This paper proposes a new robust decentralized power oscillation dampers (POD) design of doublyfed induction generator (DFIG) wind turbine for damping of low frequency electromechanical oscillations in an interconnected power system. The POD structure is based on the practical 2
^{nd}
order lead/lag compensator with single input. Without exact mathematical model, the inverse output multiplicative perturbation is applied to represent system uncertainties such as system parameters variation, various loading conditions etc. The parameters optimization of decentralized PODs is carried out so that the stabilizing performance and robust stability margin against system uncertainties are guaranteed. The improved firefly algorithm is applied to tune the optimal POD parameters automatically. Simulation study in twoarea fourmachine interconnected system shows that the proposed robust POD is much superior to the conventional POD in terms of stabilizing effect and robustness.
1. Introduction
It is well known that the advantage of power system interconnection is the augmentation of system reliability, economics and security etc. However, the inevitable problem in longitudinal interconnected power systems is the low frequency interarea oscillation with poor damping
[1]
. Under the heavy power flow condition and weak tieline, the interarea oscillation mode may be unstable. Moreover, when the severe short circuits occur in the system, the interarea oscillation may cause the system instability. To damp out the interarea oscillation, the power system stabilizer (PSS) has been successfully applied
[2
,
3]
. Nevertheless, the PSS may cause the negative impact to the voltage control of automatic voltage regulator (AVR)
[4]
.
At present, the wind generators have been installed widely in power systems. In
[5]
, an impact of wind power integration on generation dispatch in power systems is investigated. As sharing of wind generations increase, they should not only generate electrical power, but also contribute other functions. Especially, the damping of power system oscillation is significantly anticipated. For instance, the ability of power oscillation damping is included in the new Spanish grid code for wind power
[6]
.
Among of wind generators, the doublyfed induction generator (DFIG) wind turbine has been extensively used
[7]
. Since the active and reactive power outputs of DFIG can be controlled independently by the power converters based on vector control
[8]
, flux magnitude and angle control
[9]
, the DFIG can be applied to stabilize the power oscillation. The power oscillation damper (POD) is equipped with the DFIG wind turbine with the same function as PSS. The PODs with various inputs such as the angle variation
[4]
, the slip of DFIG
[10]
etc., have been presented and successfully damp out the power oscillation. Besides, the PODs are tuned in the small signal stability model of the power system by particle swarm optimization
[11]
, bacterial foraging
[12]
and differential evolution
[13]
so that the dynamic performance and fault ride through capability of DFIG are improved. In power systems, however, there are various uncertainties e.g. loading conditions, wind patterns, unpredictable network, variation of system parameters and severe disturbances etc. The PODs proposed in previous works which have been designed without taking system uncertainties into account may not be able to handle the system stability. The POD with high robustness against such uncertainties is significantly required.
In
[14]
, the robust control design of POD for DFIG has been proposed. The POD parameters optimization problem is formulated based on a mixed
H_{2}/H_{∞}
control using linear matrix inequalities (LMI). Simulation results in single machine infinite bus guarantee that the robustness and performance of the proposed POD is superior to the conventional POD. However, there are some limitations of the study in
[14]
as follows.

1) Since the objective of POD design is to stabilize the local oscillation mode in the single synchronous generator connected to an infinite bus, the proposed design cannot guarantee the stabilizing effect of POD on various oscillation modes such as interarea modes, local modes etc., in a multimachine power system. Therefore, the improvement of POD design in a multimachine power system is highly anticipated.

2) This work considers only the single POD design which is restricted to the singleinput singleoutput (SISO) system. To improve this, the design technique which can be applied to the multiple PODs case in the multiinput multioutput (MIMO) system, is expected.

3) A mixedH2/H∞control using linear matrix inequalities which is used in the POD design in this paper, has been widely applied to design power system damping controllers in the previous research works. The new design technique is significantly required.
In this paper, the new robust decentralized design of POD equipped with DFIG wind turbines for stabilization of interarea oscillation in interconnected power systems is presented. The structure of POD is specified as a practical 2
^{nd}
order leadlag compensator with single input. Without difficulty of mathematic representation, system uncertainties are modeled by an inverse output multiplicative perturbation. The POD parameters optimization problem is formulated so that the damping performance and robustness can be guaranteed. To achieve the optimal parameters of PODs, the firefly algorithm (FA) is used to solve the optimization problem. Simulation study conducted in a twoarea fourmachine power system confirms that the stabilizing effect and robustness of the proposed POD is superior to those of the conventional POD.
The organization of this paper is described as follows. First, the study system and modeling are explained in Part 2. Next, the detail of proposed design method is provided in Part 3. Subsequently, Part 4 shows simulation results. Finally, the conclusion is given.
2. Study system and modeling
 2.1 Study system
Fig. 1
depicts a twoarea fourmachine interconnected power system
[15]
which is used as a study system. Each synchronous generator is represented by a 6
^{th}
order model. The synchronous generator is equipped with an AVR type 3 and a turbine governor (TG) type 2
[16]
. To supply electric power to the system, the DFIG wind turbines equipped with POD are installed at bus 7 and bus 9. In this study, it is assumed that the power flow in a tieline (
P_{tie}
) between bus 7 and bus 9 is in a heavy condition. Besides, system disturbances such as three phase short circuit occasionally occur in the system. These conditions cause the interarea oscillation with poor damping. To handle this oscillation, these DFIGs equipped with PODs are applied.
Twoarea fourmachine system with DFIG wind turbines.
 2.2 DFIG model
The structure of DFIG wind turbine and control system is demonstrated in
Fig. 2
[16]
. The DFIG control is performed by controlling the rotor side converter based on the vector control technique. The vector control provides an independent control of active and reactive power. Here, the fluxbased rotating reference frame is used to model the DFIG. The quadrature (
q
)axis current of the rotor side converter (
i_{qr}
) is applied to control the real power output while the direct (
d
)axis current (
i_{dr}
) is used to control the reactive power output. Here, the converter is modeled as an ideal current source, where rotor currents
i_{qr}
and
i_{dr}
are used for rotor speed control and voltage control, respectively, which are depicted in
Fig. 3(a)
and
3(b)
. The active and reactive power of DFIG injected into the grid can be written in terms of rotor currents as
where
P
and
Q
are active and reactive power of DFIG, respectively,
ω_{m}
is a rotor speed of DFIG,
p
*
_{w}
is the power speed characteristic which roughly optimizes the wind energy capture,
x_{s}
is a stator selfreactance,
x_{u}
is a magnetizing reactance,
T_{e}
is the time constant of power control,
v
is a magnitude of DFIG terminal voltage,
v
^{0}
ref
is the initial reference voltage,
v_{ref}
is the actual reference voltage,
v_{SI}
is input signal of POD,
v_{s} ^{POD}
is an additional signal of POD,
K_{v}
is the voltage controller gain, and
i_{dr}
^{min}
,
i_{dr}
^{max}
,
i_{qr}
^{min}
,
i_{qr}
^{max}
are
d
and
q
axis minimum and maximum rotor currents, respectively. Here, the stabilization of power oscillations is performed by the voltage control loop via the POD signal.
Configuration of DFIG wind turbine and vector control strategy.
Rotor speed and voltage control scheme of DFIG.
 2.3 POD model
Fig. 4
shows the structure of the POD which is a 2
^{nd}
order leadlag compensator with single input. The POD consists of a stabilizer gain
K_{stab}
, a washout filter with time constant
T_{w}
=5 s, and two phase compensator blocks with time constants
T
_{1}
,
T
_{2}
,
T
_{3}
and
T
_{4}
. The washout signal ensures that the POD output is zero in steady state. The input signal
v_{SI}
is the active power flow in the representative transmission line where the interarea oscillation mode can be observed easily. Here, the input signals of POD1 and POD2 are the power flow in line 78 and line 89, respectively. The output signal
v_{s}^{POD}
is subject to an antiwindup limiter,
v_{s}
^{min}
and
v_{s}
^{max}
are minimum and maximum of
v_{s} ^{POD}
. The gain
K_{stab}
determines the amount of damping produced by POD while the phase compensator block gives the appropriate leadlag compensation of the output signal.
Structure of POD.
 2.4 Linearized power system model
The linearized system state equations in
Fig. 1
can be expressed by
where Δ
x
is a state vector [Δ
δ
Δ
ω
Δ
e’_{q}
Δ
e’_{d}
Δ
e’’_{q}
Δ
e’’_{d}
Δ
v_{m}
Δ
v_{r}
Δ
v_{f}
Δ
x_{g}
Δ
v_{w}
Δ
ω_{m}
Δ
θ_{p}
Δ
i_{dr}
Δ
i_{qr}
], Δ
δ
is a power angle deviation, Δω is a rotor speed deviation, Δ
e’_{q}
is a
q
axis transient internal voltage deviation, Δ
e’_{d}
is a
d
axis transient internal voltage deviation, Δ
e’’_{q}
is a
q
axis sub transient internal voltage, Δ
e’’_{d}
is a
d
axis sub transient internal voltage deviation, Δ
v_{m}
is a measurement voltage deviation, Δ
v_{r}
is a regulator voltage deviation, Δ
v_{f}
is a field voltage deviation, Δ
x_{g}
is an output signal of governor deviation, Δ
v_{w}
is an output signal of wind speed deviation, Δ
ω_{m}
is a rotor speed deviation of DFIG, Δ
θ_{p}
is a pitch angle deviation, Δ
i_{dr}
is a DFIG rotor current deviation in
d
axis, Δ
i_{qr}
is a DFIG rotor current deviation in
q
axis, Δ
u_{POD}
is an input vector of POD signal, Δ
y
is an output vector of power flow in tieline deviation,
n
is a number of state variables,
m
is a number of PODs,
A
is a system matrix,
B
is an input matrix,
C
is an output matrix and
D
is a feed forward matrix.
The output vector Δ
u_{POD}
consists of the
m^{th}
control signal from POD (Δ
u_{POD,m}
) which can be expressed by
where Δ
P_{tie,m}
is a tieline power deviation of the
m^{th}
POD,
K_{stab,m}
,
T
_{m1}
,
T
_{m2}
,
T
_{m3}
and
T
_{m4}
are gain and time constants of the
m^{th}
POD. These gain and time constants are optimally tuned by the proposed design. Note that the system in (2) is the multiinput multioutput (MIMO) control system and referred to as the nominal plant
G
. The multimachine power system included with PODs can be represented by an MIMO system
G
with a decentralized controller
K
, as depicted in
Fig. 5
. An MIMO system G is composed of
m
inputs and
m
outputs while a controller K consists of
m
PODs as diagonal controllers.
MIMO system G with a decentralized controller K.
3. Proposed robust design controller
 3.1. Uncertainty modeling
Robustness is a vital issue in control system design because real systems are vulnerable to system uncertainties. To improve the robustness of POD against system uncertainties such as various generating and loading conditions, wind patterns, and unpredictable network structure etc., the inverse output multiplicative perturbation is applied to represent such uncertainties without difficulty of exact equations.
Fig. 6
depicts the feedback control system with inverse output multiplicative perturbation and external disturbance
[17]
where,
G
is the nominal plant,
K
is the designed controller,
r(t)
is the reference input,
e(t)
is the error tracking,
d(t)
is the external disturbance,
y(t)
is the output of the system and Δ
_{M}
is the system uncertainties. Based on the small gain theorem
[17]
, for a stable multiplicative uncertainty, the system stable if
Then,
Define the robust stability index (
γ_{∞}
) as
As a result
In (7), the value of Δ
_{M}

_{∞}
implies the maximum boundary of uncertainties that the system can tolerate. When
γ_{∞}
increases, this boundary decreases, and the system robust stability margin becomes lower. In power systems,
P_{tie}
is the system variable which highly affects the system robust stability
[18]
. When
P_{tie}
increases,
γ_{∞}
tends to increase (lower robust stability margin). In this study, it is assumed that
P_{tie}
increases with the same amount.
Fig. 7
shows the relation between
P_{tie}
with respect to
γ_{∞}
. Note that
P_{tie,i}
, and
γ_{∞, i}
,
i
= 1,..,
n
, are the ith data of tie line power flow and the
i^{th}
data of
γ_{∞}
, respectively, and
n
is the number of power flow data. The
P_{tie,n}
which is the
n
^{th}
data of
P_{tie}
can be written in the form of the arithmetic sequence as
where
d_{as}
is a difference value between
P_{tie,i}
and
P
_{tie,i1}
,
i
=1,…
n
. It should be noted that
d_{as}
is a constant value.
Control system with inverse output multiplicative perturbation and external disturbance.
Relation between γ_{∞} and P_{tie}.
The
γ_{∞,i}
which corresponds to
P_{tie,i}
, is calculated by
where
G_{i}
is the
i^{th}
data of the nominal plant
G
at
P_{tie,i}
In
Fig. 7
,
γ
^{*}
_{∞}
which is the minimal value of
γ_{∞,i}
, implies the highest robust stability margin of the system against uncertainties. When the sum of difference between
γ_{∞,i}
and
γ
^{*}
_{∞}
is minimized,
γ_{∞,i}
is nearly equal to
γ
^{*}
_{∞}
. It means that the proposed controller can provide the system robust stability margin at any
P_{tie,i}
.
In addition, the controller is designed to move the eigenvalue corresponding to all oscillation modes to the Dstability region with the desired real part (
σ_{spec}
) and the desired damping ratio (
ζ_{spec}
) of eigenvalue as depicted in
Fig. 8
.
Dstability region.
Based on the above concept, the parameters optimization problem of PODs can be written as
Subject to
where
ζ_{h}
is the damping ratio of the
h^{th}
oscillation mode,
σ_{h}
is the real part of the
h^{th}
oscillation mode, os is the number of all oscillation modes,
γ^{*} _{∞, spec}
is the specified value of
γ_{∞}
^{*}
, which is appropriately selected by the designer,
RF
is the ranking factor,
K_{stab}
^{min}
and
K_{stab}
^{max}
are minimum and maximum gains,
T
^{min}
and
T
^{max}
are minimum and maximum time constraints.
Note that, the damping ratios and real parts of all oscillation modes in an area or between two areas can be improved so that they satisfy with the design specification.
 3.2 Firefly algorithm applied for the optimization problem
The FA is a metaheuristic algorithm inspired by the flashing behavior of fireflies
[19]
. The primary purpose of a firefly’s flash is to act as a signal system to attract other fireflies. For this work, the firefly algorithm is applied to solve the objective function (10) with parameters of POD i.e.
K_{stab,m}
and
T_{m,j}, m
=1,2 and
j
=1,…,4. Consequently, the stepbystep of the improved firefly algorithm is readjusted as the following.

1. Generate initial population of each firefly with random positions and light intensity.

2. For each firefly, ifζh≥ζspec, σh≤σspec, h=1,… ,osandγ∞*≤γ*∞, specgo to step 3. Otherwise go to step 1.

3. Check the number of firefly, if number of firefly = number of max firefly, then go to step 4. Otherwise go to step 1.

4. Evaluate the objective function in (10) by usingRFas follows;
where
In (11), the
RF
is used to determine the value of (10).
The
error
implies the difference between
γ^{*}_{∞, spec}
and
γ_{∞}^{*}
.

5. Rank the fireflies by their light intensity, i.e. the value of objective function.

6. Move all fireflies towards brighter onesxa+1by
where
α
is the randomization parameter,
rand
is the random number in (0,1),
β
_{0}
is the attractiveness at
r
=0,
r
is the distance between any fireflies
i
and
j
at
x_{a}
and
x_{b}
,
γ
is the light absorption coefficient.
where
d
is the number of tuned parameters,
z
=1,… ,
d
.
According to the objective function (10), the values of
x_{a}
and
x_{b}
which consists of the tuned parameters of the PODs, are presented by
where the subscript
a
and
b
are tuned parameters at position
x_{a}
and
x_{b}
, respectively,
x_{a,z}
and
x_{b,z}
are parameters of PODs which corresponding to the
z
series data of
x_{a}
and
x_{b}
, respectively. Note that, when the firefly moves from the current position
x_{a}
to the new position
x
_{a+1}
by substituting (15) into (13) and (14), this results in the change in parameters of PODs. Accordingly, the value of objective function in (10) is updated.

7. Make sure that the fireflies are within the range,Kstabmin≤Kstab,m≤Kstabmax, andTmin≤Tm.j≤Tmax.

8. When the maximum number of is reached, stop the process. Otherwise, go to step 4.
The flow chart of improved firefly algorithm for solving the optimization problem (10) is depicted in
Fig. 9
.
Flow chart of improved firefly algorithm.
4. Simulation results
In the simulation study, MATLAB programming and Power System Analysis Toolbox (PSAT)
[20]
are used. The parameters of FA and search parameters are set as follow; number of firefly=30, maximum iteration=350,
α
=0.2,
γ
=1,
β
_{0}
=1,
ζ_{spec}
=0.05 (or 5%),
σ_{spec}
=0.1,
P
_{tie,1}
=2.0 p.u.,
d_{as}
=0.3,
γ^{*} _{∞,spec}
=1.5,
os
=3 (two local modes and one interareamode), [
K_{stab}
^{min}
K_{stab}
^{max}
]=[0.1 15], and [
T
^{min}
T
^{max}
]=[0.1 10].
Fig. 10
depicts the convergence of the objective function (10) in case of the proposed robust POD which is referred to as “DFIGRPOD”. The DFIGRPOD is compared with the conventional POD designed without considering the robustness which is stand for “DFIGCPOD”. Based on the pole assignment method, the DFIGCPOD is designed at
P_{tie}
= 4.0 p.u. to yield the same damping ratio and real part of the dominant modes as in case of DFIGRPOD. The optimization problem of DFIGCPOD is formulated as follows;
Subject to
Convergence curve of the objective function (10).
The optimization objective in (16) is to move the dominant oscillation to the Dstability region as show in
Fig. 8
. Solving (10) and (16) by FA, the optimized parameters of DFIGCPOD and DFIGRPOD are given in
Table 1
.
Optimized parameters of DFIGCPOD and DFIGRPOD.
Optimized parameters of DFIGCPOD and DFIGRPOD.
Table 2
provides the eigenvalue and damping ratio of the dominant interarea oscillation mode. The damping ratio of oscillation mode is very poor in case of without POD. On the other hand, the damping ratio is improved as designed specification by both DFIGCPOD and DFIGRPOD.
Eigenvalue analysis result
Eigenvalue analysis result
The robustness of DFIGCPOD and DFIGRPOD is evaluated by
γ_{∞}
.
Fig. 11
depicts the variation of
γ_{∞}
against an increase in
P_{tie}
. Obviously,
γ_{∞}
is case of DFIGCPOD largely changes. The DFIGCPOD is very sensitive to the uncertainty due to the tieline power flow. On the other hand,
γ_{∞}
is case of DFIGRPOD rarely changes. The DFIGRPOD is not sensitive to variation of the tieline power flow.
The variation of γ∞ against an increase in P_{tie}.
Next, the variation of damping ratio against an increase in
P_{tie}
is depicted in
Fig. 12
. Under heavy power flow condition, the damping ratio in case of DFIGCPOD largely decreases. On the other hand, the damping ratio in case of DFIGRPOD is still greater than 5 % of the desired damping ratio.
The variation of damping ratio against an increase in P_{tie}.
The nonlinear simulation of four case studies as given in
Table 3
is carried out by PSAT. The uncertainty handing information given in
Table 3
consists of three items as follows;

1. Uncertainty due to the wind patterns applied to both DFIGs as shown inFig. 13.

2. Uncertainty due to the variation power flow levels in tieline between bus 7 and bus 9.

3. Uncertainty due to the applied faults and network structure after fault clearing.
Case studies (base 100 MVA).
Case studies (base 100 MVA).
Patterns of wind.
Figs. 14

17
depict the rotor speeds of four synchronous generators. In case 1 as shown in
Fig. 14
, without POD the rotor speeds largely oscillate. On the other hand, the oscillations are effectively damped by both DFIGCPOD and DFIGRPOD. In case 2 as depicted in
Fig. 15
, the damping effect of DFIGCPOD is much less than that of DFIGRPOD. In cases 3 and 4 as shown in
Figs.16
and
17
, respectively, the stabilizing effect of DFIGCPOD is completely deteriorated. The rotor speeds severely oscillate and the synchronous generators lose synchronism. On the other hand, the DFIGRPOD is robustly capable of damping out the oscillation. Simulation results confirm that the DFIGRPOD is very robust against the various power flow levels, wind patterns and severe faults.
Rotor speeds of synchronous generators in case 1.
Rotor speeds of synchronous generators in case 2.
Rotor speeds of synchronous generators in case 3.
Rotor speeds of synchronous generators in case 4.
5. Conclusion
In this paper, the new robust decentralized controller design of PODs equipped with DFIG wind turbines has been presented. Without exact mathematical equation, the inverse output multiplicative perturbation model is adopted to represent system uncertainties. The POD structure is the practical 2
^{nd}
order lead/lag compensator with single input. The parameters of PODs are simultaneously and automatically optimized by FA so that the robust stability margin and damping effect are improved. Simulation study confirms that the robustness and stabilizing performance of the proposed robust POD is much superior to those of the conventional POD under various faults, wind patterns and heavy tieline power flow conditions.
Acknowledgements
This work was supported by the King Mongkut’s Institute of Technology Ladkrabang Research Fund no. KREF 055706.
BIO
Tossaporn Surinkaew received B.Eng. degree in Electrical Engineering from King Mongkut’s Institute of Technology Ladkrabang (KMITL), Thailand, 2012. Currently, he is a master course student in electrical engineering department at KMITL. His research interests are robust control & power system stability.
Issarachai Ngamroo received B.Eng. degree in Electrical Engineering from KMITL in 1992. He earned his M.Eng. and Ph.D. degrees in Electrical Engineering from Osaka University, Japan in 1997 and 2000, respectively. Currently, he is a professor of electrical engineering department, faculty of engineering, KMITL. His research interests are in the areas of power system stability, dynamics & control.
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