DFPSU employs a dual-feedback configuration, enabling the selection of an appropriate magnetic control strategy to achieve rapid and precise regulation of various physical quantities. In circumstances where the stator and rotor are both equipped with permanent magnet synchronous motors, the dual-feedback electrical machine’s mathematical model is expressed as follows: (1)Te=32np(ψsdisq−ψsqisd)=32np(ψrqird−ψrdirq) (2){Ps=32(usdisd+usqisq)Qs=32(usqisd−usdisq)where subscripts s and r denote stator/rotor-side affiliation of parameters. Subscripts d and q specify dq-axis components in synchronous rotating coordinates. Fundamental variables are defined as: u (voltage), i (current), ψ (flux linkage), and np (pole count), maintaining consistency with IEEE Std 115-2019 rotating machinery nomenclature.

The water supply system consists of a water cave, a pressure regulating chamber and a pressure pipe. As the water flow through the pipe will cause a series of energy losses including friction loss in the pipe and friction loss in the valves and accessories, the required head of the device not only includes the static head determined by the difference in water level between the upper and lower tanks, but also the additional head required to compensate for this loss [7]. The characteristics of the pipe can be obtained as shown in Eq. (3). (3){Hneed=Hst+HlossHloss=KfQ2Kf=fp+1Zwhere Hneed is the head required by the device; Hst is the static head; Hloss is the head loss; Q is the flow rate; Kf is the loss coefficient, which includes the friction coefficient of the pressure pipe fp and the friction coefficient of the guide vane.

Mathematical models of water delivery systems can be divided into rigid and elastic models, depending on whether the elasticity of water hammer is taken into account. When the elasticity of the water flow and the pipe wall is taken into account, an elastic dynamic model of the water delivery system is obtained as shown in Eq. (4). Further expansion of the hyperbolic function in the equation according to an infinite series allows the dynamic model to be selected according to the requirements of the application. In general, expansion of the second term is sufficient to meet the requirements of most power system stability studies. (4){Q=1Zhptanh⁡(sTep)(Hst−H−Hloss)tanh⁡(sTep)=∑n=1∞22n(22n−1)B2n(sTep)2n−1(2n)!where s is the differential operator; H is the effective head; Zhp is the normalized hydraulic resistance of the pressure pipe; Tep is the elastic time constant of the pressure pipe; n is the expansion order; and B is the Bernoulli number.

The theory of rigid water hammer posits that, for a relatively short water supply pipeline (<800 m), the water hammer pressure can be instantaneously transmitted to the entire water supply pipeline, i.e., Tep≈0. Retaining the expansion of Eq. (4) to the first term enables the rigid model to be obtained. It is also noted that the inertial time constant of the water flow in the pressure pipe, Tω=ZhpTep, and Eq. (5) is obtained. Combining the pipe characteristics given in Eq. (3) results in the rigid dynamic model of the water delivery system, as shown in Fig. 2. (5)Q=1Tωs(Hst−H−Hloss)

When the pump-turbine is switched to pump mode, the unit is near rated operation, the guide vane opening is large, and there is serious crossing of the full characteristic curve in the corresponding area. At this time, the unit’s flow rate is basically not affected by the guide vane opening, and its mechanical power and efficiency are closely related to the speed. The adjustment of the input active power is mainly achieved by controlling the speed rather than the guide vane opening.

In this mode, the pump’s operating characteristics are characterized by fitting the full characteristic curve of the pump turbine. In this section, the head-flow (H−Q) curve, power-flow (P−Q) curve, and efficiency-flow (η−Q) curve of the pump at varying speeds are fitted. The corresponding expressions are shown in Eqs. (6)–(8). The fitted curves are presented in Fig. 3. (6)H=a0ω2+a1ωQ+a2Q2 (7)P=b0ω3+b1ω2Q+b2ωQ2+b3Q3 (8)η=c0+c1ωQ+c2ω2Q2+c3ω3Q3

In consideration of the aforementioned points, this section synthesizes the rigid water hammer model of the water supply system with the power and head expressions, respectively, to derive the pump-turbine model in the state of the pump (see Fig. 4).

When the DFPSU is in pumping operation, active power control can be achieved by adjusting the speed of the unit, and the primary objective is to require the speed to accurately and quickly track the optimal value. To this end, the speed priority control strategy shown in Fig. 5 is selected under pumping conditions. First, the active power command Pref of the unit and the current head are passed to the efficiency optimization module to obtain the corresponding optimal speed ωref and optimal guide vane opening Zref. Second, ωref and the reactive power command Qref are passed to the converter for fast control. As the speed tracking command changes, the input active power of the unit is also adjusted accordingly. At the same time, Pref and Zref are passed to the regulator at the pump-turbine unit, ensuring that the pump-turbine unit operates efficiently while keeping the active power reliably and stably near the reference value.

In the event that the DFPSU is engaged in pumping mode, the unit employs a speed priority control strategy, whereby the response speed of the active power is directly modified by adjusting the opening of the pump guide vane. However, the response is characterized by its relatively slow nature. Concurrently, the time scale of primary frequency regulation is relatively brief, thereby classifying this process as a transient phenomenon within electromechanical systems. During this process, the action of the pump efficiency optimization module is not taken into account, and the corresponding control cannot be automatically completed. Consequently, if the traditional strategy consistent with the power generation operating condition is employed at this time, it becomes challenging to ensure the speed and effectiveness of the frequency regulation response [10]. To address these challenges, this paper proposes a novel frequency control method, as illustrated in Fig. 6. This method converts the frequency deviation (Δf) into an additional speed command, thereby enhancing the converter’s rapid regulatory capabilities and ensuring a swift frequency regulation response. The proportional and differential coefficients of the frequency control module, designated as Kωp and Kωd, respectively, are of paramount importance.

The system frequency response (SFR) model is a significant method for analyzing the dynamic response of a power system to frequency fluctuations [11]. The primary focus of this model is the study of frequency fluctuations in the context of electrical processes, while simplifying other time-scale processes. This approach enables the effective reduction of other time-scale processes, while maintaining the key capabilities of the power system. The frequency response model is a significant method for analyzing the dynamic response of an electrical system to frequency changes. It focuses on the dynamic changes in the electrical system itself, while simplifying other time-scale processes. This approach can effectively reduce the dynamic response of an electrical system, while preserving its key characteristics.

The frequency response model consists of various frequency control models and the model of constant load power generation. The frequency control model in this paper is a model of frequency control and load-flow, which is used to describe the dynamic changes in power generation caused by frequency deviations.

The frequency dynamic response index, which is a measurement of the effect of frequency control, principally includes frequency deviation, the rate of change (dΔf/dt), maximum frequency deviation (Δfmax), and steady-state frequency deviation (Δfst). As demonstrated in Fig. 10 below:

The rate of frequency change, or dΔf/dt, is a crucial metric in the analysis of system behavior. It is primarily influenced by the power imbalance within the system and the inertia time constant. At the onset of frequency control, dΔf/dt attains a maximum value, denoted as dΔf/dt|max. This maximum value exerts a direct influence on the operation of the system’s protective devices. The magnitude of Δfmax is a critical indicator of the system’s frequency stability, with its specific value being influenced by the preceding dΔf/dt conditions. During the intermediate and late stages of frequency control, dΔf/dt becomes negative, impeding the restoration of the frequency. Eventually, the action of each frequency control unit leads to the attainment of a steady-state value, designated as Δfst.

Based on Fig. 8, ignoring the time constants of the frequency detection module and the low-pass filter in the DFPSU unit, the primary frequency control output under different operating conditions can be obtained as follows: (16)ΔPmot=p(Kωp+Kωds)τcs+1Δf

Considering that only DFPSU units are involved in frequency control in the system and ignoring their millisecond converter response time, the frequency response model shown in Fig. 9 can be rewritten as follows: (17)ΔPL=(2H+Kωd)sΔf+(D+Kωp)Δf

It is hypothesized that the system is subjected to a disturbance of amplitude ΔPstep, whose transfer function is ΔPL(s)=ΔPstep(s)/s. Under these conditions, Δf and dΔf/dt are given by Eqs. (18) and (19), respectively. (18)Δf(s)=ΔPsteps2(2H+Kωd)+s(D+Kωp) (19)dΔf(s)dt=ΔPsteps(2H+Kωd)+D+Kωp

Pursuant to the initial value theorem, calculate the derivative of the function dΔf/dt at the initial moment, that is to say, dΔf/dt|max: (20)dΔfdt|max=lims→∞sdΔf(s)dt=ΔPstep2H+Kωd

In accordance with the final value theorem, the steady-state frequency deviation, designated as Δfst, can be calculated: (21)Δfst=lims→0sΔf(s)=ΔPstepD+Kωp

As demonstrated in Eqs. (20) and (21), the differential control channel of the constructed frequency control module is equivalent to increasing the equivalent inertia of the system. This can rapidly suppress frequency changes, i.e., reduce dΔf/dt|max, and thereby significantly reduce Δfmax. In contrast, the proportional control channel has the same effect as the frequency control of a conventional unit. It increases the damping of the system and provides the corresponding power support with frequency changes, thereby helping to reduce Δfst.

In order to verify the rationality of the aforementioned inference, this study will substitute different values of Kωp and Kωd into the system transfer function described by Eq. (15). The typical parameters in the function are shown in Table 1. The resulting function curve is shown in Fig. 11. The specific simulation verification results will be described in detail in the following sections.

The root locus of each frequency control parameter is plotted according to Eq. (15). In the figure, the damping ratio is denoted by ξ, the overshoot is indicated by Mp and the natural oscillation angular frequency is represented by ωn.

As illustrated in Fig. 12a, when the fixed Kωp is a typical value and Kωd varies from [0,∞], stability is achieved due to the placement of all system poles and zeros in the left-hand side of the complex s-plane. When 0 < Kωd < 34.4, the system is in an underdamped state, and the magnitude of the zero pole, designated as ξ, is substantial, corresponding to an index (Mp) less than 5%. This ensures that the system’s response to frequency control can be reliably guaranteed to a certain extent. Conversely, when Kωp ranges from [0,∞], the system’s stability is contingent on the condition Kωp < 308. It is noteworthy that the magnitude of Kωp directly correlates with the system’s Mp generation, with higher values of Kωp resulting in greater Mp generation. To ensure that the resulting control Δf is minimal, the region where |ξ| > 0.707 is selected, i.e., 0 < Kωp< 23.8. Consequently, the corresponding Mp is less than 4.16%, and the value of ωn is substantial, thereby effectively reducing the frequency fluctuation time.

After optimizing the unit under several typical operating conditions, the optimal frequency control parameters can be obtained for each scenario. In this example, a 10% load increase is considered. Initially, the Support Vector Regression (SVR) method is employed to model the data, thereby addressing issues such as noise or outliers. The Support Vector Regression (SVR) method has emerged as a prominent machine learning fitting technique in recent years, adeptly capturing the nonlinear characteristics of data and yielding more precise fitting outcomes. The objective of the fitting process is to approximate the overall trend of the data through the SVR model to obtain a more accurate fitting curve. Subsequently, the data is smoothed using cubic spline interpolation to further optimize the smoothness and continuity of the curve. The efficacy of cubic spline interpolation in generating a smooth curve with continuous first and second derivatives between data points is well-documented. This approach effectively reduces oscillation during the fitting process, thereby maintaining the overall trend of the data.

The specific training process is illustrated in Fig. 14. Agent achieved substantial convergence after approximately 50 training episodes, attaining a final real-time reward value of −23.6822. This demonstrates strong online learning capability, indicating favorable convergence characteristics of the agent system. Consequently, it enables reliable simulation of frequency stabilization under disturbance scenarios across diverse DFPSU operating conditions.

Taking the frequency drop scenario as an optimization example, the resulting curve is shown in Fig. 15. Combining the frequency control characteristics of the generator set during a frequency drop, i.e., the generator set will reduce the speed to release the rotor kinetic energy, and increase the active power output or reduce the active power input to provide stable power support, it is not difficult to summarize the law of change of the final frequency control parameters: when the generator set is in the low speed stage, the value of the frequency control parameters is relatively small, so as to avoid the generator set from causing a serious speed drop due to excessive release of the rotor kinetic energy. As the speed increases, the frequency control parameter gradually increases to make full use of the rotor kinetic energy contained in the unit. However, when the speed increases to a certain value, the input/output active power of the corresponding unit will also increase to a higher level. At this time, in order to ensure that the unit does not overreact and the power exceeds the allowable range, the frequency control parameter is reduced accordingly. Since the principle of changing the limit frequency control parameter for the frequency increase scenario is the same as that for the frequency decrease scenario, it will not be presented here.

The differential control coefficient Kωd is selected to a larger value when the dΔf/dt value is large at the beginning of frequency control. This ensures that the rotor kinetic energy of the generator set is fully released to suppress the frequency change. Conversely, as dΔf/dt diminishes, Kωd undergoes a gradual reduction to decelerate the release of the rotor kinetic energy of the generator set, thereby averting an excessive decline in speed. Furthermore, when the sign of dΔf/dt undergoes a transition from positive to negative, Kωd undergoes an adjustment to a negative value, thereby facilitating frequency recovery and reducing the continuous deviation of the speed. The proportional control coefficient Kωp can be adjusted to a smaller value at the beginning of frequency control, while a larger value can be selected in the middle and late stages to complement the response and support time of the differential control.

The initial active power command of the DFPSU is set to −0.8 p.u., and the corresponding optimal speed command is 0.987 p.u. After the unit has been operating stably for 20 s, the load suddenly increases by 10%. The response of the system frequency, active power, and speed of the generator under the control mentioned in this paper, including traditional control, proportional control channel only, differential control channel only, and both channels acting, is shown in Fig. 16. In the context of DFPSU in pumping mode, the speed is identified as the primary control target of the converter. Consequently, the active power of the system is contingent upon the adjustment of the guide vane opening at the pump. Conversely, if the traditional control method, which converts Δf into a power command, is employed, the frequency regulation response speed will be considerably decelerated due to the restriction of the guide vane opening adjustment speed. Conversely, the converter maintains its original optimal speed control, which is faster, thereby ensuring the unit’s speed remains consistent and the rotor kinetic energy cannot be released to provide inertia support. In contrast, the control method outlined in this paper, which converts Δf into a speed command, utilizes the converter’s rapid response capability to ensure the speed of the frequency response. The rapid adjustment of speed enables a corresponding adjustment of input power, thereby effectively mitigating frequency fluctuations.

A subsequent examination of the system’s response during the proposed control with solely proportional and differential control reveals the efficacy of differential control during the initial phase of frequency control, characterized by a substantial value of dΔf/dt. It has been demonstrated that differential control facilitates the release of rotor kinetic energy by adjusting the speed to achieve instantaneous support of the active power. This, in turn, effectively reduces dΔf/dt. However, its efficacy gradually diminishes during the middle and late stages of frequency control. As the speed reverts to its original value, the active power of the unit gradually recovers, indicating that this control channel is incapable of providing continuous power support. In contrast, the proportional control channel maintains its effectiveness throughout the entire frequency control process. Subsequent to a change in speed, it can be utilized to regulate the active power of the generator unit stably by leveraging the characteristics of the pump-turbine. The effectiveness of this control channel directly corresponds to the magnitude of Δfst. The combination of these two channels has been shown to enhance their respective benefits, thereby effectively mitigating frequency fluctuations, reducing dΔf/dt, and significantly diminishing both Δfmax and Δfst.

As demonstrated in the preceding analysis, the differential control channel is analogous to virtual inertia control, a strategy that has been proven to effectively suppress frequency changes and, by extension, reduce dΔf/dt during the initial stage of frequency control. In comparison with the conventional approach of setting dΔf/dt as a power correction quantity to release rotor kinetic energy, the method proposed in this paper directly controls the speed to achieve adjustment of the rotor kinetic energy, and the inertia support effect is more rapid and obvious. The proportional control channel is analogous to droop control, which facilitates continuous power support throughout the entire frequency control process in response to changes in Δf. This approach offers significant advantages in terms of reducing Δfst. However, due to the inherent characteristics of the water pump under pumping conditions, the input active power of the DFPSU is proportional to the cube of the speed. While the proportional control method outlined in this paper may not be as intuitive as the conventional method of converting Δf into a power command, the control response speed and effectiveness are comparable to those of the traditional control method. In addition to the parameters determined using the conventional method for frequency control parameters, Fig. 17 presents the frequency response curves of the system under various frequency control parameters. It also compares their frequency dynamic response indicators as shown in Table 2, which further substantiates the impact of each control channel and parameter. It is evident that as the magnitude of Kωd increases, the initial dΔf/dt values decrease, thereby enhancing the suppression of frequency change. It is important to note that Δfmax is directly influenced by dΔf/dt; therefore, a larger Kωd will result in a smaller Δfmax. However, the differential control channel exerts a negative damping effect on frequency recovery during the later stages of frequency control, thereby slowing the rate of Δf reduction. Conversely, Kωp signifies the extent to which the timing group reliably provides a response to Δf. Consequently, an augmentation in Kωp is more advantageous in diminishing Δf, resulting in a substantial reduction in both Δfmax and Δfst.

In the event that the DFIM-PSH is operating in pumping mode, the dynamic response curves of the system at unit input powers of −0.7 and −0.9 p.u. are demonstrated in Figs. 18 and 19, respectively. Concurrently, a variable-parameter frequency control strategy can attain optimal frequency control outcomes. The DFIM-PSH adjusts the input active power by altering the speed, thereby providing stable power support to reduce Δfst. Concurrently, the inherent characteristics of differential control induce fluctuations in the speed of the generator set, thereby leading to increased frequency fluctuations. However, this does not compromise the overall effectiveness of the control strategy.

In comparison with conventional parameters, the improved parameter–adaptive control facilitates the unit in altering the speed significantly more rapidly at the commencement of frequency control. Consequently, this reduces Δfmax from 0.1985 to 0.1412 Hz in the sub-synchronous state and from 0.2261 to 0.1772 Hz in the super-synchronous state, representing a reduction of more than 20% in both instances. In the middle and late stages of frequency control, the modified parameter variable parameter control has been shown to be capable of reliably reducing the input active power with changes in speed, thereby promoting rapid frequency recovery. The resulting control Δfst reduction in both operating conditions has been shown to exceed 10%.

Fig. 20 shows the system’s frequency dynamic response index when DFPSU is in different operating conditions and states. As we can see from this analysis, the adaptive variable parameters determined using the method described in this paper can make the most of the rotor kinetic energy and frequency control capacity of the generator set when it is operating in different conditions. At the same time, they can also make sure that the generator set speed and power stay within the limits. In other words, the improved parameters are strong when the generator set is in different operating conditions. At the same time, since the variable parameters also adapt to changes in frequency, each frequency control channel can have a positive effect on restoring the correct frequency throughout the entire process. As well as reducing df/dt, Δfmax and Δfst, it also has a significant effect on reducing tst.