A New Non-Smooth Simplified Model for DFIG in Electromechanical Transient Analysis

The transient of the DFIG engages a lot of non-smooth elements such as switches of controllers and saturation of limiters, thus, the simplified non-smooth model of DFIG for electromechanical transient analysis is needed. This paper proposes a non-smooth simplified model of DFIG for electromechanical transient analysis, with the viewpoint of the transient response and energy accumulation. First, the general non-smooth dynamic behavior of DFIG during fault-on and post-fault periods are presented and the model is divided into four smooth stages. Then, the detailed model for the transient stage, according to without/with the trigger of the crowbar is presented. Furthermore, the simplified model of DFIG in the steady state is introduced according to without/with the trigger of the crowbar and saturation of limiters, determined by the stator voltage. Moreover, the transient model with/without the crowbar protection are simplified according to the principle of constant energy accumulation. Thus, the overall non-smooth simplified models of DFIG is obtained. Finally, the effectiveness of the proposed non-smooth simplified model is verified with the simulations.


I. INTRODUCTION
In recent years, WTG (wind power generator) represented by DFIG (doubly fed induction generator) has been connected to the power grid with a large scale [1]. The DFIG engages a different time scales than the conventional synchronous machine [2], which causes the serious transient stability problem of power system [3]- [13]. Thus, two basic questions about transient stability are concerned. The one is the transient stability of WTG itself [3], the other is the transient stability of power system due to the interconnection of WTG [4]- [13].
For the first problem, the detailed model for the WTG is used. It contains model for the wind turbine, the asynchronous generator and the control system. They were mainly modeled as multi-mass model [4], three-order model [5], and dimension-reduced model [6] respectively. The time domain simulation is the popular analysis method.
The associate editor coordinating the review of this manuscript and approving it for publication was Canbing Li . However, for the second problem, some references use the detailed model. For example, Ref [7] considers the highorder general model of electromechanical transient of DFIG. Ref. [8] applied the detailed model of WTG to the rotor angle stability of system with thermal & wind power generation and AC/DC transmission through simulation.
Although the detailed model retains the dynamic of WTG, since the system interconnected with WTGs contains many equipment, but the full-scale model will cause the ''dimensionality disaster'', thus result in difficulty to analyze the mechanism of impact of wind farms on the transient behavior of the AC system. Therefore, some references use different simplified models of the DFIG [8]- [12]. For example, Ref. [9] models the asynchronous generator with a three-order model, and simplifies the electromechanical transient model of the rotor side converter and crowbar controller by transfer function method. Ref. [10], [11] considers the DFIG as a constant negative resistance to study the influence of WTG's interconnection on the power-angle curve and the transient stability of power system. Ref. [ Ref. [13] believes that the WTG has a characteristic of constant power after fault. On the other hand, there are also work use the transient energy function (TEF) to analyze the transient stability of the WTG integrated power system [14]- [16]. For example, Ref. [14] proposes an unified method for transient stability assessment of system including WTGs based on corrected kinetic energy. Ref. [15] proposes the energy function of DFIG in the form of potential, reactive, and inertial energies. Ref. [16] considers the TEF of the system integrated with multiple DFIGs. Generally, the TEF is concerned with the energy of the WTG system in the post-fault period, but there is no analysis of the energy accumulation in the fault-on period.
The above works use a single simplified impedance or power source model, it is very convenient. However, since the actual WTG has different switched dynamic, such as trigger of crowbar corresponding to different faults, the single model may be different from the actual characters of WTG. Furthermore, with the large disturbance, a lot of limiter in the DFIG will be saturated, and this characteristic could be caught hardly by the single model. Thus, the transient of the DFIG should reflect the non-smooth elements such as switches of controllers and saturation of limiters, thus, the non-smooth model of DFIG for electromechanical transient analysis is needed.
In recognized the above problems, this paper proposed a non-smooth simplified model of DFIG for electromechanical transient analysis, with the viewpoint of the transient response and energy accumulation.
The contributions of this paper are as follows: (1) The model of DFIG for the transient stage in fault-on and post-fault periods are simplified with the principle of constant energy accumulation.
(2) A non-smooth simplified model of DFIG for electromechanical transient analysis.
The remainders of the paper are organized as follows. In Section II, the general non-smooth dynamic behavior of DFIG during fault-on and post-fault period is presented and the model is divided into four smooth stages, i.e., the transient and the steady-state stage in fault-on and postfault periods, respectively. In Section III, the detailed model for the transient stage, according to without/with the trigger of the crowbar is presented are proposed through formula deduction and analysis. In Section IV, the simplified steady-state model of DFIG is introduced according to without/with the trigger of the crowbar and saturation of limiters, determined by the stator voltage. In section V, the transient model with/ without the crowbar protection are reduced according to the principle of constant energy accumulation. In Section VI, the overall non-smooth simplified models of DFIG is presented. In Section VII, the error between the simplified model and the detailed model is analyzed by simulation, the effectiveness of simplification is verified. Finally, conclusions and discussions are presented in Section VIII.

II. GENERAL NON-SMOOTH MODEL FOR DFIG BASED ON SIMULATION
This section presents the general non-smooth dynamic behavior of DFIG based on simulation and shows that the model contains four smooth stages.  The dynamic of the above power system with respected to the fault can be divided into three periods, which are the pre-fault, fault-on and post-fault. When a fault occurs, the response of the crowbar of DFIG in the fault-on period can be divided into two cases according to the severity of fault, i.e., one is the crowbar triggered and the other is not, in the term of system, the fault-on system may switched to different smooth system according to the severity of fault.
The severity of the fault can be represented by the grounding resistance R f of the three-phase fault at MV1. Here, two three-phase faults with different R f resulting with the crowbar triggered/not triggered is considered. The fault occurs at MV1 at 1s, and then cleared at 1.5s.
Case 1: R f = 1 , the crowbar of DFIGs are triggered. Case 2: R f = 3 , the crowbar of DFIGs are not triggered. The output power of DFIG under the above two cases are shown in Fig. 2. The Fig.2 shows that no matter the crowbar is triggered or not, the output power of DFIG in fault-on and post-fault period go through a transient process before the steady state. According to the different output characteristics of the DFIG in the fault-on and post-fault periods, the dynamic of DFIG could be divide into four stages, i.e., (1) the transient stage in fault-on period (2) the steady-state stage in fault-on period (3) the transient stage in post-fault period (4) the steady-state stage in post-fault period For each stage, the DFIG is equivalent to different models according to whether the crowbar is triggered or not. They will be discussed in the following section.
Furthermore, with different R f , the terminal voltage drops of the DFIG, which determining the saturation of limiter and trigger of the crowbar protection, is different. Thus, the nonsmooth(switched) model is depended on the terminal voltage drop of DFIG.

III. DETAILED NON-SMOOTH MODEL OF DFIG AT TRANSIENT STAGE
This section presents the detail model of DFIG with/without the trigger of the crowbar.

A. WITHOUT THE TRIGGER OF THE CROWBAR
In the case of the crowbar is not triggered, the equivalent circuit of DFIG can be shown as in Fig. 3. Furthermore, its model can be described by the following equations.

Rotor shafting equation:
Voltage equation: Flux linkage equation: Grid interface equation: whereψ s (ψ r ) is the flux linkage of stator (rotor),Ĩ s (Ĩ r ) is the current of stator (rotor),Ũ s (Ũ r ) is the voltage of stator (rotor); p is derivative operator; R s (R r ) is the resistance of stator (rotor), L s (L r ) is the self-inductance of stator (rotor) and L m is the mutual-inductance between rotor and stator; ω 1 is the synchronous angular velocity, ω r is the machine angular velocity and ω s = ω 1 -ω r is the slip angular velocity; L eq is the equivalent inductance;Ũ eq is the equivalent voltage source of the network;Ĩ g is the current of the converter in grid side.

B. WITH THE TRIGGER OF THE CROWBAR
When crowbar is triggered, the rotor winding is shorted and the control system does not works, thusĨ g = 0, U r = 0 in (4), in this case, the DFIG is equivalent to a cage induction generator in this condition. What's more, according to Ref. [20], if there is a fault outside the wind farm or a fault with large grounding resistance inside the wind farm, the transient process in stator can be ignored, i.e., pψ s = 0 in (2). Meanwhile, it can be known from Ref. [18] that, the transient process of the grid connected to stator can be ignored generally when the stator transient process is neglected, i.e., p(−Ĩ s +Ĩ g ) = 0 in (4). Then, the detailed model under this condition can be described by following equations. Rotor shafting equation: Voltage equation: Flux linkage equation: Grid interface equation: The above analysis indicates that the order of the detailed fault-on models of DFIG is high. Furthermore, in the case of the crowbar is not triggered, it is closely related to the control system. Thus, the detailed model is not suitable for the analysis of the impact of the interconnection of large-scale wind power. Therefore, the simplification of the detailed models is necessary.

IV. SIMPLIFIED MODEL FOR STEADY-STATE STAGES
This section deduces the simplified model of DFIG in the steady state stage according to without/with the trigger of the crowbar and saturation of limiters, determined by the stator voltage. VOLUME 8, 2020

A. NON-SMOOTH MODEL FOR STEADY-STATE STAGE IN FAULT-ON PERIOD 1) WITHOUT THE TRIGGER OF THE CROWBAR
As stated in ref. [19] the steady-state active power output of DFIG is determined by the steady-state voltage in the faulton period, as shown in Fig. 4. when the DFIG stator voltage satisfies U s > U 2 , then the crowbar is not triggered, and the steady-state reactive power output keeps zero, but the active power output have two different models according to stator voltage. If the DFIG stator voltage satisfies U s > U 1 , the steady-state active power can maintain its output. In this case, the DFIG could be regarded as a constant active power source.
If the DFIG stator voltage satisfies U 2 < U s < U 1 , then, the steady-state active power is linear with the steady stator voltage, as the limiter of rotor current is saturated, the output equation is as follows, where L m is the excitation reactance and L s is the reactance of stator; s = (ω 1 -ω r )/ω 1 is the slip ratio; i limit rd is the rotor limiting current; U s (t) is the instantaneous value of stator voltage.
In this case, the DFIG could be regarded as a constant active power current source.

2) WITH THE TRIGGER OF THE CROWBAR
The trigger of crowbar short the rotor winding, and control system will quit. Thus, the model of the DFIG is similar to a cage induction generator, as shown in Fig. 5. where X ls is the leakage reactance of stator, X lr is the leakage reactance of rotor and X m is the excitation reactance; R c is the resistance of crowbar. If the excitation reactance is ignored, the equivalent impedance of DFIG is Furthermore, considering the speed of DFIG during the fault is approximately constant, i.e., s = const, then, the equivalent impedance of DFIG is approximately constant. Therefore, in this case, the DFIG can be simplified to a constant impedance.

B. MODEL OF STEADY-STAGE IN POST-FAULT PERIOD
In steady state after the fault, assuming the DFIG uses the maximum power point tracking control, then the DFIG active power output can be stated as follows, where K w is a constant related to wind turbine tip speed ratio, ω w is wind turbine speed.
Equation (11) indicates that the steady-state active power output of DFIG is determined by wind turbine speed. Furthermore, as the inertia of wind turbine is large and the time constant of DFIG pitch angle control system is larger compared with the electromechanical transient time scale, so the steady-state active power output of DFIG is approximately constant. Thus, the model of DFIG for steady-state state in the post-fault model can be regarded as a constant active power source.

V. MODEL REDUCTION FOR TRANSIENT STAGE
This section presents the simplified model for the transient stage in the fault-on and post-fault periods based on the energy accumulation, according to without/with the trigger of the crowbar.

A. MODEL REDUCTION WITH THE TRIGGER OF THE CROWBAR
If crowbar is triggered, the detailed model of DFIG is described by (5)- (8). Furthermore, the transient current response of DFIG can be obtained as follows.
where τ * a is the reciprocal of flux decay time constant; ω * s is the rotation frequency;ψ dq sf = C * 1 e ξ t is the free response of stator flux linkage; C * 1 is an undetermined coefficient relating to initial values,ψ dq sp = e ξ t t 0s e −ξ λ dλ is the forced response of stator flux linkage.
Finally, with the flux linkage (7), the stator current in d-q frame can be obtained as follows Equation (19) indicates that the stator current consists of two parts, one is the transient component of speed frequency which decay with time constant T b , the other is the steady-state component of power frequency.
Among them, the steady-state power frequency component is equivalent to the induction generator model as shown in Fig. 7 in the case of the excitation reactance X m and stator resistance R s are ignored. This verifies that when the crowbar is triggered, the model of DFIG in steady-state can be regarded as an induction generator. Moreover, it can be simplified as a constant impedance.
For the transient decaying component, its frequency is slip frequency both in dq frame and xy frame, so the power corresponding to this component is decaying during the fault. On the other hand, it is the energy exchange between DFIG and large power system and it contributes little to energy accumulation. Thus, this component can be ignored in electromechanical transient analysis, according to the principle that unchanging the energy accumulation in the fault process.
Thus, if the DFIG stator voltage satisfies U S < U 2 , the constant impedance model for DFIG can be used in the electromechanical transient analysis.

B. MODEL REDUCTION WITHOUT THE TRIGGER OF THE CROWBAR
In the electromechanical transient stability analysis, the energy accumulation caused by the mismatch of mechanical power and electromagnetic power of the DFIG is the primary concern. Thus, this subsection simplifies the model from the perspective of output characteristic with the principle of the constant energy accumulation.
If crowbar is not triggered, then theŨ r andĨ g of the detailed model (1)(2)(3)(4) in transient stage are determined by control system. Moreover, the control system is very complicated with several PI loops and limiters and switching dynamics. Therefore, it is difficult to analytically give out the detailed model. Here, the model is reduced with the energy accumulation.

VI. THE NON-SMOOTH SIMPLIFIED MODEL FOR DFIG
This section presents the overall non-smooth simplified models of DFIG. In general, the non-smooth simplified models at different stages can be obtained, as shown in Table 1. In detail, the proposed non-smooth simplified model for electromechanical transient analysis, is as follows.
For the fault-on period, the model is switched according to the DFIG stator voltage. if the crowbar and the current limit are not activated (U s > U 1 ), the model can be the constant power source. If the crowbar is not triggered but the current limit is saturated (U 2 < U s < U 1 ), the simplified fault-on model of DFIG can be the constant active current source. If the crowbar is triggered (U s < U 2 ), the simplified fault-on model of DFIG can be the constant impedance.
For the DFIG in the post-fault, the simplified fault-on model of DFIG can be a constant power source.

VII. VERIFICATION FOR THE PROPOSED MODEL WITH SIMULATION
In this section, the proposed non-smooth is verified with the simulations in DIgSILENT, the system is shown in Fig.1.

A. VERIFICATION OF THE SIMPLIFIED TRANSIENT MODEL WITH CROWBAR TRIGGERED
In the simulation, a three-phase fault with 1 grounding resistance occurs at 1.0s at MV 1 , and cleared at 1.5s. In this case, the crowbar is triggered. The equivalent fault-on impedance of the DFIG can be obtained as shown in Fig.7. As shown in Fig.7, when the fault occurs at 1.0s, the equivalent resistance and reactance drop rapidly from the pre-fault steady-state value (1pu-0pu.). Then, it damply oscillates near the fault-on steady-state value (0.0382pu-0.175pu.) at the slip frequency. The above transient process lasts about 300ms. After that, the equivalent impedance approaches the fault-on steady-state value and remains unchanged. The dynamic process is coincided with the result as the process analysis in section V. Meanwhile, the active and reactive power output during the fault is shown in Fig.8. Both active power and reactive power oscillate near their steady-state value. According to model in section V, the transient energy accumulations may be replaced by the steady-state energy accumulations, as shown by the dashed line in Fig.7.
Furthermore, Table 2 shows that the energy accumulations produced by the high-order detailed simulation model and the constant impedance model in fault-on period(1s∼1.5s). The result confirms that the errors between the active/reactive power accumulation produced by constant impedance model and detailed transient model are blow 6%. Therefore, the high-order simulation model and the constant impedance model have a high degree of accuracy when the crowbar is triggered. Thus, the rationality of simplification is verified.

B. WITHOUT THE TRIGGER OF CROWBAR
In the simulation, a three-phase fault occurs at 1.0s at MV 1 in Fig.1, and cleared at 1.5s. The fault grounding resistance R f is between 1.5 -5.0 . In this case, the crowbar will not be triggered. There are two cases.
Case 1: When R f is between 1.5 -3.5 , the current limit is saturated and the steady-state model of DFIG is constant active current source (represented by C in Table. 3).
Case 2: When R f is between 4.0 -5.0 , the current limit is not saturated and the steady-state model of DFIG is constant power source (represented by P in Table. 3).
Thus, simplified models are non-smooth and different according to different voltage drops. For the non-smooth simplified models, the comparison to detailed model, with the viewpoint of energy, can be obtained, as shown in Table 3. The Table 3 shows that, when the crowbar is not triggered, the error between the total energy accumulations produced by the detailed model and simplified model is less than 2%. This range of error is acceptable when analyzing electromechanical transient characteristics. Thus, it is reasonable to using the proposed non-smooth simplified model to replace the detailed transient model in the electromechanical transient analysis when the crowbar is not triggered.

VIII. CONCLUSION AND DISCUSSIONS
This paper proposed a non-smooth simplified model of DFIG for electromechanical transient analysis, with the viewpoint of the transient response and energy accumulation. Specifically, the general non-smooth dynamic behavior of DFIG is divided into four smooth stage. In the four smooth stages, the simplified model of DFIG according to without/with the trigger of the crowbar and saturation of limiters, determined by the stator voltage, are introduced.
In detail, the proposed non-smooth simplified model are as follows. For the fault-on period, the model is switched according to the DFIG stator voltage. if the crowbar and the current limit is not activated the model can be the constant power source. If the crowbar is not triggered but the current limit is saturated, the simplified fault-on model of DFIG can be the constant active current source. If the crowbar is triggered, the simplified fault-on model of DFIG can be the constant impedance. For the DFIG in the post-fault, the simplified fault-on model of DFIG can be a constant power source.
The proposed non-smooth simplified model not only retains main features of DFIGs, but also could serve as a good foundation to under the transient stability of power system.    However, the reactive power and the protection in DFIG is not fully consider in this paper, as it mainly focuses on the active power and energy. Thus, it will become the future works.