Damping of Inter-Area Low Frequency Oscillation Using an Adaptive Wide-Area Damping Controller

Abstract

This paper presents an adaptive wide-area damping controller (WADC) based on generalized predictive control (GPC) and model identification for damping the inter-area low frequency oscillations in large-scale inter-connected power system. A recursive least-squares algorithm (RLSA) with a varying forgetting factor is applied to identify online the reduced-order linearlized model which contains dominant inter-area 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 two-area four-machine power system and the New England 10-machine 39-bus power system, respectively. Simulation results show that the proposed adaptive WADC not only can damp the inter-area 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 lead-lag WADC are also provided.

1. Introduction

With the interconnection of regional grids and the increasing of line power, the inter-area 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 inter-area mode oscillations. However, the PSSs, which use local measurement as the input, can not always damp inter-area mode oscillations effectively because inter-area modes are not always observable from the local signals [2, 3].

With the technological advancements and increasing deployments of wide-area measurement system (WAMS) in power system, remote signals from WAMS become available for the design of wide-area 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 large-scale 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 large-scale power system. In addition, for a wide-area 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 inter-area 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 dead-time, but also cope with over-parameterization 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 wide-area signals feedback. Simulation studies are carried out based on the two-area four-machine power system and the New England 10-machine 39-bus 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 inter-area mode oscillations under various operation conditions and different disturbances. Moreover, it has much better performance than conventional WADC with time delay existing in the wide-area signals.

The rest of the paper is organized as follows. Section 2 presents a general architecture of wide-area 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 two-area four-machine 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 10-machine 39-bus power system in Section 5. Finally, some conclusions are drawn in Section 6.

 

2. Architecture of Wide-Area Damping Control

The wide-area stabilizing control structure shown in Fig. 1, so-called “decentralized/hierarchical” architecture, is used in this paper. In the local decentralized control level, including automatic voltage regulator and PSS (AVR-PSS) 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 inter-area modes under stressed operating conditions because of the local signals lack of good observability of some critical inter-area natural modes. Therefore, additional damping is required particularly for the these inter-area modes. As shown in Fig. 1, in the wide-area control level, the control signal of the WADCs are to provide damping for the inter-area 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 inter-area modes and under specific network configurations, no more than the few WADCs sites with the highest controllability of these inter-area modes need be involved in the wide-area control level.

Fig. 1.General architecture of the wide-area 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 wide-area 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.

Fig. 2.Overview of an adaptive generalized predictive controller

3.1 General system model

Although a real power system is a complex, nonlinear and high-order 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 auto-regressive and moving average (CARMA) linear model is utilized to avoid the off-set 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 white-noise sequence. A(z-1), B(z-1) and C(z-1) are na, nb and nc 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 least-squares algorithm (RLSA) with a varying forgetting factor is used to track the power system model parameters ai and bi 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 Σ0 are the preselected constants. The initial value P(0) = α2I , α2 = 105 ~ 1010.

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 θi(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, Nu is the control horizon, rj is a control weighting sequence and usually defined as a constant value, rj=r, for j=1, 2, ... Nu.

In order to obtain ŷ(t + j) , the following Diophantine equations are used:

where Ej(z-1) and Fj(z-1) are unique polynomials of order j-1 and na, respectively. Gj(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 gi of the matrix G, with dimensions N × Nu , are points of the plant's step response and can be computed recursively from the model. The elements fi 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 Umin and Umax denote the lower limit and upper limit of the control signal. I denotes the Nu 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(GTG+R), R=diag[r1 r2 ⋯ rNu] , b= -2fTG, f0=fTf, Aqp=[T, -T ]T, and bc=[IUmax, - IUmin ]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 receding-horizon 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 out Y0, system input U0, covariance matrix P(0), and choose the prediction horizon N, the control horizon Nu, weighting sequence r, the order of the prediction model na and nb.(2) At the sampling interval t, obtain the system out y(t) and system input u(t-1), update the measurement φ(t).(3) Update the parameters of the prediction model according to Eqs. (7-9).(4) Predict the output Ŷ using 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: Two-Area Four-Machine System

At first, case study is carried out based on the two-area four-machine 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 IEEE-type 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.

Fig. 3.The two-area four-machine 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 inter-area 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 ug is used as the input u(t) of the linear model, while the wide-area signal ω13 (per unit) is used as the output signal y(t). Moreover, in order to form a well-conditioned 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 wide-area signal ω13 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 wide-area input signals and WADC locations [20].

Fig. 4.Configuration of generator equipped with an adaptive WADC

For the estimation, the order of the linear model is chosen as: na=4, nb=5. The following parameters need to be specified for the proposed adaptive WADC: the prediction horizon N, the control horizon Nu, the weighting sequence r, and the sampling period Ts. 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, Nu=2, r=0.6, Ts=40ms, Σ0=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 three-phase-to-ground 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.

Fig. 5.Responses to three-phase-to-ground fault under nominal operating condition

Fig. 6.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 tie-line power changes to Ptie=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.

Fig. 7.Responses to three-phase-to-ground (Ptie=50MW)

Finally, the results about the wide-area 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 wide-area signal reaches 120ms. However, the conventional WADC can not keep the stability of the power system when the time delay of wide-area 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 wide-area signal delays.

Fig. 8.Responses to three-phase-to-ground with 80ms time delay

Fig. 9.Responses to three-phase-to-ground fault with 120ms time delay

 

5. Case Study II: New England 10-Machine 39-Bus System

To investigate the feasibility of the proposed WADC for a large-scale power system, a case study is undertaken based on the New England 10-machine 39-bus 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 fourth-order 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].

Fig. 10.The New England 10-machine 39-bus system

5.1 Design of adaptive WADC

The modal analysis results of this test system has a critical poor damping inter-area 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 inter-area mode. To determine WADC location and wide-area 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 inter-area 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 P3-18 is chosen as the wide-area 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 P3-18 when the operating condition of system is changed.

The order of the linear model is chosen as: na=7, nb=7. The following parameters need to be specified for the proposed adaptive WADC: the prediction horizon N, the control horizon Nu, the weighting sequence r, and the sampling period Ts. As similar with the two-area test system, the proposed adaptive WADC of the New England test system, desired response can be achieved by setting N=25, Nu=12, r=0.3, Ts=100ms, Σ0=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 three-phase-to-ground fault occurs at the end terminal of line 3-4 near bus 3 at t = 0.5s, followed by outage of faulty transmission line 3-4 at t = 0.6s, the responses of the active power P16-17 and P3-18 are shown in Fig. 11. It reveals that the proposed adaptive WADC provides slightly better damping performances than the conventional WADC under this situation.

Fig. 11.System responses to three-fault-to-ground fault under Scenario I (Outage of fault line 3-4)

For a larger disturbances, a three-phase-to-ground fault occurs at the end terminal of line 15-16 near bus 15 at t = 0.5s, followed by outage of faulty tie-line 15-16 at t = 0.6s, the responses of the active power P16-17 and P3-18 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 tie-line 15-16. 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.

Fig. 12.System responses to three-fault-to-ground fault under Scenario II (Outage of fault tie-line 15-16)

Fig. 13.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 3-4) 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.

Fig. 14.System responses to three-fault-to-ground fault with 100ms time delays under Scenario I (Outage of fault line 3-4)

 

6. Conclusion

An adaptive wide-area damping controller based on generalized predictive control and model identification is proposed in this paper. A parameter identification algorithm based on recursive least-squares 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 two-area four-machine power system and the New England 10-machine 39-bus 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 inter-area 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 wide-area signal is considered, the proposed adaptive WADC is more effective to retain the stability of power system than the conventional WADC.