Voltage Tracking Design for Electric Power Systems via SMC Approach Der-Cherng Liaw, Shih-Tse Chang and Yun-Hua Huang Abstract— This paper presents a output tracking design for regulating the load voltage of the electric power system Based on a model of electric power system proposed by Dobson and Chiang (1988), the saddle-node bifurcation and Hopf bifurcations were observed (Wang, et al, 1994) by treating the reactive power as system parameter Those bifurcations were found to lead to the appearance of the dynamic or the static voltage collapses of the power systems In construct to the state feedback design at the bifurcation point via the tuning of the Static Var Compensator as proposed in (Saad, et al, 2005), a sliding mode control (SMC) scheme using SVC was employed in this study to construct a load voltage tracking design control law for regulating the load voltage and hence providing the stability of electric power systems The numerical simulations demonstrated that the proposed control scheme not only could provide the regulation of the load voltage but also prevent and/or delay the appearance of bifurcation phenomena and chaotic behavior I INTRODUCTION In the recent years, the study of voltage collapse phenomena in the electric power systems has attracted lots of attention (e.g., [1]-[8]) It is due to the fact of facing the growing load demands in power systems but with little addition of the power generation and transmission facilities That leads the power systems to be operated near the stability limits As the load demands become too heavy to be offered, the magnitude of load voltage falls sharply to a very low level, which is referred as the so-called “voltage collapse.” A practical power system is a large electric network containing components such as generators, loads, transmission lines and voltage controllers In [2] and [3], Dobson and Chiang introduced a simple dynamical model for electric power systems, which consist of a generator, a nonlinear load and an infinite bus Based on that model, several results have been published regarding the nonlinear phenomena of electric power systems (e.g., [2]-[4], [6], [7], [9]) In addition, the occurrence of voltage collapse had been believed to be attributed to the existence of saddle-node bifurcation of electric power systems [4]-[5] However, it has been shown that voltage collapse may arise from the existence of Hopf bifurcation, which is prior to the appearance of This work was supported in part by the Nation Science Coucil, Taiwan, R.O.C under Grants NSC95-2221-E-009-339 Der-Cherng Liaw is with the Department of Electrical and Control Engineering, National Chiao Tung University, Hsinchu, 300, Taiwan, R.O.C saddle-node bifurcation [6]-[8] It is known that the voltage regulation issue is usually solved by the setting of either the tap changing ratio (e.g., [9]-[11]) or the extra adding capacitive load in practical electric power systems However, both schemes are only available for discrete tuning The Static Var Compensator (SVC) has recently been considered as a control actuator for improving system stability (e.g., [12]-[15]) For instance, direct state feedback linearization was proposed to simplify the design of controller as in [14] The study of the impact on the voltage collapse with state feedback control via the tuning of SVC was also given in [12]-[14] A washout filter-aided feedback design was proposed in [15] to delay the occurrence of the system instability and/or voltage collapse It is known that the sliding mode control (SMC) scheme possesses the advantages of fast response and less sensitivity to system uncertainties and/or disturbances than those by other methods To compensate system uncertainties and/or disturbances, several types of SMC schemes have been proposed (e.g., [16]-[18]) For instance, second-order sliding mode control scheme [16] and reliable control via sliding mode approach [18] have been proposed to enhance the performance of sliding mode control designs Due to those advantages, the sliding mode control scheme has been widely employed to design the control laws for a variety of applications (e.g., [19]-[23]) Instead of directly controlling the system behavior at the bifurcation point as proposed in [15], in this paper we consider a different approach by the load voltage regulation design of the electric power systems via the tuning of the SVC Such a design will be achieved by using sliding mode control scheme, which might also eliminate and/or delay the occurrence of bifurcation phenomena and system instabilities The organization of this paper is as follows A output tracking control scheme is proposed in Section II It is followed by the application to the load voltage control design in electric power systems in Section III Numerical simulations are given in Section IV to demonstrate the effectiveness of the proposed design Finally, Section V gives the conclusions II O UTPUT T RACKING D ESIGN Consider a class of nonlinear systems as given by dcliaw@cc.nctu.edu.tw x˙ = f1 (x) + g1 (x)η, (1) Shih-Tse Chang is with the Department of Electrical and Control Engineering, National Chiao Tung University, Hsinchu, 300, Taiwan, R.O.C η˙ = f2 (x, η) + d(x, η) + g2 · u, (2) stchang.ece94g@nctu.edu.tw Yun-Hua Huang is with the Department of Electrical and Control Engineering, National Chiao Tung University, Hsinchu, 300, Taiwan, R.O.C where x ∈ Rn , η ∈ R, and the system input u ∈ R In addition, d denote the system uncertainty The output function of system is given as y = h(x) ∈ R In the following, we consider to construct a control law for the input u such that the system output will approach a desired trajectory yd (t) That is, y → yd as time t increases to ∞ Let e = y − yd We then have e˙ = y˙ − y˙ d , = ∇h(x) · {f1 (x) + g1 (x)η} − y˙ d , (3) where ∇h(x) denotes the gradient of h(x) To follow the design procedure for sliding mode control (see e.g., [17]), we first treat the state variable η as a virtual input and find a control law η = η(x, yd ) such that the subdynamics of system as given in (1) will make the output error function e of (3) to be asymptotically stable at e = Choose V1 (e) = 12 e2 as a Lyapunov function candidate for the error dynamics as in Eq (3) We then have V˙ (e) = e · e, ˙ = e · {∇h(x) · [f1 (x) + g1 (x)η] − y˙ d }, = −ke2 , (4) with η = φ(x, yd ), where φ(x, yd ) = [∇h(x)g1 (x)]−1 {−∇h(x) · f1 (x) +y˙ d − ke}, for k > (5) By applying the Lyapunov stability criteria, we then have the next stability result Lemma 1: The origin e = of the error dynamics (3) will be asymptotically stabilizable by the virtual input η if ∇h(x) · g1 (x) = for all x ∈ D ⊂ Rn , where the subset D contains the neighborhood of e = Next, we construct control laws for the system input u such that the state variable η will approach the desired input φ(x, yd ) as defined in (5) with the appearance of system uncertainties Choose the sliding surface s = η − φ(x, yd ) We then have s˙ ˙ yd ), = η˙ − φ(x, ˙ yd ), = f2 (x, η) + d(x, η) + g2 · u − φ(x, = f2 (x, η) − f2 (x, φ(x, yd )) + d(x, η) + g2 · ure ˙ yd ) + φ(x, ˙ yd )|η=φ(x,y ) , −φ(x, (6) d where we choose u = ueq + ure with the assumption of g2 = and ueq = · {−f2 (x, φ(x, yd )) g2 ˙ yd )|η=φ(x,y ) } +φ(x, d re (7) For the case of d(x, η) = and u = 0, it is clear from the condition of s˙ = that all the states lying on the sliding surface s = will always be kept staying on the manifold s = by the choice of ueq as in (7) This provides that the sliding surface s = to be an invariant manifold when d(x, η) = and ure = Moreover, from Lemma 1, the error dynamics as in (3) will be asymptotically stable at e = as long as the system state lying on the sliding surface s = (i.e., η = φ(x, yd )) with the control input u = ueq Following the design of sliding mode control, we next focus on the dynamical behavior on the boundary layer That is, to construct the extra control input ure to drive the system state on the manifold of s = to enter the sliding surface s = Suppose the system uncertainty d(x, η) satisfy the following inequality: ||d(x, η)|| ≤ ρ(x, η), (8) where || · || denotes the norm function and ρ(x, η) is a nonnegative continuous function In order to compensate the uncertainties, we choose the extra control ure as ure = {−β(x, η) · sgn(s) − f2 (x, η) g2 ˙ yd ) +f2 (x, φ(x, yd )) + φ(x, ˙ yd )|η=φ(x,y ) }, −φ(x, d (9) where β(x, η) ≥ ρ(x, η) + γ with γ > and sgn(·) denotes the sign function Here, we also assume that g2 = Taking V2 (s) = 21 s2 as a Lyapunov function candidate for (6) with the control input u = ueq + ure as defined in (7) and (9), we then have V˙ (s) = s · {−β(x, η) · sgn(s) + d(x, η)} ≤ −γ|s|, (10) where | · | denotes the absolute value From the above discussions, we can conclude that 21 d|s| dt ≤ − γ · |s| This gives that |s(t)| ≤ |s(0)| − γt, ∀ t ≥ 0, which implies that the nonzero value of s will reach the sliding mode s = in a finite time We then have the next result for the output tracking design Theorem 1: Suppose g2 = and ∇h(x)·g1 (x) = for all x ∈ D ⊂ Rn , where the subset D contains the neighborhood of e = Then the output function y = h(x) for system (1)(2) will approach the desired output yd (t) via sliding mode control Moreover, one of the choices for the control input u = ueq + ure where ueq and ure are as given in (7) and (9), respectively Note that, it is known that the sign function used in the sliding mode control design might cause chattering problem In practical application, the sign function used in (9) can be replaced by a saturation function for relaxing the chattering issue An example is given in Section IV for numerical demonstrations of the proposed design III A PPLICATION TO E LECTRIC P OWER S YSTEMS In this section, we will apply the design as proposed in Section II to the load voltage tracking design of electric power systems First, we recall the mathematical model proposed by Dobson and Chiang ([2], [3]) for electric power systems Typical dynamical behavior of the power system will also be discussed It is followed by the load voltage tracking design for the power system via the tuning of the SVC controller A Dynamics of Electric Power Systems In the following, we recall the mathematical model proposed by Dobson and Chiang ([2], [3]) for electric power systems as given by δ˙m M ω˙ m kqω δ˙ T kqω kpv V˙ = ωm , (11) Em Ym = Pm − dm ωm + = +Em Ym V sin(δ − δm − θm ), (12) −kqv2 V − kqv V + Q(δm , δ, V ) = −Q0 − Q1 , (13) kpω kqv2 V + (kpω kqv − kqω kpv )V sin θm +kqω (P (δm , δ, V ) − P0 − P1 ) −kpω (Q(δm , δ, V ) − Q0 − Q1 ), (14) where δm , ωm , δ and V denote the generator phase angle, generator angular speed, load voltage phase angle and load voltage, respectively Here, the nonlinear PQ load are given as P (δm , δ, V ) = (Y0′ sin θ0′ + Ym sin θm )V −Em Ym V sin(δ − δm + θm ) −E0′ Y0′ V sin(δ + θ0′ ), (15) Q(δm , δ, V ) = reactive power when it is connected in parallel with the PQ load That is, the overall effective reactive load Q1 will become the summation of the original demanded reactive load Qo1 and the added reactive load Qadded from the SVC’s The mathemetical model for the SVC’s was proposed as (e.g., [15], [22]): −(Y0′ cos θ0′ + Ym cos θm )V +Em Ym V cos(δ − δm + θm ) +E0′ Y0′ V cos(δ + θ0′ ), (16) B˙ = E0 [1 + C Y0−2 − 2CY0−1 cos θ0 ]−1/2 , (17) Y0′ = Y0 [1 + C Y0−2 − 2CY0−1 cos θ0 ]1/2 , CY0−1 sin θ0 } = θ0 + tan−1 { − CY0−1 cos θ0 θ0′ (18) Qadded (20) = BV (21) Let x1 = δm , x2 = ωm , x3 = δ, x4 = V and x5 = B Then we can rewrite system (11)-(14) with SVC control as follows: x˙ = x˙ = x˙ = x˙ = (19) Detailed definitions of each system parameter and derivations of the model equations above can be referred to (e.g., [2], [3]) The typical dynamical behaviors of system (11)-(14) with respect to the variation of the extra demanded reactive load Q1 with P1 = are obtained by using the codes AUTO [24] and Matlab as depicted in Figures 2-4 It is shown in Figures 2-4 that the electric power systems might exhibit Hopf bifurcation and saddle-node bifurcation, and even the chaotic behavior as the value of Q1 varies Those results agree with the previous findings (e.g., [2]-[5], [10]-[11]) and might cause the undesired dynamical behavior and/or the drastic change of the load voltage In the next subsection, we will seek a control law for possibly eliminating the appearance of Hopf bifurcation and providing the regulation of the load voltage (KSV C · u − B), TSV C with Bmin ≤ B ≤ Bmax Here, B denotes the susceptance of the SVC, KSV C is the gain for the SVC, TSV C denotes the time constant and u denotes the control input In addition, the added reactive load by the SVC’s is given as (see, e.g., [15]) with E0′ = x˙ = x2 , (22) {Pm − dm x2 + Em Ym sin θm M +Em Ym x4 sin(x3 − x1 − θm )}, (23) {−kqv2 x24 − kqv x4 + Q(x1 , x3 , x4 ) kqω −Q0 − Q1 − x5 x24 }, (24) {kpω kqv2 x24 + (kpω kqv − kqω kpv )x4 T kqω kpv +kqω (P (x1 , x3 , x4 ) − P0 − P1 ) −kpω (Q(x1 , x3 , x4 ) − Q0 − Q1 − x5 x24 )} (25) (KSV C · u − x5 ), (26) TSV C where the extra demanded reactive load has become Q1 + x5 x24 Here, Q1 denotes the original demanded reactive load while x5 x24 is for the added reactive load created by the SVC’s B Load Voltage Tracking Design A washout filter type controller has been proposed in [15] to delay the appearance of bifurcation phenomena via the tuning of the SVC’s Instead of directly controlling the system stability at the bifurcated operating point, in this paper, we propose a different approach for the load voltage tracking design by using sliding mode control scheme Details are given as follows In the recent years, the SVC’s have been considered as a control scheme for voltage regulation in the electric power systems The configuration of the SVC’s is considered as a fixed capacitor connected in parallel with a thyristor controlled reactor (e.g., [15]), which might provide an additional Let the output of system (22)-(26) be the load voltage x4 Denote Vd (t) the desired load voltage for system (22)-(26) Comparing the mathematical model of the electric power systems as given in (22)-(26) with the form presented in (1)-(2), we have x = (x1 , x2 , x3 , x4 )T , η = x5 , h(x) = x4 A Open-Loop Dynamics and f1 (x) g1 (x) f11 (x) f12 (x) = f13 (x) , f14 (x) −1 = kqω · x4 kpω T kqω kpv f2 (x, η) = g2 = · x24 , −1 · x5 TSV C KSV C TSV C (27) (28) with f11 (x) = x2 and f12 (x) f13 (x) f14 (x) {Pm − dm x2 + Em Ym sin θm M +Em Ym x4 sin(x3 − x1 − θm )}, (29) = {−kqv2 x24 − kqv x4 + Q(x1 , x3 , x4 ) kqω −Q0 − Q1 }, (30) {kpω kqv2 x4 = T kqω kpv +(kpω kqv − kqω kpv )x4 = +kqω (P (x1 , x3 , x4 ) − P0 − P1 ) −kpω (Q(x1 , x3 , x4 ) − Q0 − Q1 )} First, we present the numerical results for the uncontrolled system (11)-(14) Bifurcation diagram for the example model is obtained by using code AUTO [24] as depicted in Fig The figure shows that the system will exhibit both Hopf bifurcation and saddle-node bifurcation, respectively, as denoted by “HB” and “SNB.” Here, the solid-line denotes the stable system equilibria while the dashed-dot-line is for the unstable ones The corresponding data for the two bifurcations is given in Table II In fact, we have observed several nonlinear dynamical behavior by using code Matlab as depicted in Fig and In those figures, we observe the period-1 oscillation as depicted in Fig 2(a) and 3(a), respectively, for the phase portraits and timing responses In addition, chaotic-like behaviors are also observed in Figs 2(b)-2(c) and Figs 3(b)-3(c) for the corresponding phase diagrams and time responses A drastic voltage change is found in Figs 2(d) and 3(d), which is very close to the saddle-node bifurcation point TABLE I D ATA FOR SYSTEM PARAMETERS Kpω = 0.4 p.u Kqv = −2.8 p.u P0 = 0.6 p.u Y0 = 3.33 p.u C = 3.5 p.u Em = 1.05 p.u Kpv = 0.3 p.u Kqv2 = 2.1 p.u Q0 = 1.3 p.u θ0 = deg Ym = 5.0 p.u Pm = 1.0 p.u Kqω = −0.03 p.u T = 8.5 p.u M = 0.01464 E0 = 1.0 p.u θm = deg dm = 0.05 p.u (31) To apply Theorem to load voltage tracking design for SV C system (22)-(26), we have the two conditions: g2 = K TSV C and ∇h(x) · g1 (x) = (kpω /T kqω kpv ) · x4 The next result for the load voltage tracking of system (22)-(26) follows readily from Theorem Theorem 2: The load voltage V of system (22)-(26) will approach any desired voltage Vd (t) via sliding mode control SV C if K TSV C = and (kpω /T kqω kpv ) · x4 = Moreover, one of the choices for the control input u = ueq + ure where ueq and ure are as given in (7) and (9), respectively, with yd = Vd (t), Remark 1: Note that in Theorem above, the parameters kpω , T , TSV C , kqω , and kpv , are generally nonzero In addition, in practical electrical power system the value of the load voltage x4 is non-negative The two conditions given in Theorem in practical application will then be reduced as KSV C = 0, while the attraction region for the exhibition of sliding motion will be x4 > IV N UMERICAL S IMULATIONS An example model is considered in this section to demonstrate the effectiveness of the proposed control design as given in Section III Here, we adopt the value of system parameter for system (11)-(14) from [15] as listed in Table I In addition, we choose P1 = and treat the extra demanded reactive power Q1 as system bifurcation parameter TABLE II L OCATION FOR BIFURCATION POINTS HB SNB Q1 2.98021 3.02578 δm (rad) 0.267426 0.30292 δ (rad) 0.0475028 0.0610163 V (p.u.) 0.873124 0.795136 B Closed-Loop Dynamics In the following, we consider to apply Theorem to the electrical power systems with the sliding mode control effort via the tuning of the SVC Here, we adopt the values from [15] as TSV C = 0.01 second and KSV C = In addition, for the control design, we choose k = and β(x, η) = To relax the undesired chattering behavior, here, we use a saturation function to replace the sign function for the sliding mode control in the numerical simulations as defined by (see, e.g., [17]) −1, for z < −1 z, for |z| ≤ sat(z) = 1, for z > Two cases are considered below to demonstrate the effectiveness of the proposed design First, we consider the case of the desired load voltage being Vd (t) = − 0.05e−t switched at time t = 30 second with the control gain β(x, η) = As shown in Fig 3, we find that the system instabilities prior to the control action are diminished and the load voltage approaches the desired load voltage Vd = 1.0 after t ≥ 30 second Note that, as shown in Fig 3(d), the voltage collapse appears at around t = 19 second for Q1 = 2.9899 In order to effectively recover from the instability, we set the control law to switch at t = 18 second for the case of Q1 = 2.9899 For the other case, we select the same desired load voltage as Vd (t) = − 0.05e−t with the system uncertainty d(x, η) = 0.01 In order to compensate the system uncertainty, the control gain β(x, η) is increased to 10 instead of The corresponding time responses of the load voltage phase angle x3 and load voltage x4 are obtained as depicted in Figs and 5, respectively As depicted in the numerical simulations above, the proposed control effort not only successfully provide the tracking of the load voltage but also eliminate the appearance of the undesired system behavior such as bifurcation type oscillation or the chaotic behavior V C ONCLUSIONS In this paper, we focus on the load voltage tracking design for a power system by using sliding mode control scheme It is achieved via the tuning of the SVC’s The simulations demonstrate the effectiveness of the proposed design for the power systems with or without the appearance of system uncertainty Comparing with other existing control schemes by using SVC’s (e.g., [15]), no exact knowledge of the operating condition is required in the proposed design This might relax the computation efforts for solving the operating point for the controller design The problem of the implementation issue of real time control is not considered in this study, however, it can be a research topic for the further study In this paper, we only consider the case of which P1 = and Q1 is varied The same technique might be applicable to the case of P1 = for the further study VI ACKNOWLEDGEMENTS The authors are very grateful to reviewers’ comments R EFERENCES [1] S Abe, Y Fukunaga, A Isono and B Kondo, “Power system voltage stability,” IEEE Transactions on Power Apparatus and Systems, vol PAS-101, no 10, pp 3830-3840, 1982 [2] I Dobson, H.-D Chiang, J S Throp and L Fekih-Ahmed, “A model of voltage collapse in electric power systems,” Proc 27th IEEE Conference on Decision and Control, Austin, Texas, pp 2104-2109, Dec 1988 [3] H.-D Chiang, I Dobson, R J Thomas, J S Throp and L FekihAhmed, “On voltage collapse in electric power systems,” IEEE Transactions on Power Systems, vol 5, no 2, pp 601-611, 1990 [4] I Dobson and H.-D Chiang, “Towards a theory of voltage collapse in electric power systems,” Systems & Control Letters, vol 13, pp 253-262, 1989 [5] H G Kwatny, A K Pasrija, and L Y Bahar, “Static bifurcation in 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0.87 0.868 0.25 0.26 0.27 x1 0.28 0.29 0.1 0.2 x 0.3 0.4 (a) 0.9 (b) 0.9 0.85 x4 x4 0.85 0.8 0.8 0.1 0.2 x 0.3 0.75 0.4 0.1 0.2 (c) x1 0.3 0.4 (d) x4 Fig Dynamical behavior in phase-diagram at the varied value of Q1 : (a) Q1 = 2.98201, (b) Q1 = 2.9891, (c)Q1 = 2.98975, and (d)Q1 = 2.9899 1.05 1.1 1 x4 0.75 Fig Time response of state x3 with SVC-control for Vd (t) = 1.0 − 0.05e−t : (a) Q1 = 2.98201, (b) Q1 = 2.9891, (c)Q1 = 2.98975, and (d)Q1 = 2.9899 0.85 0.95 0.9 0.85 0.9 20 40 Time (sec) (a) 60 20 30 Time (sec) (a) 40 0.7 50 1.1 0.8 1 0.75 20 40 Time (sec) (b) 0.9 10 20 30 40 Time (sec) (b) 50 10 20 30 Time (sec) (d) 50 0.9 60 0.8 0.8 0.7 0.9 0.8 0.85 10 20 30 Time (sec) (c) 40 50 0.7 40 x4 x4 10 1.1 0.95 Fig Time response of state x4 with SVC-control for Vd (t) = 1.0 − 0.05e−t with system uncertainty: (a) Q1 = 2.98201, (b) Q1 = 2.9891, (c)Q1 = 2.98975, and (d)Q1 = 2.9899 0.6 0.8 0.75 0.85 0.87 x4 x4 x4 0.875 0.865 0.8 0.95 x4 0.88 0.9 20 40 Time (sec) (c) 60 0.4 10 15 Time (sec) (d) 20 Fig Time response of power systems at the varied value of Q1 : (a) Q1 = 2.98201, (b) Q1 = 2.9891, (c)Q1 = 2.98975, and (d)Q1 = 2.9899