A direct analytical predetermination of PMSG based WPS steady-state values under different operating conditions

Abstract

This paper presents a direct analytical method to predetermine the steady-state values of a permanent magnet synchronous generator (PMSG) based wind power system (WPS) at each stage of power flow. A generalized structured is developed with two independent equivalent circuits, that is, PMSG and grid side. To effectively determine the converters performance numerals despite grid disturbances, steady-state model is structured with positive sequence components of grid voltage. The advantage of the proposed model is that the methods evade the requirements of d-q modeling and a dedicated controller to evaluate the system performance. Using the proposed steady-state model, the entire WPS components ratings is predicted evading time domain simulation with complicated controller design. Also, the simple controller design is proposed to aid in optimal power flow supplement with FRT requirements under all possible system operating conditions. Ultimately, validation of predetermined values with the simulated PSCAD/EMTDC response including the proposed controller is investigated.

Keywords

power regulation grid disturbances steady-state

Introduction

Recent development of power network with wind energy source integration is propitious with the direct-driven permanent magnet synchronous generator (PMSG) wind turbines (Mahela et al., 2019; Mohamad et al., 2020; Valenciaga and Fernandez, 2015; Yu et al., 2021). Wide-scale adoption of power electronics technology makes the wind energy source distribution and transmission more controllable (Abusara and Sharkh, 2013; Pei et al., 2015; Yaramasu et al., 2017; Yassin et al., 2016). Full-scale voltage source converter (VSC) based back-to-back configuration is the most promising converter topology for the grid integrated wind power system (WPS). Advanced control methodologies for high-performance operation, automated real and reactive power control, independent control of voltage and power regulation and reduced necessities for harmonic filtering are the advantages offered by this topology (Bakhtiari and Nazarzadeh, 2020; Hackl et al., 2019; Muyeen et al., 2010; Nasiri and Mohammadi, 2017). Intellectual control of the back-to-back converter maximizes the captured wind power, as well as fulfils the requirements imposed by the utility operators. Since this converter topology is the primary key for energy conversion, the main circuit design and converter performance evaluation are the essential issues to be considered before integrating the WPS to the grid. Transient simulation is usually carried out to find the converter’s adequate parameters and to evaluate the system performance with the necessary converter controller design (Alizadeh and Yazdani, 2013; Geng et al., 2011; Wang et al., 2020). However, designing an appropriate controller, estimating the controller parameters and evaluating steady-state performance using transient simulation is a time-consuming process. Besides, performance should be evaluated under all possible operating conditions, and frequent simulations should be performed consistently. Although, the power generation limit of a PMSG wind turbine is presented in (Jahanpour-Dehkordi et al., 2019), the maximum active and reactive power capability curve that can be exchanged from the wind turbine to the grid is assessed through control strategy in time-domain simulation. In (Du et al., 2020), even a complete small-signal model of the WPS was built only to design the controller’s parameters effectively. Eventually, the system performance was not predetermined analytically in the mentioned works of literature. A unique analytical description of the steady-state voltage magnitude and phase-angle quantities of VSC is necessary to efficiently evaluate the converter’s performance.

The main focus of this paper is to develop a detailed and generalized steady-state model for a grid integrated, PMSG based WPS. This paper also proposes a simple controller design for the back-to-back converter, that is, machine side converter (MSC) and grid side converter (GSC) to handle all possible system operating conditions. The simulated results of the controller response are validated with the predetermined steady-state values. This paper is structured into four major sections. Section 2 describes the entire system modeling of the WPS. The successive predetermination of steady-state system values at each stage of power flow without convert control action is detailed in Section 3. In Section 4 energy conversion efficiency calculation procedure at each stage of power flow is discussed. Section 5 presents the generalized WPS modeling and appropriate controller implementation to aid optimal power regulation and FRT requirement. This section also discusses the necessary modifications to be made with the conventional control schemes when subjected to different operating conditions of wind speed variation and grid disturbances. The generalized reference power generation algorithm for a two-level VSC controller is also elaborated in this section. The section further details the wind turbine blade pitch less control mechanism. Finally, Section 6 validates the direct analytically predetermined steady-state values with the simulated PSCAD/EMTDC WPS response with the proposed controller.

System modeling

The WPS illustrated in Figure 1 can be represented with PMSG stator induced EMF e_s, stator and GSC filter inductance and resistance, a two-level insulated gate bipolar transistor (IGBT) based VSCs, a dc-link capacitor and the grid voltage v_g as shown in Figure 2, neglecting the AC capacitor branch. It is evident from (Mahela et al., 2019), that the core of power conversion, that is, the back-to-back VSCs are controlled to enhance multiple functions like grid synchronization, dc-link voltage regulation, maximum power point tracking (MPPT), power limitation, rotor speed control for the same power regulation (conventionally pitch angle control), torsional oscillation damping, reactive power support to the grid, oscillation damping owing to various grid disturbances, etc. To achieve the mentioned strategies, the key point is to generate the reference sine wave with the corresponding magnitude and phase-angle. Assuming the constant dc-link voltage supply, VSCs can be more generically represented as a controllable voltage source as described in the following subsections.

Figure 1.

System under consideration.

Figure 2.

Detailed two-level back-to-back VSC circuit diagram.

GSC and grid voltage

To describe the steady-state modeling procedure, the dc-link, phase “a” of the grid and the corresponding GSC arms are considered as shown in Figure 3 (Hackl et al., 2019), and the GSC voltage is expressed as

v_{ia} = L_{f} \frac{d i_{ga}}{dt} + R_{f} i_{ga} + v_{ga}

(1)

Figure 3.

Equivalent circuit of phase “a” of GSC and grid.

In Figure 3, the bidirectional switches are identified by S_I1 and S_I4 which can be either ON or OFF and r_s is the switch on-state resistance. S_Ia and S′_Ia, are defined as the switching function of switches S_I1 and S_I4, respectively. S_Ia and S′_Ia, can be either 1 or 0 corresponding to ON and OFF states of the corresponding switches. Based on the principle of operation of GSC and regardless of pulse width modulation (PWM) scheme S_Ia and S′_Ia are always complementary which can be expressed as

S_{Ia} + S_{Ia}^{'} = 1

(2)

Correspondingly, v_ia can be expressed as in (3)

v_{ia} = v_{FH} + v_{Hn}

(3)

When S_I1 is ON, S_Ia =1 and S′_Ia = 0 and v_FH under this steady-state scenario can be given as

v_{FH} = (i_{ga} r_{s} + V_{dc}) S_{Ia}

(4)

Similarly, when S_I1 is OFF, S_Ia =0 and S′_Ia = 1 and v_FH can be written as

v_{FH} = (i_{ga} r_{s}) S_{Ia}^{'}

(5)

Neglecting r_s and expressing (1) in terms of S_Ia results in

(V_{dc} S_{Ia} + v_{Hn}) = L_{f} \frac{d i_{ga}}{dt} + R_{f} i_{ga} + v_{ga}

(6)

Voltage v_Hn can be obtained by adding equation of three-phases as given by

v_{Hn} = [- \frac{V_{dc}}{3}] \sum_{j = a, b, c} S_{Ij}

(7)

Substituting for v_Hn, from (7) in (6), the final expression for “a-phase” can be deduced as

V_{dc} S_{Ia} - [\frac{V_{dc}}{3}] \sum_{j = a, b, c} S_{Ij} = L_{f} \frac{d i_{ga}}{dt} + R_{f} i_{ga} + v_{ga}

(8)

The main advantage of expressing (8) in terms of the generalized switching functions S_Ia, S_Ib, and S_Ic is that the expression does not depend on the type of PWM technique. Figure 4 shows a generalized PWM switching pattern of S_Ia, where ω and ω_sw represent the fundamental and switching frequency of the GSC. S_Ia(avg) is the average of S_Ia within one period corresponding to the switching frequency, as S_Ia is a discrete periodic function of time, its Fourier series can be expressed as

S_{Ia} = a_{0} + \sum_{n} b_{n} \cos (n ω t)

(9)

a_{0} = S_{ia (avg)} and b_{n} = {(- 1)}^{n} (2 / n π) \sin (n π S_{ia (avg)})

Figure 4.

(a) A generalized PWM pattern of “a-phase”; (b) Switching function S_Ia and its average value S_Ia(avg), in one switching period.

The PWM patterns of S_Ib and S_Ic are identical to S_Ia, with an appropriate phase shift. Figure 4 also shows the variation in S_Ia(avg) within one period corresponding to the fundamental frequency. The fundamental component of S_Ia(avg) is referred as the control signal which can be expressed as

S_{Ia (fund)} = (m_{i} / 2) \cos (ω t - δ_{i}) + (1 / 2)

(10)

S_Ib _(fund) and S_Ic_(fund) of “b” and “c” phases are identical to (10) except for phase shift of 2π/3 and 4π/3, respectively. Finally, substituting for S_Ij in terms of fundamental components S_Ij_(fund), v_ia as in (3) can be expressed as

v_{ia} = (m_{i} V_{dc} / 2) \cos (ω t - δ_{i}) = V_{i} \cos θ_{vi}

(11)

Likewise, v_ib, and v_ic can be expressed as v_ia, with the phase shift of 2π/3 and 4π/3, respectively. Hence, the output AC terminal voltage of GSC at the fundamental frequency can be represented by the three-phase, balanced and controllable voltage sources v_ia, v_ib, and v_ic as shown in Figure 5c.

Figure 5.

Generalized equivalent model of WPS: (a) PMSG and MSC, (b) Dc-link, (c) GSC and Grid.

The mathematical model which governs the behavior of “a-phase” can be obtained by substituting for v_ia from (11) in (1).

V_{i} \cos θ_{vi} = L_{f} \frac{d i_{ga}}{dt} + R_{f} i_{ga} + v_{ga}

(12)

Similarly, “b” and “c” phase expressions are deduced as

V_{i} \cos (θ_{vi} - 2 π / 3) = L_{f} \frac{d i_{gb}}{dt} + R_{f} i_{gb} + v_{gb}

(13)

V_{i} \cos (θ_{vi} - 4 π / 3) = L_{f} \frac{d i_{gc}}{dt} + R_{f} i_{gc} + v_{gc}

(14)

Amplitude, phase-angle and frequency of the three-phase voltage source are respectively controlled by m_i, θ_vi, and ω_i of the control signal.

PMSG and MSC

The mathematical procedure described from (1 to 14) is also applicable to MSC and the corresponding stator voltage in phase-coordinates can be given as

V_{c} \cos θ_{vs} = - L_{sa} \frac{d i_{sa}}{dt} - R_{s} i_{sa} + e_{sa}

(15)

V_{c} \cos (θ_{vs} - 2 π / 3) = - L_{sb} \frac{d i_{sb}}{dt} - R_{s} i_{sb} + e_{sb}

(16)

V_{c} \cos (θ_{vs} - 4 π / 3) = - L_{sc} \frac{d i_{sc}}{dt} - R_{s} i_{sc} + e_{sc}

(17)

where the PMSG stator current co-efficient L_sj represents the stator inductance (Boldea, 2016). V_s and θ_vs correspondingly defines the fundamental magnitude and angle of MSC input terminal voltage.

DC-link

The dynamics of the dc link capacitor from Figures 1 and 2, can be expressed as

\frac{d V_{dc}}{dt} = (1 / C_{dc}) i_{c}

(18)

where i_c in terms of the defined switching functions can be written as

i_{c} = i_{dc 1} - i_{dc 2} = \sum_{j = a, b, c} (i_{sj} S_{Rj (fund)} - i_{ij} S_{Ij (fund)})

(19)

Equation (19) demonstrates that the effect of device switching on the dc link can be expressed by a current source. The generalized equivalent circuit of the back-to-back VSC, based on (12–19) is illustrated in Figure 5 and the corresponding state space representation is given in (20).

\overset{\cdot}{x} = Ax + Bu

(20)

A = [\begin{matrix} - \frac{R_{s}}{L_{s}} & 0 & 0 & 0 & 0 & 0 & - \frac{m_{i}}{L_{s}} \cos θ_{vs} \\ 0 & - \frac{R_{s}}{L_{s}} & 0 & 0 & 0 & 0 & - \frac{m_{i}}{2 L_{s}} \cos θ_{vs} \\ 0 & 0 & - \frac{R_{s}}{L_{s}} & 0 & 0 & 0 & - \frac{m_{i}}{2 L_{s}} \cos (θ_{vs} - 240^{o}) \\ 0 & 0 & 0 & - \frac{R_{s}}{L_{s}} & 0 & 0 & \frac{m_{i}}{2 L_{f}} \cos θ_{vi} \\ 0 & 0 & 0 & 0 & - \frac{R_{s}}{L_{s}} & 0 & \frac{m_{i}}{2 L_{f}} \cos (θ_{vi} - 120^{o}) \\ 0 & 0 & 0 & 0 & 0 & - \frac{R_{s}}{L_{s}} & \frac{m_{i}}{2 L_{f}} \cos (θ_{vi} - 240^{o}) \\ \frac{m_{i}}{2 C_{dc}} \cos θ_{vs} & \frac{m_{i}}{2 C_{dc}} \cos (θ_{vs} - 120^{o}) & \frac{m_{i}}{2 C_{dc}} \cos (θ_{vs} - 240^{o}) & \frac{m_{i}}{2 C_{dc}} \cos θ_{vi} & \frac{m_{i}}{2 C_{dc}} \cos (θ_{vi} - 120^{o}) & \frac{m_{i}}{2 C_{dc}} \cos (θ_{vi} - 240^{o}) & 0 \end{matrix}]

B = [\begin{matrix} 1 / L_{s} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 / L_{s} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 / L_{s} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 / L_{f} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & - 1 / L_{f} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 / L_{f} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 / C_{dc} \end{matrix}]

where $x = {[\begin{matrix} i_{sa} & i_{sb} & i_{sc} & i_{ga} & i_{gb} & i_{gc} & V_{dc} \end{matrix}]}^{T}$ and $u = {[\begin{matrix} e_{sa} & e_{sb} & e_{sc} & v_{ga} & v_{gb} & v_{gc} & 0 \end{matrix}]}^{T}$

Steady-state analysis

Considering constant dc-link voltage supply and knowing the wind speed and point of common coupling (PCC) voltage, the power flow model is developed based on the short transmission line theory. As the MSC and GSC are independently controlled, both the terminal voltages are expressed as a controlled voltage source with respect to PMSG stator induced emf and grid voltage (Boldea, 2016). Also, the converters performance indices can be analyzed independently as described in the following sub-sections. Since grid voltage is uncontrollable, it is taken as the reference, and the analysis starts from determining the GSC performance indices.

During steady-state operation, that is, under nominal grid voltage, only real power is injected into the grid, that is, the dc link capacitor voltage remains constant at the specified value, V_dc and the power pertaining to the prevailing wind speed is injected into the grid. Rather than d-q modeling which is specifically meant for designing the converter’s controller, a generic phase-coordinate model as described in section 5.2 is considered for the steady-state analysis. To satisfy the steady-state constraints, the respective power flow model is independently deduced for both the VSCs. As grid voltage is taken as a reference, the analysis starts from estimating the GSC performance numerals.

Grid and GSC under normal grid operation

The GSC performance numerals are independently analyzed for the given operating condition as per the single-line diagram illustrated in Figure 6. For the given wind speed neglecting the losses, the wind turbine output power, P_wt is equated to the grid active power, P_g which can be expressed as

P_{g} = P_{wt}

(21)

Figure 6.

Steady-state equivalent circuit representation of GSC.

where P_wt is defined as expressed in (Vijayapriya et al., 2018), under the steady-state condition the percentage of reactive power support to the grid is considered as zero. Hence, knowing the grid voltage, active and reactive power, the GSC performance values are determined as per the following steps:

Grid current estimation

The complex power of grid, S_g can be expressed grid real, P_g and reactive, Q_g power as

S_{g} = P_{g} + j Q_{g}

(22)

where $P_{g} = 3 V_{g} I_{g} \cos θ_{g}$ ; $Q_{g} = 3 V_{g} I_{g} \sin θ_{g}$ ; $θ_{g} = θ_{v g} - θ_{i g}$ ; θ_vg and $θ_{ig}$ are the angle of grid phase voltage V_g and current I_g, respectively.

As unity power factor is maintained at grid terminal, θ_g will be zero and for the known value of P_g and V_g, I_g can be deduced as

I_{g} = \frac{P_{g}}{3 V_{g} \cos θ_{g}}

(23)

Voltage drop across GSC filter

Using (23), the voltage drop across the filter reactor is determined as

V_{Lf} = I_{g} Z_{f}

(24)

where Z_{f} = R_{f} + j ω_{eg} L_{f}

Filter line complex power estimation

The power dissipation P_f and the reactive power absorption Q_f at the filter line can be estimated as

S_{f} = V_{Lf} I_{g}^{*} = P_{f} + j Q_{f}

(25)

GSC output terminal apparent power estimation

The active and reactive power (P_i and Q_i) at the GSC output terminal can be expressed as

S_{i} = S_{f} + S_{g} = P_{i} + j Q_{i}

(26)

where $P_{i} = 3 V_{i} I_{g} \cos θ_{i}$ ; $Q_{i} = 3 V_{i} I_{g} \sin θ_{i}$ ; $θ_{i} = θ_{vi} - θ_{ig}$ ; $θ_{vi}$ > is the angle of the GSC phase voltage V_i.

GSC phase-angle computation

Using P_i and Q_i, θ_vi is determined as

θ_{vi} = ta n^{- 1} (\frac{Q_{i}}{P_{i}}) + θ_{ig}

(27)

GSC modulation index

The modulation index m_i is determined from the GSC fundamental phase voltage magnitude V_i, which may be either computed from P_i or Q_i as

V_{i} = \frac{Q_{i}}{3 I_{g} \cos θ_{i}} = \frac{P_{i}}{3 I_{g} \sin θ_{i}}

(28)

In turn, m_i is computed as given in (Rashid, 2009)

m_{i} = \frac{2 V_{i}}{V_{dc}}

(29)

From the design steps (22)–(29) it can be deduced that to inject the significant percentage of active power, the ultimate parameters to be controlled are m_i and θ_i as given (27) and (29), respectively. Hence for the variation in wind speed, m_i and $θ$ vi of the GSC fundamental voltage must be correspondingly controlled.

MSC and PMSG under normal grid operation

Likewise, the MSC performance numerals, m_s and θ_vs are derived following the steps described in sub-section 3.1, considering the single-line diagram depicted in Figure 7. The fore most step is to determine the stator current I_s from the known quantities of E_s, active and reactive power (P_se and Q_se) due to the stator induced EMF. For the given wind speed the wind turbine output power can be equated to the PMSG rotor power (P_r) and neglecting the mechanical losses, P_se can be given as

P_{se} = P_{r} = P_{wt}

(30)

Figure 7.

Steady-state equivalent circuit representation of MSC.

E_s can be determined as per the procedure given in (Boldea, 2016), provided that the specified parameters are known, else in a simpler way, E_s can be obtained from the dynamic d-q modeling. According to the space vector representation of PMSG, rotor magnet is oriented along d-axis and rotor voltage along q-axis. Subsequently, the rotor voltage in dq-axis can be expressed as

e_{rd} = 0

(31)

e_{rq} = ω_{es} ψ_{PM}

(32)

where ω_es = p_pω_r. For the given wind speed, to extract the maximum wind power, optimum value of turbine speed ω_r can be estimated (Vijayapriya et al., 2018). As the voltage drop between the rotor and stator is zero, er can be equated to es and by knowing E_s, P_se, and Q_se, the MSC performance values are determined following the subsequent design steps.

Stator current estimation

The complex power at the stator induced EMF terminal can be expressed as

S_{se} = P_{se} + j Q_{se}

(33)

where $P_{se} = 3 E_{s} I_{s} \cos θ_{r}$ ; $Q_{se} = 3 E_{s} I_{s} \sin θ_{r}$ ; $θ_{r} = θ_{es} - θ_{is}$ ; θ_es and θ_is are the angle of E_s and I_s, respectively and I_s can be expressed as

I_{s} = \frac{P_{se}}{3 E_{s} \cos θ_{r}}

(34)

Voltage drop across stator winding

V_{Ls} = I_{s} Z_{s}

(35)

where Z_{s} = R_{s} + j ω_{es} L_{s}

Stator winding complex power estimation

S_{Ls} = V_{Ls} I_{s}^{*} = P_{Ls} + j Q_{Ls}

(36)

MSC input terminal apparent power estimation

S_{s} = S_{se} - S_{Ls} = P_{s} + j Q_{s}

(37)

where $P_{s} = 3 V_{s} I_{s} \cos θ_{s}$ ; $Q_{s} = 3 V_{s} I_{s} \sin θ_{s}$ ; $θ_{s} = θ_{vs} - θ_{is}$

MSC phase-angle computation

θ_{vs} = ta n^{- 1} (\frac{Q_{s}}{P_{s}}) + θ_{is}

(38)

MSC modulation index

m_{i} = \frac{2 V_{s}}{V_{dc}}

(39)

Irrespective of wind speed variation, the steady-state model remains the same provided that the grid voltage is at its nominal value. The overall flowchart to determine the back-to-back VSC performance values is illustrated in Figure 8.

Figure 8.

Flowchart to determine the VSCs performance values.

Based on the individual characteristics of different types of grid disturbances (Bollen et al., 2005), it is expected that the steady-state model must also be unique. As the MSC is completely decoupled from the grid, the grid dynamics majorly replicates as a reduction in active power at the MSC terminals. Hence, the steady-state model of MSC can be retained irrespective of wind speed variation as well as the grid disturbances. The steady-state model of GSC for the symmetrical fault is described in the next sub-section.

Analysis under symmetrical grid fault

Steady-state model of the back-to-back VSC power circuit in phase-coordinates can also be used to analysis the dynamic performance under symmetrical fault. During symmetrical fault the positive sequence component (PSC) of the positive synchronous frame (PSF) grid voltage v⁺_g+ gets reduced, which further sets the limits for the active power injection in to the grid. In this regard, v⁺_g+ can be referred as an available grid voltage and Figure 6 can be restructured as given in Figure 9. Considering the significant injection of grid active and reactive power as given in section 5, the steady-state performance of VSCs is obtained following the procedure presented in section 5.2.

Figure 9.

Steady-state model involving PSC in PSF.

Efficiency evaluation

The efficiency of the PMSG based WECS is evaluated accounting the VSC (conduction and switching losses) and transmission losses (stator and filter reactor losses) at each stages of power conversion. The active power evaluation at various power conversion stage input/output terminals is given as follows:

Stator output/MSC input terminal

The available active power at MSC input terminal is formulated as

P_{s} = P_{se} - P_{Ls}

(40)

The losses (P_Ls) in the stator winding are calculated using the stator current.

MSC output/ DC-link input terminal

The accessible active power at the dc-link input terminal is determined by deducting the MSC conduction and the switching losses from P_s. The power losses in IGBT based VSCs are calculated based on the information available in the device data-sheet.

Conduction losses

The conduction losses occur due to devices (IGBTs and diodes) on-state voltage drop and can be computed by averaging the conduction losses in each switching cycle. The device conduction loss can be expressed as given in (41).

P_{Lc} = V_{f 0} I_{avg} + r_{f 0} I_{rms}^{2}

(41)

where V_fo is the device forward voltage at no-load and r_fo is the device forward resistance. The values of V_fo and r_fo are evaluated from the device characteristics provided in the datasheet. r_fo is the ratio between the collector-emitter voltage difference and the collector current difference which is given as

r_{f 0} = \frac{Δ V_{CE}}{Δ I_{C}}

(42)

The average and RMS currents of IGBTs T_R1 and T_R4 as shown in Figure 2 can be given as

I_{TRIavg} = I_{sm} (\frac{1}{2 π} + \frac{m_{s} \cos θ}{8})

(43)

I_{TR 4 avg} = I_{sm} (\frac{1}{2 π} - \frac{m_{s} \cos θ}{8})

(44)

I_{TR 1 rms} = \sqrt{I_{sm}^{2} (\frac{1}{8} + \frac{m_{s} \cos θ}{3 π})}

(45)

I_{TR 4 rms} = \sqrt{I_{sm}^{2} (\frac{1}{8} - \frac{m_{s} \cos θ}{3 π})}

(46)

where I_sm represents the peak value of stator current. The average and RMS currents for the lower free-wheeling diode are similar to that of the upper IGBT device but in opposite direction and vice versa as given below

I_{DR 1 avg} = - I_{TR 4 avg} = - I_{sm} (\frac{1}{2 π} - \frac{m_{s} \cos θ}{8})

(47)

I_{DR 4 avg} = - I_{TR 1 avg} = - I_{sm} (\frac{1}{2 π} + \frac{m_{s} \cos θ}{8})

(48)

I_{DR 1 rms} = I_{TR 4 rms} = \sqrt{I_{sm}^{2} (\frac{1}{8} - \frac{m_{s} \cos θ}{3 π})}

(49)

I_{DR 4 rms} = I_{TR 1 rms} = \sqrt{I_{sm}^{2} (\frac{1}{8} + \frac{m_{s} \cos θ}{3 π})}

(50)

Eventually, MSC conduction loss P_Lc(MSC) is determined by substituting for I_T_Ravg, I_D_Ravg, I_T_Rrms and I_D _Rrms of six switches in (41)

Switching losses

The switching losses are the sum of on-state and off-state switching losses which mainly depends on the device characteristics, switching frequency, and device current. The switching loss of IGBT can be given as

P_{Lsw} = \frac{K_{e} f_{sw} I_{sm}}{π}

(51)

where the switching energy K_e is obtained from the energy graph provided in the device datasheet. Depending on the magnitude of I_sm, the MSC switching losses are evaluated using (51).

The active power at the dc-link input terminal is given by

P_{dc 1} = P_{s} - P_{Lc (MSC)} - P_{Lsw (MSC)}

(52)

DC-link output/GSC input terminal

The losses in dc cable can be calculated as

P_{Rdc} = R_{dc} {(P_{dc 1} / V_{dc})}^{2}

(53)

where R_dc is the dc-link cable resistance. The active power at the dc-link output terminal is given by

P_{dc 2} = P_{dc 1} - P_{Rdc}

(54)

As a lossless dc-line is considered in this work, P_Rdc is neglected in the further analysis.

GSC output terminals

The conduction and switching losses of GSC are calculated using (6.41) and (6.51) based on the grid phase current peak value I_gm. The active power at GSC output terminal is given as

P_{Rdc} = R_{dc} {(P_{dc 1} / V_{dc})}^{2} P_{i} = P_{dc 2} - P_{Lc (GSC)} - P_{Lsw (GSC)}

(55)

where P_Lc_(GSC) and P_Lsw_(GSC) respectively represents the GSC conduction and switching losses.

Grid terminal

Eventually, active power injection at grid terminal is evaluated as

P_{g} = P_{i} - P_{Lf}

(56)

P_Lf represents the power loss in the filter impedance Z_f.

Proposed controller

The significance and challenging controller design for the grid integrated WPS is presented in this Section. Accordingly, the GSC controller is modeled in synchronous reference frame (SRF) which aims to develop an active control technique with reactive power support during the grid fault conditions. Particularly, efficient management of active power injection and reactive power support without an inevitable pitch angle control mechanism is highly focused under severe grid fault. Above all, the controller is devised to meet the primary constraint of maintaining the system parameters within its safer operating limits. As an essence, the GSC controller is updated with the power oscillation cancellation term to minimize the grid disturbance impacts. The term is derived by subtracting the generated average power reference from the available grid power and its performance is compared with the predetermined values discussed Section 3 using the PSCAD/EMTDC software.

MSC design

The outer speed controller is framed by accounting the drive-train model and electromagnetic torque (Ramachandran et al., 2020). Similar to active power control, i^*_sq is related to the outer speed proportional integral (PI) compensator (Vijayapriya et al., 2018), as described below

(ω_{r}^{*} - ω_{r}) K_{p ω} (\frac{1 + τ_{ω}}{s τ_{ω}}) = T^{'} = T_{e} - T_{wt}

(57)

K_pω and τ_iω represents the proportional gain and time constant of the outer speed compensator and i^*_sq is derived as given in (58), by substituting the electromagnet torque expression.

i_{sq}^{*} = \frac{T^{'} + T_{wt}}{1.5 p_{p} ψ_{PM}}

(58)

The compensator gain values are derived in the same way as explained in (Blasko and Kaura, 1997).

GSC design

The dc-link voltage and grid reactive power control are implemented on the GSC as shown in Figure 10, except for direct reactive power control as described in the following subsection.

Figure 10.

Overall controller design.

Active power regulation—Vdc control

The regulated active power can be efficaciously injected into the grid by the coordinated dc-link voltage control on the GSC. Accordingly, active current reference i^*_iq is derived from the outer dc-link voltage control loop as described in (Vijayapriya et al., 2018) and it is illustrated in Figure 10 with respect to the PCC voltage. The current reference design helps to set the limits of the PI regulator at lower values rather than directly deriving the current reference from the output of the outer PI controller (Geng et al., 2011; Wang et al., 2020).

Reactive power regulation

The outer reactive power control loop is designed using grid reactive power Q_g, and d-axis GSC current is controlled for reactive power. The correlation between the outer PI compensator and the current reference i^*_id is given as

(Q_{c}^{*} - Q) K_{pQ} (\frac{1 + τ_{Q}}{s τ_{Q}}) = Q_{c}^{'} = 1.5 (v_{cq} i_{id} - v_{cd} i_{qd})

(59)

where K_pQ and τ_iQ represents the proportional gain and time constant of the outer reactive power controller with respect to GSC and i^*_id is derived to be

i_{id}^{*} = \frac{1}{v_{cq}} (Q_{c}^{'} 0.667 + v_{cd} i_{iq})

(60)

The overall description of reactive power controller and GSC in SRF is depicted in Figure 10. The real and reactive power reference are generated as given in (Ramachandran et al., 2020) prevailing to the grid voltage.

Model validation

Based on the analytical simulation model developed in Section 5, this section demonstrates some of the characteristics of the WPS parameters and control functions while subjected to wind speed and grid voltage variation. This section also verifies the accuracy of the developed steady-state model (Section 3) by comparing the predetermined values with those obtained from time simulation.

Steady-state response under nominal grid voltage

The steady-state behavior of the proposed model under nominal grid voltage for different wind speed operating condition is analyzed and compared with the simulation studies. In this regard, the VSC performance numerals are plotted against variation in wind speed.

Variation of θ and m with respect to ν_w

The analysis starts with inputting the known values of E_s, P_se, and Q_se and following the analysis given in Section 4 the individual values of θ_vs, m_s, θ_vi, and m_i are evaluated correspondingly for the variation in wind speed from 3 m/s to 12 m/s. The MSC phase voltage angle θ_vs varies in a random way for the linear increment in the wind speed depending on the rotor position, however, m_s varies in an incremental phase as shown in Figure 11a. It is also inferred that the pre-determined values using the steady-state model exactly matches with the time simulated values. Consequently, the variation is θ_vi and m_i with respect to GSC is illustrated in Figure 11b. The grid voltage phase angle is measured to be zero and the increment in leading phase-angle of θ_vi confirms the transfer of active power from GSC to the grid. Moreover, the increment in m_i ensures the increase in active power injection into the grid with the corresponding increment in the wind speed. It is further inferred that to transfer the same amount of active power, the performance numerals of MSC and GSC are different owing to the difference in PMSG induced EMF stator and grid terminal voltages.

Figure 11.

Steady-state model verification under normal grid operating condition: (a) MSC and (b) GSC.

Variation of P_g with respect to ν_w

Succeeding the evaluation of MSC performance value and losses (neglecting P_Lc and P_Lsw) calculation as discussed in Section 3 and 4, the amount of active power that can be injected into the grid is predetermined and plotted against the wind speed variation as shown in Figure 12a. The predetermined value is compared with time simulated value under normal grid operating condition. The case is considered for the lower wind speed operation and the corresponding grid active power injection is shown in Figure 12b. It can be validated that the active power corresponds to 0.18 MW and 0.02 MW for the corresponding wind speed of 6 m/s and 3 m/s which are closely matching with the predetermined values shown in Figure 12a.

Figure 12.

Grid active power for the variation in wind speed: (a) Pre-determined value and (b) Real-time stimulation.

Variation of I and V with respect to ν_w

The range of variations in VSC current and fundamental voltage as a function of wind speed is shown in Figure 13, which illustrates that variation of electrical quantities $| I_{g} |$ and $| V_{i (fund)} |$ as a result of wind speed variations are highly nonlinear compared to $| I_{s} |$ and $| V_{s (fund)} |$ . It is also inferred from Figure 13b that the proposed model closely matches the system parameter variations with the dedicated control strategy results as illustrate in Figure 14. The steady-state pre-determined I_g values are noted to be 698 A and 1221 A, whereas the time simulation (Figure 14) gives the instantaneous peak value of 1002 A and 1732 A for the wind speed of 10 m/s and 12 m/s, respectively. Furthermore, studies similar to that of Figure 13 identify the maximum steady-state values of the VSC variables (voltage and current) and helps in determining the ratings of the WECS components.

Figure 13.

Steady-state model verification under normal grid operating condition: (a) MSC and (b) GSC.

Figure 14.

Behavior of PMSG based WPS during a step change in wind speed (a) wind speed profile, m/s and (b) grid active power, W.

Steady-state response under grid disturbance

Considering the constraint of maintaining the dc-link voltage constant and significant percentage of active power injection into the grid, the VSCs performance analysis is carried out under symmetrical grid fault. The reference power for the dynamic model is accounted as per the power reference scheme proposed in Section 5, that is, reduced active power injection without and with reactive power support, respectively.

Power reference with respect to Section 5.3

The dynamic operating condition, that is, symmetrical grid fault with 85% and 15% voltage sag is considered to validate the predetermined value with the simulation result (Figure 15) as shown in Figures 16 and 17, respectively.

Figure 15.

Performance evaluation of proposed controller for the voltage profile of IEGC during symmetrical fault: (a) grid phase voltage, V and (b) GSC active power reference P^*_s and grid power P_g, W.

Figure 16.

Steady-state performance under 85% voltage sag: (a) MSC, (b) GSC, and (c) P_g vs ν_w.

Figure 17.

Steady-state performance under 15% voltage sag: (a) MSC, (b) GSC, and (c) P_g vs ν_w.

The performance analysis of MSC remains the same as the steady-state model under the nominal grid voltage except for the reduction in active power transfer for both scenarios of 85% and 15% voltage drop. This can be further verified by the approximate range of variation in m_i and δ_i for the variation in wind speed as shown in Figures 16a and 17b. However, with respect to GSC m_s and θ_vs correspondingly increases and decreases with the reduction in voltage sag as shown in Figures 16b and 17b. Similar to subsection 6.1, the predetermined P_g is compared with the time simulated result given in Figure 16 and it is inferred that for the wind speed of 9 m/s, P_g at 85% and 15% voltage sag approximated to P^*_s of 0.092 MW and 0.52 MW with respect to both the steady-state and simulation model.

The power circle diagram of the GSC (sending end) and grid (receiving end) is also plotted for the variation in wind speed, correspondingly under 85% and 15% voltage sag using the pre-determined values as depicted in Figure 18. It can be observed from the receiving end circle diagram that only active power is injected into the grid during the voltage sag. However, sending end circle compensate the reactive power requirement by the filter impedance and this illustration also validates the proposed model.

Figure 18.

Power circle diagram: (a) 85% voltage sag and (b) 15% voltage sag.

Power reference with respect to section 5.2

The steady-state model under symmetrical voltage sag, considering the input power reference scheme as discussed in section 5.2 is analyzed in this case. As much more inference cannot be observed from MSC performance values other than minor variation in m_s and δ_s though with reactive power support to the grid, the corresponding results are not brought out here. However, significant changes can be inferred from GSC primarily due to the use of LCL filter, that is, with the reduction in total inductance L_T, the value of m_i decreased though with reactive power support to the grid in addition to filter compensation. The characteristics of the grid active and reactive power for the variation in wind speed in terms of $θ$ _vi and m_i is shown in Figure 19. It can be again verified from Figure 19 that for the wind speed of 10 m/s P_g and Q_g are approximately predicted to be 0.79 MW and 1.13 VAR as given in Figure 20. Furthermore, it can be confirmed from Figure 21 that the predetermined value of i_i almost maintained at its rated value obeying the proposed power reference scheme of section 5.2.

Figure 19.

GSC performance analysis with respect to grid active power injection and reactive power support: (a) P_g and Q_g vs $θ$ _vi; (b) P_g and Q_g vs m_i.

Figure 20.

Performance analysis during symmetrical fault: grid active (W) and reactive power (VAR).

Figure 21.

GSC current characteristics for the variation in wind speed.

Conclusion

A generalized structured is developed for the grid integrated WPS with two independent equivalent circuits, that is, PMSG electrical input voltage interconnected to MSC voltage through stator impedance and MSC output terminal interconnected to the grid through the filter impedance. Design procedures are presented to determine the VSC controllable variables based on the available information. To effectively determine the VSC performance numerals despite grid disturbances, the steady-state model is structured with the PSC of the PSF grid voltages. The advantage of the proposed model is that the proposed study does not require d-q modeling of the system and a dedicated controller to evaluate the system performance subjected to the specified system parameter’s regulation. The proposed steady-state model is validated by comparing the results obtained through this model with the results of the time simulation incorporating dedicated and advanced control strategies such as dc-link voltage, grid active, and reactive power regulation, etc.

Using the proposed steady-state model, the entire WPS components ratings can be predicted without carrying out time domain simulation with complicated controller design.

Footnotes

Appendix A

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Ramachandran Vijayapriya

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A direct analytical predetermination of PMSG based WPS steady-state values under different operating conditions

Abstract

Keywords

Introduction

System modeling

GSC and grid voltage

PMSG and MSC

DC-link

Steady-state analysis

Grid and GSC under normal grid operation

Grid current estimation

Voltage drop across GSC filter

Filter line complex power estimation

GSC output terminal apparent power estimation

GSC phase-angle computation

GSC modulation index

MSC and PMSG under normal grid operation

Stator current estimation

Voltage drop across stator winding

Stator winding complex power estimation

MSC input terminal apparent power estimation

MSC phase-angle computation

MSC modulation index

Analysis under symmetrical grid fault

Efficiency evaluation

Stator output/MSC input terminal

MSC output/ DC-link input terminal

Conduction losses

Switching losses

DC-link output/GSC input terminal

GSC output terminals

Grid terminal

Proposed controller

MSC design

GSC design

Active power regulation—Vdc control

Reactive power regulation

Model validation

Steady-state response under nominal grid voltage

Variation of θ and m with respect to νw

Variation of Pg with respect to νw

Variation of I and V with respect to νw

Steady-state response under grid disturbance

Power reference with respect to Section 5.3

Power reference with respect to section 5.2

Conclusion

Footnotes

Appendix A

Declaration of conflicting interests

Funding

ORCID iD

References

Variation of θ and m with respect to ν_w

Variation of P_g with respect to ν_w

Variation of I and V with respect to ν_w