Quasi-Decentralized Functional Observers for the LFC of Interconnected Power Systems

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 3, AUGUST 2013 3513 Quasi-Decentralized Functional Observers for the LFC of Interconnected Power Systems Hieu Trinh, Tyrone Fernando, Senior Member, IEEE, Herbert H. C. Iu, Senior Member, IEEE, and Kit Po Wong, Fellow, IEEE Abstract—This letter presents a novel approach to the load-fre- quency control (LFC) of interconnected power systems. Based on functional observers theory, quasi-decentralized functional observers (QDFOs) are designed to implement any given global PI state feedback controller. The designed functional observers are decoupled from each other and also of low-order; thus, they are cost effective and easy to implement. Although the proposed approach is applicable to area power systems, an example of a two-area interconnected power system with reheat thermal turbines is considered for simplicity. Index Terms—Functional observers, load-frequency control (LFC), power systems. I. INTRODUCTION AND PRELIMINARIES T O simplify the analysis, but without loss of generality,a two-area system consists of reheat thermal turbines as shown in Fig. 1 is considered (see [1] and [2] for a list of nomen- clature and data). In Fig. 1, the dashed lines show how the area control error (ACE) is computed within each area. For Area 2, note that . Denote , , , and as the state, control, load-disturbance, and output vectors, where As a result, an eleventh-order, 2-input, 2-disturbance state- space representation for the TAIPS is obtained, where (1) In (1), the integral of ACEs are used as controlled feedback variables as this ensures zero steady-state values for tie-line power and frequencies to any step-change in load disturbances. Pole-placement or optimal control by state-feedback has been extensively covered in the literature [1], [3]. Thus, let us now assume that a PI state feedback controller, , can be designed for the TAIPS to satisfy some prescribed closed- loop system performance. Let , where is the control input signal for the th-Area and is the th-row of the feedback gain . Clearly, the critical problem with this control law is the unavailability of some of the state variables. Also, due to the global nature of the feedback control law, it requires data transfer between the two areas. Note that the well-known Luenberger state observers can be used to Manuscript received July 28, 2012; revised September 30, 2012 and January 08, 2013; accepted March 05, 2013. Date of publication April 30, 2013; date of current version July 18, 2013. Paper no. PESL-00115-2012. H. Trinh is with the School of Engineering, Deakin University, Geelong, VIC 3217, Australia (e-mail: [email protected]). T. Fernando, H. H. C. Iu, and K. P. Wong are with the School of Electrical, Electronic and Computer Engineering, University of Western Australia, Crawley, WA 6009, Australia (e-mail: [email protected]; her- [email protected]; [email protected]). Digital Object Identifier 10.1109/TPWRS.2013.2258822 Fig. 1. Block-diagram of a two-area interconnected power system (TAIPS). generate the estimated feedback control signals. However, they lead to high-order centralized observer-based control schemes. This letter proposes novel functional observers (FOs) to im- plement any global PI control law . FOs estimate linear functions of the state vector without estimating all the in- dividual states and so can reduce the order and complexity of the designed observers. Thus, FOs can play a useful role in the LFC of power systems. The theory of FOs has received considerable renewed attention and useful results have been reported [4]–[7]. Here, unlike conventional observer-based LFC control schemes which use one centralized observer, we propose two low-order FOs, one at each Area. Each FO is completely decoupled from each other; thus, there is no transferring of data between them. However, to be able to estimate the global control law, it is nec- essary that each FO uses some output information from the other Area. For this reason, we refer to both FOs as quasi-decentral- ized. Overall, the scheme reduces the complexity of the designed observers and provides some degrees of robustness to faults and attacks; thus, it increases thesystempracticalityandreliability. II. DESIGN OF QDFOS FOR THE TAIPS For any given global feedback control law , it would be most desirable to use only the local output infor- mation, , to generate the estimate . How- ever, according to the functional observability test (see [5, The- orem 5]), the triplet is not functional observable. This means that there does not exist any FO if only the local output information is used to reconstruct . What this im- plies is that decentralized output information is not possible as it is necessary to use additional outputs from the other Area. Thus, let denote the output information to be used at the th-Area, where and , . Here, contains the local outputs and additional outputs from the other Area. Note that and , only and are required to be sent over since can be constructed at the th-Area. 0885-8950/$31.00 © 2013 IEEE 3514 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 3, AUGUST 2013 Fig. 2. Block-diagram implementation of QDFOs for the TAIPS. We can prove that when the load disturbance is character- ized as a step-change of any magnitude, can in fact be ignored altogether in the design of FOs; also the resulting ob- server-based closed-loop system still ensures zero steady-state values for tie-line power and frequencies. This is a usefulfinding since it overcomes the existence problem of unknown input state observers [8] for the TAIPS. Let denote the es- timated control input of Area , where . Consider the following minimum-order FO: (2) (3) where , , , , , and are observer pa- rameters to be determined. Let the global control input matrix be partitioned as , then based on [6], the existence of the FO as described by (2)–(3) is given by the satisfaction of the following matrix equations: (4) Refer to [6] to systematically find , , (Hurwitz), , , and to satisfy (4). Replacing by in (3), the estimated control signal for the TAIPS can now be constructed by using the following equations: (5) (6) where and . Fig. 2 shows a block-diagram implementation of represented by (5)–(6). The flow of information from one Area to another Area is represented by the dotted lines. The two FOs are decoupled from each other; thus, there is no transferring of data among them. However, some partial outputs from the other Area are required to be sent over since without them, the control signal cannot be estimated [5] even when there is no disturbance, i.e., . Each FO can be regarded as one decentralized facility and observe that it is less complex than the centralized facility since each FO has lower order than the centralized one. Also, having two decentralized facilities would provide more robustness to faults and attacks, thus increasing the overall system reliability. Also, if the control law is decen- tralized, i.e., , then can be constructed by using only the local output information, . We can also show that the eigenvalues of the combined controller-observer are the union of the eigenvalues of the controller and the two QDFOs. Fig. 3. step responses to the load demand in Area 1. To illustrate the proposed QDFOs, first, a global linear- quadratic regulator is designed with , and also by imposing all the closed-loop poles of the TAIPS to have a prescribed stability margin of at least (MATLAB command ). Then, two QDFOs are de- signed; each FO is a third-order. Fig. 3 shows a comparison of the responses of for the case where a 0.1 pu step increase in the load demand from Area 1 takes place. Observe that the closed-loop responses in terms of overshoot and settling time as well as zero steady-state deviations for the frequency have been achieved. Comparing to the conventional unknown input state observers [8], an eleventh-order state observer is in fact required. Here, both decoupled FOs each of only a third-order is required to implement the global control law. III. CONCLUSION This letter has presented novel QDFOs for the LFC of power systems. The designed FOs are decoupled from each other; thus, there is no data transferring between them. They are also of low-order; hence, they are cost effective and easy to implement. This is the first instance where reduced-order characteristics of FOs have been utilized in the LFC problems. REFERENCES [1] C. E. Fosha and O. I. Elgerd, “The megawatt-frequency control problem: A new approach via optimal control theory,” IEEE Trans. Power App. Syst., vol. PAS-89, no. 4, pp. 563–571, 1970. [2] M. L. Kothari, J. Nanda, D. P. Kothari, and D. Das, “Discrete-mode automatic generation control of a two-area reheat thermal system with new area control error,” IEEE Trans. Power Syst., vol. 4, no. 2, pp. 730–738, May 1989. [3] Ibraheem, P. Kumar, and D. P. Kothari, “Recent philosophies of au- tomatic generation control strategies in power systems,” IEEE Trans. Power Syst., vol. 20, no. 1, pp. 346–357, Feb. 2005. [4] T. L. Fernando, H.M. Trinh, and L. Jennings, “Functional observability and the design of minimum order linear functional observers,” IEEE Trans. Autom. Control, vol. 55, no. 5, pp. 1268–1273, May 2010. [5] L. S. Jennings, T. L. Fernando, and H. M. Trinh, “Existence conditions for functional observability from an eigenspace perspective,” IEEE Trans. Autom. Control, vol. 56, no. 12, pp. 2957–2961, Dec. 2011. [6] T. Fernando, S. MacDougall, V. Sreeram, and H. Trinh, “Existence conditions for unknown input functional observers,” Int. J. Control, vol. 86, pp. 22–28, 2013. [7] H. Trinh and T. Fernando, Functional Observers for Dynamical Sys- tems. Berlin, Germany: Springer-Verlag, 2012. [8] M. Darouach, “Full-order observers for linear systems with unknown inputs,” IEEE Trans. Autom. Control, vol. 39, no. 3, pp. 606–609, Mar. 1994.