INTERNATIONAL TRANSACTIONS ON ELECTRICAL ENERGY SYSTEMS
Int. Trans. Electr. Energ. Syst. (2016)
Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/etep.2185
Exact feedback linearization-based permanent magnet synchronous
generator control
Sahibzada M. Ali1*,†, Muhammad Jawad2, Feng Guo3, Arshad Mehmood1, Bilal Khan1,
Jacob Glower3 and Samee U. Khan3
1
COMSATS Institute of Information Technology, Electrical Engineering Department, Abbottabad, Pakistan
2
COMSATS Institute of Information Technology, Electrical Engineering Department, Lahore, Pakistan
3
Electrical and Computer Engineering Department, North Dakota State University, Fargo, ND, 58108-6050, U.S.A.
SUMMARY
This paper presents the technical aspects, theoretical analysis, and comparisons of two new permanent magnet synchronous generator (PMSG) grid-interfaced models named PMSG boost and PMSG rectifierinverter. The exact feedback linearization (EFL) control scheme is used for the effective design and stability
analysis of the aforementioned models. The complexity of the EFL control laws is discussed in the light of
the nonlinear coordinate transformations and linearization (local and global) of the PMSG models. We also
discussed the effectiveness of the EFL control over the output Direct Current (DC) link voltage of the
PMSG during the electrical grid faults and the mechanical perturbations. Moreover, we compared the aforementioned models and concluded that the stability of the PMSG rectifier-inverter is higher compared with
the PMSG boost during the variation of wind speed values from minimum to maximum. Furthermore,
the robustness of the EFL control scheme for the PMSG rectifier-inverter is compared with the conventional
proportional and integral controller and state-feedback controller. The EFL controller reports a faster output
response, better accuracy, and quicker settling time of the output DC link voltage as compared with the proportional integral controller. Copyright © 2016 John Wiley & Sons, Ltd.
key words:
permanent magnet synchronous generator (PMSG); exact feedback linearization (EFL); wind
energy models; proportional integral (PI) control; nonlinear systems
1. INTRODUCTION
Renewable energy resources, such as solar energy, biomass energy, and wind energy are the emerging
need of today’s power market [1–7]. The performance and reliability of variable speed wind energy
conversion systems (WECS) are maintained using permanent magnet synchronous generator (PMSG).
The PMSG model for the wind energy applications needs a balanced and stabilized control in the gridconnected mode. The promising features of the PMSG are [8,9]: (i) simple mechanical and electrical
structure; (ii) low wind speed operation; (iii) self excitation; (iv) high power factor; (v) high reliability;
(vi) high efficient operation; and (vii) low maintenance costs.
For the research community, the integration of wind energy generation systems with the conventional
power grids introduces various challenges. Some of the main challenges are: (i) control of the DC link
voltage; (ii) support of the grid voltage during perturbations; (iii) sensitivity of the parameters drift during
extreme conditions; and (iv) optimized and robust performance of the PMSG even in worst-case scenarios [9]. Conventionally, DC link voltage was controlled using grid-side converter, but researchers
suggested that the aforesaid control and maximum power point tracking (MPPT) can be effectively
achieved by generator-side converter [10]. The basic issues that occur in the grid interconnections are
the voltage support during unbalanced conditions and the control of the output DC link voltage.
*Correspondence to: Sahibzada Muhammad Ali, COMSATS Institute of Information Technology, Electrical Engineering Department, Abbottabad, Pakistan.
†
E-mail: muhammadali.sahibzad@ndsu.edu
Copyright © 2016 John Wiley & Sons, Ltd.
S. M. ALI ET AL.
Several control schemes have applied for an optimized performance of variable speed WECS. In the
classical control schemes, such as the proportional integral (PI) control, the linearized expression is
obtained through the Taylor series expansion by ignoring the higher terms [8]. The final linearized
expression of the PI control can only provide the stable operation in the fixed domain of the given
parameters. By ignoring the higher order operating states of the dynamical system, the overall response
of the system is slow. The PI tuning is one of the difficult tasks for any given system specially for
nonlinear systems. The following techniques are used for the tuning of PI controllers, which are based
on certain types of systems such as, Zeigler Nicholas method and Cohen–Coon reaction curve method.
The parameter tuning of the PI control system is obtained by trial and error method, thus making the
control system problematic for practical applications [11]. The use of adaptive controllers with
variable speed WECS has several advantages, such as high tracking quality and power quality [5].
In addition to the complexity of control, the rotor dynamical characteristics must be evaluated quite
accurately. Various modern control schemes, such as fuzzy logic control, sliding mode control, and
robust control [1,5,7], are applied for resolving the WECS issues. The aforementioned control technologies vary in terms of mathematical complexity, control objectives, and applications.
The exact feedback linearization (EFL) is a well-known control scheme that transforms a nonlinear
system into a completely linear one through various techniques, such as the input–output transformation,
zero dynamics approach, disturbance decoupled analysis, and EFL algorithms [12]. All the aforesaid
schemes were applied in WECS except algorithms of EFL, which we will apply in our paper. The EFL
provides higher accuracy, optimizes performance in case of fast varying dynamics, stabilizes control of
each independent variable, and increases the robustness of the control system in case of faults and
disturbances [12]. Moreover, the EFL provides an independent control of each regulated variable
in WECS integration. Furthermore, the EFL has significant simplification in controller synthesis
and system operation. The transformation of an EFL scheme is based on differential geometry and
Lie algebra. The final linearized expression is obtained through mapping between differential spaces,
such as space X, space Y, and space Z. The authors in [13] presented the concept of the local and
global linearization of the EFL nonlinear affine systems. Moreover, the necessary and sufficient conditions for the single-input single-output and multiple-input multiple-output (MIMO) EFL systems
were presented in [14]. The linearization of the affine nonlinear EFL system was presented in
[15]. The idea of MIMO EFL systems was further elaborated by the authors in [16]. The aforementioned research work shows the development of the nonlinear affine EFL control scheme.
We present a theoretical analysis and brief comparative discussions on the nonlinear PMSG control
models applied to the wind energy systems and grid-interconnected applications.
The main contributions of our paper in the light of the above stated issues are:
• We present a robust design of the grid-connected PMSG for machine-side control and grid-side
control. Our model provides an optimized control of: (i) DC link voltage; (ii) converter current;
(iii) PMSG rotor speed; (iv) stator current; and (v) maximum power transfer to the grid.
• A stable DC link voltage control is achieved during three-phase electrical grid faults (short-time
and long-time) and mechanical disturbances (varying input wind speed from minimum to a
maximum) with perturbed generator parameters.
• Comparative analysis of the two PMSG models is discussed in detail based on the control features,
such as robustness, optimization, and local stability.
• Comparison of the PMSG rectifier-inverter is performed with the classical PI controller for the
validation of robustness and stability.
• Identification of the sensitive parameters that causes the abrupt response in the output of the
grid-interfaced PMSG boost.
The remainder of the paper is structured as follows. Section 2 discusses the related work on the
PMSG and wind energy control systems. The local and global linearization of the affine nonlinear
system is described in Section 3. The detailed description of the models, namely, PMSG wind turbine, PMSG boost, and PMSG rectifier-inverter are presented in Section 4. Section 5 describes the
simulation results of the PMSG boost model, whereas Section 6 discusses the simulation results of
the PMSG rectifier-inverter model. Section 7 concludes the paper with a summary and proposal
for future work.
Copyright © 2016 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
PMSG GRID-INTERFACED DIFFERENTIAL GEOMETRIC CONTROL
2. RELATED WORK
The use of wind energy models for online power generation and energy storage systems has resulted in
various benefits to the supplying agencies, such as the grid voltage support, power flow control, maximum power extraction, grid synchronization, and optimized performance of the inter-connected systems [1–7]. The authors designed the robust WECSs for controlling the maximum energy capture,
damping torque oscillations, and generator speed with varying wind turbulences.
The PMSG is useful for the constant frequency operation to obtain the maximum use of the wind
speed [17]. Reference [18] explains the sliding-mode voltage control for the DC–DC converters, such
as buck, boost, and buck–boost that are designed for controlling the switching frequency. In [19], the
authors describe the fuzzy logic-based small wind turbine control model for the maximum power extraction by perturbing the turbine parameters. In [20], the authors presented the feedback linearization
under grid faults with improvement in DC link voltage. The authors showed MPPT of PMSG-based
wind turbines using feedback linearization technique [21]. The EFL for induction motor control is presented in [22]. We provide the comparison of the two PMSG models for the analysis of the stability
and nonlinear control (EFL). Contrary to prior work, in this paper, we present the technical debate
on the controlling parameters of the two PMSG models that lack in the aforementioned research work
of the WECSs.
The authors in [23–28] have focused on the feedback linearization scheme for controlling various
power systems, such as the wind energy and solar energy systems. The back-to-back PWM converter is
one of the most common structures in the wind power systems. Both the generator-side and the grid-side
converter have the same main circuit that makes the control system similar in both sides of the system [29].
The design of the speed controller for the grid-connected PMSG is presented in [8] that uses the
input–output feedback linearization approach. The speed controller worked efficiently for varying
torques, but the designed system was not validated by introducing the electrical grid faults for short
and large range of time. The authors in [30] presented the control model for the variable speed wind
turbines. The objective function was the control of maximum power extraction, reactive power, and
grid voltage support. The soft-switching hysteresis current control and space vector modulation
schemes were employed by the authors in [31]. The local subsets for the stability and robustness were
not considered in the aforementioned research work. Consequently, we incorporate the stability subsets and the effects of the grid disturbances for various periods of time on the PMSG models.
3. EXACT FEEDBACK LINEARIZATION CONDITIONS FOR AN AFFINE NONLINEAR
SYSTEMS
Consider a nonlinear affine system as:
Ẋ ¼ f ðX Þ þ gðX ÞðuÞ;
(1)
y ¼ hðX Þ
In Equation 1, the parameter X ∈ Rn is the state vector, u is the control variable, and (f, g) are the
n-dimensional vector fields. To check whether or not an affine nonlinear system can be linearized into
the Brunovsky normal form (BNF), the Lie bracket, and the Lie derivative operations are performed as:
ad f g ¼ ð½ f ; gX Þ ¼ Δg:ð f Þ Δf :ðgÞ;
ad 2f g ¼ Δad f :g Δf :ad f g;
ad if g
¼
Δad fi1 g:f
(2)
ði1Þ
Δf :ad f g
Consider the Lie derivative of the scalar function λ(X) along f(X) as:
∂λðX Þ
f i ðX Þ
i ¼ 1 ∂xi
n
L f λ ðX Þ ¼ ∑
(3)
For the relative degree equal to or less than the degree of the state vector, the following conditions
must be met:
Copyright © 2016 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
S. M. ALI ET AL.
Lg Lkf hðX Þ ¼ 0; k < r 1; ∀xεΩ
Lg Lr1
f hðX Þ≠0;
i
D ¼ g; ad f g; …; ad fn1 g ðX Þ;
h
(4)
r ðDÞ ¼ n:
The matrix D must be involutive at X = X0. In Equation 4, r(D) is the rank of the matrix D and n is
the degree of the state vector. If the aforementioned conditions are all fulfilled, then the nonlinear
coordinate transformations can be successfully employed on the system model.
For the system model, given in Equation 1, the necessary conditions for the local and global linearization in a certain space can be represented by the following:
Ẋ ¼ f ðX Þ þ gðX ÞðuÞ
(5)
In Equation 5, XεS and S are the open subsets of Rn. The vector fields f(X) and g(X) are C∝ vector
fields on S. Consider the sequence of positive numbers p1, p2, …, pn, such that:
S ¼ ðs1 ; s2 ; …; sn Þ; s1 εp1 ; s2 εp2 ; …; sn εpn ;
s1 ≥s2 ≥s3 ; …; ≥sn ; p1 ≥p2 ≥p3 ; …pn ;
(6)
N
sn εpn > 0; ∑ pi ; k ¼ 1; 2; …; N:
i¼1
Suppose that C ∝s ðU Þ is the set having s dimensional vectors and gl(s, C∝(U)) be the s × s non
singular matrices combination. Now, S → U, where U opens in Rn, uε C ∝s ðU Þ ; andz ε gl s; C ∝s ðU Þ .
The following conditions must be strictly met for the linearization of an affine nonlinear system:
e ; S ¼ p1 ≥f p2 ≥p3 ≥; …; pn > 0;
e
g2 ; e
g3 ; …; e
gs εG
g1 ; e
N
ðN1Þ
∑ pi ¼ n; C ¼ ðe
g1 ; e
g2 ; …; e
gs ; Lf e
g1 ; …; Lf e
gp2 ; …; Lf
i¼1
ðN 1Þ
e
g1 ; …; Lf
e
gpn ;
ðN1Þ
ðN1Þ
e
e
g1 ; e
g2 ; …; e
gs ; Lf e
Gi ¼ Sp e
g1 ; …; Lf e
gp2 ; …; Lf
g1 ; …; Lf
gpn Þ; i ¼ 1; 2; …; N;
(7)
Di Δ SpðX 1 ; …; X i Þ; X i εC:
¯
¯
The bijective equality for X in the open subset of R is desired as X is a convex relation ϕ → X = F
(W). The conditions of the global linearization illustrate that the nonlinear coordinate transformation
will be global, if for any value of the initial condition, the solution of the Jacobian Matrix remains
non-singular [10]. Moreover, the nonlinear system is linearized in a large enough region of the space
or global space.
n
4. PERMANENT MAGNET SYNCHRONOUS GENERATOR WIND ENERGY CONVERSION
SYSTEM
The WECS composes of various electrical elements, such as wind turbine, PMSG, machine side pulsewidth modulation (PWM) inverter, and grid network. The diagram of the WECS and wind turbine
power curve is described in Figure 1.
4.1. Permanent magnet synchronous generator wind turbine model
The output mechanical power generated by the wind turbine is presented as follows:
1
P ¼ ρAV 3 C p ðβ; γÞ
2
Copyright © 2016 John Wiley & Sons, Ltd.
(8)
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
PMSG GRID-INTERFACED DIFFERENTIAL GEOMETRIC CONTROL
Figure 1. Permanent magnet synchronous generator (PMSG) boost converter and wind turbine operating
power regions.
In Equation 8, ρ is the air density of air, A is the area of the wind blades, V is the velocity of wind, Cp
is the performance coefficient, β is the pitch angle of the blades, and γ is the tip speed ratio. The expression of gamma is defined as follows:
γ¼
2:237V
ω
(9)
In Equation 9, ω is the rotor mechanical speed of the wind power generator. The simplified expression of Cp is presented by a nonlinear relation as:
Cp ¼
1
γ 0:022β2 5:6 expð0:17γÞ
2
(10)
The wind turbine operates in four basic regions. The working regions are described as follows:
• In Region A, no power will be generated by the wind turbine because of low wind speed.
• In Region B, sub-rated power is to be produced. Sub-rated region exists between cut-in speed and
rated speed.
• In Region C, rated power is produced by the wind turbine.
• In Region D, no power is produced because of the existence of stronger winds. The wind turbine
goes to shut down mode for mechanical safety.
In the PMSG wind turbine model, the generating unit is connected to the grid through back-to-back
PWM converters. Conventionally, the DC link voltage is controlled by the grid side converter, while
maximum power extraction is controlled by the generator side converter.
4.2. Nonlinear permanent magnet synchronous generator boost converter
The nonlinear PMSG boost converter is presented in Figure 1 [8]. The PMSG model is interfaced with
the rectifier-boost converter circuitry to obtain the desired DC link voltage. The boost circuitry is interlinked with the voltage source inverter for synchronization with the grid network. The input torque to
the PMSG is the wind that is blowing towards the PMSG blade section. The output is the controlled
DC link voltage in the presence of the electrical disturbances (faults, outages, etc.) and the mechanical
perturbations (varying wind speed as an input torque). The power turbine characteristic curve of the
PMSG boost is presented in Figure 2.
The objective is to obtain the steady state response of the output DC link voltage during the grid
faults and varying input torques. In our case, the maximum rated wind speed is 12 m/s, while rated
turbine power output is 2 MW. The operating region of wind turbine is between points B and C.
Copyright © 2016 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
S. M. ALI ET AL.
Figure 2. Wind turbine characteristic curve for the permanent magnet synchronous generator boost case.
The wind turbine goes to shut-down mode when wind speed exceeds 12 m/s. The MPPT of the wind
turbine occurs at rated wind speed and controlled by grid-side converter.
The stator voltages are analyzed as follows:
diA
þ ωe ψ PM cosðθe Þ
dt
diB
2π
þ ωe ψ PM cos θe uB ¼ iB Rs Ls
dt
3
diC
2π
þ ωe ψ PM cos θe þ
uC ¼ iC Rs Ls
3
dt
uA ¼ iA Rs Ls
(11)
(12)
(13)
The currents, electromagnetic torque, and rotor speed are analyzed using Park transformation. The
equations are analyzed as follows:
2
3
2 3
2π
2π
" #
iA
Sin θe þ
Sinðθe ÞSin θe 6
7
id
3
3
26
76 7
(14)
¼ 6
74 iB 5
34
2π
2π 5
iq
Cos θe þ
Cosðθe ÞCos θe iC
3
3
3
T e ¼ pn ψ PM iq
2
J dωe F
ωe
Te ¼ Tm pn dt
pn
(15)
(16)
dθe
¼ ωe
dt
(17)
Comparing Equations 15 and 16
3
J dωe F
ωe
pn ψ PM iq ¼ T m 2
pn dt
pn
ω2e
(18)
J
F
3
þ ωe T m þ pn ψ PM iq ¼ 0
pn pn
2
(19)
Solving for ωe using quadratic equation.
ωe ¼
Copyright © 2016 John Wiley & Sons, Ltd.
J d2 θe
p
3
ψ iq p2
þ Tm n þ
F dt 2
F 2F PM n
(20)
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
PMSG GRID-INTERFACED DIFFERENTIAL GEOMETRIC CONTROL
diL
¼ ðd 1Þudc þ u0
dt
u0 ¼ uAB ¼ uA uB
L
(21)
(22)
iL ¼ iA ¼ iB
(23)
The state-space model of the PMSG boost converter is developed using three parameters, namely,
inductor current, electrical rotating machine speed, and rotor electrical angle. The state-space model
can be presented as [32]:
I˙L ¼ C 1 I L þ C 2 ωe sinðΘe 60Þ þ C 3 ;
ω̇e ¼ C 4 I L sinðΘe 60Þ þ C 5 ωe þ C 6 ;
(24)
Θ̇e ¼ ωe :
In Equation 24, the constants from C1 C7 consist of the several model constant parameters, such as
stator resistance, stator inductance, inertia, and friction, that we define as follows:
2Rs
;
2Ls þ L
pffiffiffi
3ϕ PM
;
¼
2Ls þ L
U dc
;
¼
2Ls þ L
pffiffiffi 2
3pn ϕ PM
;
¼
J
F
;
¼
J
p Tm
¼ n ;
J
U dc
:
¼
2Ls þ L
C1 ¼
C2
C3
C4
C5
C6
C7
(25)
The selected state variables on which our mathematical control model will be based on is described
by the following three controlling parameters:
x1 ¼ I L ;
x2 ¼ ωe ;
(26)
x3 ¼ ðΘe 60Þ
The vector fields f(X) and g(X) are formed by performing the necessary conditions of the Lie algebra.
The control law u is the desired duty cycles for boosting the DC link voltage to a rated value and the output
function is y. The mathematical expressions for the aforementioned case can be defined as follows:
0
1
C 1 x1 þ C 2 x2 sinx3 þ C 3
B
C
C
f ðX Þ ¼ B
@ C 4 x1 sinx3 þ C 5 x2 þ C 6 A;
0
1
C7
B C
C
gðX Þ ¼ B
@ 0 A;
x2
(27)
0
u ¼ d;
y ¼ hðX Þ ¼ x3
Copyright © 2016 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
S. M. ALI ET AL.
The Lie derivative and the Lie bracket operation is performed using the transformations presented in
Section 3. We define the mapping, inverse mapping, derived mapping, and solving a set of partial
differential equations for the conversion of the nonlinear system into a linear system as described
in Figure 3. The sufficient and necessary nonlinear coordinate transforms are carried out from the
W space to the Zn1 space. Moreover, the mapping will be calculated from the X space to the Zn1
space. The coordinate transform T is defined as T = Rn1f 1. The diffeomorphic relation among
W, X and Z spaces through the nonlinear coordinate transforms is shown in Figure 3. For the BNF,
the mapping between any two spaces must exhibit a local diffeomorphic relation [30]. The exact
e 1 ðZ Þ is called the BNF. Consequently, the BNF and linearized system will
linearized system X ¼ F
become as:
Ż ¼ AZ þ BV;
ż 1 ¼ z2 ;
ż2 ¼ z3 ;
(28)
ż3 ¼ v:
The summary of control law model (through the EFL) can be observed from the sub-plot (b) of
Figure 3. The nonlinear mathematical control law model for the PMSG boost is illustrated in terms
of the vector fields, controlling parameters, and the linear control variable. The performance index
or cost function J is linear quadratic Riccati, which is described as:
1 ∞ T
dΦn ðX ÞT
dΦn ðX Þ
R
Þdt;
J ¼ ∫0 Φ ðX ÞQΦðX Þ þ
2
dt
dt
Figure 3. Coordinate transformations between W, X, and Z spaces and flow of the exact feedback linearization (EFL) control scheme. BNF, Brunovsky normal form.
Copyright © 2016 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
PMSG GRID-INTERFACED DIFFERENTIAL GEOMETRIC CONTROL
where Q is semi-positive definite n × n matrix with choice of Q = QT ⪰ 0 and R is a positive definite
m × m matrix with the choice of R = RT ⪰ 0. The choice of Q and R are:
0
1
B
Q ¼ @0
0
0
1
1
C
0A
0
0
0
1
0 0
0
0 1
and
0
B
R ¼ @0
1
C
1 0A
The control law model of the PMSG boost is presented in Figure 4. The triggering pulses from
Space Vector Pulse Width Modulation (SVPWM) are used to control the switching of the boost converter.
The expression of the linear control variable v is a linear quadratic Riccati problem that provides optimized values of the control variables, namely, k1, k2, and k3. The linear control variable v is obtained
by solving the expression V = K*Z and K* = R 1BTP*. The value of P* is obtained by solving Riccati
equation. After completely solving a set of partial differential equations, the control law u becomes:
RE ¼ AT P þ PA þ PBBT P þ Q;
ef ðX Þ þ v
u¼d¼ 1
;
e
g 1 ðX Þ
v ¼ ðx3 x3o Þ 2:29x2 2:14ðC 4 x1 sinx3 þ C 5 x2 þ C6 Þ;
(29)
ef 1 ðX Þ ¼ C 4 sinx3 ðC 1 x1 þ C 2 x2 sinx3 þ C 3 Þ þ C 5 x2 þ C4 x1 x2 cosx3 ;
e
g1 ðX Þ ¼ C 4 C 7 sinx3 :
Equation 29 is the desired control law for the stable operation of the PMSG boost. The value of u
provides the duty cycles (PWM) to the boost converter for maintaining the stable output response of
the system. The denominator term in the control law contains constants and a state variable term.
The boundaries of x3 are defined in the subset Ω as follows:
Ω ¼ x3 =x3 εu; x3 ≠ð0; π Þ; ðC 4 C 7 Þ≠0:
(30)
The electrical parameters for the PMSG boost are listed in Table I. In Equation 29, the numerator
and the denominator have the trigonometric terms, such as the sine and the cosine. Therefore, an
Figure 4. Control model of the permanent magnet synchronous generator boost. BNF, Brunovsky normal form.
Copyright © 2016 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
S. M. ALI ET AL.
Table I. Parametric values of the permanent magnet synchronous generator boost.
Model parameters
Value
Rated power (generator)
Flux linkage of generator
Stator resistance
Stator inductance
Number of pole pairs
Moment of inertia
Friction factor
DC link inductance
DC link capacitance
DC link voltage
Synchronizing frequency
Rotor speed
Sampling frequency
Rated wind speed
Rated torque
2 MW
9.7 Wb
0.1 ohms
0.835 mH
40
100 000 Kg m2
1000 Kg m2/s
50 mH
500 mF
900 V
60 Hz
32 rps
20 kHz
12 m/s
4000 N m
impression of the cotangent term is created that produces small spikes in the output response of the
system. The sensitivity of the sine parameter in the denominator of the control law is very high that
needs strict thresholds for the stable operation of the system.
4.3. Permanent magnet synchronous generator rectifier-inverter
The back-to-back PWM converter is described in Figure 5 [33]. Both the generator and the grid sections of the converter have the same main circuits that make the control system similar for both sides
of the system [9]. The stabilized DC link voltage is obtained from the output of the rectifier section.
The synchronization of the obtained voltage with the electrical grid is performed by the inverter section. The capacitor between the two sections eliminates the ripples from the DC link voltage. The input
is the wind speed that generates the mechanical torque. The wind turbine power characteristic curve is
described in Figure 6.
4.3.1. Generator-side control. In conventional current vector control, the control strategy of the
generator-side converter (GSC) includes the following two main parts:
• Current control loop and
• Conversion from current control signals to voltage control signals.
The complete generator control system using space vector PWM is shown in Figure 7. The
quadrature-axis and direct-axis voltage equations of the synchronous d-q frame of reference are
described as:
U q ¼ Rs I q þ Ls I˙q þ ωLq I d þ ωϕ PM ;
(31)
U d ¼ Rs I d þ Ls I˙d ωLd I q :
Figure 5. Permanent magnet synchronous generator (PMSG) rectifier-inverter.
Copyright © 2016 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
PMSG GRID-INTERFACED DIFFERENTIAL GEOMETRIC CONTROL
Figure 6. Wind turbine characteristic curve for the permanent magnet synchronous generator rectifierinverter case.
The quadrature-axis and direct-axis currents are used as state variables that control Vq and Vd
through EFL. Uq and Ud are in turn used to generate three-phase PWM pulses through space
vector modulation mechanism. The aforesaid triggering pulses are used to control the DC link
voltage.
4.3.2. Grid-side control. The active and reactive power grid controls are achieved by controlling the
direct and quadrature current components. The control scheme for the grid-side converter system
(GSCS) is similar to the GSC. The GSCS circuitry consists of the circuit that generates controlled
DC link voltage, DC bus, an inverter, and the grid section. The GSCS converts the Uq and Ud voltage
signals from Cartesian coordinates to polar coordinate system. The aforesaid signals are converted into
Figure 7. Generator control and grid control models. PMSG, permanent magnet synchronous generator.
Copyright © 2016 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
S. M. ALI ET AL.
three-phase triggering pulses through d-q transformations. The MPPT is achieved using the two control loops that control the active and reactive power [30].
The MPPT mechanism ensures the transfer of maximum active power and reactive power to the
grid side for synchronization. The DC link voltage control loop is used to control the d-axis current
Id. The total power coming from the rectifier is delivered to the grid by the inverter. The reference DC
voltage Udc (ref) must be set a little higher than the actual one, which keeps the inverter to deliver the
power. A reactive power control loop is setting a q-axis current reference Iq (ref) to a current control
loop that is similar to the d-axis current control loop. The grid controller ensures that all of the power
in the DC link must be delivered to the grid. The scheme of the grid-side controller is depicted in
Figure 7.
4.3.3. Mathematical model of the permanent magnet synchronous generator rectifier-inverter. The
nonlinear state-space model of the PMSG rectifier-inverter in the grid-connected mode is described
in terms of the three parameters, namely, direct axis current, quadrature axis current, and electrical
rotating speed. The nonlinear state-space model of the PMSG rectifier-inverter is linearized using
the same scheme of the EFL employed for the PMSG boost. The three controlling parameters will
be analyzed for the conversion of nonlinear system into a completely linear model, the BNF. The
simplified model is described as [34]:
ẋ 1 ¼ k1 x2 x3 þ k 2 x1 ;
ẋ 2 ¼ k 4 x2 þ k5 x1 x3 þ k6 V q þ k 7 x3 ;
(32)
ẋ 3 ¼ k 8 x2 þ k9 :
In Equation 32, the terms k1 k8 consist of various constant model parameters, such as direct axis
inductance, quadrature axis inductance, and stator resistance. These constants are mathematically presented as:
k1 ¼ p;
R
;
k2 ¼
Ld
R
;
k3 ¼
Lq
k 4 ¼ p;
1
k5 ¼ ;
Lq
λp
k6 ¼
;
Lq
(33)
1:5p2 λ
;
4J
BP
:
k8 ¼
2J
k7 ¼
The state variables will be equal to the degree of the state vector. The nonlinear PMSG rectifierinverter is of order three. Therefore, the three state variables are given as follows:
x1 ¼ I d ;
x2 ¼ I q ;
(34)
x3 ¼ ω:
The vector fields f(X) and g(X) are calculated for analyzing the Lie algebra operation. The vector
fields, control variable, and the output function are defined as:
Copyright © 2016 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
PMSG GRID-INTERFACED DIFFERENTIAL GEOMETRIC CONTROL
0
k 1 x2 x3 þ k 2 x1
1
B
C
C
f ðX Þ ¼ B
@ k 3 x2 þ k4 x1 x3 þ k 6 x3 A;
0
0
1
k7 x2 þ k 8
(35)
B C
C
gðX Þ ¼ B
@ k 5 A;
0
u ¼ U q ¼ d;
y ¼ hðX Þ ¼ x2 :s
The control law of the PMSG rectifier-inverter is derived on the same principle as that of the EFL.
Proceeding in a similar fashion, the final control equation for the PMSG rectifier-inverter is obtained
after solving a lengthy set of expressions according to the EFL model, as described in Figure 3. The
linear control variable and the control law become:
DðAÞ þ EðBÞ þ F ðCÞ þ ðz1 2:29z2 2:14z3 Þ
;
k 4 k 5 k 7 x1 þ k1 k4 k 5 x23 þ k23 k5 þ k 5 k 6 k7
z1 ¼ hðxÞ;
u¼
z2 ¼ Lf hðxÞ;
z3 ¼ L2f hðxÞ;
A ¼ k 1 x2 x3 þ k2 x1 ;
(36)
B ¼ k3 x2 þ k 4 x1 x3 þ k 6 x3 ;
C ¼ k7 x2 þ k 8 ;
D ¼ k2 k4 x3 þ k 3 k4 x3 þ k 4 k 7 x2 þ k4 k8 ;
E ¼ k 4 k 7 x1 þ k1 k 4 x23 þ k 23 þ k 6 k 7 ;
F ¼ k2 k 4 x1 þ k 3 k4 x1 þ 2k 1 k 4 x2 x3 þ k3 k6 þ k4 :
The electrical parameters for the PMSG rectifier-inverter are listed in Table II as:
The denominator term in Equation 36 of the control law is void of the sine and the cosine terms. The
sensitivity of the controlling parameters x1 and x3 is very low, as compared with the sensitivity of the
control law for the PMSG boost. Because the local subset has no trigonometric terms in Equation 36,
Table II. Parametric values of the permanent magnet synchronous generator rectifier-inverter.
Model parameters
Rated power (generator)
Flux linkage of generator
Stator resistance
Stator inductance
Number of pole pairs
Moment of inertia
Friction factor
DC link inductance
DC link capacitance
DC link voltage
Synchronizing frequency
Sampling frequency
Rotor speed
Rated wind speed
Rated torque
Copyright © 2016 John Wiley & Sons, Ltd.
Value
2 MW
9.7 Wb
0.1 ohms
0.835 mH
40
100 000 Kg m2
1000 Kg m2/s
30 mH
800 mF
1400 V
60 Hz
20 kHz
40 rps
12 m/s
5500 N m
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
S. M. ALI ET AL.
the local stability associated with the thresholds x1 and x3 will produce more stability than the PMSG
boost. The boundary subset is defined as follows:
Ω ¼ ðx1 x3 jðx1 x3 Þ≠0Þ; ðk1 ; ……:; k 9 ≠ 0Þ:
(37)
5. SIMULATION RESULTS OF THE PERMANENT MAGNET SYNCHRONOUS GENERATOR
BOOST
The PMSG boost is implemented using the EFL scheme in MATLAB/Simulink with zero initial
conditions.
The solution for the aforesaid problematic response is suggested by further linearization of the control law. The subset for the local stability will become:
Ω ¼ ðx3 jx3 εu; x3 ≠0Þ; ðC 4 C 7 ≠ 0Þ:
(38)
In windy areas, the proportionality of getting the desired mechanical input (wind) is optimum. The
high-speed wind factor is putting a safety measure on the state variable x3. But as the wind speed fluctuates, the varying parameter x3 will produce a shift in the value of sinx3.
The stability of the aforementioned designed model is analyzed in the presence of a three-phase
short-circuit line to the ground faults across the grid section. The three-phase short-circuit faults are
introduced as: (i) short-time short-circuit fault (SSCF) and (ii) long-time short-circuit fault (LSCF).
Figure 8 shows the DC link voltage response during the SSCF and LSCF. The PMSG parameters, such
as friction, inertia, stator resistance, and flux linkage of the magnets, are also changed from the nominal
to the slight off-nominal values. These PMSG machine model parameters are changed to 1.5% of the
rated machine values. The heavy LSCF will make the DC link voltage to reach the value of zero for
some duration of the clearing time. As soon as the clearing time of the fault is over, the DC link voltage
maintains a steady-state response. Because of the large reduction in the DC link voltage (Udc) magnitude during the SSCF and LSCF with perturbed generator parameters, the robustness level of the
PMSG boost against the faults is low.
The stability of the PMSG boost is further analyzed by varying the input mechanical torque
from minimum to the maximum value. We analyzed that the stability of the PMSG boost is only
limited to the mechanical torque variation of 80%. The minimum torque variation of 80% means
that the wind speed is reduced to 20% from the maximum 100% rated speed. When the torque
Figure 8. DC link voltage response during grid faults. SSCF, short-time short-circuit fault; LSCF, long-time
short-circuit fault.
Copyright © 2016 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
PMSG GRID-INTERFACED DIFFERENTIAL GEOMETRIC CONTROL
variation goes below 80% of the rated value, the output response of the DC link voltage goes below the rated value of 900 V, as defined in Table II. The PMSG boost is not robust because the
designed controller is only accepting the minimum varying input torque (wind speed) of 20%.
The stability of the PMSG boost is also compromising because of the aforementioned limitation
of the controller. When the wind speed varies to 50% and 30% of the rated speed, the optimized
DC link voltage response is unachievable. The output DC link voltage response of the PMSG
boost with varying input torques is shown in Figure 9. The DC converter current Idc during
steady-state (SS), SSCF, and LSCF with perturbed generator parameters is presented in Figure 10.
The controller takes 5 s for the current to settle down in steady-state. This output response shows
variations in peak-time and overshoot during SSCF and LSCF. Similar output response occurs for
the PMSG rotor speed during parameter variations and faults. The rotor speed response is described in Figure 11.
Figure 9. DC link voltage response during varying input mechanical torques.
Figure 10. DC link converter current (A). SS, steady-state; SSCF, short-time short-circuit fault; LSCF,
long-time short-circuit fault.
Copyright © 2016 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
S. M. ALI ET AL.
Figure 11. Permanent magnet synchronous generator (PMSG) rotor speed (RPM). SS, steady-state; SSCF,
short-time short-circuit fault; LSCF, long-time short-circuit fault.
6. SIMULATION RESULTS OF THE PERMANENT MAGNET SYNCHRONOUS GENERATOR
RECTIFIER-INVERTER
The PMSG rectifier-inverter is implemented and interfaced with the grid section in the back-to-back
converter topology. The three-phase line to ground faults (SSCF and LSCF) across the grid-side is introduced in the PMSG rectifier-inverter. The generator parameters, such as inertia, friction, and stator
resistance values, are also perturbed from the nominal values to 1.5 times of the rated machine values
to validate the stability and performance of the PMSG rectifier-inverter controller. The output DC link
voltage maintains a steady-state value as shown in Figure 12. The robustness level of the PMSG
rectifier-inverter against the SSCF and LSCF faults is more than the PMSG boost.
The stability of the PMSG rectifier-inverter is further verified by comparing the output DC link voltage response between the PI (linear) controller and the EFL (nonlinear) controller with the minimum
Figure 12. The DC link voltage during the short-time short-circuit fault (SSCF) and long-time short-circuit
fault (LSCF) faults with perturbed generator parameters. EFL, exact feedback linearization; PI, proportional
integral.
Copyright © 2016 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
PMSG GRID-INTERFACED DIFFERENTIAL GEOMETRIC CONTROL
and maximum varying input torques. The input torque was varied between 30% (maximum) and 80%
(minimum) of the rated torque values for the validation of the model. The settling time, accuracy, and
output stability of the PMSG rectifier-inverter under fast varying dynamics were improved by using
the EFL controller compared with the PI controller. The EFL controller is able to maintain the steady
DC link voltage for the aforementioned input torque variations, as shown in Figures 13–15. However,
the classical PI controller starts producing the unstable and abrupt responses to the varying torques
(Figure 16).
The PMSG rotor speed is compared between EFL scheme and PI scheme. The rotor speed is analyzed during SS, SSCF, and LSCF. The rotor speed drifts away from the rated speed during SSCF
with PI control. The graphical analysis of the rotor speed is described in Figure 15. The d-q transformation is a three-phase to two-phase transformation. The d-q axes are selected, such that they are
orthogonal to each other and hence decoupled. The d-q-reference frame is chosen in synchronous
reference frame, where the relative motion of d-q-axis and synchronous speed is zero, which results
Figure 13. The DC link voltage of the permanent magnet synchronous generator rectifier-inverter during
30% of the rated input torque (Tm = 0.3 Tmo). EFL, exact feedback linearization; PI, proportional integral.
Figure 14. The DC link voltage of the permanent magnet synchronous generator rectifier-inverter during
50% of the rated input torque (Tm = 0.5 Tmo). EFL, exact feedback linearization; PI, proportional integral.
Copyright © 2016 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
S. M. ALI ET AL.
Figure 15. The DC link voltage of the permanent magnet synchronous generator rectifier-inverter during
80% of the rated input torque (Tm = 0.8 Tmo). EFL, exact feedback linearization; PI, proportional integral.
in the best choice to design the controller for DC (constant) values of Id and Iq. The decoupled Id
and Iq currents are presented in control scheme [32,34]. The output function of the state-feedback
law is quadrature component of the stator current Iq. The variation in Iq is analyzed in subplots
(a), (b), and (c) during SS, SSCF, and LSCF with the EFL control scheme, while Iq response with
PI control is highlighted in (d). The aforementioned results are presented in Figure 17. The variation
in Id is analyzed in subplots (a), (b), and (c) during SS, SSCF, and LSCF with the EFL control
scheme, while Id response with PI control is highlighted in (d), as shown in Figure 18. During
short-circuit grid fault, voltage sag is created, which affects the inter-connected network. The WECS
must ‘ride through’ the faulty period and provide required power support. In case of SSCF, the active power and reactive power transferred to the grid through the EFL control scheme are presented
in Figure 19.
Figure 16. Permanent magnet synchronous generator (PMSG) rotor speed. The plot shows ω during steady-state
(SS), short-time short-circuit fault (SSCF), long-time short-circuit fault (LSCF) with exact feedback linearization
(EFL) control, and ω during SSCF with proportional integral (PI) control.
Copyright © 2016 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
PMSG GRID-INTERFACED DIFFERENTIAL GEOMETRIC CONTROL
Figure 17. Stator q-axis current. The subplot (a) exact feedback linearization (EFL)-based Iq, no fault
applied, (b) EFL-based Iq during short-time short-circuit fault, (c) EFL-based Iq during long-time
short-circuit fault, (d) EFL-based Iq during short-time short-circuit fault with proportional integral control,
and (e) EFl-based Iq during long-time short-circuit fault with proportional integral control.
The comparison of the PMSG boost and PMSG rectifier-inverter based on the control responses obtained through the EFL scheme is also listed in Table III. Table III highlights that control features of
the PMSG rectifier-inverter, such as stability, optimization level, placement feasibility, and robustness
level dominates the PMSG boost. The output responses of both of the PMSG models are compared
based on the local stability subsets, derived control laws, local linearization, and output response of
the DC link voltage from the respective converter systems. The comparison justifies the performance
and the effectiveness level of the PMSG rectifier-inverter for the wind energy applications. Moreover,
comparative features of the EFL and PI evaluated in this paper are summarized in Table IV.
Figure 18. Stator d-axis current. The subplot (a) exact feedback linearization (EFL)-based Id, no fault
applied, (b) EFL-based Id during short-time short-circuit fault, (c) EFL-based Id during long-time
short-circuit fault, (d) EFL-based Id during short-time short-circuit fault with proportional integral control,
and (e) EFL-based Id during long-time short-circuit fault with proportional integral control.
Copyright © 2016 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
S. M. ALI ET AL.
Figure 19. Active power and reactive power transferred to the grid during short-time short-circuit fault.
Table III. Comparison of PMSG boost and PMSG rectifier-inverter.
Control features
Robustness level
System stability
Mathematical complexity
Computational burden
Reliability
Optimization level
Self-tuning capability
Parameter sensitivity
Control law stability
Computational cost
Grid-interface performance
Placement area
Applications
PMSG boost
PMSG rectifier-inverter
Low
Low
High
High
Low
Low
High
High
Local
High
Low
Constant high windy
Local and global controls
High
High
High
High
High
High
High
Medium
Local
Medium
High
Low, medium, and high windy
Local and global controls
PMSG, permanent magnet synchronous generator.
Table IV. Comparison of PI and EFL.
Control features
Robustness level
System stability
Mathematical complexity
Computational burden
Reliability
Optimization level
Self-tuning capability
Parameter sensitivity
Control law stability
Computational cost
Grid-interface performance
Settling time
Accuracy level
Applications
PI
EFL
Low
Low
Low
Low
Low
Low
None
Low
Low
Low
Low
Slow
Low
Limited in WECS
High
High
High
High
High
High
High
High
High
High
High
Fast
High
Wide in WECS
EFL, exact feedback linearization; PI, proportional integral; WECS, wind energy conversion systems.
Copyright © 2016 John Wiley & Sons, Ltd.
Int. Trans. Electr. Energ. Syst. (2016)
DOI: 10.1002/etep
PMSG GRID-INTERFACED DIFFERENTIAL GEOMETRIC CONTROL
Table V. Symbols and notation meanings for mathematical analysis.
Symbols
Notation meanings
IL
Rs
Ls
pn
φPM
ωe
Θe
1
φD
X1
Id
Iq
Ud
Uq
Ld
Lq
Tem
J
ψf
Udc
Tm
Tmo
Inductor current
Stator resistance
Stator inductance
Number of pole pairs
Useful flux linkage
Electrical speed of the machine
Rotor electrical angle
Integral curve for mapping
Direct axis current
Quadrature axis current
Direct axis voltage
Quadrature axis voltage
Direct axis inductance
Quadrature axis inductance
Electromagnetic torque
Inertial constant
Useful rotor field flux
DC link voltage
Input mechanical torque
Rated input mechanical torque
7. CONCLUSIONS AND FUTURE WORK
The stability and robustness against faults and disturbances play a pivotal role in the efficiency of the
grid-interfaced PMSG wind energy systems. We applied the EFL control scheme for the critical and
comparative analysis of the PMSG boost and PMSG rectifier-inverter. The spikes perturbations in
the output DC link voltage response of the PMSG boost were the main cause of instability in the designed system. The involvement of the trigonometric parameters in the control model of the PMSG
boost resulted in small spikes in the output DC link voltage. The EFL control law for the PMSG boost
was further linearized and simplified that resulted in the effective performance of the system in case of
electrical grid faults. The stability of the PMSG boost was limited, as the mechanical torque variations
were only accepted up to 95% of the rated value.
In the near future, we will extend the robust affine PMSG grid-interfaced wind energy system to various MIMO-WECS using the EFL control scheme.
For the ease of understanding, the most commonly used mathematical symbols are given in Table V.
ACKNOWLEDGEMENTS
The authors are highly grateful to Osman Khalid for providing valuable suggestions for improving and modifying
the overall contents of the paper.
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