Decentralized Power System Damping Controller Design using H∞ Loop-shaping Technique Nilesh Modi, T. K. Saha & N. Mithulananthan School of Information Technology & Electrical Engineering, The University of Queensland Brisbane, Australia 4072. Email:
[email protected] The operating point of the power system always changes with the variation in the load. Such operation point deviation, often known as uncertainties in robust control terminology, is not systematically considered by conventional control methods. Robust control theory is one way to consider these uncertainties while designing the controller [6], [7]. Such theory has been applied to electric power system controller design since early 1990s. Few successful designs for excitation control system and PSS has been reported in [8], [9], [10].
Abstract—Power system low frequency oscillations influence dynamic behaviour of a complex power system and threaten its security. The dynamic performance of power system can be improved by damping low frequency oscillation with the help of supplementary controllers. This paper presents the design of supplementary controller for Static Var Compensator (SVC) to damp low frequency, in particular inter-area oscillations. The controller is designed based on H∞ loop-shaping technique in which the robust stabilization of the normalised coprime factor plant is formulated into a generalized H∞ problem. Moreover, the selection of proper feedback signal from the available measurements of the system plays vital role in designing the controller. The residue analysis is used to select the suitable locally available signal as an input signal to the proposed controller. Two power systems with varying sizes and complexities are used to test with the design of the proposed controller. The power systems include the benchmark, two-area system for low frequency oscillation studies and a modified version of a practical system. Both eigenvalue analysis and time domain simulations are used to study the performance of the proposed H∞ controller.
I.
Qihua and Jin have attempted to design a robust SVC controller for improving power system damping. The controller was designed based on H∞ mixed-sensitivity technique [11]. Majumder et. al. examined this H∞ loop-shaping design in linear matrix inequality (LMI) region for power system damping controllers [12]. He-Chen et. al. applied PMU based wide-area measurement system (WAMS) technique and LMI based robust control theory together to deal with the Interarea oscillations of power system [7]. One of the difficulties with this approach is that the selection of the mixed sensitivity weights. The selection of weights for the closed-loop transfer function is carried out without much regard to the actual limitation of the closed loop and the closed-loop specification is to be made without considering the characteristic of the nominal plant. This may lead to unrealistic design [5]. The problem can be alleviated with loop-shaping design methodology. Such technique has been applied to the design of power system stabilizer in [13], [14], and terminal voltage control in [15]. Anaparthi et. al. used coprime factorisation approach for designing multi-input stabilizer for power system damping control [16]. Cuk Supriyadi et.al. used it for designing PSS and TCSC [17].
INTRODUCTION
Power system oscillations in an interconnected system are inherent phenomena. For many years, engineers have found these low frequency oscillations, which involve groups of generators on one side of the tie-line oscillating against groups of generators on the other side of the tie-line [1]. One of the major concerns of the power system operators is secure operation of the system primarily in presence of low-frequency electro-mechanical oscillations. Such oscillation usually lies in the range of 0.1Hz to 1.0Hz [2]. Sometimes these small lowfrequency oscillations grow in amplitude that could lead to partial or full outage incidences. A number of incidents due to such small signal instability have been reported in [3].
In this work an attempt has been made to design robust controller for SVC to damp power system inter-area oscillations using explicit H∞ loop-shaping design procedure. The proposed method has been initially tested on small twoarea system and later on sufficiently large, modified practical 14- generator Australian power system. Though the placement of SVC affects the damping performance, it has been excluded from the work as the intention is to design supplementary controller on the existing SVC.
Power System Stabilizers are the most common devices used so far to improve damping of power system oscillation [4]. These PSSs are effective in damping local modes but not effective for inter-area modes because of low observability of these modes at PSS location. One possible solution is to use flexible ac transmission system (FACTS) devices at appropriate location with best control signal. The primary objective of such devices is to provide dynamic voltage support to the system, but these devices can be helpful in improving damping of the system if supported with supplementary damping controller [5] often known as Power Oscillation Dampeners (POD).
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This paper has been organized in as follows. After briefing of control design, fundamentals of H∞ loop-shaping design procedure is explained in Section II. Section III explains input signal selection for the controller. Section IV illustrates both the systems under study and the simulation results. Section V summaries the conclusions and contributions of the work.
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II.
H∞ LOOP-SHAPING DESIGN
Gp = {M + Δ M }−1{N + Δ N }−1 : Δ M Δ N < ε (3) where, ε > 0 is known as stability margin. To maximize
The controller for damping power system oscillations has been designed based on explicit loop shaping design procedure. This design procedure involves shaping the magnitude of the loop transfer function. The brief loop-shaping procedure, as explained in [18] is outlined below.
this stability margin is the robust stabilization of normalized coprime factor plant as suggested by Glover and McFarlane [18]. A controller that satisfies (4) for a specified γ > γ min is the H∞ controller
1) Using pre and/or post compensator, the nominal plant singular values are shaped, using classical loop shaping technique, to give a desired open-loop shape. A shaped plant is formed by combining the nominal plant with pre and/or post compensators.
⎡K ⎤ −1 −1 ⎢ I ⎥ ( I − GK ) M ⎣ ⎦
3) The feedback controller and shaping compensators are combined to achieve final controller. The principle idea of ’loop shaping’ is that the closed-loop transfer function magnitude for Single-Input Single-Output (SISO) system (or maximum singular value for Multiple-Input Multiple-Output - MIMO systems) can be directly determined by the singular values of the open-loop transfer function. Therefore, the design of a controller can be achieved by electing the controller, which appropriately shapes the singular value of the open-loop system [18], [19].
∞
To improve the conditioning of the design problem, the inputs and outputs should be properly scaled. The pre- and/or post-compensators should be selected in such a way that it gives desirable singular value of the shaped plant. The shaped plant should have high gain at low frequencies, roll-off rate of approximately 20dB/decade at desired frequencies. Integral action should be added at low frequencies.
A. H∞ Loop-shaping design procedure The H∞ loop-shaping design is basically the combination of classical loop-shaping technique along with H∞ robust stabilization, as proposed by McFarlane and Glover [20]. The following is the brief explanation of the procedure as discussed in [20], [21], [6].
Once the controller is designed, next task is to select the best control signal, preferably a local signal for adding maximum damping of the weak modes. The signal should be easily measurable and also the mode of interest should be observable in that signal [22].
This process involves two-stages. In the first stage, the open-loop plant is augmented by pre- and/or post-compensator to achieve the desired shape of the singular values of the openloop frequency response. In second stage, the resulted shaped plant is robustly stabilized using H∞ optimization.
III.
INPUT SIGNAL SELECTION FOR SVC CONTROLLER
Location of the device and the input signal to damping controller play a vital role in providing damping to the system to reduce oscillations. As far as the allocation of SVC is concern, it is mainly the outcome of comprehensive analysis of power system under study. This location is also subjected to other factors like feasibility, cost and environmental issues. Along with this primary objective, the supplementary damping controller is often added to SVC to damp electro-mechanical oscillations. The objective of this work is to select appropriate input (sometime called feedback) signal to the damping controller rather than location of SVC. Decentralized control scheme, sometimes, has low observability index, but its high reliability and fast response time attract engineers to use them rather than selecting the remote signal.
1) Open-loop shaping: As stated earlier, during the first stage the open-loop plant is shaped with pre- and/or postcompensators. These compensators are usually known as weights. These weights will shape the singular value of ppenloop plant. If W1 and W2 are the pre- and pre-compensator respectively, then the shaped plant Gs, is given by (1). GS = W2 GW1 (1) where G is the nominal plant. 2) Robust Stabilization: Let plant G has normalized left coprime factorization G = M −1 N . A perturbed plant model GP can be written as Where,
(4)
B. Weight Selection The loop-shaping design procedure depends on the selection of proper weights. The pre- and/or post-compensators should be carefully designed as it governs the entire process of loop-shaping design. The following guidelines suggested in [5] have been used to decide the weights [21].
2) A feedback controller is to be synthesized, which robustly stabilizes the shaped plant.
G P = ( M + Δ M ) −1 ( N + Δ N ) −1
≤γ
A. Controllability & Observability Controllability assists in deciding the location of the device in large power system. To get maximum benefit from the device, it is advisable to place the device at the bus with maximum controllability index. At this point it is evident to understand that, this location might not be the best location for damping lower frequency oscillations. These emphases on the best input signal selection for supplementary controller design.
(2)
Δ M and Δ N are stable unknown transfer
functions, which represent the uncertainty in the nominal plant model G. Now the objective is to stabilization of both family of nominal plant model G and perturbed plants as defined by (3).
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This best signal selection is subjected to its observability indices, which can be addressed with residue analysis.
A. Case Study System – I A small test power system is showing in Figure 1. Machines 1 and 2 are considered in area 1 and machines 3 and 4 are considered in area 2. The figure and system data were adopted from [4].
B. Controllability & Observability While, the state-space representation of the system gives complete input-output behaviour of the system, the transfer function gives only input-output behaviour of the selected input-output pair. For small signal stability analysis mainly depend on the system matrix but for control design we are often interested in the input-output behaviour of selected signals. To understand residue and it’s important for selecting the feedback signal for damping controller, consider the following generalized state space model with usual notations.
Figure 1. : Two-area System.
•
x = Ax1 + bu (5) y = cx1 (6) n 1×n n×1 where, x1 ∈ R , b ∈ R and c ∈ R . The transfer
The entire system was developed in PSS/E. Standard SVC model CSVGN1 available in PSS/E library was used at bus 8 [23]. After carrying out eigenvalue analysis, one pair of complex eigenvalue was found to be with low damping ratio. This eigenvalue is likely to be shifted toward right-half plane during the change in the system operating condition. The damping ratio for this eigenvalue was 0.016 and the frequency of oscillation is 0.589 Hz. 1) Residue Analysis: The interest was to improve the damping ratio of this mode by designing the damping controller for SVC. The choice was made with the signal having maximum residue. To design decentralized control choices left with the control designer are the bus voltage, line power and line current. As the bus voltage is sensitive to output of SVC, this signal was omitted and not further investigated for residue analysis. Table I shows normalized residue values for the signal P89, I89, P78, and I78. The maximum residue is from the line current of line 7 − 8 (I78).
function between output variable y and input variable u can be given by (7)
G(S ) =
y( s) = C ( sI − A) −1 B u( s)
(7)
Rewriting the above equation and using the orthogonal relationship between right and left eigenvectors V and W, i.e.VW = 1, we can derive residue of the transfer function as shown in (8) [27], [22]. n
G (s) = ∑ i =1
Ri s − λi
(8)
where, Ri is the residue of the output variable y and the input variable u with regard to the ith mode. Residue is the product of modal observability Cvi and modal controllability wBi. Residue, being the product of controllability and observability, the input signal to the controller is selected on maximum residue magnitude. In decentralized control the choice left to the control designer is the selection between the local available signals such as SVC bus voltage, line current/power of the line emitting/terminating at SVC bus. The residue of each signal is measured and accordingly the input signal to the controller has been selected. IV.
TABLE I.
RESIDUE MAGNITUDE FOR POTENTIAL SINGNALS Signal I89 P78 P89 I89
Residue Magnitude 1.00 0.94 0.90 0.93
Once the input-signal (feed-back signal) was decided the controller design was carried out with loop-shaping design in H∞ synthesis. The following weights were used as pre and postcompensators, respectively.
SIMULATIONS AND RESULTS
The proposed controller design has been tested on two power systems. Both systems were implemented in PSS/E [23] to carry out load-flow analysis. The results of load flow analysis were then taken to Mudpack [24] for dynamic analysis of the system including time-domain simulation. The controller was designed in MATLAB, and then integrated to MUDPACK dynamic data file. After carrying out eigenvalue analysis, speed eigenvector is calculated to get insight of the critical inter-area mode. Although, the speed-eigenvector is observed for both the systems under consideration, it has been plotted only for study system II here. The same approach has been used while showing the result of the controller design procedure.
W1 =
10 and W2 = 1 . s + 100
This system was of the order of 46, which is difficult to handle for sub-sequent controller design. Therefore, it has been reduced to 8th order without compromising the response of the system in the frequency range of interest. The balanced truncation and residualization via Schur’s method of decomposition of grammian product, available in Robust Control Toolbox [25], has been used to reduce the order of the system. Once the reduced order plant is available, the loopshaping design procedure has been carried out to design H∞
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controller. The value of γ achieved was 1.25. This value is sufficiently low to consider that the successful robust design has been achieved [18], [21]. The order of the achieved controller is the sum of the reduced plant and the order of the weights. In this case, the achieved controller is of the order of 9, which is high for the implementation of the controller. This controller has further been reduced to 3rd order using model reduction technique. The final transfer function of the controller, in zero-pole-gain form is given by (9).
K ∞ = 0.025149
s − 12.09 s + 0.49 s + 0.001215 (9) s − 0.0004039 s + 0.9606 s + 1.832
With the designed controller the damping ratio of interarea mode is improved from 0.016 to 0.062. Table II reflects the results achieved with damping control design via H∞ Loopshaping technique. TABLE II. Status No Controller With H∞ POD
DAMPING RAITO OF THE TEST SYSTEM I Damping Ratio 0.016 0.062
Frequency (Hz) 0.589 0.576
2) Time-Domain Simulation: After designing the controller in MATLAB, it has been integrated in MUDPACK dynamic data file. The disturbance is initiated by changing the mechanical input power reference of Generator 2, Pm2, by 5% to its current value and the output of all the generators has been observed up to 20 sec. Although the simulation was carried out for all the generators in the system, the output of Generator 1 is shown in Figure 2.
Figure 3. Simplified 14-Generator Australian System.
First, eigenvalue analysis was carried out for the system under study to find out the inter-area modes. Eigenvalue plot revealed that there were two weak modes. The damping ratio of these modes has already been improved by equipped PSS at all the generators. Table III lists the weak modes. The damping ratio of these modes is likely to be reduced when the system will be under stress condition or in case of contingencies.
Figure 2. Output Power of Generator 1: ‘- -‘ without controller, ‘-‘ with controller.
It is evident from the results that the damping controller based on H∞ loop-shaping technique improves the system damping. The oscillations of the output power reaches to the acceptable variation within 15 seconds. B. Case Study System-II The test system is shown in Figure 3, which is a simplified 14-generator, 50Hz system loosely represents southern and eastern Australian network [26].
TABLE III.
DAMPING RATIO FOR INTER-AREA MODE
Mode No. Mode 1 Mode 2
Damping Ratio 0.131 0.124
Frequency (Hz) 0.320 0.423
The speed-eigenvector plot of the mode of interest, mode 1, is plotted in Figure 4. It is observed from Figure 4 that generators located in area 1, 3 and 5 coherently oscillate against the generators in area 4.
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Figure 4. Speed-Eigenvector plot of Mode 1.
Figure 5. Unshaped and Shaped Plant
1) Residue analysis: The residue analysis has been carried out for the mode of interest. Care has been taken to select the local available signal to design decentralized control. From the residue analysis three potential signals have been identified. The normalized residues of these signals are tabulated in TableIV, along with the SVC location. TABLE IV. Input Signal Pa507-509 Ia411-142 Ia507-509
POTENTIAL FEEDBACK SIGNALS Residue 1.00 0.84 0.94
SVC PSVC_5 BSVC_2 PSVC_5
The power of the line 507-509, Pa507−509, has been selected as feedback signal for the damping controller. With this input signal, the H∞ loop-shaping design is carried out to design damping controller. First the order of the plant, which was of the order of 174, is reduced to 9th order without compromising its characteristic in the frequency range of interest, as described before. Figure 5 shows the result of the H∞ loop-shaping procedure. The value of achieved γ was 1.41. With the achieved controller, the eigenvalue analysis has been carried out and the result is tabulated in Table V. TABLE V. Damping Ratio without Controller With H∞ POD
Figure 6. Output Power of GPS_4 with (solid line) and without (dashed line) controller.
DAMPING RATIO WITH POD Mode 1 0.131 0.213
Mode 2 0.124 0.244
2) Time-Domain Simulation: After designing the controller in MATLAB, it has been integrated in MUDPACK dynamic data file. The disturbance was initiated by changing the mechanical input power reference of Generators in area 4 by 5% and that of generators in area 5 with -5% to its current value and the output of all the generators has been observed. Figures 6 to 8 shows the output response of the generator in presence and absence of the POD.
Figure 7. Output Power of TPS_4 with (solid line) and without (dashed line) controller.
It is evident from Figures 6 to 8 that the damping controller based on H∞ loop-shaping technique improves the system damping. The oscillations of the output power reaches to the acceptable variation within 10 seconds.
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[6] [7]
[8]
[9]
[10]
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Figure 8. Output Power of PPS_5 with (solid line) and without (dashed line) controller.
V.
[13]
CONCLUSIONS
The paper proposed a decentralized SVC controller based on H∞ loop-shaping technique for damping inter-area oscillations. The best control input signal for the controller was judiciously selected based on the outcome from the residue analysis which consists of controllability and observability information of a mode to locations and control input signals. The controller has been successfully implemented in the benchmark two-area test system for low frequency oscillation studies and a modified practical power system. Numerical results from eigenvalue analysis and time domain simulation show that the proposed controller is successful in damping low frequency oscillation in both the systems. In future work, a centralized controller design based on wide-area measurement signal will be evaluated and compared with the decentralized control along with checking the robustness of these controllers for different operating conditions.
[14]
[15]
[16]
[17]
[18]
ACKNOWLEDGEMENT The authors would like to acknowledge the support from ARC and Powerlink Queensland for this work, through an ARC Linkage Grant.
[19]
[20]
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