Efficient modeling of DFIG- and FSC-based wind turbines for frequency stability analysis

The overall control system of a variable-speed WT based on a DFIG is depicted in Fig. 1. As shown, the rotor terminals are coupled to the grid through a back-to-back frequency converter comprising two separately controlled voltage source converters (MSC and GSC), which are rated for a fraction of the total generator power (slip power \({{s}_{\text{G}}}{{P}_{\text{ag}}}\)), where \({{s}_{\text{G}}}\) is the slip and \({{P}_{\text{ag}}}\) is the air-gap power. The power flow fed into the grid via the rotor circuit depends on the generator speed, whose ratio to the stator angular frequency divides the operating points of the WT into sub-synchronous, synchronous and super-synchronous operating points.

The overall control system of the variable-speed FSC-based WT is illustrated in Fig. 5. As shown, the stator terminals are coupled to the grid through a back-to-back frequency converter comprising two independently controlled voltage source converters (MSC and GSC). As a result, the entire power of the generator is fed into the grid via the frequency converter, which results in a higher rated power compared to converters of DFIG-based WTs. The aerodynamic and drive train model, the induction generator, the pitch and speed controller as well as the GSC controller remain unchanged and are implemented in the same way as explained in the previous sections for the DFIG-based WT model. The only difference is setting \({{\underline{u}}_{\text{R}}}\) in the induction generator equations to zero (squirrel-cage induction generator SCIG), which is why the slip power \({{s}_{\text{G}}}{{P}_{\text{ag}}}\) is wasted as rotor losses in the rotor circuit (\({{P}_{\text{R}}}={{P}_{\text{RV}}}={3}/{2}\;{{R}_{\text{R}}}{{\lvert{{\underline{i}}_{\text{R}}}\rvert}^{2}}\)). The MSC controller is the only model-specific sub-module that is treated in the following section.

In RMS simulation, the WT models are represented in the overall network nodal equation system by their Norton equivalent circuits, whose current sources depend on the state variables of the differential equations. The overall equivalent circuit of the DFIG-based WT consists of the Norton equivalent circuits of the DFIG and the GSC, as shown in Fig. 7a, where \(\underline{u}_{\text{S}}^{\prime}=\text{j}{{\omega}_{\text{S}}}{{k}_{\text{R}}}{{\psi}_{\text{R}}}\), \(\underline{Z}_{\text{S}}^{\prime}={{R}_{\text{S}}}\text{+j}{{\omega}_{\text{S}}}L_{\text{S}}^{\prime}\) and \({{\underline{Z}}_{\text{GSC}}}=\text{j}{{\omega}_{\text{0}}}{{L}_{\text{GSC}}}\). Since all sub-modules of the FSC-based WT except the GSC are fully decoupled from the grid via the DC-link of the frequency converter, the equivalent circuit of the FSC-based WT is represented by the Norton equivalent circuit of the GSC, as shown in Fig. 7b.

Since the PI controllers of the MSC controller in Fig. 3 are assumed to be quasi-stationary, the rotor current reference value \({{\underline{i}}_{\text{R}\text{,ref}}}\) can be calculated using the feed forward terms of the outer power controller loop retransformed from the stator-voltage-oriented reference frame into the original reference frame of the induction generator in complex notation as follows: Taking into account the quasi-stationary assumption of the inner current controller, the rotor current is set to the rotor current reference value, i.e., \({{\underline{i}}_{\text{R}}}={{\underline{i}}_{\text{R}\text{,ref}}}\). Furthermore, the rotor voltage can be derived from the feed forward terms of the inner current controller loop retransformed from the stator-voltage-oriented reference frame into the original reference frame of the induction generator in complex notation: The rotor flux linkage, which is needed to calculate the electrical torque using (10), can be obtained from (8). Considering the stator current as a function of the apparent stator power reference, the current source of the Norton equivalent circuit of the DFIG can be calculated as follows: Since the DC voltage and thus the DC voltage controller are assumed to be also quasi-stationary, i.e., \({\dot{u}}_{\text{DC}}=0\) and \(u_{\text{DC}}^{\text{meas}}={{u}_{\text{DC}\text{,ref}}}\), and thus the transients of the DC voltage are assumed to be decayed in the time frame of interest, a power equilibrium at the DC node can be permanently assumed. This implies that the MSC active power provided by the rotor circuit is equal to the GSC active power \({{P}_{\text{G}}}={{P}_{\text{R}}}\). Considering the GSC current as a function of the rotor power and reactive power reference value, the current source of the Norton equivalent circuit of the GSC can be determined: The model structure of the simplified DFIG-based WT is depicted in Fig. 8. The interface variables to the aerodynamic model as well as to the pitch and speed controller are written in orange.

Since the PI controllers of the MSC controller in Fig. 6 are assumed to be quasi-stationary, the d‑component of the stator current reference value \({{i}_{\text{Sd}\text{,ref}}}\) can be calculated using the feed forward term of the outer power controller loop retransformed from the transient-voltage-oriented reference frame into the original reference frame of the SCIG. Since the SCIG is operated in the constant flux region, the q‑component of the stator current is controlled at a constant value, which is known from the initialization procedure [8]. Taking into account both the d‑ and q‑components, the stator current reference value can be determined in complex notation as follows: Considering the quasi-stationary assumption of the inner current controller, the stator current is set to the stator current reference value, i.e., \({{\underline{i}}_{\text{S}}}={{\underline{i}}_{\text{S}\text{,ref}}}\). The slip frequency \({{s}_{\text{G}}}{{\omega}_{\text{S}}}\) can be calculated from (6) under consideration of \({{\underline{u}}_{\text{R}}}=0\) and \({{\underline{{\dot{\psi}}}}_{\text{R}}}=0\): Substituting \({{\underline{\psi}}_{\text{R}}}\) in (8) using \({{\underline{\psi}}_{\text{R}}}={\underline{u}_{\text{S}}^{\prime}}/{\text{j}{{\omega}_{\text{S}}}{{k}_{\text{R}}}}\;\) and then rearranging the resulting equation leads to: Substituting (18) and \({{\underline{\psi}}_{\text{R}}}={\underline{u}_{\text{S}}^{\prime}}/{\text{j}{{\omega}_{\text{S}}}{{k}_{\text{R}}}}\;\) in (17) and adding it to the rotor angular frequency \({{\omega}_{\text{R}}}\) leads to the equation of the stator angular frequency \({{\omega}_{\text{S}}}\): The stator voltage can be obtained from the feed forward terms of the inner current controller loop retransformed from the transient-voltage-oriented reference frame into the original reference frame of the induction generator in complex notation: The rotor current as a function of the rotor flux linkage and the slip frequency can be obtained by rearranging (17). After substituting the resulted equation in (8) the equation of the rotor flux linkage, which is needed to calculate the electrical torque, can be obtained: The same underlying assumptions made regarding the GSC of the DFIG-based WT, are equally valid for the FSC based WT. Considering the GSC current as a function of the stator power and reactive power reference value, the current source of the Norton equivalent circuit of the GSC can be determined: The model structure of the simplified FSC-based WT is depicted in Fig. 9. The interface variables to the aerodynamic model as well as to the pitch and speed controller are written in orange.

Equation (23) demonstrates the positive sequence of the overall network nodal equation system, where the fundamental oscillations of voltages and currents are considered through their RMS values [8]: The WT models as well as other active devices (e.g. generating units and their respective control systems, dynamic loads, etc.) are considered by their Norton equivalent circuits, whose admittance equations in the matrix notation are represented on the right side of (23), where \(\underline{\mathbf{Y}}\) is a diagonal matrix whose diagonal elements are occupied by the internal admittances of the active devices. The quantities \(\underline{\mathbf{u}}\) and \({{\underline{\mathbf{i}}}_{\text{q}}}\) represent the nodal voltage and the current source vectors. The passive electrical network connects the active device models and is considered by its steady-state model, whose admittance equation in matrix notation is represented on the left side of (23), where \({{\underline{\mathbf{Y}}}_{\text{N}}}\) indicates the network admittance matrix including the static loads. The network nodal voltages at each time step are calculated by rearranging (23), as demonstrated in (24). While the current sources of the detailed WT models depend on the state variables of the differential equations at each time step, the current sources of the simplified WT models at each time step are calculated using the terminal voltages of the last time step (see 14, 15, 22), which are obtained from (24). The resulting current sources are used to obtain the terminal voltages at the current time step, which serve as input variables of the WT model structures, as illustrated in Figs. 8 and 9. This algebraic dependency between the current sources and the terminal voltages should be considered utilizing appropriate methods. Two possible approaches are taken into account in this paper. The first approach is to reduce the time step size to such an extent that the voltage difference between two time steps is imperceptibly small, so that they have no influence on the results. Adoption of this solution leads to a higher computational time, which is not purposeful when simulating large grid areas in the time frame of interest of the frequency stability analysis. In the second approach, the algebraic loops are solved iteratively at each time step. Considering this solution, the time step size can even be increased considerably, which will lead to a significant reduction in computational time. The results of both methods are illustrated and compared to each other in Sect. 6.

The first examples consist of step-response simulations of a 2 MW DFIG-based and a 2 MW FSC-based WT model each as a detailed and a simplified model and coupled to a 20 kV passive equivalent grid via a transformer (0.69/20 kV). The model parameters of both WT models used in this paper are identical and can be found in [7]. The initialization procedure of each WT model is performed based on the terminal output apparent power, which are known from a Newton-Raphson-based power flow calculation [15]. The initialization approaches used in this paper are extensively discussed in [8].

Some characteristic variables of step response simulations of both detailed WT models to different deterministic wind speeds are compared to those of both simplified WT models in Figs. 12 and 13. In each case, the wind speed is increased from 10 m/s to 14 m/s, with 1 m/s steps every 50 s, as shown in Figs. 12a and 13a. After 250 s, the operating points return to the initial values with the same step sizes. At wind speeds lower than 12 m/s, the speed controller controls the rotor speed according to the tracking characteristic curve (see Figs. 12c and 13c). The MSC controller of the detailed WT models controls the WT terminal active power to its reference value provided by the speed controller.

As mentioned in Sect. 3, taking into account the quasi-stationary assumption of the converter controllers of the simplified WT models, the reference values of the WT terminal active power are directly and without any delay considered in the calculation of the current sources of the Norton equivalent circuit of the MSC and GSC (see 14, 15 and 22). At wind speeds equal to or higher than 12 m/s, the pitch controller adjusts the rotor blade angle (Figs. 12a and 13a) by controlling the rotor speed and limits the mechanical torque and consequently the output active power to the WT rated power (Figs. 12b and 13b). The output active power of the FSC-based WT is equal to the stator active power neglecting the converter losses. Its output reactive power is kept constant at the reference value of the GSC that is equal to the initial value of the WT terminal reactive power. The output active and reactive power of the DFIG-based WT are composed of the stator portion and the rotor circuit portion. The reactive power of the rotor circuit is kept at zero and the reactive power of the stator is kept constant at the initial value of the WT terminal reactive power over the whole simulation time. As shown in Figs. 12 and 13, the conducted comparison of the step response simulation results demonstrates an acceptable level of model accuracy of the simplified WT models against the detailed RMS WT models.

In order to demonstrate the comparability of the contributions of the simplified WT models against the detailed WT models in improving the frequency performance of power systems, the WT models are extended with the droop-based FFR controller introduced in Sect. 4. Furthermore, the simplified as well as the detailed WT models are programmed in the MATLAB environment and are implemented in a MATLAB-based RMS simulation tool, which is discussed in [16, 17] and is referred to as a “distributed-rotating-mass based model”.

The simulation tool does not utilize an infinite busbar and is therefore well suited for performing frequency studies. In this simulation tool, the synchronous generators are represented through a fifth-order state space model commonly called model 2.1, according to the IEEE Std 1110-2019 [11]. The excitation systems are modelled using a ST1C excitation system with a PSS1A stabilizer from the IEEE Std 421.5-2016 [12]. The prime mover models and their control systems, which are also discussed in [16], enable the simulation of primary and secondary control power activation in accordance with the specific regulatory requirements [3, 4]. The implemented composite load model comprises of a dynamic and a static component. The dynamic component is represented by a third-order induction motor and the static component by a ZIP model [1]. The passive network components (i.e., transformers and transmission lines) are represented by their steady-state models. The model parameters of all devices and their corresponding controllers used in this paper can be found in [17]. Fig. 14 shows the single line diagram of a 220 kV test system.

Four loads (L) are supplied by three synchronous generators (G) and two wind parks (WP) through a transmission grid. WP1 comprises 80 DFIG-based WTs and WP2 consists of 80 FSC-based WTs each with a rated active power of 2 MW. No aggregation of the WT models is performed in this paper. To determine the nodal voltages and powers at the considered operating point, a Newton–Raphson-based power flow calculation is conducted. The initial terminal apparent powers of the active devices (i.e., synchronous generators, loads and wind power plants) resulting from the power flow calculation are shown in Table 2.

Four scenarios with constant wind speed are simulated based on an unscheduled increase in load L1 by 45 MW after 10 s. In the scenario 1 the FFR controllers of all WTs are deactivated. Thus, only the synchronous generators offer their inertia to control the ROCOF and the FN. In the three remaining scenarios, fifty percent of the WTs in each WP provide FFR capabilities. These three scenarios differ with respect to the modeling depth, i.e., detailed (scenario 2) or simplified (scenario 3 and 4). In order to solve the algebraic loop resulting in the case of utilizing the simplified WT models, in scenario 3, the time step size reduction is applied, while the iterative approach is used in the scenario 4.

Figs. 15a and 16a show the power and rotor speed response of an exemplary DFIG-based WT from WP1 and an exemplary FSC-based WT from WP2, respectively, to the above-mentioned disturbance in the scenarios 2, 3 and 4 by means of the power–speed characteristic curves. Point A in each figure represents the stationary operating point before the perturbation in each scenario. If the grid frequency exceeds the threshold, the FFR controller is triggered and thus the sum of \(P_{\text{WT}}^{\text{activation}}\) and \({{P}_{\text{add}}}\) is commanded to the MSC controller as the active power set point. In the overproduction phase (i.e., the solid curves), since the electrical output power plus the WT losses are higher than the available mechanical power extracted from the wind, the rotating mass of the WT decelerates, and the rotor speed decreases. The slope of the solid curves can be modified by setting the minimum frequency limit \({{f}_{\text{min}}}\). The higher this frequency is set, the steeper the solid curves become. Once the activation time period \({{t}_{\text{FFR}}}\) is over, the speed controller is activated and the recovery phase (i.e., the dashed curves) begins with the goal of accelerating the WT back to the desired operating point on the tracking characteristic curve, which depends on the current wind speed. As the wind speed is constant in all scenarios, the desired rotor speed is the same as the pre-fault rotor speed (i.e., point A). Fig. 15b and c as well as Fig. 16b and c demonstrate the mechanical power, the output active power and the WT rotor speed of both selected WTs in the overproduction phase and the recovery phase in the time domain.

As shown in the time domain illustrations, the output active power curves in the overproduction phase and their peak values of both WT models in all three scenarios are identical. Furthermore, the output active power curves of the DFIG-based WT models in the recovery phase are very good comparable in all three scenarios. On the other hand, the output active power curve of the FSC-based WT model falls slightly lower in the recovery phase in scenario 2 and continues with a higher slope compared to scenarios 3 and 4. As illustrated in Fig. 15a and c as well as in Fig. 16a and c, the rotor speed of both selected WTs exhibits a more dynamic behavior in scenario 2 and falls insignificantly lower compared to scenarios 3 and 4. As it is observed, this dynamic behavior has no effect on the output variable of the FFR controller of both WT models, which directly sets the reference value of output active power to the MSC controller.

Fig. 17 depicts the center of inertia frequencies in all four scenarios mentioned above. As shown in the figure, very good comparable frequency responses with respect to the FN are observed in scenarios 2, 3 and 4, i.e. similar improvements of the FN are observed in all three scenarios. However, the frequency responses differ slightly regarding the undesirable but unavoidable second frequency drop caused by the power reduction in the recovery phase, which is deeper the higher the maximum additional power achieved in the overproduction phase. This observed difference is due to the fact that the output active power curve of the FSC-based WT model falls slightly lower in the recovery phase in scenario 2 compared to scenarios 3 and 4, as depicted in Fig. 16. This deviation is not relevant for the evaluation of the test system’s frequency response, because it has no influence on the two important frequency stability indicators, i.e., ROCOF and FN.