Wide-Area Robust H2/H∞ Control with Pole Placement for Damping Inter-Area Oscillation of Power System

The damping of inter-area oscillations is an important problem in electric power systems (Klein et al., 1991; Kundur, 1994; Rogers, 2000). Especially in China, the practices of nationwide interconnection and ultra high voltage (UHV) transmission are carrying on and under broad researches (Zhou et al., 2010), bulk power will be transferred through very long distance in near future from the viewpoints of economical transmission and requirement of allocation of insufficient resources. The potential threat of inter-area oscillations will increase with these developments. If inter-area oscillations happened, restrictions would have to be placed on the transferred power. So procedures and equipments of providing adequate damping to inter-area oscillations become mandatory. Conventional method coping with oscillations is by using power system stabilizer (PSS) that provides supplementary control through the excitation system (Kundur, 1994; Rogers, 2000; Larsen et al., 1981), or utilizing supplementary control of flexible AC transmission systems (FACTS) devices (Farsangi et al., 2003; Pal et al., 2001; Chaudhuri et al., 2003, 2004). Decentralized construction is often adopted by these controllers. But for inter-area oscillations, conventional decentralized control may not work so well since they have not observability of system level. Maximum observability for particular modes can be obtained from the remote signals or from the combination of remote and local signals (Chaudhuri et al., 2004; Snyder, et al., 1998; Kamwa et al., 2001). Phasor measurement units (PMUs)-based wide-area measurement system (WAMS) (Phadke, 1993) can provide system level observability and controllability and make so-called wide-area damping control practical. On the other hand, power system exists in a dynamic balance, its operating condition always changes with the variations of generations or load patterns, as well as changes of system topology, etc. From control theory point of view, these changes can be called uncertainty. Conventional control methods can not systemically consider these uncertainties, and often need tuning or coordination. Therefore, so-called robust models are derived to take these uncertainties into account at the controller design stage (Doyle et al., 1989; Zhou et al., 1998). Then the robust control is applied on these models to realize both disturbance attenuation and stability enhancement.


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In robust control theory, H 2 performance and H ∞ performance are two important specifications. H ∞ performance is convenient to enforce robustness to model uncertainty, H 2 performance is useful to handle stochastic aspects such as measurement noise and capture the control cost. In time-domain aspects, satisfactory time response and closed-loop damping can often be achieved by enforcing the closed-loop poles into a pre-determined subregion of the left-half plane (Chilali et al., 1996). Combining there requirements to form so-called mixed H 2 /H ∞ design with pole placement constrains allows for more flexible and accurate specification of closed-loop behavior. In recent years, linear matrix inequalities (LMIs) technique is often considered for this kind of multi-objective synthesis (Chilali et al., 1996;Boyd et al., 1994;Scherer et al., 1997Scherer et al., , 2005. LMIs reflect constraints rather than optimality, compared with Riccati equations-based method (Doyle et al., 1989 ;Zhou et al., 1998), LMIs provide more flexibility for combining various design objectives in a numerically tractable manner, and can even cope with those problems to which analytical solution is out of question. Besides, LMIs can be solved by sophisticated interior-point algorithms (Nesterov et al., 1994). In this chapter, the wide-area measurement technique and robust control theory are combined together to design a wide-area robust damping controller (WRC for short) to cope with interarea oscillation of power system. Both local and PMU-provided remote signals, which are selected by analysis results based on participation phasor and residue, are utilized as feedback inputs of the controller. Mixed H 2 /H ∞ output-feedback control design with pole placement is carried out. The feedback gain matrix is obtained through solving a family of LMIs. The design objective is to improve system damping of inter-area oscillations despite of the model changes which are caused mainly by load changes. Computer simulations on a 4-generator benchmark system model are carried out to illustrate the effectiveness and robustness of the designed controller, and the results are compared with the conventional PSS. The rest of this chapter is organized as follows: In Section 2 a mixed H 2 /H ∞ output-feedback control with pole placement design based on the mixed-sensitivity formulation is presented. The transformation into numerically tractable LMIs is provided in Section 3. Section 4 gives the benchmark power system model and carries out modal analyses. The synthesis procedures of wide-area robust damping controller as well as the computer simulations are presented in Section 5. The concluding remarks are provided in Section 6.

H ∞ mixed-sensitivity control
Oscillations in power systems are caused by variation of loads, action of voltage regulator due to fault, etc. For a damping controller these changes can be considered as disturbances on output y (Chaudhuri et al., 2003(Chaudhuri et al., , 2004, the primary function of the controller is to minimize the impact of these disturbances on power system. The output disturbance rejection problem can be depicted in the standard mixed-sensitivity (S/KS) framework, as shown in Fig. 1, where sensitivity function S(s)=(I-G(s)K(s)) -1 . An implied transformation existing in this framework is from the perturbation of model uncertainties (e.g. system load changes) to the exogenous disturbance. Consider additive model uncertainty as shown in Fig. 2, The transfer function from perturbation d to controller output u, T ud , equals K(s)S(s). By virtue of small gain theory, ǁT ud ∆(s)ǁ ∞ <1 if and only if ǁW 2 (s)T ud ǁ ∞ <1 with a frequency-depended weighting function |W 2 (s)|>|∆(s)|. So a system with additive model uncertain perturbation (Fig. 2) (Fig. 1) if the weighted H ∞ norm of transfer function form d to u is small than 1, and the weighting function W 2 (s) is the profile of model uncertainty.
In (1), the upper inequality is the constraint on nominal performance, ensuring disturbance rejection, the lower inequality is to handle the robustness issues as well as limit the control effort. Knowing that the transfer function from d to y, T yd , equals S(s). So condition (1) is equivalent to The system performance and robustness of controlled system is determined by the proper selection of weighting function W 1 (s) and W 2 (s) in (1) or (2). In the standard H ∞ control design, the weighting function W 1 (s) should be a low-pass filter for output disturbance rejection and W 2 (s) should be a high-pass filter in order to reduce the control effort and to ensure robustness against model uncertainties. But in some cases, there would be a low-pass requirement on W 2 (s) when the open-loop gain is very high by applying standard lowerpass design, which will result in a conflict in the nature of W 2 (s) to ensure robustness and minimize control effort (Pal et al., 2001). So the determination of W 2 (s) should be careful.

H 2 performance for control cost requirement
It is known that the control cost can be more realistically captured through H 2 norm, see (Pal et al., 2001) and its reference, this enlightens directly adding H 2 performance on controller output u at the design stage, i.e. consider constraint to constrain the control effort and mitigate the burden of selection of W 2 (s). The weighting function W 3 (s) is used to compromise between the control effort and the disturbance rejection performance, as shown in Fig

Pole placement constraint
H 2 /H ∞ design deals mostly with frequency-domain aspects and provides little control over the transient behavior and closed loop pole location. Satisfactory time response and closedloop damping can often be achieved by forcing the closed-loop poles into a suitable subregion of the left-half plane, and fast controller dynamics can also be prevented by prohibiting large closed-loop poles. Therefore, besides H ∞ and H 2 norm constraint, pole placement constraint that confine the poles to a LMI region is also considered. A LMI region S(α, r, θ) is a set of complex number x+jy such that x<-α<0, |x+jy|< r, and tan(θ)x<-|y|, as shown in Fig. 4. Confining the closed-loop poles to this region can ensure a minimum decay rate α, and minimum damping ratio ζ=cos(θ), and a maximum undamped natural frequency ω d = rsin(θ). The standard mathematical description of LMI region can be found in (Chilali et al., 1996). The multiple-objective design including H ∞ /H 2 norm and pole placement constrains can be formulated in the LMIs framework and the controller is obtained by solving a family of LMIs.

Multiple-objective synthesis using LMI method
General mixed H 2 /H ∞ control with pole placement scheme has multi-channel form as shown in Fig. 5. G(s) is a linear time invariant generalized plant, d is vector representing the disturbances or other exogenous input signals, z ∞ is the controlled output associated with H ∞ performance and z 2 is the controlled output associated with H 2 performance, u is the control input while y is the measured output. The state-space description of above system can be written as The goal is to compute a output-feedback controller K(s) in the form of By virtue of bounded real lemma (Boyd et al., 1994) and Schur's formula for the determinant of a partitioned matrix, matrix inequality condition (3) is equivalent to the existence of a symmetric matrix X ∞ >0 such that The closed-loop poles lie in the LMI region (see Fig. 4) S(0, 0, θ) if and only if there exists a symmetric matrix X D such that (Chilali et al., 1996): For H 2 performance, ǁW 3 (s)T ud (s)ǁ 2 does not exceed 2 if and only if D c2 =0 and there exist two symmetric matrices X 2 >0 and Q>0 such that This condition can be deduced from the definition of H 2 norm (Chilali et al., 1996 ;Scherer et al., 1997). The multiple-objective synthesis of controller is through solving matrix inequality (8) to (10). But this problem is not jointly convex in the variable and nonlinear, for example nonlinear entry A c X ∞ i n ( 8 ) , s o t h e y a r e n o t n u m e r i cally tractable. Choosing a single Lyapunov matrix X=X ∞ =X 2 =X D and linearizing change of variables can cope with this problem. Choosing a single Lyapunov matrix makes the resulting controller not globally optimal, but is not overly conservative from the practical point of view. The linearizing change of variables is important for multiple-objective output feedback robust synthesis based LMIs. The details can be found in (Chilali et al., 1996 ;Scherer et al., 1997) This standard LMI problem (Boyd et al., 1994) is readily solved with LMI optimization software. An efficient algorithm for this problem is available in hinfmix() function of the LMI control toolbox for Matlab (Gahinet et al., 1995).

Low frequency oscillation in power system
One of the major problems in power system operation is low frequency (between 0.1 and 2 Hz) oscillatory instability. Normally no apparent warning can be identified for the occurrence of such kinds of growing oscillations caused by the changes in the system's operating condition or by improper-tuned sustained excitation. The change in electrical torque of a synchronous machine following a perturbation can be resolved as ΔT e =T S Δδ+ T D Δω, where T S Δδ is the component of torque change in phase with the rotor angle perturbation Δ and is referred as the synchronizing torque component, T S is the synchronizing torque coefficient. Lack of sufficient synchronizing torque will result in aperiodic drift in rotor angle. T D Δω is the component of torque in phase with the speed deviation Δω and is referred to as the damping torque component, T D is the damping torque coefficient. Lack of sufficient damping torque will result in oscillatory instability. In next section, an example will be used to illustrate the low frequency oscillation of a weaktied system and the design of a wide-robust damping controller (WRC) to effectively increase the damping ratio of inter-area mode.

System model and modal analysis
A 4-generator benchmark system shown in Fig. 6 is considered. The system parameters is from (Klein et al., 1991) or (Kundur, 1994). However some modifications have been made to facilitate the simulations. The generator G2 is chosen as angular reference to eliminate the undesired zero eigenvalues. Saturation and speed governor are not modeled. Excitation system is chosen by thyristor exciter with a high transient gain. All loads are represented by constant impedance model and complete system parameters are listed in Appendix. After linearization around given operating condition and elimination of algebraic variables, the following state-space representation is obtained.
where x is state vector; u is input vector, y is output vector; A is the state matrix depending on the system operating conditions, B u and C y are input and output matrices, respectively. The number of the original state variables is 28, since generator 2 has been chosen as angular reference, 2 sates are eliminated, so the number of state variables is 26. Following the small-signal theory (Kundur, 1994), the eigenvalues of the test system and corresponding frequencies, damping ratios and electromechanical correlation ratios are calculated. The results are classified in Table 1. It can be found that mode 3 is undamped, which means that the disturbed system can not hold transient stability.
The electromechanical correlation ratio in Table 1 is determined by a ratio between summations of eigenvectors relating to rotor angle and rotor speed and summations of other eigenvectors. If the absolute value of one entry (correlation ratio) is much higher than 1, the corresponding mode is considered as electromechanical oscillation. A conception named participation phasor is used to facilitate the positioning of controller and the selection of remote feedback signal. Participation phasor is defined in this easy way: its amplitude is participation factor (Klein et al., 1991;Kundur, 1994) and its phase angle is angle of eigenvector. The analysis results are shown in Fig. 7, in which all vectors are originated from origin (0, 0) and vector arrows are omitted for simplicity. It can be seen that  Mode 1 is a local mode between G1 and G2. The Participation phasor of G3 and G4 are too small to be identified;  Mode 2 is a local mode between G3 and G4. The Participation phasor of G1 and G2 are too small to be identified;  Mode 3 is an inter-area mode between G1, G2 and G3, G4. Wide-area controller is located in G3, which has highest participation factor than others. Even if using local signal only, the controller locating in G3 will have more effects than locating in other generators. Often the residue indicates the sensitivity of eigenvalues to feedback transfer function (Rogers, 2000), that is to say if residue is 0 then feedback control have no effects on controlled system, so residue is used to select suitable remote feedback signal provided by PMU. The residue corresponding to the transfer function between rotor speed output of G1 and excitation system input of G3 is 1.58 (normalized value), while the residue corresponding to the transfer function between rotor speed output of G2 and excitation system input of G3 is 1 (normalized value). So the remote signal is chosen from G1. (1) Reduce the original system model through Schur balanced truncation technique (Zhou et al., 1998), a reduced 9-order system model can be obtained. The frequency responses of original and reduced model are compared in Fig. 9, it shows that reduced system has proper approximation to original system within considered frequency range.

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The weighting functions are in accordance with the basic requirements of mixed-sensitivity design. W 1 (s) is a low-pass filter for output disturbance rejection, W 2 (s) is a high-pass filter for covering the additive model uncertainty, and W 3 (s) is a weight on H 2 performance.
(3) Controller design by using the Robust Control Toolbox in Matlab. The solution is numerically sought using suitably defined objectives in the arguments of the hinfmix() function of the Robust Control Toolbox. The LMI region is chosen as a conic sector with inner angle equals 2*acos(0.17) (corresponding damping ratio 17%) and apex at the origin.
(4) Controller reduction through Schur balanced truncation technique. A 4-order 2-input 1output controller is obtained. The state-space representation of the designed controller is A washout filter 10s/(10s+1) is added in each feedback channel as shown in Fig. 8. This is a standard practice to prevent the damping controllers from responding to very slow www.intechopen.com Wide-Area Robust H 2 /H∞ Control with Pole Placement for Damping Inter-Area Oscillation of Power System 341 variations in the system conditions (Kundur, 1994). A limit of [-0.15, 0.15] (pu) is imposed on the output of the designed controller.

Computer simulations and robustness validation
Computer simulations are carried out to test the effectiveness and performance of the designed controller and validate the robustness in different operating conditions. The simulation is carried out by Matlab-Simulink. A 5%-magnitude pulse, applied for 12 cycles at the voltage reference of G1, is used to simulate the modes of oscillation. For comparison, one conventional PSS is also considered. The PSS has one gain, one washout and two phase compensations, the block diagram representation of the conventional PSS is shown in Fig. 10. The parameters are adopted directly from (Kundur, 1994   The pulse response with the designed WRC is shown in Fig. 12, which is compared with the response with one conventional PSS located in G3. The state variable is the tie line active power. Both of the damping controllers can ensure the system asymptotic stable but better damping performance is achieved by the WRC.  To test the robustness of the designed controller to changes of operating conditions (or model uncertainties), load changes are considered. Eight different operation conditions are considered, corresponding load L 1 and L 2 in normal conditions and change between ±5%and ±10%, respectively. The load change, making the tie line power change, is the primary factor affecting the eigenvalues of the matrix A (also the damping ratios) in system model (12), and also used to select the weighting function W 2 (s). Fig. 15 shows the frequencies and damping ratios corresponding to these changes. The horizontal axis is the load changes Fig. 15. Damping ratios and frequency corresponding to load change for mode 1 to mode 3 (including the nominal operating condition) for mode 1 to mode 3. The upper vertical axis is the damping ratios corresponding to each load change, the lower vertical axis is the frequencies corresponding to each damping ratio. For inter-area mode, mode 3, the damping ratios are higher than 0.15 in all these load levels. The damping rations of the whole system is higher than 0.08 in all cases. The controlled system has proper damping performance and keeps robustness against the variations of system loads.

Conclusion
This chapter applies robust control theory to power system, to design wide-area robust damping controller to cope with inter-area oscillation. Both local signal and suitable chosen PMU-provided remote signal are utilized to construct the feedback loop. A conception named participation vector is used for facilitating the positioning of controller, and the residue is utilized to select suitable remote signal. The controller is designed based on mixed H 2 /H ∞ output-feedback control with pole placement, and the controller parameters are obtained through solving a family of linear matrix inequalities. The designed controller is applied on a 4-generator power system model. The computer simulations are performed for pulse disturbance as well as system operating changes. The designed controller shows better damping than conventional PSS and keeps robustness with load variations.