Nonlinear Adaptive Backstepping Excitation Controller Design for Higher-Order Models of Synchronous Generators

Proceedings of the 20th World Congress Proceedings of the 20th World Congress The International Federation of Congress Automatic Control Proceedings of 20th Proceedings of the the 20th World World The International Federation of Congress Automatic Control Toulouse, France, July 9-14, 2017 The International Federation of Automatic Control Available online at www.sciencedirect.com Proceedings of the 20th World Congress The International Federation of Automatic Control Toulouse, France, July 9-14, 2017 Toulouse, France, July The International of Automatic Control Toulouse, France,Federation July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017

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IFAC PapersOnLine 50-1 (2017) 4368–4373 Nonlinear Adaptive Backstepping Nonlinear Nonlinear Adaptive Adaptive Backstepping Backstepping Excitation Controller Design for Nonlinear Adaptive Backstepping Excitation Controller Design Excitation Controller Design for for Higher-Order Models Synchronous Excitation Controller for Higher-Order Models of of Design Synchronous Higher-OrderGenerators Models of Synchronous Higher-OrderGenerators Models of Synchronous Generators T. K. Roy ∗∗ , M. A. Generators Mahmud ∗∗ , Amanullah M. T. Oo ∗∗ ,

T. K. Roy ∗∗ , M. A. Mahmud ∗∗ , Amanullah M. T. Oo ∗∗ , ∗∗ T. Mahmud ,, Amanullah M. H. ∗∗ T. K. K. Roy Roy ,, M. M. A. A.and Mahmud Amanullah M. T. T. Oo Oo ,, and H. R. R. ∗Pota Pota ∗∗ ∗ ∗∗ H. T. K. Roy , M. A.and Mahmud , Amanullah M. T. Oo ∗ , and H. R. R. Pota Pota ∗ and H.University, R. Pota ∗∗Waurn Ponds, VIC 3216, ∗ School of Engineering, Deakin ∗ School of Engineering, Deakin University, Waurn Ponds, VIC 3216, ∗ School Engineering, Deakin University, Waurn Ponds, Australia. (tkroy, apel.mahmud, and aman.m)@deakin.edu.au School of of Email: Engineering, University, Ponds, VIC VIC 3216, 3216, Email: (tkroy,Deakin apel.mahmud, and Waurn aman.m)@deakin.edu.au ∗Australia. ∗∗ Australia. Email: (tkroy, apel.mahmud, and aman.m)@deakin.edu.au School of Engineering, Deakin University, Waurn Ponds, VIC School of Engineering and Information Technology (SEIT), The ∗∗ Australia. Email: (tkroy, apel.mahmud, and aman.m)@deakin.edu.au School of Engineering and Information Technology (SEIT), 3216, The ∗∗ ∗∗ School of Engineering and Information (SEIT), The Australia. (tkroy, apel.mahmud, and Technology aman.m)@deakin.edu.au University New South Wales, Canberra, ACT 2600, Australia. School Email: ofof Engineering and Information Technology (SEIT), The University of New South Wales, Canberra, ACT 2600, Australia. ∗∗ University New South Wales, Canberra, ACT 2600, Australia. School ofof and Information Technology (SEIT), The [email protected] University ofEngineering New E-mail: South Wales, Canberra, ACT 2600, Australia. E-mail: [email protected] [email protected] University of New E-mail: South Wales, Canberra, ACT 2600, Australia. E-mail: [email protected] E-mail: [email protected] Abstract: Abstract: In In this this paper, paper, a a nonlinear nonlinear excitation excitation control control scheme scheme is is proposed proposed to to enhance enhance transient transient Abstract: In this paper, a nonlinear excitation control scheme is proposed to enhance transient stability and voltage regulation of higher-order synchronous generators. The is Abstract: In this paper, a nonlinear excitation control scheme is proposed to enhance transient stability and voltage regulation of higher-order synchronous generators. The controller controller is stability and voltage regulation of higher-order synchronous generators. The controller is Abstract: In this a nonlinear control scheme proposed to enhance transient designed based on an backstepping method and the dynamical model stability voltage regulation of excitation higher-order synchronous generators. The controller is designed and based onpaper, an adaptive adaptive backstepping method and the isfifth-order fifth-order dynamical model of of designed based on backstepping method and the fifth-order dynamical model of stability andgenerators voltage regulation higher-order synchronous generators. The controller is synchronous is to the controller. controller is designed designed based on an an adaptive adaptive andThe the proposed fifth-order dynamical of synchronous generators is used used backstepping toofdesign design the method controller. The proposed controller is model designed synchronous generators is used to design the controller. The proposed controller is designed designed based on an adaptive backstepping method and the fifth-order dynamical model of recursively to adapt some unknown stability sensitive parameters of synchronous generators and synchronous generators is used to design the controller. The proposed controller is designed recursively to adapt some unknown stability sensitive parameters of synchronous generators and recursively to adapt some sensitive parameters of synchronous generators and synchronous used to stability designthrough the controller. Thelaws proposed controller is designed these parameters are adaptation on formulation of recursively togenerators adapt someisunknown unknown stability sensitive parameters of based synchronous and these unknown unknown parameters are estimated estimated through adaptation laws based on the the generators formulation of these unknown parameters are estimated through adaptation laws based on the formulation of recursively to adapt some unknown stability sensitive parameters of synchronous generators and control Lyapunov functions (CLFs). The proposed control scheme is also capable to overcome these parameters estimated laws based the formulation of controlunknown Lyapunov functionsare (CLFs). The through proposedadaptation control scheme is alsoon capable to overcome control Lyapunov functions (CLFs). The proposed control is capable to these unknown parameters are estimated through adaptation laws based the formulation of the of Finally, the effectiveness of control Lyapunov functionsproblems (CLFs). The proposedparameters. control scheme scheme is also also capable to overcome overcome the over-parameterization over-parameterization problems of unknown unknown parameters. Finally, theon effectiveness of the the the over-parameterization problems of unknown parameters. Finally, the effectiveness of the control Lyapunov functions (CLFs). The proposed control scheme is also capable to overcome proposed scheme is evaluated on a test single machine infinite bus system by applying a symthe over-parameterization problems of unknown parameters. Finally, the effectiveness of the proposed scheme is evaluated on a test single machine infinite bus system by applying a symproposed scheme on test single machine infinite bus system by applying aaofsymthe over-parameterization problems of at unknown parameters. effectiveness the metrical three-phase short-circuit different locations the test system and proposed scheme is is evaluated evaluated on a afault test single machine infiniteof bus by applying symmetrical three-phase short-circuit fault at different locations ofFinally, the system testthe system and compared compared metrical three-phase short-circuit fault at different locations of the test system and compared proposed scheme is evaluated on a test single machine infinite bus system by applying a symto that of an existing adaptive backstepping controller. Simulation results demonstrate the metrical three-phase short-circuit fault at different locations of the test system and compared to that of an existing adaptive backstepping controller. Simulation results demonstrate the to that an existing adaptive backstepping controller. Simulation results demonstrate the metrical three-phase short-circuit fault at different of the test compared superiority of proposed control scheme over the existing controller in terms of settling time to that of of an existing adaptive backstepping results the superiority of the the proposed control scheme overcontroller. the locations existingSimulation controller in system termsdemonstrate ofand settling time superiority of proposed control scheme over the controller in of to of an existing adaptive backstepping controller. the and providing additional damping into the system. superiority of the the proposed control scheme the existing existingSimulation controller results in terms termsdemonstrate of settling settling time time andthat providing additional damping into the over system. and additional damping the system. superiority of the proposed control into scheme the existing controller in terms of settling time and providing providing additional damping into the over system. © 2017, IFAC (International Federationinto of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. and providing additional damping the system. Keywords: Keywords: Adaptive Adaptive backstepping backstepping controller, controller, excitation excitation system, system, Lyapunov Lyapunov function, function, stability stability Keywords: Adaptive backstepping backstepping controller, controller, excitation excitation system, system, Lyapunov Lyapunov function, function, stability stability sensitive parameters Keywords: Adaptive sensitive parameters sensitive parameters parameters Keywords: Adaptive backstepping controller, excitation system, Lyapunov function, stability sensitive sensitive parameters 1. (2004)) 1. INTRODUCTION INTRODUCTION (2004)) in in order order to to eliminate eliminate the the low-frequency low-frequency oscillaoscilla1. INTRODUCTION INTRODUCTION (2004))However, in order orderthe to LQR eliminate the low-frequency low-frequency oscillations. controller as proposed proposed oscillain (Ko (Ko 1. (2004)) in to eliminate the tions. However, the LQR controller as in 1. INTRODUCTION tions. However, the LQR controller as proposed proposed inof(Ko (Ko (2004)) in order to eliminate the low-frequency oscillaet al. (2004)) cannot guarantee voltage stability tions. However, the LQR controller as in et al. (2004)) cannot guarantee the voltage stability of the the Power et al. al. (2004)) (2004)) cannot guarantee theasvoltage voltage stability of(Ko the However, thethe LQR controller as proposed inof system. Moreover, controllers proposed in (Kundur Power systems systems are are large-scale large-scale nonlinear nonlinear and and interconintercon- tions. et cannot guarantee the stability the system. Moreover, the controllers as proposed in (Kundur Power are large-scale nonlinear and interconnected systems which require to operate close their staPower are large-scale and their interconsystem. Moreover, theguarantee controllers asvoltage proposed in (Kundur (Kundur al. (2004)) cannot the stability of the (1994); Ko et al. (2004)) are mainly designed based on nected systems which require tononlinear operate close sta- et system. Moreover, the controllers as proposed in (1994); Ko et al. (2004)) are mainly designed based on the nected systems systemsorder which require tononlinear operate close close their sta- system. Power are to large-scale and demands. interconbility meet increased load nected which require to operate their sta(1994); Ko Ko et al. al. (2004)) (2004)) aresystems mainly designed based on the the Moreover, controllers asdesigned proposed in (Kundur linearized model of power which are capable to bility limits limits in in order to meet the the increased load demands. (1994); et are mainly on linearized model ofthe power systems which arebased capable to bility limits in order to meet the increased load demands. nected systems which to increased operate close their staWhen such power systems disturbances, the bility in order torequire meetexperience the load demands. linearized model of operation power systems which arebased capable to Ko et al. (2004)) are mainly designed on the provide satisfactory over a fixed set of operating Whenlimits such power systems experience disturbances, the (1994); linearized model of power systems which are capable to provide satisfactory operation over a fixed set of operating When such power systems experience disturbances, the bility limits in order to meet the increased load demands. stability margin of the whole system deteriorates and When such power systems experience disturbances, the provide satisfactory operation over a fixed set of operating linearized model of power systems which are capable to points and unable to cope the changes in operating points stability margin of the whole system deteriorates and provide satisfactory a fixed set of operating points and unable tooperation cope the over changes in operating points stabilitysuch margin ofsystems thebecomes whole system deteriorates and When power experience disturbances, the provide sometimes, the system unstable. In stability margin of the whole system deteriorates and points and unable tooperation cope as thethe changes in operating points satisfactory over a fixed set ofmodels operating due to large disturbances power system are sometimes, the system becomes unstable. In power power syssyspoints and unable to cope the changes in operating points due to large disturbances as the power system models are sometimes, the system system becomes unstable. In are power sysstability of the whole deteriorates and points tems, problems related to the stability mainly sometimes, the becomes unstable. In power sysdue to to and largeunable disturbances as the power in system models are to cope the changes operating points highly nonlinear. tems, the themargin problems related to system the stability are mainly due large disturbances as the power system models are highly nonlinear. tems, the problems related to the stability are mainly sometimes, thethe system becomes unstable. In are power sysdominated by dynamics of generators where the tems, the problems related to the the stability mainly highly nonlinear. to nonlinear. large disturbances as the power system models are dominated by the dynamics of the generators where the due highly The of dominated by theare dynamics of the generators where the highly tems, theof related of to the the stability Under are mainly majority these synchronous generators. these dominated by the dynamics generators where the The limitation limitation of operating operating points points of of linear linear controllers controllers can can nonlinear. majority ofproblems these are synchronous generators. Under these Theovercome limitationby of using operating points controllers of linear linear controllers controllers can be nonlinear majority of of by these are synchronous generators. Under these dominated the dynamics of the generators where the The limitation of operating points of can circumstances, the excitation systems of synchronous genmajority these are synchronous generators. Under these be overcome by using nonlinear controllers and and aa great great circumstances, the excitation systems of synchronous gen- The be overcome overcome by using nonlinear controllers and a great great limitation of operating points of linear controllers can deal of attention has been paid over the last few decades circumstances, theare excitation systems of synchronous synchronous gen- be majority of these synchronous generators. Under these by using nonlinear controllers and a erators are used to provide damping into the system circumstances, the excitation systems of gendeal of attention has been paid over the last few decades to to erators are used to provide damping into the system be dealovercome ofthis attention has been paid over overcontrollers theexcitation last few few decades to by has using nonlinear and a great solve problem. Inbeen the literature of controller erators are used to provide damping into the the system circumstances, the excitation systems of synchronous gendeal of attention paid the last decades to and the automatic voltage regulators (AVRs) are used to erators are used to provide damping into system solve this problem. In the literature of excitation controller and the automatic voltage regulators (AVRs) are used to deal solve this problem. In the literature of excitation controller of attention has been paid over the last few decades to design for power systems, there exist several nonlinear conand the automatic voltage regulators (AVRs) are used to erators are used to provide damping into the system solve this problem. In the literature of excitation controller regulate the voltage to maintain the stability of power and the automatic voltage regulators to design for power systems, there exist several nonlinear conregulate the voltage to maintain the(AVRs) stabilityareofused power designthis for problem. powersuch systems, there exist exist several nonlinear consolve Inas literature ofseveral excitation controller trol feedback linearization (FBL) (Mahregulate the voltage to maintain maintain the(AVRs) stability ofused power and the automatic voltage regulators to for power systems, there nonlinear consystems. The effectiveness of the stability of regulate the to the stability of power trol techniques techniques such asthe feedback linearization (FBL) (Mahsystems. The voltage effectiveness of preserving preserving the are stability of design trol techniques such as feedback linearization (FBL) (Mahdesign for power systems, there exist several nonlinear conmud et al. (2014a)), sliding mode controller (SMC) (Dash systems. the The voltage effectiveness of preserving the stability stability of trol regulate to on maintain the of stability of power such as feedback linearization (FBL) (Mahpower depends the types controllers used systems. The effectiveness of the of mudtechniques et al. (2014a)), sliding mode controller (SMC) (Dash power systems systems depends on thepreserving types of controllers used mud et(1996)), al. (2014a)), (2014a)), sliding mode controllerapproach (SMC) (Dash trol techniques suchadaptive as feedback linearization (FBL) (Mahet al. and backstepping (Roy power systems depends on(Mahmud thepreserving types(2014)). of controllers controllers used systems. The effectiveness of the stability of mud et al. sliding mode controller (SMC) (Dash for the excitation systems power systems depends on the types of used et al. (1996)), and adaptive backstepping approach (Roy for the excitation systems (Mahmud (2014)). et al. (1996)), and adaptive backstepping approach (Roy mud et al. (2014a)), sliding mode controller (SMC) (Dash al. (2015b)). These nonlinear controllers are designed by for the excitation systems (Mahmud (2014)). power systems depends on the types of controllers used et al. (1996)), and adaptive backstepping approach (Roy for the excitation systems (Mahmud (2014)). et al. (2015b)). These nonlinear controllers are designed by Power stabilizers are used et al. al. (2015b)). These nonlinear controllers are designed by al. (1996)), and adaptive backstepping approach (Roy considering one-axis model of the synchronous generator for the system excitation systems (PSSs) (Mahmud Power system stabilizers (PSSs) are(2014)). used to to damp damp out out et (2015b)). These nonlinear controllers are designed by considering one-axis model of the synchronous generator Power system stabilizers (PSSs) are used to damp out the low-frequency oscillations in power systems and the Power system stabilizers (PSSs) are used to damp out considering one-axis model of the synchronous generator et al. (2015b)). These nonlinear controllers are designed by which does not clearly reflect its actual behaviors. the low-frequency oscillations in power systems and the considering one-axis model theactual synchronous generator which does not clearly reflectofits behaviors. the low-frequency low-frequency oscillations inthrough power systems and out the considering Power system stabilizers (PSSs) are used to damp voltage stability is maintained AVRs (Kundur the in power systems and the which does does not not clearly reflectofits its actual behaviors. one-axis model theactual synchronous generator voltage stability isoscillations maintained through AVRs (Kundur which clearly reflect behaviors. FBL is the employed excitation voltage stability isoscillations maintained through AVRs (Kundur the low-frequency power systems and the which (1994)). These are lead-lag compensators voltage stability is maintained through AVRs (Kundur FBL technique technique is one one of of the mostly mostly employed excitation does not clearly reflect its actual behaviors. (1994)). These PSSs PSSs are mainly mainlyinthe the lead-lag compensators FBL technique technique is one one of systems. the mostly mostly employed excitation control scheme power FBL can (1994)). stability These PSSs are mainly mainly the lead-lag compensators voltage is maintained through AVRs (Kundur is of the employed excitation which the difficulties of transient stabil(1994)). These are the lead-lag compensators control scheme for for power systems. FBL can be be categorized categorized which have have the PSSs difficulties of maintaining maintaining transient stabil- FBL control scheme for power systems. FBL can be categorized FBL technique is one of the mostly employed excitation into three different categories for power system which have the difficulties of maintaining transient stabil(1994)). These PSSs are mainly the lead-lag compensators control scheme for power systems. FBL can be categorized ity severe disturbances. A regulator which have the difficulties of maintaining transient stabil- into three different categories for power system applicaapplicaity under under severe disturbances. A linear linear quadratic quadratic regulator into three different categories for power system applicacontrol power systems. FBL cansystem be categorized tions in terms of the excitation systems. These ity under severe disturbances. A linear quadratic regulator which have the difficulties of maintaining transient stabildifferent categories power applica(LQR)-based control scheme is proposed in (Ko et ity under severe disturbances. quadratic regulator tionsthree in scheme terms offorcontrolling controlling thefor excitation systems. These (LQR)-based control scheme Aislinear proposed in (Ko et al. al. into tionsthree in terms terms of controlling controlling thefor excitation systems. These different categories power system applica(LQR)-based control scheme Ais islinear proposed in (Ko (Ko et al. al. into ity under severe disturbances. quadratic regulator tions in of the excitation systems. These (LQR)-based control scheme proposed in et tions in terms of controlling the excitation systems. These (LQR)-based control scheme is proposed in (Ko et al. Copyright © 2017, 2017 IFAC 4464Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 4464 Copyright ©under 2017 responsibility IFAC 4464 Peer review© of International Federation of Automatic Copyright 2017 IFAC 4464Control. 10.1016/j.ifacol.2017.08.879 Copyright © 2017 IFAC 4464

Proceedings of the 20th IFAC World CongressT.K. Roy et al. / IFAC PapersOnLine 50-1 (2017) 4368–4373 Toulouse, France, July 9-14, 2017

categories are direct feedback linearization (DFBL), exact feedback linearization (EFBL), and partial feedback linearization (PFBL). When DFBL is employed the power system is linearized into a system of the similar order and a new state variable in terms of active power delivered into the system is introduced (Chapman et al. (1993)). EFBL is similar to that of the DFBL where the order of feedback linearized system equals to the order of the system (Verrellia and Damm (2010)). In both cases, the rotor angle of the synchronous generators, which is not available from direct measurement, appears in excitation control laws though an observer-based controller can solve this problem (Mahmud et al. (2012)). However, the use of an observer is not a cost-effective and straightforward solution. In such cases, PFBL can be used as it uses the speed deviation of the power system as the output function and the feedback linearized system becomes a reducedorder system (Mahmud et al. (2014b,a)). However, the main drawback of these controllers is that the performance of the controller heavily relies on the parameters of systems and in practice, it is very difficult to know the exact parameters of the system. So far, the feedback linearizing excitation controller is designed based on the third-order model of synchronous generators and it may not applicable (especially DFBL and EFBL) for higher-order models. Sliding mode controllers (SMCs) are useful to overcome the parameter sensitivity problems the feedback linearizing controllers. Sliding mode excitation controllers are proposed in (Ribeiro et al. (2015)) which are less sensitive to the variations in parameters and external disturbances. In (Colbia-Vega et al. (2008)), a robust sliding mode excitation controller is proposed by considering external uncertainties. However, the main difficulty to implement the SMC is the selection of the sliding surface with respect to the possible measurement errors and external disturbances. In (Ribeiro et al. (2015); Colbia-Vega et al. (2008)), the controller are still designed based on the oneaxis model of synchronous generators and the dynamics of the AVR are neglected. This limitation is overcome by employing a higher-order block SMC in (Loukianov et al. (2011)) where the robustness is analyzed with respect to the unmodeled dynamics of the exciters. Again, the selection of sliding surface and reduction of chattering effects is very challenging for higher-order system model, especially in the presence of unmodeled exciter dynamics. Adaptive backstepping controllers have the ability to dynamically estimate the unknown parameters of the system. An adaptive backstepping controller is designed in (Wang et al. (1994)) for a single machine infinite bus (SMIB) system by considering the infinite bus voltage and transmission line parameters are as unknown parameters. However, the infinite bus voltage is always known. A similar approach is proposed in (Mitra et al. (2015)) in order to design the excitation controllers for power system by considering the damping coefficients of synchronous generators as unknown parameters. However, the other stability sensitive parameters considered as exactly known and this is not usual the case. Recently, a nonlinear robust adaptive backstepping controller is proposed in Roy et al. (2015a) where all stability sensitive parameters such as the damping coefficient, direct-axis open-circuit time constant, transient reactance, and external disturbances are taken into

4369

Vt

Infinite Bus

G Transformer

Transmission line

Vs

Fig. 1. Test System account. Although the adaptive backstepping controllers as proposed in (Mitra et al. (2015); Roy et al. (2015a)) are effective for ensuring power systems stability with some specific limitations, the voltage regulation problem is not clear as these controllers are designed based on the thirdorder model of a synchronous generator and the AVR model is completely neglected. The aim of this paper is to design a nonlinear adaptive backstepping controller to improve the voltage regulation and transient stability by considering a two-axis model of the synchronous generator along with the dynamics of the AVR. The stability sensitive parameters of the synchronous generator are estimated through the adaptation laws. Control Lyapunov functions (CLFs) are formulated at different stages of the controller design process to verify the stability of the system. The performance of the controller is tested on an SMIB system under a three-phase short-circuit fault at different locations and compared with an existing adaptive backstepping controller. The significant advantage of the proposed controller is that it can ensure the voltage regulation and transient stability in a much faster way which will be obvious from the simulation results as presented in a later section of this paper. 2. SMIB SYSTEM MODEL For designing the proposed controller, a simple power system, namely, SMIB system as shown in Fig. 1, is used in this paper. This system comprises a synchronous generator which is connected to an infinite bus through a transformer and two parallel transmission lines. The twoaxis dynamical model of a synchronous generator is used in this paper. The complete dynamical model of the synchronous generator connected to an infinite bus with an IEEE Type II excitation system can be written as follows (Mahmud et al. (2017)): δ˙ = ω − ω0 D ω0 ω0 (ω − ω0 ) + Pm − (E ′ Iq + Ed′ Id ) ω˙ = − 2H 2H 2H q (xd − x′d ) 1 1 E˙ q′ = − ′ Eq′ − Id + ′ Ef d ′ Tdo Tdo Tdo (1) ′ (xq − xq ) 1 Iq E˙ d′ = − ′ Ed′ + ′ Tqo Tqo Ef d KA + (Vref + Vc − Vt ) E˙ f d = − TA TA where the symbols have their usual meanings and the definitions of these symbols can be found in (Mahmud et al. (2017)). Based on this model, the nonlinear adaptive excitation controller will be designed using backstepping method in the following section. However, before design the proposed controller the control problem formulation is discussed in the next section.

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3. CONTROL PROBLEM FORMULATION

Step 1: To satisfy the control objective, the rotor angle tracking error can be defined as:

From the dynamical model of an SMIB system as described in Section II, it is clear that the power system model is highly nonlinear and the main reason of these nonlinearities is the interconnections as represented by the currents Id and Iq . Power systems are inherently stable and the synchronism as well as the stability margin of the system are affected due to the presence of any small or large disturbances which are not desirable for the practical operation. Moreover, it is difficult to exactly know the parameters of power systems (especially the parameters of synchronous generators which are dependent on the transient characteristics). The nominal values of these parameters are considered based on the steady-state operation. However, the steady-state operating points change continuously in power systems due to fluctuating load demand. The most stability sensitive parameters of synchronous generators are the d- and q-axes open-circuit transient time constants and the variations of these parameters can significantly affect the stability of power systems. Moreover, the time constant of the voltage regulator varies over time as the AVR continuously adjust the terminal voltage during the transient. Therefore, it is worth to consider these parameters as unknown and with these unknown parameters, some terms in the dynamical model of the synchronous generator as represented by (1) can be written as follows: (xq −x′ ) (x −x′ ) θ1 = T1′ , θ2 = − dT ′ d , θ3 = − T1′ , θ4 = T ′ q , θ5 = do

− T1A ,

and θ6 =

do

qo

e1 = x1 − x1d Taking the time derivative of (3), yields

e˙ 1 = x˙ 1 = x2 (4) Here, it is assumed that x2 is a virtual control variable to stabilize the dynamics of rotor angle tracking error as described by (4). To find the stabilizing function for x2 , a control Lyapunov function (CLF) is defined as follows 1 2 e 2 1 ˙ 1 is Using the value of e˙ 1 from (4), W W1 =

If all these unknown parameters are incorporated within the SMIB model as represented by (1), it can be written as: x˙ 1 = x2 D ω0 ω0 x2 + Pm − (Iq x3 + Id x4 ) 2H 2H 2H x˙ 3 = −θ1 x3 + θ2 Id + θ1 x5

α = −k1 e1 (7) where k1 is a positive design constant which guarantees ˙ 1 is converted to ˙ 1 ≤ 0. Then W W ˙ 1 = −k1 e21 W

x˙ 4 = θ3 x4 + θ4 Iq

(8)

It clearly shows that (8) is negative semi-definite which is the main aim of this step and practically feasible for synchronous generators. Step 2: The stabilizing function α may not be the exact one, as it is a virtual control input. So, let’s define the second error variable e2 as

x˙ 2 = −

(2)

(5)

˙ 1 = e1 x2 (6) W At this point, the stabilizing function for x2 needs to be ˙ 1 would be negative semichosen in such a way that W ˙ definite, i.e., W1 ≤ 0 which makes the system stable. In this case, the stabilizing function x2 = α is chosen as

qo

KA TA .

(3)

e2 = x2 − α

(9)

Its derivative is D ω0 ω0 x2 + Pm − (Iq x3 + Id x4 ) + k1 x2(10) 2H 2H 2H Since α˙ = −k1 x2 . Now by augmenting the first CLF with a quadratic term in the error variable e2 , the following CLF is considered e˙ 2 = −

x˙ 5 = θ5 x5 + θ6 (Vref + Vc − Vt ) where δ, ω − ω0 , Eq′ , Ed′ , and Ef d are considered as x1 , x2 , x3 , x4 , and x5 respectively. In this paper, the main objective is to design an adaptive excitation controller to enhance the voltage regulation and overall transient stability of the power system. Another objective is to guarantee the robustness of the proposed scheme with respect to the unknown parameters θ1 , θ2 , θ3 , θ4 , θ5 , and θ6 . The following section shows the necessary steps to design a nonlinear adaptive excitation controller for a synchronous generators used in an SMIB system. 4. ADAPTIVE CONTROLLER DESIGN FOR SMIB SYSTEM

1 (11) W2 = W1 + e22 2 ˙ 1 from (8) and e˙ 2 from (10), Substituting the values of W the derivative of W2 can be written as ˙ 2 = −k1 e21 + e2 ( ω0 Pm − D x2 − ω0 Id x4 − W 2H 2H 2H (12) ω0 Iq x3 + k1 x2 ) 2H At this point, the stabilizing functions for x3 = α1 , and ˙2≤0 x4 = α2 are chosen as follows to make W

The section is aimed to obtain the control law Vc in order to stabilize the SMIB system as represented by (2) under severe disturbances while using an adaptive backstepping controller. The design procedure of the proposed scheme is elaborately discussed in the following steps: 4466

2H ω0 D Pm − x2 + k1 x2 ) ( ω0 Iq 2H 2H 2H α2 = k2 e2 ω0 Id α1 =

(13)

Proceedings of the 20th IFAC World CongressT.K. Roy et al. / IFAC PapersOnLine 50-1 (2017) 4368–4373 Toulouse, France, July 9-14, 2017

where k2 is a positive design constant. Inserting (13) into (12) yields ˙ 2 = −k1 e21 − k2 e22 W

Step 3: However, since the stabilizing functions as represented by (13) are not the real control inputs, therefore, the third and fourth error variables are defined as follows e3 = x3 − α1

(15)

e4 = x4 − α2

(16)

The time derivative of e3 by using the value of x˙ 4 can be expressed as (17)

In order to handle the unknown parameters θ1 , and θ2 as appeared in (17); it is essential to define these parameters in term of their estimated values θˆ1 , and θˆ2 , respectively. Now the estimation errors of these parameters can be defined as θ˜i = θi − θˆi with i = 1, 2. Thus in terms of estimation errors, (17) can be rewritten as e˙ 3 = (θˆ1 + θ˜1 )(x5 − x3 ) + (θˆ2 + θ˜2 )Id − α˙ 1

parameters adaptation laws will not be selected at this step to update θˆ1 , θˆ2 , θˆ3 , and θˆ4 . Instead, the following tuning functions are defined:

(14)

˙ 2 is negative semi-definite From (14), it can be seen that W for k1 > 0 and k2 > 0. The remaining error dynamic of this design procedure are discussed in the following steps.

e˙ 3 = −θ1 x3 + θ2 Id + θ1 x5 − α˙ 1

τ1 = γ1 e3 (x5 − x3 ), τ2 = γ2 e3 Id , τ3 = γ3 e4 x4 , τ4 = γ4 e4 Iq

˙ 3 = −k1 e2 − k2 e2 − k3 e2 − k4 e2 − 1 θ˜1 (θˆ˙1 − τ1 )− W 1 2 3 4 γ1 (25) 1 ˜ ˆ˙ 1 ˜ ˆ˙ 1 ˜ ˆ˙ θ2 (θ2 − τ2 ) − θ3 (θ3 − τ3 ) − θ4 (θ4 − τ4 ) γ2 γ3 γ4 It is clear that the first, second, third, and fourth terms in the right side of (25) is negative semi-definite. Since the decision about the estimation of unknown parameters has not been finalized yet, the remaining terms on the right side of (25) are tolerated at this step. Now it is essential to obtain the time derivative of α3 in order to complete the next step which can be written as α˙ 3 = A + B1 θ1 + B2 θ2

1 1 W3 =W2 + (e23 + e24 + θ˜12 2 γ1 (20) 1 1 1 + θ˜22 + θ˜32 + θ˜42 ) γ2 γ3 γ4 where γi with i = 1, 2, 3, 4 is the adaptation gain parameter. Now by using (14), (18), and (19), the derivative of W3 can be written as ˙ 3 = −k1 e2 − k2 e2 + e3 (θˆ1 x5 − θˆ1 x3 + θˆ2 Id − α˙ 1 ) W 1 2 θ˜1 ˙ + e4 (θˆ3 x4 + θˆ4 Iq − α˙ 2 ) − [θˆ1 − γ1 e3 (x5 − x3 )] γ1 (21) ˜ ˜ θ3 ˙ θ2 ˙ − (θˆ2 − γ2 e3 Id ) − (θˆ3 − γ3 e4 x4 ) γ2 γ3 ˜ θ4 ˙ − (θˆ4 − γ4 e4 Id ) γ4 The derivative of W3 as represented by (21) should be negative semi-definite in order to stabilize the system. Under this circumstance, the stabilizing function x5 = α3 can be chosen as 1 ˆ (θ1 x3 − θˆ2 Id + α˙ 1 − k3 e3 ) ˆ θ1 θˆ3 x4 + θˆ4 Iq − α˙ 2 = −k4 e4

α3 =

1 1 ˙ ˙ (¨ α1 − θˆ2 Id − θˆ2 I˙d ) − 2 (α˙ 1 − θˆ2 Id − k3 e3 )θˆ1 , θˆ1 θˆ1 k3 B1 = B(x5 − x3 ), B2 = BId , and B = 1 − α˙ 1 . θˆ1 A=

The derivation of final control law along with parameters adaptation laws and the stability analysis of the whole system is presented in the next step. Step 4: In this step, the final error is defined as e5 = x5 − α3

(27)

Now the objective is to asymptotically vanish the error variables (e1 , e2 , e3 , e4 , e5 ). To this end, the final error dynamics using (2) and (26) can be written as e˙ 5 = θ5 x5 + θ6 (Vref + Vc − Vt ) − A − B1 θ1 − B2 θ2

(28)

and in terms of the estimation errors (28) can be rewritten as e˙ 5 = (θˆ5 + θ˜5 )x5 + (θˆ6 + θ˜6 )(Vref + Vc − Vt ) − A − B1 (θˆ1 + θ˜1 ) − B2 (θˆ2 + θ˜2 )

(29)

The original control input Vc appears in (29) and the main purpose is to calculate the Vc so that the errors e1 , e2 , e3 , e4 , and e5 converge to zero as t → ∞. The final CLF can be considered to design the actual control law and parameters adaptation laws as:

(22)

1 1 1 W4 = W3 + (e25 + θ˜52 + θ˜62 ) 2 γ5 γ6

(23)

And to avoid the over-parameterization caused by appearance of θ1 , θ2 , θ3 , and θ4 in the subsequence steps, the

(26)

where

(18)

(19)

(24)

Using (22), (23), and (24), the derivative of W3 can be rewritten as

Similarly, (16) can be expressed as e˙ 4 = (θˆ3 + θ˜3 )x4 + (θˆ4 + θ˜4 )Iq − α˙ 2 For this case, consider the following CLF

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(30)

˙ 3 , and e˙ 5 , the derivative of After inserting the values of W W4 can be written as

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Proceedings of the 20th IFAC World CongressT.K. Roy et al. / IFAC PapersOnLine 50-1 (2017) 4368–4373 4372 Toulouse, France, July 9-14, 2017

4 Speed deviation (pu)

1 NABC ENABC

0.8 0.6 0.4 0.2 0 9.5

NABC ENABC

0 -2 -4

10

10.5

11

11.5 12 Time (s)

12.5

13

13.5

9.5

14

Fig. 2. Terminal voltage in case of a short-circuit fault at the terminal of the generator ˙ 5 = −k1 e2 − k2 e2 − k3 e2 − k4 e2 − 1 θ˜1 (θˆ˙1 − τ1 + γ1 W 1 2 3 4 γ1 1 ˜ ˆ˙ 1 ˙ e5 B1 ) − θ2 (θ2 − τ2 + γ2 e5 B2 ) − θ˜3 (θˆ3 − τ3 )− γ2 γ3 (31) 1 1 1 ˜ ˆ˙ ˙ θ4 (θ4 − τ4 ) − θ˜5 (θˆ5 − γ5 e5 x5 ) − θ˜6 γ4 γ5 γ6 ˙ˆ [θ6 − γ6 e5 (Vref + Vc − Vt )] + e5 [θˆ6 (Vref + Vc − Vt )+ θˆ5 x5 − A − B1 θˆ1 − B2 θˆ2 ]

5. SIMULATION RESULTS To show the effectiveness of the designed excitation controller an SMIB system is used as shown in Fig. 1. The physical parameters of the SMIB system used in simulation studies are as follows: ω0 = 314.59 rad/s, Vs = 1 pu, Vref = 1 pu, H = 3.5 s, D = 3 pu, xd = 1.863 pu, x′d = 0.296 pu, xq = 0.474 pu, x′q = 0.18 pu. The tuning parameters for the controller are selected as: k1 = k2 = 5, k3 = 2, and k4 = 4; and the adaptation gain parameters are chosen as follows: γ1 = γ2 = 0.5, γ3 = 5, γ4 = 2, γ5 = 6, and γ6 = 4. The performance of the designed nonlinear adaptive backstepping controller (NABC) is evaluated by considering two cases: i) threephase short-circuit fault at the terminal of the generator and ii) the same at the middle of a transmission line. The performance is then compared with that of an existing NABC (ENABC) as proposed in (Roy et al. (2015c)). In both cases, the fault is applied at t=10 s and cleared at t=10.2 s. • Case 1: Controller performance in case of a three-phase short-circuit fault at the terminal of the synchronous generator

11

11.5 12 Time (s)

12.5

13

13.5

14

NABC ENABC

1.2 1 0.8 0.6 0.4 0.2 9.5

10

10.5

11

11.5 12 Time (s)

12.5

13

13.5

14

Fig. 4. Output power in case of a short-circuit fault at the terminal of the generator Terminal voltage (pu)

1 NABC ENABC

0.8 0.6 0.4 0.2 9.5

10

10.5

11

11.5 12 Time (s)

12.5

13

13.5

14

Fig. 5. Terminal voltage in case of a short-circuit fault at the middle of a transmission line
10-3 2 Speed deviation (pu)

˙ 4 = −k1 e2 − k2 e2 − k3 e2 − k4 e2 − k5 e2 (34) W 1 2 3 4 5 which is negative definite or semi-definite. Therefore, the derived adaptive backstepping control law will stabilize the whole system. Simulation studies are presented in the following section to show the effectiveness of the designed controller.

10.5

1.4

θˆ6 Ve + θˆ5 x5 − A − B1 θˆ1 − B2 θˆ2 + k5 e5 (32) θˆ6

˙ ˙ ˙ θˆ1 = τ1 − γ1 e5 B1 , θˆ2 = τ2 − γ2 e5 B2 , θˆ3 = τ3 , (33) ˙ ˙ ˙ θˆ4 = τ4 , θˆ5 = γ5 e4 x5 , θˆ6 = γ6 e4 (Vref + Vc − Vt ) where Ve = Vref − Vt . Combining (31)-(33), the derivative of W4 becomes

10

Fig. 3. Speed deviation in case of a short-circuit fault at the terminal of the generator

The actual control law and parameters adaptation laws can be designed as follows Vc = −


10-3

2

P e (pu)

Terminal voltage (pu)

1.2

NABC ENABC

1 0 -1 -2 -3 9.5

10

10.5

11

11.5 12 Time (s)

12.5

13

13.5

14

Fig. 6. Speed deviation in case of a short-circuit fault at the middle of a transmission line A three-phase short-circuit fault is the most severe disturbance in power systems. In this case, a three-phase shortcircuit fault is applied at the terminal of the synchronous generator. When a three-phase short-circuit fault is occurred at the generator terminal, the terminal voltage is zero during the faulted condition which can be seen from Fig. 2. From Fig. 2, it is obvious that in spite of strong disturbance, the designed NABC (black line) stabilizes the terminal voltage in a faster way as compared to an ENABC (blue line). The stability of power systems with synchronous generator depends on the operating speed. At normal operating condition, the synchronous generator operates at synchronous speed, i.e., the speed deviation will be zero. The speed deviation response of the synchronous generator is shown in Fig. 3 from where it can be seen that both controllers provide the zero speed deviation but convergence speed of the designed NABC (black line) is much faster than the ENABC (blue line). The designed NABC (black line) provides better damping as compared to the ENABC (blue line). The active power response of the synchronous generator is shown in Fig. 4 from where it is obvious that there is less oscillation with the designed NABC (black line) as compared to ENABC (blue line).

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Proceedings of the 20th IFAC World Congress T.K. Roy et al. / IFAC PapersOnLine 50-1 (2017) 4368–4373 Toulouse, France, July 9-14, 2017 2.5 NABC ENABC

2 P e (pu)

1.5 1 0.5 0 -0.5 9.5

10

10.5

11

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12.5

13

13.5

14

Fig. 7. Output power in case of a short-circuit fault at the middle of a transmission line • Case 2: Controller performance in case of a three-phase short-circuit fault at the middle of one of the transmission lines The performance of the designed control scheme is also tested when a three-phase short-circuit fault is occurred at the middle of one of the two parallel transmission lines. In this case, the terminal voltage will not be zero during the fault period as shown in Fig. 5. From Fig. 5, it can be observed that the designed NABC (black line) acts much faster way as compared to the ENABC (blue line). The corresponding speed deviation response is shown in Fig. 6. It can be seen that when designed NABC (black line) is used, the speed deviation response has much smaller oscillations and the oscillations are damped out in a faster way than the ENABC (blue line). The corresponding output power response of the synchronous generator is shown in Fig. 7. 6. CONCLUSION A nonlinear adaptive backstepping controller based on the higher-order model of the synchronous generator has been designed for ensuring the transient stability and voltage regulation of power systems. The controller is designed by considering the main stability sensitive parameters of synchronous generators as unknown and adaption laws are used to estimate these unknown parameters based on the Lyapunov stability criterion. The effectiveness of the designed controller is evaluated by applying a symmetrical three-phase short-circuit fault at different locations of the SMIB system. Simulation results clearly demonstrate that the designed adaptive backstepping controller is more effective as compared to an existing adaptive backstepping controller in terms of providing damping, regulating voltage, and settling the system to the pre-fault steady-state operating point. Future work will be devoted on designing a similar adaptive backstepping controller for higher-order model of a multimachine power system. REFERENCES Chapman, J.W., Ilic, M., King, C., Eng, L., and Kaufman, H. (1993). Stabilizing a multimachine power system via decentralized feedback linearizing excitation control. IEEE Trans. on Power System, 8(3), 830–839. Colbia-Vega, A., Leon-Morales, J. de, F.L., Pena, O.S., and Mata-Jimenez, M.T. (2008). Robust excitation control design using sliding-mode technique for multimachine power systems. Electric Power Systems Research, 78(9), 1627–1634. Dash, P., Elangovan, S., and Liew, A. (1996). Sliding mode control of a static controller for synchronous generator stabilization. Int. J. Elect. Power Energy Syst., 18(1), 55–6.

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Ko, H.S., Lee, K.Y., and Kimc, K.C. (2004). An intelligent based LQR controller design to power system stabilization. Electric Power Systems Research, 71(1), 1–9. Kundur, P. (1994). Power System Stability and Control. McGraw-Hill, New York. Loukianov, A.G., Canedo, J.M., Fridman, L.M., and SotoCota, A. (2011). Robust excitation control design using sliding-mode technique for multimachine power systems. IEEE Trans. on Ind. Electron., 58(1), 337– 347. Mahmud, M.A. (2014). An alternative LQR based excitation controller design for power system to enhance small-signal stability. Int. Journal Elect. Power Eng. Syst., 63, 1–7. Mahmud, M.A., Hossain, M.J., Pota, H.R., and Oo, A.M.T. (2017). Robust partial feedback linearizing excitation controller design for multimachine power systems. IEEE Transactions on Power Systems, 32(1), 3– 16. Mahmud, M.A., Hossain, M.J., Pota, H.R., and Roy, N.K. (2014a). Robust nonlinear excitation controller design for multimachine power systems. in Proc. of IEEE PES General Meeting, 27–31. Mahmud, M.A., Pota, H., and Hossain, M.J. (2014b). Transient stability enhancement of multimachine power systems using nonlinear observer-based excitation controller. Int. Journal of Electrical Power & Energy Systems, 58, 57–63. Mahmud, M.A., Pota, H.R., and Hossain, M.J. (2012). Full-order nonlinear observer-based excitation controller design for interconnected power systems via exact linearization approach. Int. Journal of Electrical Power & Energy Systems, 41(1), 54–62. Mitra, A., Mukherjee, M., and Naik, K. (2015). Enhancement of power system transient stability using a novel adaptive backstepping control law. in Proc. of IEEE Third ICCCCIT, 1–5. Ribeiro, R.L.A., Neto, C.M.S., Costa, F.B., Rocha, T.O.A., and Barreto, R.L. (2015). A sliding-mode voltage regulator for salient pole synchronous generator. Electric Power Systs. Res., 129, 178–184. Roy, T.K., Mahmud, M.A., Shen, W., and Oo, A.M. (2015a). Robust adaptive backstepping excitation controller design for simple power system models with external disturbances. in Proc. of IEEE Conf. on Control Applications, 715–720. Roy, T.K., Mahmud, M.A., Shen, W., and Oo, A.M.T. (2015b). A nonlinear adaptive backstepping approach for coordinated excitation and steam-valving control of synchronous generators. in Proc. of the 10th Asian Control Conf., 1–6. Roy, T.K., Mahmud, M.A., Shen, W., and Oo, A.M.T. (2015c). Nonlinear excitation control of synchronous generators based on adaptive backstepping method. in Proc. of the 10th Int. Conf. on Industrial Electronics and Applications, 11–16. Verrellia, C.M. and Damm, G. (2010). Robust transient stabilisation problem for a synchronous generator in a power network. Int. J. of Control, 83(4), 816–828. Wang, Y., Hill, D.J., Middleton, R.H., and Gao, L. (1994). Transient stabilization of power systems with an adaptive control law. Automatica, 30(9), 1409–1413.

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