Fast-Frequency-Response Control Method for Electrode Boilers Supporting New Energy Accommodation

: With the large-scale integration of new energy generation, represented by wind and photovoltaic power, into the power grid, the intermittency, randomness, and fluctuations of their output pose significant challenges to the safe and stable operation of the power system. Therefore, this paper proposes a control method for electrode boiler systems participating in rapid grid frequency response based on a fuzzy control strategy. This method improves the traditional electrode boiler control strategy, giving it characteristics similar to those of synchronous generators in terms of active power–frequency droop, allowing it to actively adjust active power based on system frequency disturbances. Furthermore, it optimizes its control performance indicators using fuzzy algorithms. The simulation results on the Matlab/Simulink platform demonstrate that the modified electrode boiler control system, when applying this method, can effectively address power disturbances in the system, reduce system frequency deviations, and contribute to enhancing the grid frequency regulation capability and system stability.


Introduction
In recent years, new energy generation technologies represented by wind power and photovoltaics have rapidly developed, and the installed capacity has been increasing year by year.As of 2022, China's newly added installed capacity of wind power and photovoltaic power generation reached 125 million kilowatts, breaking through 100 million kilowatts for three consecutive years, reaching a historical high.The electricity generation from wind power and photovoltaics reached 11.9 trillion kilowatt-hours, an increase of 207.3 billion kilowatt-hours compared to 2021, representing a year-on-year growth of 21%.This accounted for 13.8% of the total electricity consumption in the entire country, an increase of two percentage points compared to the previous year.
However, due to the random, intermittent, and fluctuating nature of renewable energy generation, there has been a significant increase in the need for flexible adjustments in the power system.This has led to a substantial occurrence of the curtailment of wind and solar power in some regions.At the same time, with the large-scale development of renewable energy, a significant number of conventional power generation units have been replaced by renewable energy units.However, these renewable energy units, when connected to the grid through power electronic components, cannot provide effective rotational inertia.This results in a decrease in the overall system inertia, leading to a deterioration in the system's frequency and voltage regulation capabilities.This situation presents a serious challenge to the safe and stable operation of the power system [1].
To enhance the integration capacity of high proportions of renewable energy, clean heating has been vigorously promoted in northern China, implementing electricity substitution to increase the space for accommodating renewable energy generation.Researchers have conducted extensive studies on this topic.In [2], a dynamic optimization scheduling method was proposed for an electric-thermal combined heat and power system that considers the optimal control of combined heat and power units along with electric boilers.This method aims to reduce the thermal-electric coupling and enhance the system's regulation capability.By establishing models for the electric and thermal output of combined heat and power units that consider the optimal control process, the impact of load variations on the thermal-electric characteristics and unit pressure safety was analyzed.Based on this analysis, optimization scheduling was carried out with the constraint of minimizing the system's operational cost.The article demonstrated that considering the optimal control of combined heat and power units and the optimal scheduling of electric boilers can promote wind power consumption and improve the overall economic performance of the system.In [3], an optimal economic scheduling model for a cogeneration energy system was proposed to minimize the total operating cost and the amount of abandoned air of an electric boiler.The proposed model considers the thermal characteristics of the building and the constraints of electrothermal balance, cogeneration units, heat storage tanks, electric boiler power, and wind power generation.Based on this model, a linear optimization model was established to analyze the different scheduling strategies under three working conditions: cogeneration; cogeneration and heat storage; and cogeneration, a heat storage unit, and an electric boiler.Taking a six-node cogeneration system as an example, the effectiveness of the model and the solution method was verified, and the influence of the heat storage tank and the electric boiler on wind abandonment and coal consumption was illustrated.In [4], a multisource system unit combination model considering high wind power penetration with thermal storage electric boilers was presented.This model harnesses the energy storage capabilities of thermal storage electric boilers to perform load shifting, thereby improving the wind power integration capacity and the economic operation of the system.In [5], a scheme for accommodating excess wind power using energy coupling devices in a comprehensive energy system was studied.The use of thermal storage electric boilers as energy coupling devices decouples the heating network from the electrical grid.Models for energy storage electric boilers and control strategies were established to support combined heat and power plants in meeting their heat demand while reducing their electrical output, thus increasing the utilization of wind power.In [6], to address the issue of challenging wind power integration in China's Three-North region, a multiobjective optimization model for wind power regulation was established, based on a wind power-thermal storage electric boiler heating system.The objectives of this model were to maximize wind power regulation capacity while minimizing the number of electric boiler regulation events and operational costs.An optimized solution was designed using an improved multiobjective particle swarm algorithm.This proposed approach was validated based on real engineering data and demonstrated its ability to reduce operational costs and enhance the capacity for wind power integration.In [7], in order to improve the acceptability of curtailment, a scheduling model of indirect heating coordination between a regenerative electric boiler and energy storage battery was established.The selection process of the electric boiler electrode was optimized according to the characteristics of curtailment, and the optimal operation strategy of the hybrid energy storage system was proposed.Based on the operation data of a real wind farm, different control methods were compared and analyzed.The results show that the ability of the power system to accept wind power can be further improved under the requirement of meeting heating demand and reducing the amount of electric boiler electrode regulation.
The abovementioned studies primarily focus on using electrode boilers for heating to increase the integration capacity of renewable energy generation.However, they do not address the active participation of electrode boilers in power system frequency response and control.At the same time, in view of the frequency stability of the power system under the high proportion of new energy access, researchers try to transform the power electronic converter to give it a certain frequency support ability.In [8], an improved differential sag plus damping control strategy based on an island microgrid was proposed.The dynamic sag mechanism was established to realize the flexible sharing of transient power among distributed generation, so that the response of the distributed generation to disturbance is flexible and stable.In [9], an optimal droop control of an H-Infinity controller based on an improved artificial bee colony algorithm was proposed, which improves power sharing among parallel inverters, realizes power response optimization and the voltage and frequency stability of a microgrid, and makes the performance of the network more robust to external interference.Compared with traditional droop control and the traditional particle swarm optimization algorithm, the improvement and accuracy of the method were verified.
Based on the above ideas, this paper further studies the feasibility and technical scheme of using the electrode boiler to rapidly adjust the control characteristics and participate in the fast frequency response of the power system.By applying fuzzy control to enhance the control performance of electrode boilers, this paper aims to imbue them with active power-frequency droop characteristics under certain control strategies.This enables them to actively adjust active power based on system frequency deviations, thereby supporting the safe and stable operation of the power system with a high proportion of renewable energy integration.

The Working Principle of Electrode Boiler Primary Frequency Control
Primary frequency control in a power system refers to the automatic operation of the governing systems of generating units when the system's frequency deviates from its rated frequency.This automatic operation adjusts the active power output of generators to keep the system frequency within an acceptable range.Primary frequency control in a power system is characterized by its rapid response and proportional control, primarily addressing short-term power fluctuations with quick response times.Electrode boilers, as large-capacity loads, can exhibit adjustable droop characteristics through specific control strategies, allowing them to participate in primary frequency control and contribute to frequency stabilization within the system.The working principle of their involvement in primary frequency control is illustrated in Figure 1.In Figure 1, a f is the upper limit of the system frequency adjustment dead zone; b f is the lower limit of the system frequency adjustment dead zone; The relationship between the electrode boiler power response and frequency deviation is shown in Equation ( 1): where f  represents the frequency deviation; From the above figure, it can be seen that when ab f f f  , the system assumes that the frequency deviation is within the normal range.The electrode boiler operates in a conventional state and does not participate in system frequency regulation.When La f f f  , according to Equation (1), the power setpoint is obtained, and the electrode boiler reduces its corresponding power, delivering power to the grid.When L ff  , the electrode boiler stops working, and the heating is supplied by hot water stored in the heat storage tank, with all the power previously supplied by the electrode boiler now directed to the grid.When bH f f f  , based on Equation (1), the power set-point is obtained, and the electrode boiler increases its corresponding power, absorbing power from the grid.When H ff  , the electrode boiler operates at maximum power, absorbing power from the grid.If there is excess heat, it can be stored in the heat storage tank.During this time, it can be used in conjunction with market bidding strategies to lower heating prices, incentivize heat consumption by users, and enhance response flexibility [10].

The Rapid Response Characteristics of Electrode Boilers
The power of the electrode boiler is primarily influenced by the electrical conductivity of the water and the water level inside the boiler.When the electrical conductivity of the water in the boiler is constant, the power can be controlled through water level adjustments.Based on the deviation between the power and the setpoint, a controller acts on the valve after the circulation pump, regulating the makeup water flow to control the water level inside the inner drum.This achieves the goal of power control.The formula for calculating the power of the electrode boiler is as follows [11]: By optimizing the control parameters of the electrode boiler, the system can reach the frequency-stable state at about 17 s.According to the "Implementation Rules for Power Grid Operation and Management in the East China Region", in systems primarily consisting of thermal power units, the response time for primary frequency control, i.e., the time it takes for the primary frequency control power increase to reach 90% of the target power increase, should be less than 30 s.Therefore, it can be inferred that the power response time of the electrode boiler meets the timeliness requirements for primary frequency control.
Due to the inherent time lag in thermal systems, when the frequency fluctuations cross the frequency control dead band, making short-term adjustments to the power of the electrode boiler to participate in frequency control does not significantly impact the heating supply of the thermal system.This capability allows the electrode boiler to possess primary frequency control characteristics while still meeting the constraints of the thermal system.

Fuzzy Controller
Fuzzy control is a control method based on fuzzy logic designed to address the complexity, uncertainty, and vagueness in systems.It does not rely on precise mathematical relationships between system inputs and outputs.Instead, fuzzy control defines fuzzy control sets, fuzzy relations, and fuzzy rules to simulate the process of fuzzy reasoning, thereby achieving control over the system.In this paper, the fuzzy control method is employed to achieve the online tuning of control parameters for electrode boilers, enhancing the dynamic performance indicators of the response.
The algorithm flowchart of the fuzzy controller is illustrated in Figure 2. In Figure 2, it can be seen that the control process of the fuzzy controller mainly includes fuzzification, fuzzy inference, and defuzzification.This paper takes power deviation e and the rate of change in deviation c e as in- puts.The output variable is obtained through quantification factors, fuzzy processing, and de-fuzzification processing by a fuzzy controller, along with a proportional factor.The calculation formulas for the adjusted K , K , K are as follows [12]: ZO represents zero, PS represents slightly positive, PM represents moder- ately positive, and PB represents strongly positive [13].Their membership functions are Gaussian-type.
Fuzzy control rules are typically summarized based on expert knowledge, with the aim of ensuring a high level of control accuracy.For electrode boiler power control, this paper adopts a total of 49 rules based on expert experience [14,15].The fuzzy control rules are shown in Table 1.By using the maximum membership degree method to defuzzify this control rule table, precise control parameter variation values can be obtained.The control parameter modification values are calculated using scaling factors and then combined with the initial parameters to obtain the final control parameters.

Control Strategy for Electrode Boiler Based on Fuzzy Control
The use of fuzzy control algorithms allows for the rapid adjustment of the power output of an electrode boiler, enabling a quick frequency response.The specific idea behind this is as follows: when a power disturbance occurs, leading to a frequency deviation in the system, the first step is to determine whether the frequency deviation exceeds the system's frequency control dead band.Then, the deviation and the frequency control coefficient are used to calculate the power setpoint value.The electrode boiler takes the power setpoint value as a control signal, and finally, the electrode boiler's rapid action, using the fuzzy control algorithm, balances the system's active power fluctuations in a short time.
The specific process of the electrode boiler control strategy based on fuzzy control is shown in Figure 3.

Electrode Boiler Power Response Model
When an electrode boiler participates in the primary frequency regulation of the system, it rapidly adjusts its output based on the frequency deviation occurring in the system, balancing the active power fluctuations in the power grid to stabilize the frequency within a reasonable range.
Currently, modeling of the electrode boiler mainly involves the study of temperature and power [17].Since the stable control of temperature takes a long time and cannot meet the time requirement of primary frequency modulation, this study models the power response of electrode boiler.The power simulation model of the electrode boiler can be approximated using a pure lag element and a first-order inertia element, represented by a transfer function as follows: where k represents the relationship between the electrode boiler's power and frequency deviation, T represents the inherent characteristics of the electrode boiler, and  rep- resents the system delay effect.According to the literature [18], the amplification factor is set as k 0.8 = , the time delay is

T 15s =
, and 2  = .During the frequency regulation process, the grid frequency deviation f  is used as the control signal, and the active power output P  of the electrode boiler is the output signal, as shown in Figure 4.

System Frequency Response Model
The fundamental reason for the frequency deviation in the power system is the mismatch between the active power output of generation units in the system and the power demand on the load side.Power systems experience relatively small power disturbances during stable operation; so, linearization is performed near the system's stable operating point to construct a linear model for dynamic analysis.Following the modeling approach proposed in the literature [19], the dynamic model of the power system, including the electrode boiler, is established as shown in Figure 5.Because the new energy generation with power electronic converters as the interface does not have the ability to participate in primary frequency modulation without an additional control strategy transformation, it is assumed in this paper that the primary frequency modulation of the whole system is mainly provided by the conventional synchronous unit and the modified electrode boiler.
In Figure 5

Simulation Analysis
To validate the impact of a high proportion of new energy sources on the grid frequency stability and the supporting effect of the fast-frequency-response control method based on fuzzy control for electrode boilers in the context of new energy integration, a simulation example system considering different proportions of new energy and electrode boiler frequency regulation is established in Matlab/Simulink-R2022a.The system includes a synchronous generator model, a photovoltaic model, a wind power model, and an electrode boiler model.Nodes 2 and 3 are, respectively, connected to photovoltaic and wind power; node 1 is connected to the synchronous motor and electrode boiler device; and nodes 5, 6, and 8 are connected to the load.Among them, the installed capacity of the synchronous generator is 80 MW, the wind power installed capacity is 10 MW, the photovoltaic installed capacity is 10 MW, it is equipped with a 15 MW electrode boiler, and the maximum heat storage capacity of the heat storage tank is 150 GJ.The operating parameters of the system are shown in Table 2.The system's base frequency is set to 50 Hz.To minimize unnecessary actions of the electrode boiler, the frequency control dead band is set to 50 ± 0.03 Hz.The model in Figure 6

Frequency Response Characteristics of Systems with Different Shares of Renewable Energy
According to the method proposed in reference [20] for calculating the penetration rate of renewable energy, different generator inertial constants M can be used to simu- late the proportion of renewable energy in the system.It can be calculated that in the model described in this paper, when M8 = and M6 = are used, they can simulate the cases where renewable energy accounts for 20% and 40%, respectively.To analyze the frequency response characteristics of systems with different proportions of renewable energy, a step power disturbance of 0.01 p.u. is introduced into the system, and the system's frequency response characteristics are shown in Figure 7.In Figure 7, it can be observed that when a 0.01 p.u. power disturbance occurs in the system, with a 0% proportion of renewable energy, the maximum frequency deviation is 0.06 Hz.With a 20% proportion of renewable energy, the maximum frequency deviation is 0.064 Hz, and with a 40% proportion of renewable energy, the maximum frequency deviation is 0.069 Hz.Moreover, as the proportion of renewable energy increases, the slope of the frequency response curve becomes steeper.It is evident that as the penetration rate of renewable energy increases, the system's frequency rises more rapidly and reaches higher peaks.The frequency response to the same level of power disturbance is more pronounced in high-renewable-energy penetration grids.Therefore, power systems need to have corresponding primary frequency control capabilities to support the continuous integration of renewable energy.

Frequency Control Characteristics of Electrode Boilers under Step Power Disturbance
In order to analyze the role of the fast frequency response of the electrode boiler and the frequency dynamic response of the system, a power generation disturbance with a duration of 40 s and an amplitude of 0.01 p.u. is added to the frequency simulation model with a new energy penetration rate of 20% in Section 5.1, as shown in Figure 8.The power system frequency and power response curves are shown in Figures 9 and 10 respectively.At the same time, to analyze the optimization effect of the fuzzy control of electrode boiler power, the power response curve of traditional control methods is compared with that of fuzzy control electrode boilers, as shown in Figure 11.The controller parameters for traditional electrode boiler control are selected as follows:     When a power disturbance of 0.01 p.u. occurs in the power system, the comparison of the system response characteristics when electrode boilers participate in frequency control or when they do not is shown in Table 3.It can be observed that the participation of electrode boilers in system frequency regulation rapidly provides active support within a short time, suppressing continuous frequency fluctuations.This leads to a reduction in the peak value of system frequency deviation by 0.003 Hz, a 5% decrease; the steady-state value of frequency deviation decreases by 0.005 Hz, a 19% decrease; and the system power deviation decreases by 0.0011 p.u., an 11% decrease.As shown in Figure 11, the overshoot of electrode boiler conventional control is 67%, with a settling time of 20 s.For the fuzzy control of electrode boilers, the overshoot is 64%, with a settling time of 17 s.It is evident that the application of fuzzy control reduces the overshoot of the electrode boiler power response by 3% and reduces the settling time by 3 s, making the response process faster and smoother and causing a lower power impact on the grid.

Frequency Control Characteristics of Electrode Boilers under Time-Sequential Random Power Disturbance
To validate the rapid frequency response method of electrode boilers in supporting the intermittent output of renewable energy, this section introduces a time-sequential random power disturbance with a maximum amplitude of 0.015 p.u. and a duration of 10 min [21] into the frequency simulation model with a renewable energy penetration rate of 20%, as shown in Figure 12.The grid frequency and power response curves are presented in Figures 13 and 14      As shown in Figure 13, the system frequency substantially fluctuates when there is a random power disturbance in the new energy generation.The comparison of the system response characteristics when electrode boilers participate in frequency control or when they do not is presented in Table 4.It can be observed that the application of electrode boilers in frequency control reduces the maximum system frequency deviation by 0.003 Hz, a decrease of 4.2%, and it decreases the active power fluctuation by 0.002 p.u., a reduction of 16.7%.As shown in Figure 15, the power response curve of electrode boilers indicates that the maximum power response of conventional control electrode boilers reaches 0.0031 p.u., while the maximum power response of electrode boilers with fuzzy control is 0.0028 p.u.This suggests that the oscillation amplitude is lower, allowing for more flexible adjustment of the power in response to fluctuations in renewable energy generation.

Conclusions
In this work, considering the impact and challenges brought by the high proportion of new energy access to the power grid on the safe and stable operation of the power system, the fast frequency response control method of the electrode boiler to support the new energy consumption is studied.The improved sag control strategy is applied to the electrode boiler so that it can adjust the output power independently according to the frequency deviation and cope with the fluctuation in the power generated by new energy sources; meanwhile, the fuzzy algorithm is applied to the electrode boiler to optimize the control parameters to shorten the power response time and improve the frequency response performance index.This study shows that the electrode boiler can not only heat the excess power generated by the new energy for heating, increasing the space for new energy consumption, but it can also quickly respond to the instantaneous fluctuation in the power system frequency under the high proportion of intermittent new energy access based on the control method in this paper to provide fast power support and improve the primary frequency regulation support capability of the power system.The follow-up work of this paper is the multipoint distribution of the electrode boiler aggregation control method to meet the high proportion of intermittent new energy access under the secondary frequency regulation needs of power system AGC.

Figure 1 .
Figure 1.Principle of electrode boiler participation in primary frequency control.
minimum frequency that the electrode boiler power can support; H f is the maximum frequency that the electrode boiler power can support; N P is the normal operating power of the electrode boiler; and max P is the maximum power of the electrode boiler.

DK
is the frequency regulation effec- tiveness coefficient of the electrode boiler; and B P  represents the change in power of the electrode boiler.

1 R
2) where P represents the power of the electric boiler; I represents the input current of the electric boiler; L represents the length of the cylindrical electrode in water;  rep- resents the resistivity of the solution; b represents the distance from the electrode axis to the center; d represents the diameter of the electrode cross-section; R represents the radius of the cylindrical container; 0 R represents the constant resistance value in the en- tire circuit; and represents the resistance of the solution.
of the proportional coefficient modification; I K  is the value of the integral coefficient modification; and d K is the value of the differential coefficient modification.Power deviation and its rate of change are divided into different fuzzy sets, and the membership functions are used to calculate the degree of membership of the input variables to each fuzzy set.The finer the fuzzy set division, the more precise the control.Therefore, variables can be divided into the following levels of fuzzy subsets:

Figure 4 .
Figure 4. Transfer function diagram of the electrode boiler.
, g T represents the governor's frequency control time constant, t T repre- sents the boiler's reheating time constant, R T represents the reheater's time constant, M represents the inertia of the generation unit, D represents the load damping coeffi- cient, f  represents the frequency deviation, frequency regulation effect coefficient of the electrode boiler, and R represents the droop coeffi- cient.

Figure 5 .
Figure 5. Frequency-response-equivalent model of the power system including the electrode boiler.
is subjected to unit scaling, and the parameter selection is as follows: M 10 = ,

Figure 7 .
Figure 7. Frequency response characteristics of systems with different renewable energy proportions.

Figure 8 .
Figure 8. Power disturbance of renewable energy.

Figure 9 .
Figure 9. Frequency response of the whole system.

Figure 10 .
Figure 10.Active power response of the whole system.

Figure 11 .
Figure 11.Active power response of electrode boiler.
respectively.Additionally, the power response curves of electrode boilers with and without the application of fuzzy control algorithms are compared in Figure 15.

Figure 13 .
Figure 13.Frequency response of the whole system.

Figure 14 .
Figure 14.Active power response of the whole system.

Figure 15 .
Figure 15.Active power response of electrode boiler.

Table 1 .
Regulation patterns for

Table 3 .
Comparison of electrode boiler participation in frequency control before and after step power disturbance.

Table 4 .
Comparison of electrode boiler participation in frequency control before and after timesequential random disturbance.