IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 24, NO. 5, SEPTEMBER 2016
1573
Evaluating the Novel Application of a Class of Sampled Regulators for Power System Control Shadi Khaleghi Kerahroudi, Student Member, IEEE, Fan Li, Gareth A. Taylor, Senior Member, IEEE, Maysam Abbod, and Martin E. Bradley
Abstract— The focus of this paper is on the nonparametric system design approach using a class of sampled regulators. Based on the review and evaluation of two stability design methods that were originally established for this class of sampled integral regulators, this paper has extended the stability theory and design algorithms in order to additionally consider generalized proportional–integral–derivative regulators. The link between the two original design methods has been revealed, based on which the whole benefit of the class of sampled regulator design methods can be embraced in a single framework. Furthermore, the suitability of the proposed design algorithms has been demonstrated in several power system applications. Index Terms— Computer-based stability control, high-voltage dc (HVdc) link, sampled regulator, thyristor controlled series capacitor (TCSC).
I. I NTRODUCTION ASED on the time-domain approach, Åström [1], Lu and Kumar [2], and Mossaheb et al. [3] have introduced a class of sampled integral regulators and ensured the stability of the closed-loop systems with their designs in a series of developments. These design methods are developed for unknown systems and are based on their open-loop step responses. This approach is therefore termed as nonparametric, which is in contrast to any model-based, hence parametric, methods. Because the models are nonstructural time responses, these algorithms can be applied to the design of control schemes for large, nonlinear, and complex power system applications without the requirement for any model simplification. However, the tradeoff to the dimensionality in this nonparametric approach is the length of the time series that would be needed to capture a dynamic process. For a parametric approach when the stability is obtained with simplification, it is critical to justify its validity with regard to the original
B
Manuscript received January 23, 2015; revised July 23, 2015; accepted October 17, 2015. Date of publication December 11, 2015; date of current version August 4, 2016. Manuscript received in final form October 27, 2015. This work was supported in part by National Grid plc, U.K., and in part by the Engineering and Physical Sciences Research Council. Recommended by Associate Editor P. Korba. S. K. Kerahroudi is with the Brunel Institute of Power Systems, Brunel University London, Uxbridge UB8 3PH, U.K., and also with National Grid, Market Operation, Wokingham RG41 5BN, U.K. (e-mail:
[email protected]). F. Li and M. E. Bradley are with National Grid, Market Operation, Wokingham RG41 5BN, U.K. (e-mail:
[email protected]). G. A. Taylor and M. Abbod are with the Brunel Institute of Power Systems, Brunel University London, Uxbridge UB8 3PH, U.K. (e-mail:
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2015.2502558
system, while in the nonparametric time response approach, an estimate of the truncation errors can be established with relative ease. Between these two approximations, quantifying the impacts of structural simplification often can be more challenging than formulating the truncation error from a wellconstructed sequence. Developments of this class of sampled regulators have paved a novel approach to the designs of the computercontrolled systems, achieving both stability and optimality by exploring time-domain input and output properties of an unknown system. With some extensions from the original sampled integral regulators, the general form of the feedback control law of u(nT ) that we shall explore further in this paper can be written as u(nT ) = C P e(nT ) + C I e(nT ) n b(r T )u((n − r )T ) +C D 2 e(nT ) +
(1)
r=1
where is the differencing operator. Applied to the system error e(nT ) = y 0 (nT ) − y(nT ), it gives e(nT ) = e(nT ) − e((n − 1)T ) and 2 e(nT ) = e(nT ) − 2e((n − 1)T ) + e((n − 2)T ). C’s and b’s are the regulator parameters and T is the sampling rate, which will be defined in the next section. It is clear that this class of the sampled regulators is the extension of the traditional proportional–integral– derivative (PID) control law (in the discrete form) where it is essential to incorporate additional control terms of u(nT ) in order to achieve both stability and dynamic performance in this formulation. In this paper, we shall focus first on the theories of this class of sampled regulators and the stability design methods that were originally established for a class of sampled integral regulators will be reviewed [1]–[3]. Based on these earlier developments, the main contribution and focus of this paper is first on the extension of these algorithms to the generalized PID regulators in the form of (1). We shall then establish how the whole wealth of these different design methods can be embraced by a single framework and implemented in a computer-based control program. Subsequently, the applications of this class of sampled regulators for power systems will be presented in Section II. Various developments have been reported on the control of flexible ac transmission systems (FACTS), including thyristor controlled series capacitor (TCSC), static var compensator (SVC), and high-voltage dc (HVdc) link.
1063-6536 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
1574
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 24, NO. 5, SEPTEMBER 2016
Chang et al. [4] investigated a case of coordination of HVdc and SVC control for damping improvement. They reported interactions between SVC and HVdc when these control systems are designed and operated separately. Based on the simplified classical model of the synchronous generators, they formulated the state-space models and further demonstrated the use of wide-area signals for the projective control to achieve the coordination. Wang et al. [5] applied a passivitybased control design to a TCSC power flow control problem. They have showed that the system stability can be ensured by this design. While the enlightening concept has been demonstrated, the development is based on the simplified classical synchronous machine and subsequent state-space models. Before these research results can be applied to practical implementations, one would need to address that how a fullscale power system can be efficiently modeled and to justify that the design based on simplified models is valid for the original system. Phulpin et al. [6] applied model predictive control (MPC) to HVdc power flow control in multimachine systems. Due to the adaptive nature of the MPC algorithm, the accuracy of the model is less critical, though the proof of stability at each step of the control point still remains as an open question. Examples of the applications of the sampled integral regulators to some advanced FACTS devices have been reported in [7] and [8]. The works are based on the design methods as given in [3] using a sampled integral regulator. In this paper, following the new theoretical development, this class of sampled PID regulators are then applied to two power system control problems. 1) Design of a TCSC Control System: In this example, in order to meet a wide range of the commonly used performance indices, different design methods of this class of regulators are examined. 2) Coordination Between Voltage Regulation and Power Flow Control Using SVC and HVdc: Multiple control functions and objectives within the SVC, i.e., voltage regulation and power oscillation damping (POD), are addressed along with the power flow control objective of the HVdc. Postfault stabilizing control strategy is also demonstrated in this example. II. A SSUMPTIONS , N OTATIONS , AND D EFINITIONS Consider a stable time-invariant plant P [which can be either single-input and single-output (SISO) or multi-input and multioutput (MIMO)], and assume that it can be linearized in the neighborhood of its operating point. The time response of this system is denoted by H (t) = (Hi j (t)) ∈ R m×m , where Hi j (t) is the step responses of the i th row and j th column entry of H (t), with u j (t) and yi (t) as its input and output, respectively. u j (t) and yi (t) are j th and i th components of u(t) and y(t) ∈ R m . Since P is assumed to be stable, its step response is bounded exponentially. In terms of a pair of parameters K , β > 0, and with any matrix norm defined on R m×m that is induced from the measure of linear vectors defined in R m , this bound can be expressed as H (t) − H (∞) ≤ K exp(−βt), t ≥ 0.
The relationship between the output and the input of the linearized plant can be expressed in terms of the system ∞ impulse response, h, as y(t) = h(t) ∗ u(t) = 0 h(s)u(t − s)ds, where ∗ denotes the convolution operator. In the discrete form with the sampling rate of T , the system input–output can be related by its unit step response [1] y(nT ) =
n
H (r T )u((n − r )T )
(2)
r=1
where n = 0, 1, 2, . . . and H (r T ) = H (r T ) − H ((r − 1)T ) with H (r T ) = 0 for r ≤ 0. The input vector u(t) to the plant is defined as 0, t I being the solution to the matrix Lyapunov equation (8) BK + K B = −(K + T K ) N T T H , and where = = H∞ ∞ n=1 (Hn Hn ), K N T = n=1 Hn (Hn−1 − H∞ ). The optimal solution may not always satisfy A < I in this calculation, but it does not affect the stability. For stability, sequence {ϕn } needs to satisfy only the conditions of Lemma 1. When an optimal solution exists, this choice of A ensures both stability and the optimal error tracking, hence to achieve the better performance that is featured by decoupling and improved damping. The steps for the passivity design method can be summarized as follows. 1) Choose a sampling control period T and obtain the unit step responses of {Hn }.
1575
2) Truncate time series sequence at the Nth term. 3) Obtain an optimal r.s.p.s.d. parameter matrix A, according to (8). 4) Calculate C = (I − A)−1 (H N )−1 to ensure the zero-error and decoupled steady state. 5) Calculate {bn } with b0 = 0, bn = CHn − An , n ≥ 1. Then implement the control law of (5). Discussions: If different performance criteria are used and separate tuning of parameters C and {bn } are required, this method can be inflexible, since both parameters C and {bn } are functions of a single parameter A. Since the optimisation in the passivity design is defined in terms of the rms system tracking error, the control may not directly meet certain commonly used engineering criteria, e.g., damping ratio, settling time, and overshoot, of the system time response. Lu and Kumar [2] adopted a slightly conservative approach in the proof of the system stability, prior to the development of the passivity method. Nevertheless, since their method is based on the concept of pole placement, it provides a different design approach to this class of sampled regulators, which can just be a useful alternative to the passivity design to overcome the fore-mentioned inflexibility issues. B. Lu–Kumar Methods Lu and Kumar developed their algorithm based on slightly different form of this class of sampled linear integral regulators n (Pr − Pr+1 )u n−r . u n = P1−1 yn0 − yn + P1−1
(9)
r=1
The design algorithm is summarized in the following lemma. Lemma 2 (Lu–Kumar MIMO System Design [2]): For a MIMO linear system defined in (4) the closed loop system with the regulator given by (9) is stable if the sampling rate T , the truncation length N, and parameters {Pn } are chosen such that the following conditions are satisfied. 1) (H N )−1 exists and exp(−τ N)H N−1 is bounded uniformly in N for some 0 ≤ τ ≤ βT . 2) Pr = Pr+1 + Hr − Hr−1 , 1 ≤ r ≤ N. 3) γ exp(−β N T )H N−1 < 1, where γ = K (1 + exp(−βT ))/(1 − exp(−βT )). The proof of the lemma can be found in [2], but it is interesting to see how this regulator in Lemma 2 is related to the sampled regulator of (5) of Lemma 1. We shall establish the following corollary to answer this question. Corollary 1 (A Common Framework of the Sampled Integral Regulator): It is shown in [2] if {Pn } is defined by Condition 2 in Lemma 2 above the sampled linear integral regulator then becomes| n −1 0 −1 u n = H∞ y n − y n + H∞ H r u n−r
(10)
r=0
which is in the same form as that obtained from the Passivity design algorithm of Lemma 1, when the parameter sequence of {ϕn } is set to {0}. Furthermore, the control law of (10) is reduced to a single-step control action when tracking a −1 y 0 , n ≥ 0. constant step response, i.e., u n = H∞
1576
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 24, NO. 5, SEPTEMBER 2016
Proof: In Lemma 1, when {ϕn } = {0}, bn = CH ≥ 1), Property 1 leads to C = (I + ∞ n , (n −1 −1 ϕ )H = H n ∞ ∞ . Then, (5) becomes (10). Furthermore, n=1 Property 2 of the 2-step control law is reduced to a single-step control action for step inputs as stated in Property 2. The proof of Corollary 1 may suggest an alternative to the proof of Lemma 2 as in [2], however, this proof is based on the infinite sequence of {Pn }. In Lemma 2, Condition 3 has established a sufficient criterion for a minimum N that can guarantee both stability and exponential decay rate in the design. As pointed out in [2] that a desired exponential decay rate μ (0 ≤ μ ≤ β) can always be obtained when N is sufficiently large. This indicates that the sampled regulator may achieve a similar exponential decay rate to that of the open loop plant. For SISO systems, Condition 2 of the parameter selection in the design method of Lu and Kumar can be relaxed to a set of inequalities of Pr − Pr+1 ≥ H r − Hr−1 , 1 ≤ r ≤ N. With choices of parameters b0 = 0, b1 = 1, and bn = 0, (n ≥ 2), (5) reduces to classical sampled integral regulator (11) u n = C yn0 − yn + u n−1 . Åström [1] first proved that for a stable SISO plant with monotonic step response, the closed-loop system with the integral regulator of (11) will be stable if its parameters T and C are chosen such that C −1 ≥ H1 ≥ H∞ /2. IV. E XTENSION OF THE S AMPLED I NTEGRATORS TO G ENERALIZED PID R EGULATORS In this section, we extend Lemmas 1 and 2 to the class of sampled PID regulators under the general framework of (3). For the Passivity design, when P- and D-terms are incorporated, Lemma 1 can be extended as follows. Theorem 1 (Passivity Design for Sampled PID Regulators): When the linear system of (4) is under the control of the sampled regulators of (3), with the selection of the parameters in the sequence of {bn } such that b0 = 0, bn = a1 Hn + a2 Hn−1 + a3Hn−2 − ϕn , (n ≥ 1), the closed loops transfer function Y 0 → u can be obtained as ∗ u = a1 {yn0 } + 0 } + a {y 0 } where = ϕ = {ϕ }, n ≥ 1. If {ϕ } a2 {yn−1 3 n−2 n n is selected to be a convex sequence and ϕn converges, then is a strictly passive operator. Consequently, the closed loop system that maps y 0 → y is stable. Furthermore, a zero steady state is ensured when the system is subject to a constant step −1 change, if condition C I = a1 + a2 + a3 = (I + ∞ n=1 ϕn )H∞ is satisfied. Parameters ai (i = 1, 2, 3) are derived from the PID parameters as follows: ⎧ ⎨ a1 = C P + C I + C D a2 = −C P − 2C D (12) ⎩ a3 = C D . Proof: The delayed outputs of the linearized system in n the form of (4) can be written as yn−1 = r=1 Hr−i u n−r , i = 0, 1, 2. In the sampled PID regulator of (3), one can
rearrange the terms of the same time instance together, and introduce new parameters ai (i = 1, 2, 3) as defined in (12) u n = C P en + C I en + C D 2 en +
n
br u n−r
r=1
= a1 en + a2 en−1 + a3 en−2 +
n
br u n−r .
r=1
Note that en = (yn0 − yn ) and substitute the expressions of delayed system outputs into the equation above, one obtains un +
n (a1 Hr + a2 Hr−1 + a3 Hr−2 − br )u n−r r=1
0 0 . = a1 yn0 + a2 yn−1 + a3 yn−2
Taking the z-transform on convolution sequence equations of 0 } + a {y 0 } y(t) = h(t) ∗ u(t) and ∗ u = a1 {yn0 } + a2 {yn−1 3 n−2 one obtains that: Y (z) = H (z)U (z) and (z)U (z) = (a1 + a2 z −1 + a3 z −2 )Y 0 (z). Substituting U (z) into the expression of Y (z), one obtains the transfer function of the closed loop system Y (z) = H (z) −1(z)(a1 + a2 z −1 + a3 z −2 )Y 0 (z). The proof of stability in [3] is again applicable here to the closed-loop sequence operator . Proof in [3] shows that with the selection of the parameter sequences of {bn } and {ϕn } according to Theorem 1, and that the open loop system is stable, the closed loop system mapping y 0 → y will be guaranteed to have no poles outside the unit disc by the z-transform definition adopted here, hence stable. The condition to ensure the zero-steady state error when the system of (4) is subject to a constant step change then can be established by applying the final value Theorem of z-transform to the transfer function of closed loop system obtained above. By setting the steady state error y 0 − y∞ = [I − H (1) −1(1)(a1 + a2 + a3 )]y 0 = 0, also by noticing that H (1) = H∞ and (1) = (I + ∞ n=1 ϕn ), one obtains the condition for C I . Remarks: 1) With the incorporation of P- and D-terms and the choice of the convex sequence to be {A, A2 , . . . , A N , . . .}, the original 2-step control of the Passivity design algorithm becomes a 4-step control for a linear system when it is subject to a constant step change of y 0 ⎧ 0 a1 y , n=0 ⎪ ⎪ ⎪ ⎨[(I − A)a + a ]y 0 , n=1 1 2 un = 0 ⎪[(I − A)(a1 + a2 ) + a3 ]y , n = 2 ⎪ ⎪ ⎩ (I − A)(a1 + a2 + a3 )y 0 , n ≥ 3. 2) C P and C D are selected before the determination of C I . 3) For the selection of the convex sequence {ϕn } = {A, A2 , . . . , A N , . . .}, 0 ≤ A < I will guarantee stability but optimal choice is open for further research. The Lu–Kumar design can be extend similar to the sampled PID regulators of (3). Theorem 2 (Lu–Kumar Design for Sampled PID Regulators): For a MIMO linear system defined in (4), the closed loop system with the sampled regulator given by (3) is stable if the sampling rate T , the truncation length N,
KERAHROUDI et al.: EVALUATING THE NOVEL APPLICATION OF A CLASS OF SAMPLED REGULATORS
and parameters {Pn } are chosen such that the following conditions are satisfied. 1) (H N )−1 exists and exp(−τ N)H N−1 is bounded uniformly in N for some 0 ≤ τ ≤ βT . 2) bn = (A1 Hn + A2 H n−1 + A3 H n−2 ), 1 ≤ n ≤ N. 3) γ exp(−β(N − 2)T )(A1 + A2 + A3 ) < 1, where γ = K (1 + exp(−βT ))/(1 − exp(−βT )) where ⎧ ⎪ ⎨ A1 = C P + C I + C D (13) A2 = −C P − 2C D ⎪ ⎩ A3 = C D . Remarks: 1) Condition 2 implies that C P H N + C I H N + C D 2 H N = I , which is the extension of P1 = H N as in the original Lu–Kumar’s integral regulator where C P = C D = 0. 2) The relationship between the parameters in (3) and that of the Lu–Kumar regulator is given by bn = P1−1 (Pn − Pn+1 )1 ≤ n ≤ N. 3) When N → ∞, Condition 3 can always be satisfied. 4) In the SISO case, Condition 2 can be relaxed to an inequality. Proof: The development is similar to that in Lu and Kumar [2] n=1
an ≤
N
A1 Hn + A2 Hn−1 + A3 Hn−2 − bn
n=1
+ +
∞ n=N+1 ∞
∞
A1 Hn +
A2 Hn−1
n=N+1
Fig. 1.
1577
SMIB system for the TCSC control system design.
3) Select C I according to −1 0 < C I ≤ H∞ (1 − C P H ∞ − C D 2 H ∞ ).
4) Checking length of the time series N satisfying, otherwise GOTO Step 2 γ exp(−β(N −2)T )
0. 2) For a selection of sampling rate T , obtain the time response sequence of {Hn } with sufficient length.
In this example, both Passivity design (Lemma 1) and Lu–Kumar algorithm (Lemma 2) have been explored. The TCSC study is based on a single machine-infinite bus (SMIB) system. The single-line diagram is as shown in Fig. 1. In this example, a TCSC provides a connection, alongside with an ac circuit, between two generation groups. The TCSC is modeled as a variable inductance. Initially, the TCSC provides 30% compensation to the reactance of Line1 and Line2. This TCSC can achieve up to 75% compensation to the original line reactance. TCSC is a highly nonlinear power flow control device. The control system regulates the power flow to minimize interarea oscillation under the normal operation. Postfault, the controller changes the reactance of the TCSC to achieve fast restoration of power flow, following the fault clearance, to keep the system in synchronism. The design based on Passivity algorithm produced an optimal and stabilizing solution, A = 0.7311, by solving Lyapunov equation (8) in Algorithm 1. C I = 0.5545 is obtained to ensure the zero-steady state. Assuming no truncation a large N = 3000 is used. The stable operating range under this set of parameters is found between 750–790 MW, with the base case operating point being 770 MW, which is sufficient for the regulation purposes. For Lu–Kumar method, based on the open-loop unit step response of the system, H∞ = 6.7, K = 1, and β = 0.917.
1578
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 24, NO. 5, SEPTEMBER 2016
Fig. 3. TCSC control of Line1 and Line2 power flow. Blue curve: Passivity design [case (a)]. Red curve: case (b) of Algorithm 2. Black curve: Case (c) of Algorithm 2.
Fig. 2. Comparison between the Passivity design and Lu–Kumar design for the TCSC control system. −1 H∞
CI = = 0.149. When C P = C D = 0, with the sampling rate T = 0.01 s N ≥ 400 will satisfy the condition in Algorithm 2 γ exp(−β(N −2)T ) = 5.6 < = 6.7
C P H N + C I H N + C D 2 H N 2C P + C I + 4C D
which is sufficient to ensure stability for the design. Study results are as shown in Fig. 2. Fig. 2(a) shows the power flow on Lines1 and 2. The Lu–Kumar design shows the robustness of this class of sampled regulators when the time series is heavily truncated. Its control is slightly slower than that of the Passivity design. Fig. 2(b) shows the control signals. The 2-step action from the Passivity design (Property 2) and the single-step action from Lu–Kumar design (Corollary 1) to a constant step change in power reference point of the TCSC. Using Algorithm 2, one can achieve a balance between performances of: 1) the fast response to minimize the tracking error and 2) the better damping with nonovershooting responses. When C P = C D = 0, C I = 0.002 is selected that satisfies Condition 3 in Theorem 2, −1 = 0.149. The stable control range i.e., 0 < C I ≤ H∞ with this design is widened to 500–1000 MW, with the base case operating point at 770 MW. In order to see the effects of the P-and D-control, one resets C P = 0.14, C D = 0.5,
Fig. 4.
SMIB test system for the coordination control of HVdc and SVC.
and keeps C I = 0.002. N ≥ 1200 will ensure both stability conditions satisfied, by Step 4 in Algorithm 2. It is important to note that since this is only the sufficient condition, when N < 1200 the system under the control is not necessarily unstable. However, there would be no guarantee of stability in the control system by this design if these conditions are violated. The time responses of the following three design cases are overlaid in Fig. 3: 1) passivity design; 2) Algorithm 2 with C P = C D = 0 and C I = 0.002; 3) Algorithm 2 with C P = 0.14, C D = 0.5, and C I = 0.002. While the Passivity control gives minimum error response, Algorithm 2, on the other hand, allows freedom of choosing integral time (C I ) such that both fast and slow (but nonovershooting) performances can be achieved. It is also noticed that under sampled regulator controls the damping time constant is ∼0.95 s, which is slightly better than that of the open loop system (1.09 s). This again shows that estimates in Lu–Kumar theory are only a sufficient condition. VI. D ESIGN OF C OORDINATED C ONTROL FOR HVdc AND SVC The SVC and embedded HVdc link are fast and automatically controlled devices. Primarily HVdc is for flow control and SVC for voltage regulation. Both HVdc and SVC can also incorporate additional POD function to improve the damping control. The HVdc–SVC test system is shown in Fig. 4.
KERAHROUDI et al.: EVALUATING THE NOVEL APPLICATION OF A CLASS OF SAMPLED REGULATORS
1579
Fig. 5. HVdc and SVC–POD control system design using a MIMO-sampled regulator. Blue curves: HVdc and SVC are under the individual SISO control. Red curves: HVdc and SVC are under the MIMO control, using the sampled regulator of (10). Green curves: set points.
The initial transfer level is set at 2000 MW. The POD of the SVC senses system oscillation from power flow on ac lines B101, B102, CX02, and CX03. These two control systems, i.e., HVdc and SVC–POD, if running separately, could have interactions with both trying to adjust power flows on the same set of circuits. In the nonparametric formulation, this is a 2 × 2 MIMO system. Dimension of the control system is determined by the number of the system input and output variables (e.g., voltage and power flow of the circuits under control), rather than the number of the state variables in the system (e.g., the dynamics that determined by the order of the generators and controllers). This can be a significant advantage when considering a fullscale power transmission system. It is common to have thousands of state variables in a full-scale power system model, while the number controlling and controlled quantities would be much lower, and normally in the same order of the number of the FACTS devices. Kundur [9] suggested that for stability the high speed current control with a superimposed power control is preferable. In this study, the control system of HVdc is placed in cascade controls voltage, where the POD power feedback signal is with its internal control system. The SVC control system superimposed on the voltage signal. The HVdc power control subsystem has much faster dynamics compared with that of SVC. For a unit step change in the power control
channel the step response settles under 10 s, while the step response in the voltage control channel can take ∼50 s to settle (more precisely the smallest β ∼ 0.086). It is a stiff system by the definition used in numerical analysis. When the control system design is based on the Passivity method (Lemma 1), there is no optimal solution that satisfies A < I . With a postcompensation of a gain to the SVC voltage control channel optimal solutions can be obtained. However, it is found that static gain compensation to the system can lead the control signals hitting the physical limits of HVdc or SVC. Alternatively, Lu–Kumar method of (10) can be used with the integral time H∞ = diag(0.1302, 1.0). For the sampling time T = 0.01 s, N = 12 000 would be required, which ensures the stability in Lu–Kumar method, i.e., γ exp(−β N T )H N−1 = 0.52 < 1. In the test, it is found that N can be reduced to 6000 without causing significant deterioration of control. Interaction can be seen when the HVdc link and the SVC are operated under their individually designed SISO control system (using continuous PI controllers). With the MIMO control system, this interaction is significantly reduced, as shown in Fig. 5, especially in the case of disturbance to SVC voltage [Fig. 5(c)] that is due to the HVdc power changes. To test the interactions between the SVC and HVdc, the control reference points of each of these devices are subject to a step change in turn while the set-point of the other device
1580
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 24, NO. 5, SEPTEMBER 2016
Fig. 6. Postfault study case for the HVdc–SVC control system: Blue curves—without HVdc postfault action. Red curves—with HVdc postfault actions, under the MIMO sampled regulator control.
is kept unchanged. Fig. 5(a) and (b) shows the responses of the voltage of SVC and the active power of the HVdc when a 10% step change is imposed on the SVC voltage control set-point, where the HVdc set-point is kept constant at the original operating point. Fig. 5(c) and (d) shows the responses of the system when a 100-MW step change is added to the HVDC control set-point, where the SVC setpoint is constant. With this arrangement, the SVC voltages are shown in the top plots, where Fig. 5(a) shows voltage change by self-excitation and Fig. 5(c) shows voltage change excited by the HVdc disturbance. The HVdc line flows are the bottom plots, with Fig. 5(b) being the response of HVdc link power to disturbance from the SVC’s set-point change and Fig. 5(d) is the HVdc link step response following the change of its set-point. It is evident from the blue curve shown in Fig. 5(c) that the step change in the HVdc flow control causes substantial disturbance in both the SVC voltage and the POD damping control signal, when the system is under the SISO control. Whereas, when the system is under the MIMO-sampled regulator control [as shown by the red curve in Fig. 5(c)], the interaction between HVdc power flow control and the SVC–POD control is significantly reduced. In the case of network faults, if there is no coordinated control, HVdc would keep its prefault flow, while the loss of a circuit due to the fault clearance would reduce the postfault transfer capability across the constraint boundary circuits. As a
consequence, stability is reduced. Under the MIMO control the embedded HVdc can provide a rapid postfault action, picking up the flow from the lost ac circuit. As a result, loading to the accelerating machines are quickly restored and system stability maintained. This is demonstrated in postfault study case presented in Fig. 6. In this case, a 3-phase fault is placed on line B101 at the SVC terminal. Postfault, B101 is tripped after the fault clearance. In addition, the MIMO-sampled regulator redispatched the HVdc power flow to increase to 980 MW from the prefault set-point of 500MW at the fault clearance, which picked up a significant portion of the prefault flow from B101. Fig. 6 shows the results of the postfault system, with and without postfault stabilizing control actions. Where, Fig. 6(a) represents the power flow on the HVdc controlled circuit. And Fig. 6(b) represents the HVdc control signals. The MIMO control signal has not been significantly clipped by the HVdc physical limit (modeled as 1200 MW in this study). As can be seen in Fig. 6(c), the voltage under the SVC control postfault is kept to its prefault set-point. The MIMO control system regulates the bus voltage through SVC, while minimizing interference by the HVdc postfault step change to the POD. The SVC control signals are also shown in Fig. 6(d). The SVC limit on the control variable is [−0.75, 1.5] p.u. and with the postfault stabilizing action, the SVC control signal is less restricted, hence more effective, than that of nonpostfault control.
KERAHROUDI et al.: EVALUATING THE NOVEL APPLICATION OF A CLASS OF SAMPLED REGULATORS
The limiting effects can be seen as in Fig. 6(c). In the postfault period, system has undergone through various nonlinear system stages therefore this case has demonstrated robustness of the sampled regulators. The postfault system is different from the prefault system with both the line switching and operating point changes. Desirably, the parameters of the controller could be updated postfault. With the proposed time domain design approach, this class of sampled regulators can be further developed into adaptive schemes, using recursive techniques for the nonparametric system identification as described, e.g., in [10]. VII. C ONCLUSION In this paper, the essential relationship between the two different major designs for a class of sampled regulators has been revealed, which enables both to be incorporated under a single computer-based control system framework. The special features of Passivity design and Lu–Kumar methods have been further explored. Both methods have then been extended to a more general form incorporating the proportional and derivative actions, which enhances the original single integral control function for more flexibility and better performance. The designs have been demonstrated through two power system applications, including power flow control using TCSC, coordination of HVdc flow control and the SVC voltage regulation with the power damping control. These examples also illustrated techniques covering both SISO and MIMO system designs. A typical power system fault case is used to test the robustness of this class of sampled regulator. This study confirms that the sampled regulators are applicable to power system control, even at a postfault situation when such a highly nonlinear system is subjected to severe disturbances. Further to the theoretical development as presented in this paper, the design methods are being tested with the fullscale Great Britain (GB) transmission system for the coordinated control of various types of FACTS devices, including TCSC, HVdc links, and SVC and associated auxiliary POD functions. The advantage of the nonparametric time-domain design approach is to be further confirmed in these large scale system applications. Forming stability control strategies in view of both the energy balancing mechanism and special protection schemes for the GB transmission system as operated by National Grid will be the focus of the next stage of work. ACKNOWLEDGMENT The authors would like to thank Dr. R G. Cameron for his invaluable suggestions and advice. The authors would also like to thank their colleagues at National Grid (United Kingdom) for their technical input. R EFERENCES [1] K. J. Åström, “A robust sampled regulator for stable systems with monotone step responses,” Automatica, vol. 16, no. 3, pp. 313–315, 1980. [2] W.-S. Lu and K. S. P. Kumar, “A staircase model for unknown multivariable systems and design of regulators,” Automatica, vol. 20, no. 1, pp. 109–112, 1984. [3] S. Mossaheb, R. G. Cameron, and F. Li, “Passivity and the design of sampled regulators for uncertain systems,” Int. J. Control, vol. 45, no. 6, pp. 1941–1952, 1987.
1581
[4] Y. Chang, Z. Xu, G. Cheng, and J. Xie, “Coordinate damping control of HVDC and SVC based on wide area signal,” in Proc. IEEE Power Eng. Soc. General Meeting, Jun. 2006, pp. 1–7. [5] Z. Wang, S. Mei, J. Ye, X. Pang, and M. Li, “Study on passivitybased control of TCSC,” in Proc. IEEE Int. Conf. Power Syst. Technol., Oct. 2002, pp. 1918–1922. [6] Y. Phulpin, J. Hazra, and D. Ernst, “Model predictive control of HVDC power flow to improve transient stability in power systems,” in Proc. IEEE Int. Conf. Smart Grid Commun., Oct. 2011, pp. 593–598. [7] H. F. Wang, F. Li, and R. G. Cameron, “FACTS control design based on power system nonparametric models,” IEE Proc. Generat., Transmiss. Distrib., vol. 146, no. 5, pp. 409–415, Sep. 1999. [8] H. F. Wang and F. Li, “Multivariable sampled regulators for the co-ordinated control of STATCOM AC and DC voltage,” IEE Proc. Generat., Transmiss. Distrib., vol. 147, no. 2, pp. 93–98, Mar. 2000. [9] P. Kundur, Power System Stability and Control. New York, NY, USA: McGraw-Hill, 1993. [10] T. C. Hsia, System Identification: Least-Squares Methods, 1st ed. Lexington, MA, USA: Lexington Books, 1977. [11] MATLAB Toolbox Release 2009a, MathWorks, Inc., Natick, MA, USA, 2009.
Shadi Khaleghi Kerahroudi (S’12) received the B.Sc. degree in electrical power engineering from the University of Tehran, Tehran, Iran, and the M.Sc. (Hons.) degree in sustainable electrical power from Brunel University London, Uxbridge, U.K., in 2011, where she is currently pursuing the Ph.D. degree in industrial engineering, based in the industry with the electricity transmission system operator, National Grid, Wokingham, U.K. Her current research interests include stability control system based on flexible ac transmission devices, high-voltage dc links, and thyristor-controlled series compensation. Fan Li received the B.Sc. degree in control engineering from the Beijing University of Chemical Technology, Beijing, China, in 1982, and the Ph.D. degree in control engineering from the University of Bradford, Bradford, U.K., in 1987. He is currently a Power System Engineer with National Grid, Wokingham, U.K. Dr. Li received the Corporate Membership of The Institution of Engineering and Technology and was the Chartered Electrical Engineer in 1992. He was the co-author of the IEE Crompton Premium Award winning paper of a sampled regulator design for flexible ac transmission devices in 2000. Gareth A. Taylor (SM’06) received the B.Sc. degree from the University of London, London, U.K., in 1987, and the M.Sc. and Ph.D. degrees from the University of Greenwich, London, in 1992 and 1997, respectively. He was the National Grid U.K. Post-Doctoral Scholar with Brunel University London, Uxbridge, U.K., from 2000 to 2003, where he is currently a Professor and the Director of the Brunel Institute of Power Systems. His current research interests include transmission system operation, smart grids, and power system modeling and analysis. Maysam Abbod received the Ph.D. degree in control engineering from The University of Sheffield, Sheffield, U.K., in 1992. He is currently a Senior Lecturer of Intelligent Systems with the Department of Electronic and Computer Engineering, Brunel University London, Uxbridge, U.K. He has authored over 50 papers in journals, nine chapters in edited books, and over 50 papers in refereed conferences. His research work is on a multidisciplinary nature ranging from biomedical applications to industrial applications. His current research interests include intelligent systems for modeling and optimization. Martin E. Bradley received the Degree in electrical engineering from Imperial College London, London, U.K., in 1983, and the Ph.D. degree in high-voltage direct current (HVdc) thyristor valve technology from Aston University, Birmingham, U.K., in 1988. He was involved in the HVdc and static var compensator technology for General Electric Company, Rugby, U.K., from 1983 to 1989. Since then, he has been with National Grid, Wokingham. U.K., where he has been involved in various roles mainly relating to the development of software tools for planning and operations. He is currently the Strategy and Innovation Manager, overseeing programs of work in transmission analysis.
