Application of Filippov Theory to the IEEE Standard Anti-windup PI Controller

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1 Application of Filippov Theor to the IEEE Stanar Anti-winup PI Controller Mohamme Ahsan Aib Mura School of Electrical & Electronic Eng. Universit College Dublin (UCD) Dublin, Irelan. Brenan Haes School of Electronic Engineering Dublin Cit Universit (DCU) Dublin, Irelan. Feerico Milano School of Electrical & Electronic Eng. Universit College Dublin (UCD) Dublin, Irelan. Abstract This paper applies the Filippov theor of ifferential equations with iscontinuous right-han sie to moel anti-winup PI controllers. The propose approach solves the ealock issue that arises in the PI control moel recommene b the IEEE Stanar An illustrative example as well as a case stu base on a one-machine infinite-bus network show how the propose approach works an iscusses the effects of the propose approach on the transient response of power sstems. Inex Terms PI control, ealock, anti-winup, Filippov theor, sliing surface. A. Motivation I. INTRODUCTION Proportional Integral (PI) controllers are commonl emploe in power sstem applications []. The IEEE Stanar is the recommene moel. It proposes an antiwinup (AW) or non-winup PI control [2] moel for namic analsis of power sstems. The ifferential equation of this moel is iscontinuous. Due to iscontinuit, simulations with the IEEE moel can lea to failure of a numerical metho or trajector ealock known as chattering Zeno [3]. This work applies Filippov theor (FT) to evelop a metho for the continuation of trajectories beon such a ealock point. B. Literature Review Integral winup phenomenon of PI controllers leas to excess energ to be issipate in the sstem, which in turn results in a poor controller performance [4]. Thus AW control structures are often use an several solutions are propose [5], [6]. Among all possible AW implementations, the efinition of the IEEE Stanar is a conitional integration tpe [2]. This IEEE AW moel poses several challenges for both software implementation an numerical integration an one such issue is the ealock. The ealock behavior that prevents the continuation of trajectories is iscusse in [3] an [7] for PI controllers in win turbines. To prevent such ealock two approaches are propose in [3], [7]: using an extra feeback loop an emploing a eaban or hsteresis. The feeback solution has two isavantages: it requires one extra parameter to be tune an, epening on the isturbance, it can result in This work is supporte b Science Founation Irelan, b funing Mohamme Ahsan Aib Mura an Feerico Milano, uner Investigator Programme Grant No. SFI/5/IA/374. a significantl ifferent namic behavior compare to the IEEE AW moel as iscusse in []. The application of the eaban approach introuces chattering of the solution on the iscontinuous surface. The breakown of numerical integration techniques relate to the IEEE AW moel at the ealock point is ientifie ue to iscretization in [8]. To alleviate this problem an auxiliar iscrete variable base on the semi-implicit approach [9] is propose in [8]. However, such semi-implicit formulation cannot be aopte b most power sstem simulation tools. Another technique consiere in [8] is to emplo a limite integrator. But this ma result in a elae response compare to the IEEE AW metho. All the propose methos mentione above to overcome ifficulties associate with the IEEE AW are base on a hoc approaches. This work proposes a continuation technique base on FT []. Due to the conitions that ictate the non-smoothness of the IEEE AW moel, the solution can enter into a constraine subset of the state space, tpicall known as sliing []. The formalism introuce b Filippov in [] is a powerful tool to efine a vector fiel on the sliing surface an to hanle iscontinuities. This has been applie in other fiels, e.g., in power electronics [2]; an energ harvesters [3]. However, attempts to appl the FT to power sstem namic analsis have not been conucte thus far. C. Contributions The main contributions of the paper are as follows: The application of FT to the IEEE tpe AW PI control that leas to smooth continuation of trajectories. A comparison of the propose approach with the approaches iscusse in [3] an [8]. D. Organization The remainer of this paper is organize as follows. Section II presents the IEEE Stanar AW PI moel along with the numerical issues associate with it. Section III introuces FT an explains its solution concept. A simple example of an AW PI controller with a time epenent input an a Single Machine Infinite Bus (SMIB) power sstem network are simulate b appling FT an compare with two existing solutions in Section IV. Finall, in Section V, conclusions an future work irections are rawn.

2 II. PROBLEM FORMULATION This section first presents the IEEE stanar AW PI controller moel an then illustrates the trajector ealock problem b introucing a relevant example. A. Anti-winup PI control The conitional integration AW metho switches off the integration to avoi winup effects epening on certain conitions an there exists several implementations [5]. The IEEE stanar proposes one of such AW metho, epicte in Fig.. Mathematicall, the moel is [2]: If ě w max : w w max an 9x, If ď w min : w w min an 9x, Otherwise : w k p u `x an 9x k i u. () Time [s] u w x 9x.2 u k i k p s x w min w max Fig. : Proportional-integral block with an anti-winup limiter in accorance with the IEEE Stanar [2]. w u w x 9x B. Numerical Issues of the IEEE Stanar We use a simple example to explain the ealock phenomenon that can occur when using the IEEE stanar moel. Let the following signal be the input signal to the PI controller: if t ă 3 then: 9u else: 9u, an the parameters consiere are, k i 3, k p, w max.2, w min.2 an the initial values at t are x.5, an u. The sstem is simulate for 6.5 s with a time step. s. Simulation results are shown in Fig. 2. The input u increases in the first 3 secons of the simulation, hence an w increase. For ą w max.2, at t.427 s w becomes constant an 9x switches to. So, the integrator is locke to prevent winup. For t ą 3 s, u an starts to ecrease. At t s, ă.2 an the right-han sie of 9x unlocks. However, at the same time, u ą an, hence, 9x ą. Then x will increase, thus causing to increase again towars w max. Depening on the time step of the integration an on the value of 9x an on the rate of change of the input u, a ealock (ccling) situation can arise which consists in locking an unlocking the state variable x preventing the numerical integration from converging. Reucing the time step of the integration scheme oes not solve this issue because of chattering at that iscontinuous point. At this point, some kin of continuation process in necessar. Using the eaban (b) base solution metho with b =.3, the ealock oes not appear in the trajector shown in (2) Time [s] Fig. 2: An example to explain the ealock phenomenon that occurs with the IEEE Stanar anti-winup PI control moel. Fig. 2. The b is implemente as in [3]. The trajector obtaine with the b moel chatters between the bouns generate b the eaban an a reasonable value for b cannot be known as a priori. III. FILIPPOV THEORY Filippov sstems form a subclass of iscontinuous namical sstems which can be escribe b a set of first-orer orinar ifferential equations (ODEs) with a iscontinuous right-han sie []. Consier the following switche namical sstem: # f pxq when hpxq ă 9x fpxq (3) f 2 pxq when hpxq ą where, the event function h : R n Ñ R an an initial conition xpt q x are known. The state space R n is split into two regions R an R 2 separate b a hper-surface Σ where R, R 2 an Σ are characterize as, R tx P R n hpxq ă u, R 2 tx P R n hpxq ą u, Σ tx P R n hpxq u, such that R n R YΣYR 2, assuming that the graient of h at x P Σ never vanishes, h x pxq for all x P Σ. (4)

3 The vector fiel on Σ is efine b Filippov continuation approach, known as Filippov convex metho []. This metho states that the vector fiel on the surface of iscontinuit is a convex combination of the two vector fiels in the ifferent regions of the state-space: $ & f pxq, x P R 9x fpxq cotf pxq,f 2 pxqu, x P Σ (5) % f 2 pxq, x P R 2 where, copf,f 2 q is the minimal close convex set containing f an f 2, i.e. cotf,f 2 u tf F : x P R n Ñ R n : f F p αqf `αf 2 u, (6) where α P r,s. Definition : An absolutel continuous function x : r,τ s Ñ R n is sai to be a solution of (3) in the sense of Filippov, if for almost all t P r,τ s it hols that 9x P F pxptqq where F pxptqq is close convex hull in (6). Now, the question is what happens when the trajector of 9x f pxq, with xpq x reaches at Σ in finite time. The possibilities are: (a) transversal crossing, (b) attractive sliing or repulsive sliing an (c) smooth exit. Filippov formulate a first orer theor to ecie what to o in such kin of situation, summarize in the following. A. Filippov First Orer Theor Filippov first orer theor efines the vector fiel if the solution approaches the iscontinuous surface. Let x P Σ an npxq is the unit normal to Σ at x i.e. npxq hxpxq h where, xpxq h x pxq hpxq an B Bx ; the components of f pxq an f 2 pxq onto the normal to the Σ are n T pxqf pxq an n T pxqf 2 pxq respectivel. ) Transversal Crossing: If at x P Σ, pn T pxqf pxqq.pn T pxqf 2 pxqq ą, (7) the trajector leaves Σ, an two cases are possible. The sstem will move to R 2 with f f 2, if n T pxqf pxq ą or it will enter to R with f f, if n T pxqf pxq ă. 2) Sliing moe: Sliing occurs, at x P Σ if, pn T pxqf pxqq.pn T pxqf 2 pxqq ă. (8) The sliing moe can be an attracting or a repulsive one. An attracting sliing moe will occur if, pn T pxqf pxqq ą an pn T pxqf 2 pxqq ă, x P Σ. (9) Repulsive sliing falls outsie the scope of this work. Intereste reaers are pointe to [] for further etails of this form of sliing. While sliing along Σ, time erivative f F is given b: f F pxq p αpxqqf pxq `αpxqf 2 pxq, () where, αpxq is given b [proof, see []]: αpxq n T pxqf pxq n T pxqpf pxq f 2 pxqq () The sliing moe continues until one of the vector fiels starts to point awa. When this happens, the solution can be continue above or below the sliing surface. The exit point is calculate numericall b fining either the root αpxq or αpxq as appropriate. The following remarks are relevant: If f F pxq f pxq, f F pxq f 2 pxq such a solution is often calle a sliing motion. A solution having an attractive sliing moe exists an is unique, in forwar time. If at the point of iscontinuit, conition (8) becomes ď an f pxq f 2 pxq then a continuous vectorvalue function f F pxq is given which etermines the velocit of motion 9x f F pxq along the iscontinuit line. If n T pxqf pxq then f F pxq f pxq; if n T pxqf 2 pxq then f F pxq f 2 pxq. IV. CASE STUDY In orer to emonstrate the application of Filippov theor on IEEE AW PI controller two case stuies are consiere. The first case stu re-calls the example from Section II-B an the secon one consiers an SMIB power sstem network. The algorithm escribe in [4] is applie for numerical simulation. A. Case Stu I Consier the example iscusse in Section II-B with the same input an parameters. The mathematical moel for upper limit becomes: $ «ff 9u & f pxq f ns when hpxq ă k i x 9x fpxq «ff 9u % f 2 pxq f s when hpxq ą, where f ns an f s are the ifferential equations when the controller is not saturate an saturate, respectivel an u varies accoring to (2). The controller output signal k p x `x 2. The switching manifol is given b: hpxq.2. So, h x pxq r Bhpxq Bhpxq Bx Bx 2 s T rk p s T, an the normal to the switching surface is: n T pxq rk p s. The simulation results are shown in Fig. 3 an how FT is applie at each iscontinuous point escribe below. t (s): With initial conitions r;.5s, hpxq ă, thus the sstem starts in the non-saturate region an is moele with f f ns. t.4268 (s): The sstem has struck the switching manifol i.e. hpxq with u At the switching surface, calculating, j n T pxqf pxq r s p.4268q j n T pxqf 2 pxq r s

4 x u w Fig. 3: Response of trajectories using Filippov theor..4.2 w,ft x,ft w,s x,s w,s2 x,s Fig. 4: Comparision of trajectories using Filippov theor (FT), eaban approach (S) an limite integrator technique (S2). The sstem unergoes a transversal intersection since rn T pxqf pxqs.rn T pxqf 2 pxqs ą. Since n T pxqf pxq ą, the sstem moves region with f f s. t 3 (s): a time epenent switching occurs, exactl at that moment for 9u, an hpxq ą so, the sstem continues with f f s. For values of t greater than 3, the input signal now starts to ecrease. t (s): The sstem has again struck the switching manifol with u At this point, calculating, j n T pxqf pxq r s.284 3p.4268q j n T pxqf 2 pxq r s. Gen v =θ jx 3 v 3 =θ 3 p l `jq l Fig. 5: A single generator connecte to an infinite bus. u in k s `T s `T 2s Fig. 6: Block iagram of AVR an PSS. c 3 v ref _ v jx 23 k a `T as v a v 2 =θ 2 k p + v min v max k i s v f rn T pxqf pxqs.rn T pxqf 2 pxqs.284 ă an accoring to (9), the sstem slies along Σ. Next, the sliing vector fiel on Σ is calculate using (,): n T pxqf pxq 3u αpxq n T pxqpf pxq f 2 pxqq 3u f F pxq p αpxqqf pxq `αpxqf 2 pxq j j x. x 2 t (s): At this point, αpxq for u 3 an the trajector leaves Σ with vector fiel f ns. ) Comparison of Solutions: This section compares the Filippov solution approach with the b solution metho an the limite integration technique (LIT). The limite integrator [8] moel is as follows: If ě w max : w w max, If ď w min : w w min, Otherwise : w k p u `x, If x ě x max an 9x ě : x x max an 9x, If x ď x min an 9x ď : x x min an 9x, Otherwise : 9x k i u. (2) (3) The b value use is.3 an the integrator is limite. for LIT. The simulation results are shown in Fig. 4. Using FT an LIT the trajector continues smoothl before an after each event; the b approach will alwas results in chattering whenever a ealock conition appears. On the other han, LIT oes not show the ealock an provies some flexibilit to choose the integrator limits inepenentl. However, correct values for limits of the integrator state are often unknown an a common choice is to use the same values as the output limit. Therefore, in most of the situations using LIT compare to FT an b, will result in a ifferent convergence rate to stea state after the output leaves the limit. B. Case Stu II Consier the SMIB sstem shown in Fig. 5, the generator is equippe with an Automatic Voltage Regulator (AVR) an a Power Sstem Stabilizer (PSS) as epicte in Fig. 6. The generator moel is a thir orer tpe; the PSS consists of a stabilizer gain an a lea lag block an the AVR is an static tpe with PI control [5]. The namics of the sstem is escribe b a set of ifferential-algebraic equations (DAEs)

5 in the following form [5], 9x fpx,q, gpx,q, (4) where x an are the vector of state an algebraic variables respectivel. For this test sstem, x rδ ω e q v a x i s s T, rv v 3 θ θ 3 v f c c 2 c 3 s T, where δ, ω, e q are the rotor angle, rotor spee an q-axis transient voltage respectivel; v a, x i an s are the state variables of AVR an PSS; v, v 3, θ, θ 3 are the bus voltages an angles respectivel; v f is the generator fiel voltage; c, c 2, c 3 are the algebraic variables of PSS. The algebraic equations of the SMIB sstem are given b, p e `b 3 v v 3 sinpθ θ 3 q, b 3 v 3 v sinpθ 3 θ q `b 23 v 3 sinpθ 3 q `p l, q e `b 3 rv 2 v v 3 cospθ θ 3 qs, b 3 rv 2 3 v 3v cospθ 3 θ qs `b 23 rv 2 3 v 3cospθ 3 qs `q l, v f `k p v a `x i, c u in k s, c 2 c p T T 2 q, c 3 c p T T 2 q s, where b 3 {x 3 an b 23 {x 23 are known line parameters; k p, k s, T an T 2 are the control parameters of AVR an PSS; input to the PSS is u in ω; p l p l p v3 v 3 q, q l q l p v3 v 3 q 2, v 3 is known from power flow calculation; p l an q l are the active an reactive power of the loa respectivel; the reactive an active power of the generator sinpδ θ q respectivel. Note that, the voltage an angle of the infinite bus are v 2 an θ 2 respectivel. The PI controller in AVR (see Fig. 6) is an IEEE Stanar tpe. Lets consier the switching manifol for an upper limit, hpxq k p v a `x i v max. When hpxq ă, the ifferential equations of the SMIB sstem are given b, are: q e x rv 2 e q v cospθ δ qs, p e e q v x 9δ ω (5) 9ω M pp m p e Dωq (6) 9e q T pv f x x e q ` x x x v cospδ θ qq (7) 9v a pk a pv ref `c 3 v q v a q{t a (8) 9x i k i v a (9) 9s T 2 pc 2 s q, (2) where x, x are the -axis snchronous an transient reactance respectivel; T, M, D an p m are the -axis open circuit transient time constant, the mechanical starting time, the amping coefficient an the mechanical power input to the generator respectivel; v ref is the reference voltage; T a, k i an k a are the control parameters of AVR an PSS. TABLE I PARAMETERS OF THE COMPONENTS OF THE SMIB NETWORK Name Values Generator M 8, D, x.25, x, p m, T 6 Line x 3.3, x 23.5 Loa p l.7, q l. AVR k a 2, T a.5, k p 5.5, k i 35, v max.58, v min.5, v ref PSS k s.5, T.23, T Fig. 7: Response of the state an fiel voltage using Filippov theor. When hpxq ą i.e. the fiel voltage reaches to its upper limit (v max ) then (7) an (9) will be switche an all other states will remain same, as follows: r Bhpxq Bx 9e q T pv max x x e q ` x x x v cospδ θ qq (2) 9x i (22) We consier f px,q is (5)-(2) an f 2 px,q is (5), (6), (2), (8), (22) an (2). Calculating, h x pxq... Bhpxq Bx 6 s T r k p s T, an the normal Bhpxq Bx 2 to the switching surface is: n T pxq r k p s. The initial values of the state variables an algebraic variables are calculate from the power flow solution an are: x rδ ω e q v a x i s s T r s T, rv v 3 θ θ 3 v f c c 2 c 3 s T r s T. All the parameters of ifferent components of the SMIB sstem are given in Table I. ) Simulation Results: The SMIB test sstem is simulate b appling a step increase to loa (p l.7,q l.5) an voltage reference set-point of AVR (v ref.) at 5 s. The response of the PI controller state, fiel voltage an limite fiel voltage (v f ) using FT are shown in Fig. 7. To explain how FT is applie uring each event, hpxq; r n T pxqf px,q an r 2 n T pxqf 2 px,q are shown in Fig. 8. Simulation results clearl show that a piece-wise smooth solution is achieve using Filippov solution technique. Relevant remarks on the simulation results are given below: For the initial operating point of the sstem hpxq ă, so the sstem simulation starts with f px,q. x i v f v f

6 .5 hpxq r r hpxq x i, FT x i, S x i, S Fig. 8: Response of the switching manifol hpxq, r n T pxqf px,q an r 2 n T pxqf 2px,q using Filippov theor. Fig. 9: Comparision of trajectories using Filippov theor (FT), eaban approach (S) an limite integrator technique (S2). At 5 s the isturbance is applie an a little bit after that the sstem reaches to the switching manifol i.e. hpxq. Due to the conition (7) an r ą (see Fig. 8) a transversal crossing is happene. The sstem switches to f 2 px,q an the integrator state an the fiel voltage become constant (see Fig. 7). The sstem continues with f 2 px,q as long as hpxq ą. At t 5.53 s, hpxq again (see the arrow in Fig. 8) an the conitions (8) an (9) are met, so an attracting sliing moe occurs on hpxq. The vector fiel (f F px,q) an αpx,q are calculate numericall using () an (). Therefore uring that sliing hpxq remains at, the sstem continues with f F px,q (see Figs. 7-8). The sstem moves to f px,q when hpxq ă. The theor of Filippov assumes sstems are moele using ODEs. However in this work, our sstem moel emplos DAEs to simulate power sstems. The application of FT was possible ue to the fact that hpxq epene onl on the state variables an not the algebraic ones. In aition, the case stuies shows the effectiveness of FT for upper limit of the IEEE AW PI controller but it is trivial to appl for lower limit too. For completeness Fig. 9 compares the b an LIT with the FT (onl the integrator state is shown). V. CONCLUSIONS This paper stuies the trajector ealock issue of the IEEE Stanar AW PI controller moel. To solve this ealock problem Filippov theor is propose. The case stuies prove that an effective trajector continuation can be achieve using convex combination efine b Filippov. Other alternative solution techniques were also compare with the FT. We are currentl activel working on implementing a sstematic formulation of the FT in a software tool for power sstem analsis. We believe that the results of this paper are promising. However there is still much research to o to make FT suitable for commercial grae software tools. In particular, we aim at stuing how to appl FT to sstems where hpx, q epens on both state an algebraic variables. REFERENCES [] M. A. A. Mura, Á. Ortega, an F. Milano, Impact on power sstem namics of PI control limiters of VSC-base evices, in Power Sstems Computation Conference (PSCC), June 28, pp. 7. [2] IEEE recommene practice for excitation sstem moels for power sstem stabilit stuies - reline, IEEE St (Revision of IEEE St ) - Reline, pp. 453, Aug 26. [3] I. A. Hiskens, Dnamics of tpe-3 win turbine generator moels, IEEE Transactions on Power Sstems, vol. 27, no., pp , Feb 22. [4] K. J. Åström an T. Hägglun, Avance PID Control. ISA - The Instrumentation, Sstems an Automation Societ, 26. [5] A. Visioli, Moifie anti-winup scheme for PID controllers, IEE Proceeings - Control Theor an Applications, vol. 5, no., pp , Jan 23. [6] S. Tarbouriech an M. Turner, Anti-winup esign: an overview of some recent avances an open problems, IET Control Theor Applications, vol. 3, no., pp. 9, Januar 29. [7] I. A. Hiskens, Trajector ealock in power sstem moels, in 2 IEEE International Smposium of Circuits an Sstems (ISCAS), Ma 2, pp [8] D. Fabozzi, S. Weigel, B. Weise, an F. Villella, Semi-implicit formulation of proportional-integral controller block with non-winup limiter accoring to IEEE Stanar , in Bulk Power Sstems Dnamics an Control Smposium (IREP), 27, pp. 7. [9] F. Milano, Semi-implicit formulation of ifferential-algebraic equations for transient stabilit analsis, IEEE Transactions on Power Sstems, vol. 3, no. 6, pp , Nov 26. [] A. F. Filippov, Differential Equations with Discontinuous Righthan Sies. Kluwer Acaemic Publishers, 988. [] M. i Bernaro, P. Kowalczk, an A. Normark, Bifurcations of namical sstems with sliing: erivation of normal-form mappings, Phsica D: Nonlinear Phenomena, vol. 7, no. 3, pp , 22. [2] D. Giaouris, S. Banerjee, B. Zahawi, an V. Pickert, Stabilit analsis of the continuous-conuction-moe buck converter via Filippov s metho, IEEE Transactions on Circuits an Sstems I: Regular Papers, vol. 55, no. 4, pp , Ma 28. [3] P. Harte, E. Blokhina, O. Feel, D. Fournier-Prunaret, an D. Galako, Electrostatic vibration energ harvesters with linear an nonlinear resonators, International Journal of Bifurcation an Chaos, vol. 24, no., p. 433, 24. [4] P. T. Piiroinen an Y. A. Kuznetsov, An event-riven metho to simulate Filippov sstems with accurate computing of sliing motions, ACM Trans. Math. Softw., vol. 34, no. 3, pp. 3: 3:24, Ma 28. [5] F. Milano, Power Sstem Moelling an Scripting, ser. Power Sstems. Springer Berlin Heielberg, 2.