the dispatch of frequency regulation resources while ensuring grid frequency stability has become a hot topic in current research [1]. As a flexible load device, electrode boilers can rapidly respond to regulation commands and have gradually become some of the most important frequency regulation devices in power systems in recent years. During peak load periods, electrode boilers can absorb excess electrical energy, reducing pressure on the power system, and provide auxiliary services through appropriate power-regulation strategies when grid frequency fluctuates [2]. Additionally, electrode boilers offer advantages such as compact size, high thermal efficiency, low noise, and energy-saving environmental benefits, making them highly applicable in residential areas, schools, and other locations. When electrode boilers participate in power system regulation, it is essential to consider the time requirements and response accuracy for power adjustment while meeting the heating demands of the region where the boiler is located. Furthermore, the economic benefits of electrode boilers participating in power system regulation must be comprehensively evaluated. Therefore, the rational control of electrode boiler operation is of significant research and practical importance for enhancing the frequency regulation capability of power systems, improving frequency response efficiency, and achieving economic benefits.

Existing research predominantly focuses on single-objective optimization, such as optimizing the frequency regulation performance or economic benefits of electrode boilers, thereby neglecting the potential of electrode boilers in multi-objective collaborative control. For instance, reference [3] introduces an electrode boiler-assisted load-balancing system for thermal power units operating in islanded mode, designing decision-making and load-switching control strategies for islanded operation and optimizing turbine and boiler control strategies to ensure a stable transition to islanded operation during grid faults, demonstrating a robust load-impact resistance and accurate adjustments under various fault conditions, thereby ensuring grid safety and stability. Reference [4] employs scenario reduction methods to simulate the randomness and uncertainty of wind and photovoltaic power. It then minimizes the total operating cost as the objective function, conducting energy flow analysis on an integrated energy system comprising the IEEE 14-node power system and an 8-node thermal system. The YALMIP toolbox and CPLEX solver are used to determine the optimal capacity of electrode boilers and thermal storage devices. By comparing scenarios where electrode boilers are connected at different locations, it is concluded that placing electrode boilers and thermal storage devices on the heat source side of the thermal network optimizes system economics. Reference [5] constructs a thermal storage electrode boiler-load model based on thermal balance principles and analyzes the adjustable characteristics of a single thermal storage electrode boiler. It then investigates the mechanism and combined control methods for thermal storage electrode boilers to participate in wind power curtailment absorption. Finally, a model and control strategy for large-scale thermal storage electrode boilers to absorb curtailed wind power are proposed. Case studies show that, compared to control strategies constrained by the building’s heat demand, the proposed strategy not only maximizes wind power curtailment absorption but also achieves better long-term economic benefits. Reference [6] establishes a distributed unit model for an integrated energy system and proposes an optimal dispatch method considering carbon and green certificate trading parameters. It introduces a government-led, multi-party-participation trading mechanism and incorporates thermal storage electrode boilers to achieve thermoelectric decoupling through nighttime wind power-curtailment heating. Additionally, it improves the dispatch method for electric vehicles by shifting peak charging loads to off-peak periods, thereby enhancing wind power absorption and achieving peak shaving and valley filling, enabling joint source-load dispatch. Reference [7] develops an integrated wind storage–electric boiler system, considering the temporal energy shifting capability of energy storage and electrical–thermal complementarity. It establishes mathematical models for thermal units, CHP plants, energy storage, and thermal storage boilers, constructing a cost-minimized scheduling model with wind-curtailment penalties. The solution is obtained through particle swarm optimization under system constraints. Reference [8] proposes a wind-curtailment mitigation strategy for heating seasons using thermal storage concentrated solar power (CSP) plants integrated with electric boilers. By coordinating CSP plants (utilizing their thermal storage to shift heat supply) with electric boilers and CHP units, the strategy enhances CHP flexibility to improve wind power integration. Case studies on the IEEE-30 system and Gansu Hexi grid demonstrate that the approach effectively reduces wind curtailment, increases solar thermal utilization in CSP plants, and lowers total system costs. Reference [9] takes three typical electric heating loads—centralized heat-storage electric heating, direct electric heating, and distributed electric heating—as research subjects. Using a time-series production-simulation method, it constructs an economic evaluation model for wind power-curtailment accommodation based on real-time peak-shaving ancillary services. With actual operational data from electric heating projects in northern China as case studies, the paper compares the economic performance of these typical heating loads in accommodating wind power curtailment. It further analyzes the impacts of key factors such as ancillary service prices, wind-curtailment levels, and operational characteristics of heating loads on economic feasibility.

However, these single-objective optimization strategies struggle to comprehensively balance multiple objectives, such as system frequency stability, equipment operational efficiency, and economic benefits, and fail to fully leverage the role of electrode boilers in power system frequency regulation. Building on existing research, this paper proposes a multi-objective collaborative optimization control method. This method focuses on electrode boilers participating in grid power regulation and constructs a multi-objective optimization model that takes into account factors such as power system frequency stability, equipment operational efficiency, and economic benefits. Compared to traditional single-objective optimization methods, this approach achieves a better multi-objective balance during power support, providing a more efficient power-regulation solution and offering robust technical support for the flexible load dispatch of electrode boilers.

2. Multi-Point Collaborative Control Architecture for Electrode Boilers

2.1. Typical Control Architecture of Electrode Boilers

In a power system, frequency stability is the foundation for safe and reliable operation, while frequency instability is often caused by an imbalance of active power within the system. Specifically, when the active power supplied by the generation side does not match the demand on the consumption side, a frequency deviation occurs. Maintaining active power balance is a key component of the power system’s stability. Generation facilities need to adjust in real time according to load changes to ensure that generation matches load demand. For example, when there is a sudden increase in load (such as a rise in electricity demand during peak hours), if the generation side cannot promptly increase power output, the system frequency will drop, potentially triggering protection mechanisms in electrical equipment and even leading to power outages. Conversely, if the generation output is not reduced accordingly when the load decreases, the frequency will rise, which also threatens the system’s stability [10].

The typical regional power grid studied in this paper includes traditional units, renewable energy-generation units, and electrode boilers with distributed unified scheduling. A simplified dynamic model, as shown in Figure 1, is employed to simulate the operation of the electrode boilers, replacing the full-device model. Literature [11] has demonstrated the feasibility of this simplified dynamic model through a comparison of simulation results between the simplified and full-device models. It is assumed that the control area contains five electrode boilers, with n representing the electrode boiler index and t representing the frequency regulation moment. In Figure 1, $\mathrm{\Delta }{\mathrm{P}}_{\mathrm{G}}$ represents the frequency regulation output of traditional units; ${\mathrm{P}}_{\mathrm{sn}}$ represents the adjustment power of electrode boiler n participating in system frequency regulation; $\mathrm{\Delta }{\mathrm{f}}_{\mathrm{t}}$ represents the system frequency deviation; and ${\mathrm{P}}_{\mathrm{A}\mathrm{G}\mathrm{C}}$ represents the secondary frequency regulation demand signal, calculated by integrating the regional control error [12]. Based on the operational status of wind turbines and electrode boilers—including the installed capacity of power plants, ramp rate, rated power of the electrode boilers, and rated capacity of thermal storage tanks—the real-time operational conditions of the system are communicated to the control unit. Utilizing a genetic algorithm to solve the optimization mathematical model allows for the real-time distribution of power-adjustment commands.

2.2. Dynamic Response Model of Electrode Boilers and Generation Units

The dynamic response model of the unit is primarily designed to accurately describe the unit’s power dynamic response process after receiving power-adjustment commands. For different types of units, the dynamic response model includes common components such as regulation capacity limits, ramp rate, and frequency regulation delay [13], as well as transfer components that reflect the unit’s energy conversion characteristics. Currently, Automatic Generation Control (AGC) units commonly use frequency-domain models to describe the dynamic response process, as shown in Figure 2. Here, Td represents the secondary frequency regulation delay of the unit, and G(s) represents the unit’s power response transfer function, which typically takes the form shown in

Table 1. Dynamic response transfer function of different types of AGC units.

Type	Types of Transfer Functions
Synchronous Units	${\displaystyle \frac{1+{\mathrm{T}}_1\mathrm{s}}{(1+{\mathrm{T}}_2\mathrm{s})(1+{\mathrm{T}}_3\mathrm{s})(1+{\mathrm{T}}_4\mathrm{s})}}$
Wind and Solar Renewable Energy	${\displaystyle \frac{1}{1+{\mathrm{T}}_5\mathrm{s}}}$

. T1–T5 are the known parameters of the transfer function [14]. Consequently, by performing the inverse Laplace transform on the frequency-domain transfer function using the ilaplace function in MATLAB 2022a, the actual output of the regulating power in the time domain can be computed based on the input power, as follows:

$$\mathrm{\Delta }{\mathrm{P}}^{\mathrm{out}}(\mathrm{t})={\mathrm{L}}^{-1}[{\displaystyle \frac{{\mathrm{G}}_{\mathrm{i}}(\mathrm{s})}{\mathrm{s}(1+{\mathrm{T}}_{\mathrm{d}}^{\mathrm{i}}\mathrm{s})}}\cdot {\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{N}}[{\mathrm{e}}^{-\mathrm{\Delta }\mathrm{T}\cdot (\mathrm{k}-1)\mathrm{s}}\cdot {\mathrm{D}}_{\mathrm{i}}^{\mathrm{i}\mathrm{n}}(\mathrm{k})]}]$$

(1)

$${\mathrm{D}}_{\mathrm{i}}^{\mathrm{in}}(\mathrm{k})=\mathrm{\Delta }{\mathrm{P}}_{\mathrm{i}}^{\mathrm{i}\mathrm{n}}(\mathrm{k})-\mathrm{\Delta }{\mathrm{P}}_{\mathrm{i}}^{\mathrm{i}\mathrm{n}}(\mathrm{k}-1)$$

(2)

$$\mathrm{\Delta }{\mathrm{P}}_{\mathrm{i}}^{\mathrm{out}}(\mathrm{k})=\mathrm{\Delta }{\mathrm{P}}_{\mathrm{i}}^{\mathrm{out}}(\mathrm{t}=\mathrm{k}\cdot \mathrm{\Delta }\mathrm{T})$$

(3)

In the equation, i represents the i-th AGC unit; k represents the k-th discrete control cycle; ${\mathrm{P}}_{\mathrm{i}}^{\mathrm{i}\mathrm{n}}$ and ${\mathrm{P}}_{\mathrm{i}}^{\mathrm{out}}$ represent the input adjustment power command and the actual output of the adjustment power for the i-th AGC unit, respectively; and ΔT is the AGC control cycle, typically ranging from 1 to 16 s.

The modeling of electrode boilers primarily involves the study of temperature and power [15]. In this paper, we focus on modeling the power response of the electrode boiler. The power-simulation model of the electrode boiler can be approximately represented by a pure delay element and a first-order inertia element, with the transfer function expressed as follows:

$$\mathrm{G}(\mathrm{s})={\displaystyle \frac{\mathrm{k}}{1+{\mathrm{T}}_{\mathrm{s}}}}{\mathrm{e}}^{-\mathrm{\tau }\mathrm{s}}$$

(4)

In the equation, the time constant T represents the inherent characteristics of the electrode boiler, while the delay time accounts for the system’s delay effect. To verify the fast response characteristics of the electrode boiler and its ability to support the power system, the simulation of the electrode boiler’s response to Automatic Generation Control (AGC) regulation commands is as follows:

As shown in Figure 3, when the power-adjustment command is issued to the electrode boiler, it can adjust its power within the specified frequency regulation time to meet the power balance requirements of the power system.

3. Multi-Point Coordinated Control Model of Electrode Boilers

As a fast and flexible load device, the electrode boiler effectively contributes to power adjustments in the grid. However, in practical operation, it must not only meet daily heating demands but also consider the impacts of participating in power system adjustments. To ensure optimal frequency regulation performance, the multi-point coordinated control strategy for the electrode boiler should prioritize responding to frequency fluctuations in the grid to maintain system stability. Nevertheless, accurately tracking power-adjustment values necessitates continuous adjustments to the boiler’s power output, which increases losses due to frequent start-ups and shut-downs. Therefore, the multi-objective optimization mathematical model—considering both wind turbines and electrode boilers—primarily focuses on factors such as frequency regulation benefits, operating costs, heat loss, and frequency deviation.

3.1. Benefit Model

The frequency regulation compensation benefit encompasses both frequency regulation mileage compensation and frequency regulation capacity compensation, with frequency regulation mileage compensation serving as the primary source of the overall frequency regulation benefit [16]. This compensation depends on three factors: frequency regulation performance indicators, frequency regulation mileage, and compensation standards. That is,

$$\mathrm{I}={\mathrm{K}}_{\mathrm{p}}\mathrm{D}{\mathrm{B}}_{\mathrm{f}\mathrm{r}\mathrm{e}}$$

(5)

In Equation (5), D represents the frequency regulation mileage, which refers to the absolute value of the difference between the unit’s actual output and the output value at the time the AGC command was issued. The frequency regulation mileage D of the unit within a billing period is the sum of the adjustment amounts responding to the AGC control commands during that period, as shown in Figure 4. In a billing period, the total mileage is D = D1 + D2 + D3, where D1 = P1 − P0, D2 = P1 − P2, and D3 = P0 − P2. The mileage compensation standard ${\mathrm{B}}_{\mathrm{fre}}$ is the clearing price in the frequency regulation ancillary services market.

3.2. Cost Model

The operating cost of the electrode boiler primarily depends on factors such as its maximum capacity, the capacity of the thermal storage tank, and the cost of electricity purchases.

$${\mathrm{C}}_{\cos \mathrm{t}}={\alpha }_{\mathrm{i}}{\mathrm{P}}_{\mathrm{e}\mathrm{h}}^{\mathrm{o}\mathrm{u}\mathrm{t}}(\mathrm{t})+{\beta }_{\mathrm{i}}|{\mathrm{P}}_{\mathrm{c}\mathrm{h}}^{\mathrm{o}\mathrm{u}\mathrm{t}}(\mathrm{t})-{\mathrm{H}}_{\mathrm{e}\mathrm{l}}(\mathrm{t})|$$

(6)

In the equation, ${\alpha }_{\mathrm{i}}$ and ${\beta }_{\mathrm{i}}$ represent the operation cost coefficients for the electric heating part and the thermal storage device of the i-th electrode boiler, respectively; ${\mathrm{P}}_{\mathrm{e}\mathrm{h}}^{\mathrm{o}\mathrm{u}\mathrm{t}}(\mathrm{t})$ represents the output heat power of the electric heating part of the electrode boiler during the t-th period, in MW; ${\mathrm{H}}_{\mathrm{e}\mathrm{l}}(\mathrm{t})$ represents the thermal load allocated to the electrode boiler, in MW; and ${\mathrm{P}}_{\mathrm{c}\mathrm{h}}^{\mathrm{o}\mathrm{u}\mathrm{t}}(\mathrm{t})$ represents the operating power of the thermal storage device of the i-th electrode boiler during the t–th period, in MW.

The electricity purchase cost comprises two parts: the cost of wind power curtailment and the cost of purchasing low-cost electricity from the grid when the heat generated from the consumed wind power fails to meet the system’s heat-load demand. Therefore, the electricity purchase cost can be expressed as follows:

$${\mathrm{F}}_{\mathrm{g}}={\mathrm{Q}}_1{\mathrm{C}}_{\mathrm{q}\mathrm{f}}+{\mathrm{Q}}_2{\mathrm{C}}_{\mathrm{g}}$$

(7)

In the equation, ${\mathrm{Q}}_1$ represents the consumed curtailed wind power, in MW·h; ${\mathrm{C}}_{\mathrm{q}\mathrm{f}}$ represents the price of curtailed wind power; ${\mathrm{Q}}_2$ represents the amount of low-cost electricity purchased from the grid; and ${\mathrm{C}}_{\mathrm{g}}$ represents the price of low-cost electricity.

The construction cost of the electrode boiler is mainly related to the maximum heating power of the electrode boiler and the maximum capacity of the thermal storage device.

$${\mathrm{F}}_{\mathrm{j}}={\mathrm{f}}_1{\mathrm{P}}_{\mathrm{e}\mathrm{b},\max }^{\mathrm{i}\mathrm{n}}+{\mathrm{f}}_2{\mathrm{S}}_{\mathrm{c}\mathrm{h},\max }$$

(8)

In the equation, ${\mathrm{f}}_1$ represents the construction cost coefficient of the electrode boiler heating device; ${\mathrm{P}}_{\mathrm{e}\mathrm{b},\max }^{\mathrm{i}\mathrm{n}}$ represents the maximum heating power of the electrode boiler; ${\mathrm{f}}_2$ represents the construction cost coefficient of the thermal storage device; and ${\mathrm{S}}_{\mathrm{c}\mathrm{h},\max }$ represents the maximum capacity of the thermal storage device.

3.3. Loss Model

Under a fixed temperature difference, heat loss is proportional to time. As the output power of the electrode boiler changes, the amount of hot water stored in the electrode boiler and thermal storage tank will also increase, while the basic heat demand remains unchanged. This results in an increased response time in the thermal storage tank which, in turn, leads to greater thermal losses.

$${\mathrm{Q}}_{\mathrm{loss}}=\mathrm{U}\cdot \mathrm{A}\cdot \mathrm{\Delta }\mathrm{T}\cdot \mathrm{t}$$

(9)

where U represents the heat transfer coefficient of the pipeline (W/m²·°C), which is related to the pipe material and thickness; A represents the surface area of the pipeline; $\mathrm{\Delta }\mathrm{T}$ represents the temperature difference between the hot water and the environment; and t represents the residence time of the hot water in the pipeline.

3.4. Frequency Model

The frequency stability control objective of this paper is to quantify the effectiveness of frequency control by analyzing the mean squared error (MSE) between the given power-adjustment command and the actual response power. A smaller MSE indicates a better power balance between the generation and consumption sides, helping maintain the system frequency close to the preset standard value. By implementing reasonable control strategies to reduce the MSE, the frequency stability of the power system can be effectively improved, thereby minimizing operational risks.

4. Multi-Point Coordinated Optimization Model of Electrode Boilers

4.1. Optimization Objective Function

In optimization problems, multiple conflicting objectives often need to be addressed simultaneously. To simplify this, the optimization algorithm used in this paper applies a weighted sum method to convert multi-objective optimization into single-objective optimization. The original problem includes three optimization objectives: frequency response deviation ${\mathrm{f}}_1$, economic cost ${\mathrm{f}}_2$, and heat loss ${\mathrm{f}}_3$. These objectives are in conflict with each other; for example, reducing heat loss may increase operational and maintenance costs. To convert the above multi-objective problem into a single-objective problem, a weighted sum method is used. By assigning weights to each objective, multiple objectives are combined into a single objective function:

$$\mathrm{F}={\mathrm{w}}_1{\mathrm{f}}_1+{\mathrm{w}}_2{\mathrm{f}}_2+{\mathrm{w}}_3{\mathrm{f}}_3$$

(10)

In this case, ${\mathrm{w}}_1$, ${\mathrm{w}}_2$, and ${\mathrm{w}}_3$ are non-negative weights representing the relative importance of each objective, with the sum of the three weights equal to 1. By adjusting these weights, the priorities between different objectives can be balanced. In the optimization problem discussed in this paper, according to electricity market rules and contractual agreements, an electrode boiler cluster that fails to fulfill secondary frequency regulation tasks as required will face economic penalties. Sensitivity analysis reveals that the power response deviation is most sensitive to weight variations, with maximum fluctuations reaching ±50%. Heat loss is the second most sensitive factor (±28%), primarily influenced by weights w3 and w1. The total cost demonstrates relative robustness (±6.6%), slightly affected by w2 and w3. By setting the weight values to 0.4, 0.3, and 0.3, respectively, the system can achieve frequency regulation targets while balancing other optimization objectives (

Table 2. Weight sensitivity analysis.

Weight Combination	Power Deviation (kW)	Total Cost (¥)	Heat Loss (MJ)
Base Weight [0.4, 0.3, 0.3]	5.2	6800	2.5
w1↑ [0.48, 0.26, 0.26]	3.1 (−40%)	7120 (+4.7%)	2.8 (+12%)
w1↓ [0.32, 0.34, 0.34]	7.9 (+52%)	6520 (−4.1%)	2.1 (−16%)
w2↑ [0.34, 0.36, 0.30]	6.5 (+25%)	6380 (−6.2%)	2.6 (+4%)
w2↓ [0.46, 0.24, 0.30]	4.0 (−23%)	7250 (+6.6%)	2.9 (+16%)
w3↑ [0.34, 0.30, 0.36]	6.8 (+31%)	6950 (+2.2%)	1.9 (−24%)
w3↓ [0.46, 0.30, 0.24]	2.8 (−46%)	7180 (+5.6%)	3.2 (+28%)

).

Based on the actual operational constraints of the electrode boiler, the power-adjustment requirements of the power grid, and factors such as pipeline heat loss, the objective function is formulated with the goals of minimizing power response deviation, maximizing economic benefits, and reducing heat loss, as follows:

$$\left\{\begin{array}{l}\min {\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}({\displaystyle \sum _{\mathrm{t}=1}^{\mathrm{T}}{\mathrm{P}}_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{d},\mathrm{i}}-{\mathrm{P}}_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{e},\mathrm{i}}{)}^2}}\\ \max {\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}({\displaystyle \sum _{\mathrm{t}=1}^{\mathrm{T}}{\mathrm{S}}_{\mathrm{sell},\mathrm{i}}+{\mathrm{S}}_{\mathrm{subsidy},\mathrm{i}}-{\mathrm{C}}_{\cos \mathrm{t},\mathrm{i}}})}\\ \min {\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{Q}}_{\mathrm{loss},\mathrm{i}}}\end{array}\right.$$

(11)

In the equation, ${\mathrm{P}}_{\mathrm{coomand},\mathrm{i}}$ represents the response power value of each electrode boiler; ${\mathrm{P}}_{\mathrm{response},\mathrm{i}}$ represents the power adjustment command value issued by the control center; ${\mathrm{S}}_{\mathrm{sell}}$ represents the heating revenue from the electrode boiler; ${\mathrm{S}}_{\mathrm{subsidy},\mathrm{i}}$ represents the subsidy for supporting grid frequency regulation; ${\mathrm{C}}_{\cos \mathrm{t},\mathrm{i}}$ represents the operating cost of the electrode boiler; and ${\mathrm{Q}}_{\mathrm{loss},\mathrm{i}}$ represents the thermal loss in the thermal storage tank and pipeline of the electrode boiler.

As shown in Section 3.3, the heat loss of the electrode boiler is proportional to time. To minimize heat loss, power-adjustment command values should be distributed as evenly as possible among the electrode boilers. Alternatively, based on historical heat-demand data, areas with higher heat demand should be assigned more power-adjustment commands.

4.2. System Operational Constraints

In addition to considering the dynamic response transfer process of the units, the power-distribution process must also account for power balance constraints, electrode boiler capacity constraints, and ramp rate constraints, as detailed below.

$${\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{P}}_{\mathrm{a}\mathrm{f}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{d},\mathrm{i}}}={\mathrm{P}}_{\mathrm{command}}$$

(12)

$${\mathrm{P}}_{\mathrm{b}\mathrm{o}\mathrm{i}\mathrm{l},\mathrm{i},\min }\le {\mathrm{P}}_{\mathrm{b}\mathrm{o}\mathrm{i}\mathrm{l},\mathrm{i}\mathrm{ni}\mathrm{t}}+{\mathrm{P}}_{\mathrm{b}\mathrm{o}\mathrm{i}\mathrm{l}\mathrm{e}\mathrm{r},\mathrm{a}\mathrm{f}\mathrm{f}\mathrm{o}}\le {\mathrm{P}}_{\mathrm{b}\mathrm{o}\mathrm{i}\mathrm{l},\mathrm{i},\max }$$

(13)

$${\mathrm{P}}_{\mathrm{b}\mathrm{o}\mathrm{i}\mathrm{l},\mathrm{i}}(\mathrm{t})-{\mathrm{P}}_{\mathrm{b}\mathrm{o}\mathrm{i}\mathrm{l},\mathrm{i}}(\mathrm{t}-1)\le \mathrm{R}\mathrm{amp}\_\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}$$

(14)

$${\mathrm{Q}}_{\mathrm{i}\mathrm{n}}=\eta {\mathrm{P}}_{\mathrm{b}\mathrm{o}\mathrm{i}\mathrm{l},\mathrm{a}\mathrm{f}\mathrm{f}\mathrm{o}}$$

(15)

$${\mathrm{Q}}_{\tan \mathrm{k},\min }\le {\mathrm{Q}}_{\mathrm{in}}+{\mathrm{Q}}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}\le {\mathrm{Q}}_{\tan \mathrm{k},\max }$$

(16)

$${\mathrm{S}}_{\mathrm{subsidy}}={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{P}}_{\mathrm{a}\mathrm{f}\mathrm{f}\mathrm{o}}\cdot \mathrm{K}}$$

(17)

$${\mathrm{C}}_{\cos \mathrm{t}}={\mathrm{C}}_{\mathrm{base}}+{\mathrm{P}}_{\mathrm{a}\mathrm{f}\mathrm{f}\mathrm{o}}\cdot \mathrm{t}+{\mathrm{P}}_{\mathrm{cons}}$$

(18)

$${\mathrm{Q}}_{\mathrm{loss}}={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{T}}(\mathrm{Q}\cdot {\eta }_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}})}$$

(19)

In the equation, $\mathrm{N}$ represents the number of electrode boilers; ${\mathrm{P}}_{\mathrm{affo},\mathrm{i}}$ represents the power value allocated to each electrode boiler; ${\mathrm{P}}_{\mathrm{command}}$ represents the power-adjustment command of the power system; ${\mathrm{P}}_{\mathrm{boil},\mathrm{i},\min }$ represents the minimum daily heating value guaranteed by the electrode boiler; ${\mathrm{P}}_{\mathrm{boil},\mathrm{i},\max }$ represents the maximum power of the electrode boiler; $\mathrm{R}\mathrm{amp}\_\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}$ represents the ramp rate of the electrode boiler; ${\mathrm{P}}_{\mathrm{boil},\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}$ represents the initial power value of the electrode boiler; $\eta$ represents the electrical-to-thermal conversion efficiency of the electrode boiler; ${\mathrm{Q}}_{\mathrm{i}\mathrm{n}\mathrm{it}}$ represents the initial capacity of the electrode boiler; ${\mathrm{Q}}_{\mathrm{i}\mathrm{n}}$ represents the heat generated by the electrode boiler in response to power adjustment; ${\mathrm{Q}}_{\tan \mathrm{k},\min }$ represents the minimum thermal value required by the thermal storage tank to meet the heat demand; ${\mathrm{Q}}_{\tan \mathrm{k},\max }$ represents the maximum thermal value required by the thermal storage tank to meet the heat demand; ${\mathrm{S}}_{\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{y}}$ represents the subsidy obtained by the electrode boiler for participating in power adjustment; $\mathrm{K}$ represents the subsidy coefficient; ${\mathrm{C}}_{\cos \mathrm{t}}$ represents the cost of the electrode boiler participating in power adjustment; ${\mathrm{C}}_{\mathrm{base}}$ represents the basic operating cost of the electrode boiler; ${\mathrm{P}}_{\mathrm{cons}}$ represents the operation and maintenance cost of the electrode boiler; ${\mathrm{Q}}_{\mathrm{loss}}$ represents the heat loss in the heating pipeline and thermal storage tank; and ${\eta }_{\mathrm{loss}}$ represents the heat loss coefficient.

5. Optimization Algorithm Design

5.1. Basic Principle of the Algorithm

The genetic algorithm (GA) [17] is an optimization method based on natural selection and genetic principles. It simulates the biological evolution process, searching for optimal solutions within the solution space through operations such as selection, crossover, and mutation. The basic steps of the genetic algorithm include individual initialization, fitness evaluation, selection, crossover, mutation, and termination condition assessment. In the GA, each individual represents a potential solution, and the fitness function evaluates the quality of that solution. Through iterative evolution, the GA generates a new population in each generation, gradually improving solution quality via selection, crossover, and mutation mechanisms.

The fitness function of the genetic algorithm can be expressed as follows:

$$\mathrm{F}\mathrm{itness}={\displaystyle \frac{1}{1+\mathrm{F}}}$$

(20)

Here, F represents the objective function value. The goal of this fitness function is to maximize the fitness value, which corresponds to minimizing the power-distribution error, thereby optimizing the power-distribution strategy.

5.2. Algorithm Application Design

This study addresses the AGC frequency regulation scenario involving five electrode boilers and thermal storage tanks, requiring coordinated multi-device power allocation under second-level dynamic constraints. Traditional methods struggle to resolve the highly nonlinear, multi-constrained problem formed by the thermal inertia of the boilers, charge/discharge rates of the thermal storage tanks, and grid frequency regulation requirements. By designing a GA encoding strategy based on feasible solution initialization and incorporating dynamic penalty functions to handle ramp rate constraints, the GA can generate feasible solutions within 30 s. Additionally, the GA supports flexible extension of the objective function, laying the foundation for multi-objective optimization.

In the implementation of the genetic algorithm, the first step is to determine the encoding method for the individuals. In the context of the electrode boiler power-optimization allocation problem, each individual represents a power-allocation scheme. The power value of each electrode boiler can be represented using real-number encoding. For example, if there are N electrode boilers in the system, the power value for each boiler can be expressed as a real-number vector:

$$\mathrm{X}=[{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4,\dots ,{\mathrm{x}}_{\mathrm{N}}]$$

(21)

where ${\mathrm{x}}_{\mathrm{i}}$ represents the power value of the i-th electrode boiler. To ensure the rationality of the power distribution, a power range constraint can be set for each electrode boiler to ensure that each ${\mathrm{x}}_{\mathrm{i}}$ stays within a certain range. The fitness evaluation of each individual is based on the electrode boiler power-distribution error function. For each individual, the power distribution error FFF is calculated based on its corresponding power-allocation value, and then the fitness is evaluated using the aforementioned fitness function.

The selection operation in the genetic algorithm can utilize either roulette wheel selection or tournament selection. Roulette wheel selection allocates selection probabilities based on the fitness values of individuals, with those having higher fitness being more likely to be selected. Tournament selection, on the other hand, randomly selects a group of individuals and chooses the one with the best fitness for reproduction. The crossover operation can employ either single-point or multi-point crossover, which involves exchanging segments of genes between two parent individuals to generate new offspring. The mutation operation introduces random changes to an individual’s gene values, enhancing population diversity, typically with a low probability of mutation. The termination condition of the genetic algorithm can be set to either reach a predetermined maximum number of iterations or to stop when the fitness value of the best individual in the population exceeds a certain threshold. Throughout the implementation process, the algorithm continuously iterates and updates the population until the termination condition is met. Ultimately, the algorithm returns the individual with the highest fitness as the optimization result, providing the optimal power-distribution plan for the electrode boilers and achieving optimized power distribution for AGC commands.

Due to the power constraints of electrode boilers located at different positions and the constant cycle capacity of thermal storage devices, coupled with the randomness of the scheduling plan’s population, it is crucial to adjust the electrode boiler heat-supply ratio and heat storage plan. Without these adjustments, there may be instances of constraint violations, such as excessive heat generation by an electrode boiler in a specific location. Therefore, this paper aims to filter out populations from the initial set that meet the system’s constraints and generate an initial population based on historical heating data, guiding it toward the desired outcomes. This selected population will produce valid electrode boiler power-distribution plans and heating ratios, which will subsequently be optimized through scheduling. To address the multi-objective optimal scheduling problem, this paper employs an improved genetic algorithm that incorporates an elitism strategy. This strategy ensures that the best solution from each generation is preserved and carried over to the next generation without undergoing crossover or mutation. As a result, the improved genetic algorithm converges faster [18], and the overall process is illustrated in Figure 5.

The proposed GA has a time complexity of $\mathrm{O}(\mathrm{G}\cdot \mathrm{N}\cdot \mathrm{D})$, where $\mathrm{G}$, $\mathrm{N}$, and $\mathrm{D}$ represent generations, population size, and variables, respectively. This aligns with the theoretical complexity of the canonical GA [19] but benefits from vectorized operations to reduce constant factors. On a 14-core CPU platform, a genetic algorithm (GA) with a population size of 30 and 200 iterations took 28.5 s, where fitness evaluation accounted for 92% of the total time. By introducing historical data to pre-warm the population, the computation time was further reduced to 18 s, meeting the real-time requirements for AGC.

Pseudo-code is as Algorithm 1.

Algorithm 1. Genetic Algorithm (GA) for Optimizing Power Allocation of Electrode Boilers

Input: P_AGC, P_init, P_max, Q_max, C_base, c, alpha, weights

Output: best_position: Optimal power allocation best_value: Optimal objective function value

Step 1: Initialize Genetic Algorithm Parameters Initialize population size (n_population), number of generations (n_generations), etc.

Step 2: Generate Initial Population Create n_population individuals, randomly initialize power values, and normalize according to P_max

Step 3: Define Fitness Evaluation Function 1. Calculate objective functions (f1: power difference, f2: operation and maintenance cost, f3: heat loss) 2. Normalize the objective functions 3. Combine the objective functions into a comprehensive fitness value F based on weights 4. Clip power values to ensure they are within the [P_init, P_max] range

Step 4: Selection Operation 1. Calculate selection probabilities based on fitness 2. Select individuals for the next generation based on fitness proportion

Step 5: Crossover Operation 1. Randomly perform crossover on parent individuals 2. Exchange genes to generate offspring individuals

Step 6: Mutation Operation 1. Randomly mutate power values of some individuals

Step 7: Update Population Replace the current population with the new offspring individuals

Step 8: Repeat Steps 3–7 until the maximum number of iterations is reached

Step 9: Return the optimal solution (best_position, best_value)

6. Case Study Analysis

6.1. Case System Description

In this paper, simulations are conducted on the load frequency control model of the IEEE 14-bus system. One synchronous generator unit is replaced with a wind turbine unit, and an electrode boiler, along with a thermal storage device, is added to another synchronous generator unit. The specific topology is illustrated in Figure 6.

Table 3. Parameters of the dynamic response transfer function for AGC units.

Type	Transfer Function Parameters
Synchronous Units	${\mathrm{T}}_1$ = 2, ${\mathrm{T}}_2$ = 0.05, ${\mathrm{T}}_3$ = 5, ${\mathrm{T}}_4$ = 0.2
Wind Turbine Units	${\mathrm{T}}_5$ = 0.01

presents the main parameters for energy transfer and power regulation of the units [20].

The system’s thermal load is jointly supplied by an electrode boiler and a thermal storage tank. The heating area covers 2 million square meters, with a daily heating duration of 10 h, resulting in a total heating load of 1960 MWh. The operation and maintenance cost ratio for the electric thermal storage device is 1.27%, and the power-conversion efficiency is 98%. The parameters for the electrode boiler and thermal storage device are sourced from a manufacturer of electrode boilers and are detailed in

Table 4. Main technical parameters of the electrode boiler and thermal storage tank.

Parameter ZHP-2850	Electrode Boiler Model	Parameter RITEC-C50000-95	Thermal Storage Tank Model
Rated Power/MW	50	Thermal Storage Power/MW	60
Maximum Heat Supply/MW	50	Heat Release Power/MW	60
Minimum Heat Supply/MW	50	Maximum Thermal Storage Capacity/MWH	600
Thermal Efficiency/%	98	Minimum Thermal Storage Capacity/MWH	100

.

6.2. Frequency Response Characteristics

Assume that the power system dispatch center in the region issues a power-regulation command with ΔPL = 60 MW, and the wind turbine unit and electrode boiler respond according to their operating conditions. Figure 7 presents the convergence curve of the objective function value under the AGC power command using the genetic algorithm. As shown in the figure, excessively small population sizes may lead to premature convergence, while overly large populations increase computational costs. When the population size is set to 30, the algorithm achieves a high-quality optimal solution within 15 iterations, with subsequent iterations demonstrating only marginal refinements. This evidences the algorithm’s rapid convergence capability.

The grid dispatch center allocates the total Automatic Generation Control (AGC) command using either an equal distribution method or a proportional distribution method based on unit capacity. The adjusted load for each unit, obtained through both the equal distribution method and the performance-based distribution method for the electrode boiler, is presented in

Table 5. Load-allocation results.

Allocation Strategy	Equipment Type	AGC Adjustment Command/MW
Proportional Allocation	Electrode Boiler 1	10
	Electrode Boiler 2	5
	Electrode Boiler 3	5
	Electrode Boiler 4	10
	Electrode Boiler 5	9.46
	Generating Unit	20
Multi-point Cooperative Optimization	Electrode Boiler 1	5.08
	Electrode Boiler 2	15.32
	Electrode Boiler 3	15.4
	Electrode Boiler 4	12.08
	Electrode Boiler 5	1.81
	Generating Unit	10

.

In Table 5, when the electrode boiler and wind turbine share the power-adjustment command using the proportional allocation method, the command value does not account for the current operating conditions of the equipment. This oversight may result in scenarios where the rated power is high, but the adjustable power is relatively small. Furthermore, during periods of high wind and photovoltaic power generation, reducing their output could decrease economic efficiency, thereby failing to meet the optimization objectives. The data in the table represent the optimal solution obtained by the optimization algorithm, which comprehensively considers multiple optimization objectives.

The comparison of the electrode boiler AGC command response before and after optimization demonstrates that, with the introduction of the optimized command distribution proposed in this paper, the control algorithm achieves a smaller power deviation while effectively preventing overshoot in the total power command. Additionally, the electrode boiler, recognized for its rapid response capability, can manage more power disturbances during the disturbance phase and quickly restore balance to the system (Figure 8).

As shown in Figure 9, the system frequency stabilizes around 45 s. With the proportional allocation method, the lowest point of the system frequency reaches 49.93 Hz, while the lowest point using the optimized strategy proposed in this paper is 49.91 Hz, indicating a smaller impact on the system. The rapid adjustment capability of the electrode boiler allows the system to provide effective, fast support under high-power loss disturbance conditions, thereby reducing the duration of low-frequency operation. In contrast, traditional scheduling algorithms typically rely on proportional power allocation, which limits the electrode boiler’s full potential for rapid power support. Consequently, the frequency recovery rate in traditional scheduling algorithms is slower compared to the strategy proposed in this paper.

To further evaluate the performance of the algorithm, the optimization results of different algorithms are compared, as shown in

Table 6. Comparison of optimization algorithm effects.

$\mathbf{\Delta }{\mathit{P}}_{\mathbf{L}}/\mathbf{MW}$	Optimization Algorithm	Power Deviation/MW	$\mathbf{\Delta }\mathbf{f}/\mathbf{HZ}$
60	PROP	0.54	0.09
	PSO	0.35	0.075
	GA	0.3	0.07

. Here, ACE and Δf represent the average values over the simulation time, while power deviation refers to the total deviation over the simulation time. Accuracy is used to measure the closeness between the actual adjustment output and the adjustment command curve during the simulation period. From Table 6, the following can be observed: (1) Compared with PROP, which has no optimization method, both optimization methods significantly reduce power deviation, thereby improving the system’s dynamic response performance indicators; (2) Compared with the PSO algorithm, the GA algorithm yields better online optimization results, indicating that the GA algorithm is more suitable for solving the multi-point coordinated control problem of electrode boilers.

6.3. Economic Benefit Comparison

Based on the frequency regulation ancillary service market “Trading Rules” pricing of a certain province, this paper assumes the upper and lower limits of the quoted price to be 15 ¥/MW and 5.5 ¥/MW, respectively. Considering the current market clearing price, the revenue calculation in this paper uses 10 ¥/MW. During this case study, the real-time frequency regulation compensation price, electricity price, heat price, and coal price are shown in

Table 7. Market prices during the frequency regulation process.

Type	Frequency Regulation Compensation Price/¥	Electricity Price/¥	Heat Price/¥	Coal Price/¥
Price	10	0.4	44	700

.

Based on the electrode boiler revenue model and construction cost model presented in Section 3, the supply electricity revenue, heat revenue, frequency regulation income, and power-generation cost are calculated both before and after the optimized allocation of the electrode boiler. This analysis evaluates the impact of the proposed optimized allocation strategy on economic benefits. Among these, the ancillary frequency regulation revenue is positively correlated with the frequency regulation compensation price, heat price, and coal price, while negatively correlated with the electricity price and the coefficient of equipment operating costs. According to the calculation model referenced in [21], the unit electricity operating cost for wind turbine units after several years of operation is ¥0.166 per kWh.

As shown in

Table 8. Heat revenue of the electrode boiler.

Type	Frequency Regulation Revenue/¥	Operating Cost/¥	Single Frequency Regulation Revenue/¥
PROR	600	360	240
PSO	600	335	265
GA	600	333	267

, the optimized strategy proposed in this paper allocates more Automatic Generation Control (AGC) frequency regulation commands to the electrode boilers under suitable operating conditions. The heat revenue generated by the electrode boiler is higher than that of traditional units, primarily due to its efficient use of electrical energy, which enables rapid conversion of electricity into heat with quick response times, thereby reducing overall energy costs. Additionally, the electrode boiler features a simple structure and lower maintenance costs, contributing to reduced repair expenses. Its flexible operating mode allows for quick responses to load changes, and if heat supply is provided during periods of lower electricity prices, operational costs can be further minimized.

6.4. Heat Loss Analysis

The heat loss of the electrode boiler is proportional to time. Based on an analysis of historical heat-consumption data, this paper selects areas with higher heat demand to assume more power adjustment commands. This approach ensures that the heat generated by the power adjustments of the electrode boiler is maximally utilized, thereby minimizing heat loss.

In Figure 10, in the initial solution of the optimization algorithm, a higher initial power is allocated to electrode boilers located in areas with known higher heat demand. This approach guides the optimization algorithm to start from a biased solution set. Simulation results indicate that, after undertaking the same frequency regulation commands, the proposed strategy significantly reduces heat loss. Overall, the proposed allocation method effectively minimizes heat loss.

7. Conclusions

To mitigate the impact of large-scale integration of renewable energy on the frequency stability of power systems, this paper examines the differences in power support characteristics between wind turbine units and electrode boilers. A mathematical model is developed to meet system frequency regulation requirements, economic benefits, and operational constraints of the units. The model is solved using a genetic algorithm, achieving fast, efficient, and accurate distribution of power-adjustment commands. Simulation results demonstrate that the multi-point coordinated control strategy proposed in this paper—considering the operating conditions of both wind turbines and electrode boilers—can quickly yield high-quality control solutions with strong convergence stability. This approach not only enhances the frequency stability of the power grid but also significantly improves the operational efficiency and economic benefits of electrode boilers. While ensuring regional heating, it also bolsters the dynamic response performance of the entire regional power grid. Future research will explore the linkage of cross-regional power resources to further enhance the overall benefits of the power system.