This paper presents an adaptive widearea damping controller (WADC) based on generalized predictive control (GPC) and model identification for damping the interarea low frequency oscillations in largescale interconnected power system. A recursive leastsquares algorithm (RLSA) with a varying forgetting factor is applied to identify online the reducedorder linearlized model which contains dominant interarea low frequency oscillations. Based on this linearlized model, the generalized predictive control scheme considering control output constraints is employed to obtain the optimal control signal in each sampling interval. Case studies are undertaken on a twoarea fourmachine power system and the New England 10machine 39bus power system, respectively. Simulation results show that the proposed adaptive WADC not only can damp the interarea oscillations effectively under a wide range of operation conditions and different disturbances, but also has better robustness against to the time delay existing in the remote signals. The comparison studies with the conventional leadlag WADC are also provided.
1. Introduction
With the interconnection of regional grids and the increasing of line power, the interarea low frequency oscillation between different control areas is becoming a more and more serious problem which limits the enhancement of the transmission capacity of power grids or even breaks up of whole power system
[1]
. Conventionally, the power system stabilizers (PSSs) are used to damp interarea mode oscillations. However, the PSSs, which use local measurement as the input, can not always damp interarea mode oscillations effectively because interarea modes are not always observable from the local signals
[2
,
3]
.
With the technological advancements and increasing deployments of widearea measurement system (WAMS) in power system, remote signals from WAMS become available for the design of widearea damping controller (WADC) to solve the above problem
[2

5]
. Many classical control approaches, such as residue method
[2
,
3]
, robust control approach
[4
,
5]
, have been adopted for design of WADC which require a reasonably accurate model of the system at a nominal operating condition. However, lack of availability of accurate and updated information about each and every dynamic component of a largescale interconnected system and especial the model variations caused by the change of operation conditions and unavoidable faults put a fundamental limitation on such classical control design approaches. To overcome the inherent shortcomings of classical method based damping controllers, adaptive control method had been adopted to design an adaptive damping controller for power systems
[6

11]
. These controllers use a linear model with online updated parameters to design an adaptive controller to cope the model variation and uncertainty of largescale power system. In addition, for a widearea damp control system, the impact of the time delay existing in remote signals must be considered
[12
,
13]
. As the delay can typically vary from tens to several hundred milliseconds and is comparable to the period of some critical interarea modes, it should be taken into account at the design stage for a satisfactory damp performance
[14
,
15]
.
Model predictive control (MPC) is one of the major adaptive control strategies which has attracted many attentions and has been applied in process control successfully. Although there are several formulations of the MPC strategy, they explicitly use a model of system to obtain the control input signal by minimizing an object function
[16]
. The generalized predictive control (GPC) is one of the most popular control methods of MPC. The GPC approach can not only deal with variable deadtime, but also cope with overparameterization as it is a predictive method
[17]
. The GPC has been applied successfully to power system such as design of controllers for flexible AC transmission systems (FACTS) and the generator excitation system
[18
,
19]
.
In this paper, the GPC integrating model identification is proposed for the synthesis of an adaptive WADC, which can cope with variation of operating conditions, model uncertainties and robustness against time delay existing in widearea signals feedback. Simulation studies are carried out based on the twoarea fourmachine power system and the New England 10machine 39bus power system, respectively. Comparison results with the conventional WADC are also given. The results demonstrate that the proposed adaptive WADC can provide effective damping of interarea mode oscillations under various operation conditions and different disturbances. Moreover, it has much better performance than conventional WADC with time delay existing in the widearea signals.
The rest of the paper is organized as follows. Section 2 presents a general architecture of widearea damping control system. An adaptive GPC strategy and the procedure of the proposed adaptive WADC based on this GPC are described briefly in Section 3. Section 4 designs an adaptive WADC for a twoarea fourmachine benchmark power system and verifies its effectiveness by detailed simulation. Then, the proposed adaptive WADC is employed to design a WADC for the New England 10machine 39bus power system in Section 5. Finally, some conclusions are drawn in Section 6.
2. Architecture of WideArea Damping Control
The widearea stabilizing control structure shown in
Fig. 1
, socalled “decentralized/hierarchical” architecture, is used in this paper. In the local decentralized control level, including automatic voltage regulator and PSS (AVRPSS) at generators, main controls of FACTS or high voltage direct current (HVDC) transmission system, provides damping for local modes. Thus, the local mode will be very highly damped. However, these local controllers are not capable of damping interarea modes under stressed operating conditions because of the local signals lack of good observability of some critical interarea natural modes. Therefore, additional damping is required particularly for the these interarea modes. As shown in
Fig. 1
, in the widearea control level, the control signal of the WADCs are to provide damping for the interarea modes, controlled from the selected generator, FACT or HVDC using a global input signal from the WAMS. As few global signals are required only for some critical interarea modes and under specific network configurations, no more than the few WADCs sites with the highest controllability of these interarea modes need be involved in the widearea control level.
General architecture of the widearea damping control system
The above specific architecture was first discussed in
[3]
where it was preferred for its higher operational flexibility and reliability, especially in the event of loss of communication links or a failure that makes the widearea control signal unavailable. Under such circumstances, the controlled power system is still viable (although with a reduced performance level), owing to the fact that a fully autonomous and decentralized layer without any communication link is always present to maintain a standard performance level
[2]
.
3. Adaptive Generalized Predictive Control
The structure of the adaptive generalized predictive controller is shown in
Fig. 2
. It is an indirect type controller and mainly composes of two parts: system identification and controller synthesis based on generalized predictive control scheme.
Overview of an adaptive generalized predictive controller
 3.1 General system model
Although a real power system is a complex, nonlinear and highorder dynamic system, it can be represented by a relatively simple linear model with fixed structure but whose parameters vary with the operating conditions. This linear model is accurate enough for the purpose of design a damping controller
[8]
. In the application of designing an adaptive WADC, the following controlled autoregressive and moving average (CARMA) linear model is utilized to avoid the offset of the control signal
[19]
.
where,
y
(t) and
u
(t) are the output and input of the system, respectively, and
e
(t) is a discrete whitenoise sequence.
A
(z
^{1}
),
B
(z
^{1}
) and
C
(z
^{1}
) are
n
_{a}
,
n
_{b}
and
n
_{c}
order polynomial, respectively. They are given as follows
The polynomial
C
(z
^{1}
) may either represent the external noise components affecting the output (in which case its coefficients have to be estimated) or a design polynomial interpreted as a fixed observer for the prediction of future outputs. For simplicity, the
C
(z
^{1}
) polynomial is usually chosen to be 1.
 3.2 Model identification algorithm
In this paper, a recursive leastsquares algorithm (RLSA) with a varying forgetting factor is used to track the power system model parameters
a
_{i}
and
b
_{i}
shown in Eq. (1)
[6]
. As the RLSA is a well known method, only a general formulation is presented in the following, without demonstration.
Given the vector
of parameter estimates by:
and the measurement vector
φ
(t) by:
The update of the estimates is obtained by:
where,
P
(t) is the covariance matrix,
I
is an identity matrix,
K
(t) is the vector of adjustment gains, λ is the forgetting factor used to progressively reduce the effects of old measurements, and Σ
_{0}
are the preselected constants. The initial value
P
(0) =
α
^{2}
I
,
α
^{2}
= 10
^{5}
~ 10
^{10}
.
In addition, moving boundaries are introduced for every parameter to protect the parameters from large modeling errors which are caused by a variety of sudden disturbances in the power system
[11]
. The mean values of the estimated parameters at the sampling instant
t
are of the form:
where
θ
_{i}
(t) is the element in
, and
T
>1, with a value chosen to ensure stability of the parameters. The larger the value of
T
, the more stable and less adaptable the parameter boundaries become.
The high and low boundaries for each parameter are defined as follows:
where 0< γ <1, the larger the value γ, the more likely it is for the parameters to vary. At each sampling instant, each estimated parameter is bounded by its corresponding high and low boundaries.
 3.3 Generalized predictive control
The GPC is a long range predictive control approach that adopts an explicit parameterized system model to predict the process future output, which depends on both currently available input/output data and present/future control values. The latter are determined by minimizing a cost function, which is chosen in such a way that satisfying the controlled system dynamics and constraints, penalize system output deviation from the desired trajectory and minimize control efforts
[17]
.
As the design of WADC is a positional control problem, the cost function to be minimized is defined as follows:
where
E
{.} is the expectation operator,
ŷ
(
t + j
) is an optimal
j
step ahead prediction of the system output up to time
t. N
is the prediction horizon,
u
(
t
+
j
1) is an optimal
j
step ahead prediction of the system input,
N
_{u}
is the control horizon,
r
_{j}
is a control weighting sequence and usually defined as a constant value,
r
_{j}
=
r
, for
j
=1, 2, ...
N
_{u}
.
In order to obtain
ŷ
(
t + j
) , the following Diophantine equations are used:
where
E
_{j}
(z
^{1}
) and
F
_{j}
(z
^{1}
) are unique polynomials of order
j
1 and
n
_{a}
, respectively.
G
_{j}
(z
^{1}
) is an unique polynomials of order
j
1. Hence,
where
f _{j}
=
F _{j}
(
z
^{−1}
)
y
(
t
) +
H _{j}
(
z
^{−1}
)
u
(
t
− 1)
The
N
steps
j
ahead predictions can be represented by the following matrix equation:
where
The elements
g
_{i}
of the matrix
G
, with dimensions
N × N_{u}
, are points of the plant's step response and can be computed recursively from the model. The elements
f
_{i}
of the matrix
f
can be computed similarly.
One of the major advantages of GPC is its ability to handle constraints online in a systematic way. The algorithm does this by optimizing predicted performance subject to constraint satisfaction. In practice, the constraints of the control signal should be considered and can be expressed as follows:
where
U
_{min}
and
U
_{max}
denote the lower limit and upper limit of the control signal.
I
denotes the
N
_{u}
identity vector.
According to the Eqs. (10), (13) and (14), the implementation of GPC with bounded signals can be represented as a inequality constrained quadratic programming (QP) problem, which can be stated as:
where
P
=2(
G
^{T}
G+R
),
R
=
diag
[
r
_{1}
r
_{2}
⋯
r_{Nu}
] ,
b
= 2
f
^{T}
G
,
f
_{0}
=
f
^{T}
f
,
A
_{qp}
=[
T
, 
T
]
^{T}
, and
b
_{c}
=[
IU
_{max}
, 
IU
_{min}
]
^{T}
.
T
is the identity matrix.
The future control input sequence
U
can be obtained by solving the QP problem shown as Eq. (15). Because GPC is a recedinghorizon control method, only the first element of
U
is actually applied. Therefore, it is only necessary to calculate the first row of
U
at each sampling interval.
 3.4 Procedure of WADC based on adaptive GPC
In order to implement the control algorithms of the proposed adaptive WADC based on GPC, the essential procedure may be followed:

(1) Initialize the system outY0, system inputU0, covariance matrixP(0), and choose the prediction horizonN, the control horizonNu, weighting sequencer, the order of the prediction modelnaandnb.

(2) At the sampling interval t, obtain the system out y(t) and system input u(t1), update the measurement φ(t).

(3) Update the parameters of the prediction model according to Eqs. (79).

(4) Predict the outputŶusing Eq. (13) and the new model parameters updated in Step (3), over the prediction horizon N.

(5) Solve the QP problem shown as Eq. (15) with respect to control input sequence U over the control horizon Nu, satisfying the constraints.

(6) Apply the first element of the future control input sequence U obtained from the optimization procedure as the control input u(t) until new measurements are available.

(7) Let t=t+1, fetch the next sampling data, then go back to Step (2) again.
4. Case Study I: TwoArea FourMachine System
At first, case study is carried out based on the twoarea fourmachine system, as shown in
Fig. 2.
, to illustrate the principle and effectiveness of the proposed adaptive WADC using the MATLAB/Simulink software. The subtransient generator model and the IEEEtype DC1 excitation system are used for each generator. All of the loads are used the impedance model. In addition, in order to damp the local low frequency oscillations, G1 and G3 are equipped with a PSS respectively with local rotor speed as input and its transfer function is shown in Appendix. The output of PSS is limited as [0.15~0.15]pu. Details of the system data are given in Ref.
[1]
. The proposed adaptive WADC is compared with the conventional WADC proposed in
[3]
, whose parameters can be found in Appendix.
The twoarea fourmachine power system
 4.1 Design of adaptive WADC
For the test system, the generator G1 is selected to locate the proposed adaptive WADC in order to damp the only interarea oscillation mode. The detailed structure of the proposed adaptive WADC is illustrated in
Fig. 4
. The linear model of the power system shown as Eq. (1) is used as the prediction model for GPC. The output of the WADC
u_{g}
is used as the input
u
(
t
) of the linear model, while the widearea signal
ω
_{13}
(per unit) is used as the output signal
y
(
t
). Moreover, in order to form a wellconditioned optimization problem for GPC, keeping identified system model parameters at the same order of magnitude is important. Proper scaling of the output signals will help to improve the identification accuracy. Due to the widearea signal
ω
_{13}
used in this paper, 100 is the suitable scaling number according to the value used in the simulation. It should be noted that, for a relatively large scale power system, the geometric measures of observability and controllability are used to select the most effective widearea input signals and WADC locations
[20]
.
Configuration of generator equipped with an adaptive WADC
For the estimation, the order of the linear model is chosen as:
n
_{a}
=4,
n
_{b}
=5. The following parameters need to be specified for the proposed adaptive WADC: the prediction horizon
N
, the control horizon
N
_{u}
, the weighting sequence
r
, and the sampling period
T
_{s}
. Although their values are normally guided by heuristics, there are some general guideline for choosing these parameters to ensure the optimization is well proposed in
[8]
. For the proposed adaptive WADC, desired response can be achieved by setting
N
=7,
N
_{u}
=2,
r
=0.6,
T
_{s}
=40ms, Σ
_{0}
=0.02,
T
=10, γ=0.04. To avoid the excessive interference of the adaptive WADC on the local control, ±0.05pu is limited to the output of the proposed adaptive WADC.
 4.2 Simulation and analysis
When a threephasetoground fault occurs on bus 8 at
t
=1s and lasts for 0.1s, the system responses are shown in
Fig. 5
. The results show that the proposed adaptive WADC is able to achieve slightly better performances than those of conventional WADC. Note that the conventional WADC is tuned and tested under similar operation point. In addition, the identified parameters of the prediction model (1) are shown in
Fig. 6
. The identified parameters are updated fast enough to track the change of operation condition and external disturbance. After the disturbance, the parameters move to the new steady values.
Responses to threephasetoground fault under nominal operating condition
The variation of the identified parameters of the prediction model
Moreover, the response of the adaptive WADC under a new operation point (i.e. the tieline power changes to
P
_{tie}
=50MW), is shown in
Fig. 7
. The results show that the adaptive WADC has better damping ability than the conventional WADC. That is because the conventional WADC is tuned based on a nominal operating condition and its performance will be degrades when the operating condition changes.
Responses to threephasetoground (P_{tie}=50MW)
Finally, the results about the widearea signals with the time delay are shown in
Fig. 8
and
Fig. 9
. The results show that the proposed adaptive WADC has much better performances than those of conventional WADC. The proposed adaptive WADC can keep the stability of the power system when the time delay existing in the widearea signal reaches 120ms. However, the conventional WADC can not keep the stability of the power system when the time delay of widearea signal reaches 80ms. This is because the proposed adaptive WADC can absorb the change of the time delay into the prediction model and is more effective to retain the stability of power system considering widearea signal delays.
Responses to threephasetoground with 80ms time delay
Responses to threephasetoground fault with 120ms time delay
5. Case Study II: New England 10Machine 39Bus System
To investigate the feasibility of the proposed WADC for a largescale power system, a case study is undertaken based on the New England 10machine 39bus system, as shown in
Fig. 10
. This test system consists of 10 generators, 39 buses, and 46 transmission lines. Each generator is modeled as a fourthorder model and equipped with a IEEE ST1A excitation system. The mechanical power of each generator is assumed as constants for simplicity. The detailed parameters and operating conditions are given in
[21]
.
The New England 10machine 39bus system
 5.1 Design of adaptive WADC
The modal analysis results of this test system has a critical poor damping interarea mode, which has lowest damping ratio 0.0442 and oscillation frequency 0.6273Hz. Therefore, the WADC should be designed for providing damping for this critical interarea mode. To determine WADC location and widearea feedback signals, geometric measures of modal controllability/observability is employed to evaluate the relative strength of candidate signals and the performance of controllers at different locations with respect to a given interarea mode in
[5
,
20]
. WADC should be located on the generator which has larger geometric controllability with respect to the mode concerned, while smaller geometric controllability with respect to other modes in order to reduce the effect on other modes. Therefore, G4 is selected as controller location and deviation of
P
_{318}
is chosen as the widearea feedback signal for the proposed WADC. In this case, 0.1 is the suitable scaling number according to the value used in the simulation. Moreover, a washout filter is added to eliminate constant deviation of
P
_{318}
when the operating condition of system is changed.
The order of the linear model is chosen as:
n
_{a}
=7,
n
_{b}
=7. The following parameters need to be specified for the proposed adaptive WADC: the prediction horizon
N
, the control horizon
N
_{u}
, the weighting sequence
r
, and the sampling period
T
_{s}
. As similar with the twoarea test system, the proposed adaptive WADC of the New England test system, desired response can be achieved by setting
N
=25,
N
_{u}
=12,
r
=0.3,
T
_{s}
=100ms, Σ
_{0}
=0. 2,
T
=10, γ=0.04. For comparison purpose, the performances of the conventional WADC proposed in
[3]
, whose parameters can be found in Appendix, are also provided.
 5.2 Simulation and analysis
Simulation studies are carried out based on detailed nonlinear model to verify the effectiveness of the proposed adaptive WADC under a wide range of operating conditions. However, only a few typical case results are given in this paper due to the space restriction.
A threephasetoground fault occurs at the end terminal of line 34 near bus 3 at
t
= 0.5s, followed by outage of faulty transmission line 34 at t = 0.6s, the responses of the active power
P
_{1617}
and
P
_{318}
are shown in
Fig. 11
. It reveals that the proposed adaptive WADC provides slightly better damping performances than the conventional WADC under this situation.
System responses to threefaulttoground fault under Scenario I (Outage of fault line 34)
For a larger disturbances, a threephasetoground fault occurs at the end terminal of line 1516 near bus 15 at
t
= 0.5s, followed by outage of faulty tieline 1516 at t = 0.6s, the responses of the active power
P
_{1617}
and
P
_{318}
are shown in
Fig. 12
. It is observed that the proposed adaptive WADC effectively damps the power oscillations, while the conventional WADC cannot maintain the whole system stability due to the large change of operation condition caused by the outage of tieline 1516. Moreover, the identified parameters of the prediction model (1) are shown in
Fig. 13
. It shows that the identified parameters update fast enough to track the dynamic of the power system and then converge to a new operating condition.
System responses to threefaulttoground fault under Scenario II (Outage of fault tieline 1516)
The Variations of the identified parameters
To test the robustness of the proposed WADC to time delays, system responses to fault scenario I (Outage of fault line 34) with 100ms are shown in
Fig. 14
. It can be observed that the conventional WADC can not maintain the stability of the whole system when the time delay reaches 100ms, while the proposed adaptive WADC can provide satisfactory damping performance. However, in comparison with the situation without time delay, the damping performances provided by adaptive WADC under 100ms time delay decreases slightly.
System responses to threefaulttoground fault with 100ms time delays under Scenario I (Outage of fault line 34)
6. Conclusion
An adaptive widearea damping controller based on generalized predictive control and model identification is proposed in this paper. A parameter identification algorithm based on recursive leastsquares algorithm with a varying forgetting factor is applied to identify system prediction model online. The validity and effectiveness of the proposed adaptive WADC is evaluated by simulation studies on a twoarea fourmachine power system and the New England 10machine 39bus power system, respectively. Simulation results are compared with those of the conventional WADC and without WADC. The comparison results show that the performances of the adaptive WADC to damp the interarea oscillation is better than those of the conventional WADC under a wide range of operating conditions and different disturbances. Moreover, when the time delay existing in widearea signal is considered, the proposed adaptive WADC is more effective to retain the stability of power system than the conventional WADC.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (51177057, 51207063), and the National High Technology Research and Development of China (863 Program) (2011AA05A119).
BIO
Wei Yao He received the B.Sc. degree and the Ph.D. degree in electrical engineering from Huazhong University of Science and Technology (HUST), Wuhan, China, in 2004 and 2010, respectively. Subsequently, he worked as a Postdoctoral Researcher in Department of Electrical Engineering, HUST, from 2010 to 2012. He was a Visiting Student in the Department of Electrical Engineering & Electronics, The University of Liverpool, UK, from June 2008 to May 2009. Currently, he is a Lecturer in the College of Electrical and Electronics Engineering, HUST. His research interests include power system stability analysis and control.
L. Jiang He received the B.Sc. and M.Sc. degrees in electrical engineering from Huazhong University of Science and Technology (HUST), China, in 1992 and 1996, respectively; and the Ph.D. degree from The University of Liverpool, UK, in 2001. Currently, he is a Senior Lecturer in the Department of Electrical Engineering & Electronics, The University of Liverpool, UK. His current research interests are control and analysis of power system, smart grid and renewable energy.
Jiakun Fang He received the Ph.D. degree in electrical engineering from Huazhong University of Science and Technology (HUST), Wuhan, China in 2012. Currently, he is working as a Postdoctoral Researcher in Department of Energy Technology, Aalborg University, Denmark. His research interests include power system dynamic stability control and power grid complexity analysis.
Jinyu Wen He received his B.Eng. and Ph.D. degrees in electrical engineering from Huazhong University of Science and Technology (HUST), Wuhan, China, in 1992 and 1998, respectively. He was a visiting student from 1996 to 1997, and Research Fellow from 2002 to 2003 at the University of Liverpool, UK. In 2003, he joined the HUST, where he presently serves as professor. His current research interests include smart grid, renewable energy, energy storage, FACTS, HVDC, and power system operation and control.
Shaorong Wang He graduated from the Zhejiang University, Hangzhou, China in 1984 and received his Master of Engineering Degree from the North China Electric Power University, Baoding, China in 1990. He earned his Ph.D. from the Huazhong University of Science and Technology (HUST), Wuhan, China in 2004. All three degrees are in the field of Electrical Engineering. Currently, he is a full professor at the HUST. His research interests are power system control and stability analysis.
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