Steady-State Reactive Power Capability Analysis of Doubly-Fed Variable Speed Pumped Storage Unit Considering the Unit’s Operating Characteristics

Abstract

Based on the actual data of a 300 MW doubly-fed variable speed pumped storage units (DFVSPSUs) in China, the reactive power characteristics of both the stator side and the grid-side converter are analyzed, and the reactive power regulation capability of the unit is discussed. First, the power coupling relationship is analyzed, demonstrating that the reactive power-regulation capability is jointly composed of the stator side and the grid-side converter, without direct coupling between them. Next, we determine the doubly-fed induction generator (DFIG) capacity, explaining that the capacity of the DFIG exceeds the rated capacity of the unit. Then, we note that the stator-side reactive power regulation capability is limited by prime mover power, stator current, and rotor current, while the grid-side converter regulation capability is influenced by converter capacity and rotor-side real power. Furthermore, the stator-side, grid-side converter and total reactive power-regulation capabilities of the unit under different water heads and real power conditions are determined. The results demonstrate that fully considering the grid-side converter can increase the unit’s reactive power regulation capability by 12% to 26%. Finally, by comparing the reactive power operating ranges of fixed-speed and variable-speed units, the reactive power advantages of the variable-speed unit are quantified.

1. Introduction

With the world’s energy landscape gradually transforming towards clean energy, a large amount of new energy will be introduced into the power grid [1,2]. In the new electric power system, the energy structure has changed, and the introduction of high-proportion new energy grid integration and long-distance power transmission requires the system to have a wider reactive power regulation range [3,4], to address the challenges of load fluctuations and voltage stability [5,6,7,8]. Therefore, pumped storage units (PSUs) are widely adopted due to their flexible operating characteristics, rapid operating-condition-switching capability, and excellent reactive power regulation performance [9,10,11].

Compared with fixed-speed pumped storage units (FSPSUs), doubly-fed variable speed pumped storage units (DFVSPSUs) control output power by adjusting the speed, offering greater flexibility and efficiency [12,13]. The advantages of DFVSPSUs lie in their strong grid support, fast power response, high hydraulic efficiency, and ability to meet the fast power regulation needs of the new power system [14,15,16,17,18]. Additionally, doubly-fed units feature adjustable power under pump conditions and bidirectional power flow between the stator and rotor [19,20,21], making them superior in reactive power regulation. Due to the complex operating characteristics of doubly-fed units, analyzing their reactive power-regulation capability is critical for ensuring the stable operation of the power system, improving power transmission efficiency, and enhancing power quality.

Existing research primarily focuses on simulating and verifying the reactive power capability of doubly-fed induction generators (DFIGs) themselves [22,23,24], without comprehensively considering the operating characteristics of the pump–turbine system or applying them to pumped storage units. Some scholars have proposed that decoupled power control can quickly and accurately absorb reactive power without compromising stability [25,26]. Singh et al. [27] proposed a unified control strategy for a variable-speed pumped storage plant based on a DFIG under different grid conditions; by injecting a low-voltage DC power source during the startup of the stator circuit, slip losses were reduced, and the magnetizing current was lowered, effectively saving electrical energy. Huang et al. [28] analyzed the reactive power capability of a pumped storage unit motor and demonstrated the reactive voltage regulation range under pump conditions. Further, some scholars analyzed the stator-side reactive power characteristics of the DFIG using the V-curve method, exploring the reactive power-regulation capability of the DFVSPSU under different operating conditions, proving that doubly-fed asynchronous machines have reactive power-regulation characteristics similar to synchronous machines [29,30,31]. Chen et al. [32] presented an electromagnetic–electromechanical transient multi-time scale model to determine the operational characteristics of pumped storage units, studying the stable operating range and the response speed of active/reactive power regulation under various conditions. Li et al. [33] established a reactive power balance equation to confirm the strong reactive power-regulation capabilities of DFVSPSUs. However, previous studies have only explored the reactive power boundaries from the DFIG perspective, without considering the pump–turbine system or coupling the two for a comprehensive analysis. Additionally, the grid-side converter’s reactive power-regulation capability is often overlooked. Therefore, current research on the DFVSPSU’s reactive power regulation is still incomplete and lacks in-depth analysis of the unit.

Therefore, this study analyzes, in depth, the reactive power-regulation performance of DFVSPSUs by combining the operating characteristics of pump–turbines. On the basis of adopting a T-equivalent circuit analysis, the power boundary conditions on the generator stator side are derived, and the power operating range is clearly defined in the P-Q diagram. Through measured data and theoretical derivation, the reactive power-regulation capabilities of the stator-side and grid-side converters are quantitatively analyzed, thereby obtaining the overall reactive power regulation range of DFVSPSUs; furthermore, through a comparison with FSPSUs of the same capacity in China, it is first confirmed that variable-speed units can extend the reactive power-regulation range by 12–26%. This core discovery fills the gap in the quantitative evaluation of reactive power performance for variable-speed units, providing critical theoretical support for the stable operation of high-penetration renewable energy power grids.

2. Power Coupling Analysis of the Doubly-Fed Variable Speed Pumped Storage Unit

2.1. Operating Principle of the DFVSPSU

The DFVSPSU is a complex system with strong coupling among hydraulic, mechanical, electrical, and control subsystems. It mainly consists of a water intake system, pump–turbine system and its control system, and DFIG and its control system, as shown in the topology diagram in Figure 1. The stator of the DFIG is directly connected to the power grid, while the rotor is connected to the grid through a converter. Both the stator and rotor can exchange energy with the grid. The DFIG control system adjusts the rotor-side voltage to control the generator’s power output. The pump–turbine control system regulates the flow of the water intake system by changing the guide vane opening, thus controlling the pump–turbine system’s power output. The pump–turbine system is connected to the generator through the shaft, enabling the conversion between mechanical energy and electrical energy. These subsystems cooperate to ultimately achieve energy balance and stable operation of the DFVSPSU.

In Figure 1, we can clearly see the coupling relationship between the three speeds, as shown in (1):

$$\begin{array}{cc}\hfill {{\displaystyle \omega }}_{\mathrm{s}}& ={{\displaystyle \omega }}_{\mathrm{r}}+{{\displaystyle \omega }}_2\hfill \\ & ={{\displaystyle \omega }}_{\mathrm{r}}+s{{\displaystyle \omega }}_{\mathrm{s}}\hfill \end{array}$$

(1)

In the equation, ω_s represents the synchronous speed of the stator rotating magnetic field of the DFIG, ω_r represents the rotor speed, ω₂ represents the excitation magnetic field speed of the DFIG, and s represents the slip rate of the DFIG. When ω_r < ω_s and s > 0, the motor operates in a sub-synchronous state; when ω_r > ω_s and s < 0, the DFIG operates in a super-synchronous state.

Further,

$${{\displaystyle \omega }}_{\mathrm{s}}=60\hspace{0.17em}{{\displaystyle f}}_1/p$$

(2)

$${{\displaystyle \omega }}_2=60\hspace{0.17em}{{\displaystyle f}}_2/p$$

(3)

It can be derived that

$${{\displaystyle f}}_1={\displaystyle \frac{{{\displaystyle \omega }}_{\mathrm{r}}p}{60}}+{{\displaystyle f}}_2$$

(4)

where f₁ and f₂ are the frequencies of the stator and rotor currents, respectively, and p is the number of pole pairs. During operation, since the synchronous speed ω_s remains constant, the rotor speed ω_r is adjusted by regulating the frequency of the rotor excitation current f₂, thus achieving constant-frequency variable-speed operation of the unit.

2.2. Power Coupling Relationship of the DFVSPSU

The DFVSPSU is capable of both real power output and reactive power output. Based on the topology and operating principles described above, for the real power part, it includes the total real power output P of the unit, the stator real power output P_s, the rotor real power output P_r, and the real power output of the grid-side converter P_c. For the reactive power part, it includes the total reactive power output Q of the unit, the reactive power output from the stator side Q_s, the reactive power output from the rotor side Q_r, and the reactive power output from the grid-side converter Q_c. The power flow directions are shown in Figure 2.

Due to the fact that the DC bus side of the back-to-back converter only exchanges real power, under the assumption of no power loss and line loss, the real power output of the grid-side converter is equal to the real power output of the rotor, as shown in (5). However, there is no equal relationship between the reactive power output of the grid-side converter and the reactive power output of the rotor side (Q_c ≠ Q_r).

$$P_{\mathrm{c}}=P_{\mathrm{r}}$$

(5)

As shown in Figure 2, for the real power output of the unit, the mechanical power output P of the unit is equal to the sum of the stator real power output P_s and the real power output of the grid-side converter P_c. Combining with (5), the following coupling relationship is obtained:

$${P=P_{\mathrm{s}}+P}_{\mathrm{c}}={{\displaystyle P}}_{\mathrm{s}}+{{\displaystyle P}}_{\mathrm{r}}$$

(6)

In this context, the real power output of the stator and rotor is also coupled with the slip rate:

$${{\displaystyle P}}_{\mathrm{r}}=-s{{\displaystyle P}}_{\mathrm{s}}$$

(7)

Therefore, the mechanical power P of the unit can be directly represented by the stator real power output P_s:

$$P=(1-s)P_{\mathrm{s}}$$

(8)

Regarding the reactive power output of the unit, as described earlier, since the DC bus side of the back-to-back converter only exchanges real power, the reactive power output of the grid-side converter does not equal the reactive power output of the rotor side (Q_c ≠ Q_r). Furthermore, the total reactive power output Q equals the sum of the stator reactive power output Q_s and the grid-side converter reactive power output Q_c and is not equal to the sum of the stator reactive power output Q_s and the rotor-side reactive power output Q_r, as shown in the following equation:

$$\begin{array}{cc}\hfill Q& ={{\displaystyle Q}}_{\mathrm{s}}+{{\displaystyle Q}}_{\mathrm{c}}\hfill \\ & \ne {{\displaystyle Q}}_{\mathrm{s}}+{{\displaystyle Q}}_{\mathrm{r}}\hfill \end{array}$$

(9)

Stator reactive power Q_s couples with rotor reactive power Q_r, while the grid-side converter’s reactive output Q_c remains decoupled from Q_s. Q_c is constrained solely by the converter capacity and its real power output P_c. Consequently, the unit’s total reactive power Q cannot be derived directly from the stator output, unlike its mechanical power P.

Therefore, to study the reactive power characteristics of the DFVSPSU, it is necessary to analyze the reactive power characteristics of both the stator-side and the grid-side converter.

3. Reactive Power Characteristic Analysis of the Stator Side of the Doubly-Fed Variable Speed Pumped Storage Unit

In studying the reactive power-regulation capability of the stator side of the DFVSPSU, the following steps are taken: first, the operational constraints of the power output of the doubly-fed unit are explored, and then, the capacity of the DFIG is determined based on the pump–turbine data. Next, the reactive power limitation of the stator of the DFVSPSU is derived from the mathematical model, which leads to the P−Q power operating range of the DFVSPSU. Finally, the reactive power limits for both emitting and absorbing are analyzed based on the optimal operating data of the pump–turbine.

3.1. Stator Side Power Operating Constraints of the DFVSPSU

The power operating constraints on the stator side of the DFVSPSU mainly include four factors: prime mover power limitation, stator current limitation, rotor current limitation, and rotor voltage limitation:

• Prime Mover Power Limitation (P_m):

The prime mover power limitation is primarily constrained by two factors: first, the prime mover’s power limitation usually does not exceed the rated capacity of the DFIG, and it is generally restricted by the rated real power of the machine. Second, the prime mover’s power limitation is also restricted by the pump-turbine, which can be further divided into the following: maximum and minimum output limits in turbine mode; maximum and minimum input limits in pump mode.

2.

Stator Current Limitation (I_smax):

The stator current limitation is determined by the design and materials of the motor, including the stator windings and core heating, which establish the safe operating limits. Under constant stator voltage, the rated capacity of the unit determines the allowable value of the stator current.

3.

Rotor Current Limitation (I_rmax):

The rotor current refers to the current flowing through the rotor windings of the motor, which is typically determined by the heating of the rotor. The maximum rotor current I_rmax of the DFVSPSU is the current flowing through the rotor windings when the unit is operating at the rated working point.

4.

Rotor Voltage Limitation (U_rmax):

The maximum rotor voltage of the DFIG depends on the maximum operating voltage of the rotor and the maximum output voltage of the converter. Additionally, the slip ratio s also affects the rotor voltage. If the design parameters of the unit and converter are appropriate, the power operating range is generally not limited by the maximum rotor voltage.

As mentioned above, in order to obtain the stator-side reactive power characteristics of the DFVSPSU, the first step is to determine the capacity of the DFIG. Then, the study was conducted on the stator current limitation, rotor current limitation, and rotor voltage limitation using an equivalent circuit model.

3.2. Determination of the DFIG Capacity

The rated capacity of a pumped storage unit is determined by its maximum total power, while the capacity of the motor is determined by the maximum stator power output. For synchronous motors, the stator power equals the total power; hence, the motor’s capacity is equivalent to the unit’s rated capacity. However, for the DFIG, the total power is the sum of the rotor and stator powers. According to the operational characteristics of pump–turbines, during turbine modes, the unit operates in a sub-synchronous state, where the rotor power flows in the opposite direction to the total power. As a result, the stator power exceeds the total power, necessitating a DFIG capacity that is greater than the unit’s rated capacity.

This section selects the capacity of a DFIG based on actual data from a domestic 300 MW variable-speed pumped storage unit. Based on the operational characteristics of the pump–turbine for this unit, the real power output on the stator side under turbine mode and pump mode can be obtained, as shown in Figure 3.

From Figure 3, under turbine conditions, stator power increases with both higher reference output at a fixed head and a higher head at fixed output. At a 462 m head and 300 MW reference output (412 rpm), maximum stator power reaches 315.38 MW. For pump conditions, stator power rises with increased reference input at a fixed lift but decreases with a higher lift at fixed input. At a 400 m lift and 330 MW input, the peak stator power is 316.20 MW.

Therefore, the capacity of the DFIG can be selected as the maximum real power output of the stator sides under turbine operating conditions, which is 315.38 MW, rounded to 316 MW. Taking the power factor cosφ as 0.9, the capacity of the DFIG can be calculated as 351.1 MW, which is consistent with the actual situation of a domestic variable-speed unit.

3.3. Voltage and Current Limitations on the Stator Side of the DFVSPSU

To analyze the limiting relationships between different electrical quantities and the operating power during the steady-state process of the unit more clearly, this section mainly uses the conventional T-type equivalent circuit to derive the three reactive power limitations of the stator of the DFIG. These include the stator current limitation (I_smax), rotor current limitation (I_rmax), and rotor voltage limitation (U_rmax).

The conventional T-type equivalent circuit is a commonly used simplified circuit model that converts complex magnetic field relationships into intuitive circuit models, helping to understand the internal working principles of motors and thereby deriving key operating constraints. The T-type equivalent circuit of DFIG is shown in Figure 4, with the left side representing the stator side, the right side reflecting the rotor side, and the middle part simulating the energy transfer of the air gap magnetic field, where the positive directions of stator and rotor currents point towards the interior of the DFIG and the electrical quantities have been converted to the stator side.

According to Figure 4, the mathematical model of the DFIG can be derived as shown in (10):

$$\left\{\begin{array}{l}\stackrel{.}{{{\displaystyle E}}_m}=\stackrel{.}{{{\displaystyle I}}_m}\cdot {{\displaystyle jX}}_{\mathrm{m}}\\ \stackrel{.}{{{\displaystyle U}}_s}=\stackrel{.}{{{\displaystyle E}}_m}+\stackrel{.}{{{\displaystyle I}}_s}\cdot \left({{\displaystyle R}}_{\mathrm{s}}+{{\displaystyle jX}}_{\mathrm{s}}\right)\\ \stackrel{.}{{{\displaystyle U}}_r}{\displaystyle /}s=\stackrel{.}{{{\displaystyle E}}_m}+\stackrel{.}{{{\displaystyle I}}_r}\cdot \left({{\displaystyle R}}_{\mathrm{r}}/s+{{\displaystyle jX}}_{\mathrm{r}}\right)\\ \stackrel{.}{{{\displaystyle I}}_m}=\stackrel{.}{{{\displaystyle I}}_s}+\stackrel{.}{{{\displaystyle I}}_r}\end{array}\right.$$

(10)

In this equation, $\dot{{\mathit{U}}_{\mathbf{s}}}$ is the voltage phasor at the stator side of the DFIG, $\dot{{\mathit{U}}_{\mathbf{r}}}/s$ is the rotor voltage phasor converted to the stator side of the DFIG, $\dot{{\mathit{E}}_{\mathbf{m}}}$ is the induced electromotive force phasor in the air gap magnetic field, $\dot{{\mathit{I}}_{\mathbf{s}}}$ is the stator current phasor of the DFIG, $\dot{{\mathit{I}}_{\mathbf{r}}}$ is the rotor current phasor converted to the stator side of the DFIG, $\dot{{\mathit{I}}_{\mathbf{m}}}$ is the excitation current phasor of the DFIG, R_s is the stator resistance of the DFIG, X_s is the stator leakage reactance of the DFIG, R_r/s is the rotor resistance converted to the stator side of the DFIG, X_r is the rotor leakage reactance converted to the stator side of the DFIG, and X_m is the excitation reactance of the DFIG.

To provide clearer visualization of the operational limitations for the DFVSPSU in the P-Q diagram, the following methodology will be adopted: based on the traditional T-type equivalent circuit model, the active and reactive components of relevant parameters will be decomposed along the d-axis and q-axis. This approach enables the stator-side real power (P_s) and stator-side reactive power (Q_s) to intuitively depict the operational constraints for the DFIG—namely the stator current limitation (I_smax), rotor current limitation (I_rmax), and rotor voltage limitation (U_rmax). The detailed derivation is presented from (11) to (16).

Assuming $\dot{{\mathit{U}}_{\mathbf{s}}}$ as the reference quantity, i.e., $\dot{{\mathit{U}}_{\mathbf{s}}}=U_{\mathrm{s}}\mathrm{\angle }0°$, then the stator and rotor current, as well as the rotor voltage of the DFIG, can be decomposed as follows:

$$\left\{\begin{array}{l}\stackrel{.}{{{\displaystyle I}}_s}={{\displaystyle I}}_{\mathrm{ds}}+{{\displaystyle jI}}_{\mathrm{qs}}\\ \stackrel{.}{{{\displaystyle I}}_r^{\prime }}=u\stackrel{.}{{{\displaystyle I}}_r}=u\left({{\displaystyle I}}_{\mathrm{dr}}+{{\displaystyle jI}}_{\mathrm{qr}}\right)\\ \stackrel{.}{{{\displaystyle U}}_r^{\prime }}={\displaystyle \frac{\stackrel{.}{{{\displaystyle U}}_r}}{u}}=\left({{\displaystyle U}}_{\mathrm{dr}}+{{\displaystyle jU}}_{\mathrm{qr}}\right){\displaystyle /}u\end{array}\right.$$

(11)

where u represents the transformation ratio of the DFIG, $\dot{{\mathit{I}}_{\mathit{r}}^{\mathit{\prime }}}$ is the rotor current phasor of the DFIG, and $\dot{{\mathit{U}}_{\mathit{r}}^{\mathit{\prime }}}$ is the rotor voltage phasor of the DFIG.

The active and reactive power generated by the stator of the DFIG can be represented by the d-q axis components of the stator current, as shown in (12):

$$\left\{\begin{array}{l}{P}_{\mathrm{s}}=\mathrm{Re}\left(m\stackrel{.}{{{\displaystyle U}}_s}\stackrel{.}{{{\displaystyle I}}_s}\right)=m{U}_{\mathrm{s}}\cdot I_{\mathrm{ds}}\\ Q_{\mathrm{s}}=\mathrm{Im}\left(m\stackrel{.}{{{\displaystyle U}}_s}\stackrel{.}{{{\displaystyle I}}_s}\right)=-m{U}_{\mathrm{s}}\cdot I_{\mathrm{qs}}\end{array}\right.$$

(12)

From (12), it can be seen that the decomposed stator current can be represented by the active and reactive power at the stator side. Therefore, the stator current limitation (I_smax) can be directly expressed by the stator-side active and reactive power, as shown in (13):

$$\sqrt{{P_{\mathrm{s}}}^2+{Q_{\mathrm{s}}}^2}\le m{U}_{\mathrm{s}}{I}_{\mathrm{smax}}$$

(13)

where m is the number of stator winding phases and I_smax is the maximum stator current.

By combining (10), (11), and (12), the rotor current limitation (I_rmax) can be directly expressed by the stator-side active and reactive power, as shown in (14):

$$\sqrt{{P_{\mathrm{s}}}^2+{(Q_{\mathrm{s}}+{\displaystyle \frac{m{U_{\mathrm{s}}}^2}{X_{\mathrm{s}}+X_{\mathrm{m}}}})}^2}\le {\displaystyle \frac{m{U}_{\mathrm{s}}{X}_{\mathrm{m}}}{u\cdot (X_{\mathrm{s}}+X_{\mathrm{m}})}}{I}_{\mathrm{rmax}}$$

(14)

Similarly, the rotor voltage limitation (U_rmax) can be obtained as shown in (15), where U_rmax is the maximum rotor voltage. The intuitive relationship between rotor voltage and stator-side active and reactive power is shown in (16).

$$\sqrt{{{\displaystyle U}}_{\mathrm{dr}}^2+{{\displaystyle U}}_{\mathrm{ds}}^2}\le {{\displaystyle U}}_{\mathrm{rmax}}$$

(15)

$$U_{\mathrm{r}}=\sqrt{{\left(K_1{\displaystyle \frac{P_{\mathrm{s}}}{m{U}_{\mathrm{s}}}}+K_2{\displaystyle \frac{Q_{\mathrm{s}}}{m{U}_{\mathrm{s}}}}+K_3{U}_{\mathrm{S}}\right)}^2+{\left(K_4{\displaystyle \frac{P_{\mathrm{s}}}{m{U}_{\mathrm{s}}}}+K_5{\displaystyle \frac{Q_{\mathrm{s}}}{m{U}_{\mathrm{s}}}}+K_6{U}_{\mathrm{S}}\right)}^2}$$

(16)

where

$$\left\{\begin{array}{l}{K}_1=-R_{\mathrm{s}}\cdot s-\left[(X_{\mathrm{s}}+X_{\mathrm{m}})\cdot R_{\mathrm{r}}\right]/X_{\mathrm{m}}-\left(R_{\mathrm{s}}\cdot X_{\mathrm{r}}\cdot s\right)/X_{\mathrm{m}}\\ K_2=s\cdot X_{\mathrm{s}}-\left(R_s\cdot R_{\mathrm{r}}\right)/X_{\mathrm{m}}+\left[(X_{\mathrm{s}}+X_{\mathrm{m}})\cdot X_{\mathrm{r}}\cdot s\right]/X_{\mathrm{m}}\\ K_3=s+\left(X_{\mathrm{r}}\cdot s\right)/X_{\mathrm{m}}\\ K_4=s\cdot X_{\mathrm{s}}-\left(R_{\mathrm{s}}\cdot R_{\mathrm{r}}\right)/X_{\mathrm{m}}+\left((X_{\mathrm{s}}+X_{\mathrm{m}})\cdot X_{\mathrm{r}}\cdot s\right)/X_{\mathrm{m}}\\ K_5=R_{\mathrm{s}}\cdot s+\left[(X_{\mathrm{s}}+X_{\mathrm{m}})\cdot R_{\mathrm{r}}\right]/X_{\mathrm{m}}+\left(R_{\mathrm{s}}\cdot X_{\mathrm{r}}\cdot s\right)/X_{\mathrm{m}}\\ K_6=R_{\mathrm{r}}/X_{\mathrm{m}}\end{array}\right.$$

From (16), it can be observed that the rotor voltage limitation is also affected by the slip rate s. Considering that the slip rate of the DFVSPSU consistently remains below the critical slip value corresponding to the rotor voltage constraint during normal operation, the rotor voltage limitation (U_rmax) is omitted in the analysis of its reactive power-regulation capability.

In summary, using actual data from a domestic power station, the operational limits of the stator-side power of the DFVSPSU are reflected in the P-Q power chart, as shown in Figure 5. In the figure, the vertical axis represents real power, the horizontal axis represents reactive power, and the yellow region indicates the normal operating range of the DFVSPSU.

In Figure 5, the four horizontal lines parallel to the reactive power axis represent the prime mover power limitation (P_m). Due to the constraints of the pump–turbine, the maximum output limit under the turbine operating condition is 315.99 MW, while the minimum output limit is set to be not less than 30%, which is 105.33 MW. Under the pump operating condition, the maximum input limit is 344.078 MW, and the minimum input limit is set to be not less than 70%, which is 245.77 MW. According to (13), the stator current limitation (I_smax) is represented as a blue circle with its center at the origin and a radius of mU_sI_smax in Figure 5. According to (14), the rotor current limitation (I_rmax) is also represented as a red circle. However, due to the iron losses in the motor, the center of the circle is no longer at the origin but is located at a point shifted towards the negative reactive power axis. The maximum rotor current used in this study is I_rmax = 6764A.

Although the P-Q diagram of the DFIG shows the stator-side reactive power range, it does not reflect its changes under random operating conditions. In fact, the stator-side reactive power output depends on real power and head (or lift). Therefore, the following section analyzes the actual reactive power limits of the stator side under different operating conditions using optimal pump–turbine data, further investigating its reactive power-regulation capability.

3.4. Reactive Power-Regulation Capability of the Stator Side of the DFVSPSU

This section utilizes the optimal operation data of the pump–turbine to determine the operating conditions under both turbine and pump modes and further combines actual data to derive the reactive power regulation range for the stator side of the DFVSPSU.

By combining the stator-side real power output (shown in Figure 3) with the P-Q power operation limitation diagram (shown in Figure 5), the reactive power output of the stator side under turbine and pump modes can be obtained, as shown in Figure 6 and Figure 7.

From Figure 6, it can be observed that under turbine mode, for the same head, the limits of the stator-side reactive power output and absorption decrease as the reference output increases. When the reference output is 124 MW and the head is between 385 and 462 m, the stator’s maximum reactive power output is 267.16 MVA. When the reference output is 300 MW and the head is 385 m, the minimum reactive power output is 147.74 MVA. When the reference output is 124 MW and the head is 385–462 m, the stator’s maximum reactive power absorption is 324.79 MVA, and when the reference output is 300 MW at a head of 385 m, the minimum reactive power absorption is 153.3 MVA.

From Figure 7, under pump mode, for the same lift, the limits of the stator-side reactive power output and absorption decrease as the reference input increases. When the reference input is 180 MW and the lift is 425 m, the maximum reactive power output of the stator is 242.845 MVA. When the reference input is 330 MW and the lift is 400 m, the minimum reactive power output is 147.37 MVA. When the reference input is 180 MW and the lift is 425 m, the maximum reactive power absorption is 294.08 MVA. When the reference input is 330 MW and the lift is 400 m, the minimum reactive power absorption is 152.61 MVA.

4. Reactive Power Characteristic Analysis of the Grid-Side Converter

The power limits of the grid-side converter are primarily constrained by the converter’s maximum capacity (S_cmax):

$${P_{\mathrm{c}}}^2+{Q_{\mathrm{c}}}^2\le {S_{\mathrm{cmax}}}^2$$

(17)

where P_c and Q_c are the active and reactive power of the grid-side converter and S_cmax is the maximum capacity of the grid-side converter. Further transformation of this formula yields the reactive power limitation range for the grid-side converter, as shown in (19):

$$\left\{\begin{array}{l}{Q}_{\mathrm{cmin}}=-\sqrt{{S_{\mathrm{cmax}}}^2-{P_{\mathrm{c}}}^2}\\ Q_{\mathrm{cmax}}=\sqrt{{S_{\mathrm{cmax}}}^2-{P_{\mathrm{c}}}^2}\end{array}\right.$$

(18)

From (5) and (18), it is clear that to obtain the reactive power characteristics of the grid-side converter, the real power on the rotor side of the DFVSPSU needs to be determined first.

4.1. Determining the Real Power on the Rotor Side of the DFVSPSU

This section uses the actual data of a domestic variable-speed unit as an example. Based on the pump–turbine’s operating characteristics, the real power output on the rotor side under turbine and pump modes can be derived, as shown in Figure 8.

As shown in Figure 8, under turbine mode, for the same head, the rotor power decreases as the reference output increases, but when the reference output reaches 270 MW, the rotor power starts to increase. When the head is between 385 and 462 m and the reference output is 124 MW, the maximum rotor power is −9.34 MW. When the reference output is 310 MW at a head of 385 m, the minimum rotor power is −20.29 MW.

It can be seen that under pump mode, for the same lift, the rotor power increases as the reference input increases. At the same reference input, the rotor power increases with lift. When the lift is 405 m and the reference input is 200 MW, the minimum rotor power is −14.7 MW. When the lift is 470 m and the reference input is 330 MW, the maximum rotor power is 18.87 MW.

4.2. Reactive Power Characteristic Analysis of the Grid-Side Converter in the DFVSPSU

In this study, the maximum capacity of the grid-side converter for a domestic DFVSPSU is 42 MVA. By combining the rotor-side real power output (shown in Figure 8) with the theoretical analysis provided earlier, the reactive power regulation capacity of the grid-side converter under turbine and pump modes is shown in Figure 9 and Figure 10.

From Figure 9, under turbine mode, when the reference output is 124 MW and the head is between 385 and 462 m, the maximum reactive power output of the grid-side converter Q_cmax is 41.04 MVA. When the reference output is 270 MW at a head of 462 m, the minimum reactive power output Q_cmin is 36.88 MVA. Similarly, when the reference output is 124 MW and the head is between 385and 462 m, the maximum reactive power absorption Q_cmax is 41.04 MVA, and at a 270 MW output with a head of 462 m, the minimum absorption Q_cmin is 36.88 MVA.

From Figure 10, under pump mode, when the reference input is 260 MW and the lift is 450 m, the maximum reactive power output of the grid-side converter Q_cmax is 42.09 MVA. When the reference input is 330 MW at a lift of 470 m, the minimum reactive power output is Q_cmin 37.62 MVA. Similarly, when the reference input is 260 MW at a lift of 450 m, the maximum reactive power absorption is Q_cmax 42.09 MVA, and when the reference input is 330 MW at a lift of 470 m, the minimum absorption Q_cmin is 37.62 MVA.

5. Total Reactive Power Characteristics of the Doubly-Fed Variable Speed Pumped Storage Unit

The total reactive power characteristics of the DFVSPSU consist of two components: the stator-side and grid-side converter’s reactive power characteristics. Combining the conclusions from Section 3 and Section 4, the total reactive power regulation capability is summarized in Figure 11 and Figure 12.

In the figures, the gray surface represents the reactive power output and absorption on the stator side, the yellow surface represents the reactive power output and absorption on the grid-side converter, and the blue surface represents the total reactive power output and absorption of the unit.

Most existing studies focus on the stator side but neglect the grid-side converter. This study incorporates both, finding that considering the grid-side characteristics significantly improves the total reactive power regulation capability of the unit.

From Figure 11, under turbine mode, when the reference output is 124 MW and the head is between 385 and 462 m, the total reactive power output of the unit reaches a maximum value of 308.2 MVA. When the reference output is 300 MW and the head is 385 m, the minimum value of the total reactive power output is 186.7 MVA. When the reference output is 124 MW and the head is between 385 and 462 m, the total reactive power absorption of the unit reaches a maximum value of 365.83 MVA, and when the reference output is 300 MW and the head is 385 m, the minimum value of the total reactive power absorption is 192.28 MVA.

Table 1. Reactive power regulation capability of the DFVSPSU considering Q_c (unit: MVA).

Q	Qs	Qc	(Q − Qs)/Qs
308.2	267.16	41.04	15.36%
186.7	147.74	38.98	26.37%
365.83	324.79	41.04	12.54%
192.28	153.3	38.98	25.43%

takes the maximum and minimum total reactive power output and absorption of the unit under the turbine mode as examples, along with the corresponding head and reference output for the stator-side and grid-side reactive power. The calculation results show that by considering the reactive power characteristics of the grid-side converter, the total reactive power-regulation capability of the unit increases by approximately 12% to 26% compared to considering only the stator-side reactive power.

From Figure 12, under pump mode, when the reference input is 180 MW and the lift is 425 m, the maximum total reactive power output of the unit is 283.24 MVA. When the reference input is 330 MW and the lift is 400 m, the minimum total reactive power output is 187.12 MVA. When the reference input is 180 MW and the lift is 425 m, the maximum total reactive power absorption is 334.48 MVA, and when the reference input is 330 MW and the lift is 400 m, the minimum total reactive power absorption is 192.37 MVA.

6. Comparison of Reactive Power-Regulation Capabilities Between Fixed-Speed Pumped Storage Unit and Doubly-Fed Variable Speed Pumped Storage Unit

To more clearly demonstrate the superiority of variable-speed units in reactive power regulation, this section analyzes the power operating range of fixed-speed pumped storage units (FSPSUs) and compares the reactive power-regulation capabilities of fixed-speed and variable-speed units. The analysis shows that fixed-speed units, due to their fixed speed characteristics, have relatively weaker reactive power-regulation capabilities. In contrast, variable-speed units, by adjusting their speed and power factor, have relatively stronger reactive power-regulation capabilities.

6.1. Power Operating Constraints of the FSPSU

The power operating range of the FSPSU is analyzed similarly to the DFVSPSU. The reactive power operating range of the fixed-speed unit is derived in the P-Q diagram based on its operating constraints, and then, the actual reactive power regulation range of the unit is analyzed under different operating conditions using actual data from a domestic fixed-speed unit.

• P-Q Power Operating Constraints of the FSPSU:

The power operating range of the FSPSU is constrained by the prime mover power, stator winding temperature, excitation winding temperature, and other factors:

• Stator winding temperature rise constraint: The stator winding temperature rise depends on the stator current, which is influenced by the generator’s apparent power. The unit’s rated capacity is limited by the thermal capacity of the stator winding and core, ensuring safe operation, represented as a blue circle in the P-Q diagram;
• Prime mover power constraint: The rated power of the prime mover of a pumped storage unit is usually equal to the rated real power of the generator it is paired with. Therefore, the prime mover power constraint is shown in the P-Q diagram by four horizontal lines. With a rated power factor of cosφ = 0.9, the minimum output constraint for turbine operation is 50%, and the minimum input constraint for pump operation is 75% [34];
• Excitation winding temperature rise constraint: The excitation winding temperature rise constraint limits the temperature increase during operation, primarily dependent on the excitation current. Due to iron losses, the power circle’s center shifts from the origin to the negative reactive power axis, forming a purple circle;
• Other constraints: When the excitation electromotive force is constant, the maximum achievable real power and its corresponding power angle represent the static stability constraint. In practice, there is an additional 10–15% margin for the static stability constraint. When the excitation current is zero, the unit is in an excitation loss state, at which point the unit is subject to excitation loss constraints.

Figure 13 illustrates the P-Q diagram of operational limitations for the FSPSU, where the yellow and green areas represent the normal operating ranges under the turbine and pump modes, respectively.

2.

Reactive Power Regulation Range of the FSPSU:

This section provides a detailed analysis using the actual data of the domestic FSPSU, exploring the reactive power regulation range under both turbine and pump modes.

As shown in Figure 14, under turbine mode, the reactive power output and absorption of the fixed-speed unit decrease as the reference output increases at the same head. When the reference output is 150 MW, the maximum reactive power output (absorption) is 209.4 MVA (300.66 MVA). When the reference output is 300 MW, the minimum reactive power output (absorption) is 146.36 MVA (151.3 MVA).

As shown in Figure 15, under pump mode, with the same lift, the real power is non-adjustable. The reactive power output limit is represented by a curve with a downward trend, while the reactive power absorption limit is represented by a curve with an upward trend. When the lift is 400 m and the reference input is 283 MW, the minimum reactive power output (absorption) is 156.09 MVA (181.07 MVA). When the lift is 465 m and the reference input is 249.9 MW, the maximum reactive power output (absorption) is 173 MVA (224.61 MVA).

6.2. Comparison of Reactive Power-Regulation Capabilities Between FSPSU and DFVSPSU

By summarizing the conclusions from Section 5 and Section 6.1, the differences in reactive power-regulation capabilities between two domestic pumped storage units, both with a capacity of 300 MW (one fixed-speed and one variable-speed), are shown in Figure 16 and Figure 17.

As shown in Figure 16, under turbine mode, the DFVSPSU has higher reactive power output and absorption limits than the FSPSU, with a maximum total output of 308.2 MVA and absorption of 365.83 MVA, compared to 209.4 MVA and 300.66 MVA for the FSPSU. Under pump mode (Figure 17), the DFVSPSU’s reactive power regulation range forms a surface, adjustable by speed, while the FSPSU’s range forms a curve, limited by fixed speed.

In order to more clearly illustrate the differences in reactive power-regulation performance between DFVSPSU and FSPSU, the results of Figure 16 and Figure 17 are summarized in a table for comparison, as shown in

Table 2. Comparison of reactive power-regulation capabilities between DFVSPSU and FSPSU (unit: MVA).

	Turbine Mode		Pump Mode
	The Maximum Reactive Power Output	The Maximum Reactive Power Absorption	The Maximum Reactive Power Output	The Maximum Reactive Power Absorption
DFVSPSU	308.2	365.83	283.24	334.48
FSPSU	209.4	300.66	173	224.61

.

As shown in Table 2, under turbine and pump modes, the maximum values of reactive power output or absorption by the DFVSPSU are greater than those of the FSPSU. The comparison confirms that the DFVSPSU provides a wider reactive power-regulation range than the FSPSU in both turbine and pump modes, demonstrating superior reactive power-regulation capabilities.

7. Conclusions

This study investigates the reactive power-regulation capability of a 300 MW DFVSPSU in China. The DFVSPSU’s reactive power characteristics are divided into stator-side and grid-side converter components, which are not coupled and should be analyzed separately. The DFIG capacity is often greater than the unit’s rated capacity due to its sub-synchronous operation under turbine mode, providing stronger reactive power regulation. For instance, the 300 MW unit has a rated speed of 412 rpm, with a DFIG capacity of 351.1 MW, exceeding the rated unit capacity. Factors limiting stator-side reactive power regulation include the prime mover power, stator current, and rotor current, with the grid-side converter’s regulation limited by converter capacity and rotor-side real power. Based on the variable-speed characteristic curve, the unit’s reactive power-regulation capability can be improved by 12% to 26% by fully utilizing the grid-side converter. Compared to the FSPSU, the DFVSPSU offers superior reactive power regulation, with higher output and absorption limits under turbine mode and a more flexible regulation range under pump mode.

In summary, this study reveals that DFVSPSUs possess broader reactive power-regulation characteristics than FSPSUs of the same capacity. This capability enables the units to actively absorb excess reactive power during periods of low grid load, effectively improving the operational efficiency of power plants during grid imbalances. Additionally, it provides a foundation for reactive power support and dynamic optimization design for the grid, thereby supporting grid voltage stability. Furthermore, this study has further explored the reactive power-regulation capabilities of grid-side converters, which can effectively support the design of overall reactive power-regulation strategies for DFVSPSUs. Future research can focus on the allocation of reactive power characteristics between grid-side converters and the stator side, thereby optimizing a dynamic control strategy design.