Design of a Single Input Fuzzy Logic Controller Based SVC for Dynamic Performance Enhancement of Power Systems

Transcription

1 Design of a Singe Input Fuzzy Logic Controer Based SC for Dynaic Perforance Enhanceent of Power Systes DR.D. PADMA SUBRAMANIAN PROFESSOR AND DEAN, DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING AMET UNIERSITY, ANATHUR CHENNAI-6311, INDIA subraanianpads@gai.co Abstract: This paper presents a design of a Singe Input Fuzzy Logic Controer (SFLC) based Static AR Copensator (SC) for Dynaic perforance enhanceent of power systes. The SFLC uses ony one input which is the signed distance and has the advantage of reduced nuber of rues. Iproveent of dynaic response by the controer is iustrated in a bifurcation perspective. Bifurcation diagras of steady state as we as periodic soutions are constructed using continuation ethod. Fro the bifurcation diagras, the existence of various bifurcation points such as, unstabe Hopf bifurcation (UHB), stabe Hopf bifurcation (SHB), sadde node bifurcation (SNB) and period doubing bifurcation (PDB) are identified. With the use of toos of noninear dynaics, votage coapse points, and chaotic soutions due to period doubings are unearthed. The effectiveness of the SFL controer over the conventiona controer for SC in deaying the incidence of Hopf bifurcation (HBF), SNB and hence increasing the oadabiity iit is iustrated for the test syste. eywords: SFLC, SC, votage coapse,hbf, SNB I. Introduction It is recognised in the iterature that syste instabiity generay eerges fro two types of events: gradua paraeter variations and contingencies. This paper focusses on the anaysis of power systes subject to gradua paraeter variations such as oad variation. This anaysis can be ade through oca bifurcation anaysis in which the syste is studied by the eigen vaues of inearization around an equiibriu point. This anaysis is suitabe for probes raated to secure power syste operation and design with particuar attention to stabiity argins and contro actions required to postpone a specific type of instabiity. With increasing popuarity of Fexibe AC transission syste (FACTS) devices and SC being the weunderstood and widey accepted FACTS device, it is worth exporing the roe of SC in iproveent of dynaic stabiity of power systes. L.Ji and Z. XueSong [1], presented continuation based approach for tracing equiibriu curve of a power syste using differentia agebraic ode, anaysis of paraeter effect on votage stabiity and effect of SC on the occurrence of Hopf bifurcation. Effectiveness of PSS, SC and STACOM in daping interarea osciations fro the point of view of Hopf Bifurcations using Eigen vaue anaysis is presented by Mithuananthan. et a.[]. Wang. Y et a. [3] discussed votage stabiity enhanceent by a non-inear SC controer designed through direct feedback inearisation technique. Gracia kasuki et a. [4] had shown the appication of SC to dap out sustained votage osciations for a given eve of reactive power deand as we as increasing the axiu point of syste s oadabiity using a first order deay ode of SC. Mohaed S. Sad et a [5] introduced washout aided feedback in the design of SC to increase the range of stabe operation of a power syste susceptibe to votage coapse. However, to provide an iproved votage reguation and daping syste osciations and hande paraeter uncertainties and unodeed dynaics, contro ethods based on fuzzy ogic need to be deveoped. Byung Jae Choi et a.[6], suggested a sipe but powerfu FLC design ethod using a new variabe caed signed distance which is equivaent to pseudo siding ode controer. Londhe P.S et a. [7], proposed a sipified schee to design a singe input Fuzzy ogic Controer for contro of Advanced Heavy Water Reactor. This paper addresses the perforance of singe input fuzzy ogic based controer for SC in a bifurcation perspective. Bifurcation diagras of equiibriu as we as periodic branch are obtained fro the coputer ipeentation of the agorith proposed by Pada Subraanian. D. et a.[8]. The iproveent of dynaic ISSN : o 6 No 5 Oct-Nov 14 34

2 response by the proposed controer is iustrated though deaying the occurrence of HBF, SNB hence an increase in oadabiity iit and eiination of dynaic votage coapse. This paper is organized as foows: Section presents odeing of various power syste coponents incuding SC. A brief expanation about the singe input fuzzy ogic controer strategy and the design procedure for ipeentation of SFL controer for SC is presented in Section 3. In Section 4, the resuts of ipeentation of the proposed SFLC for SC are presented. The effectiveness of SFLC based SC is iustrated in this Section by coparing the resuts of SFLC based SC with the resuts of conventiona SC. Concuding rearks are presented in Section. II. Power Syste Description and odeing The description of various power syste odes used in this study are as foows. A. Power Syste Mode The power syste considered for investigation is shown in Fig. 1. Y θ π δ Y θ π/ C LOAD BSC δ Fig. 1. A sipe power syste It consists of two generator buses and a oad bus. The SC is paced at the oad bus. One of the generator buses is treated as a sack bus and the other generator is described by the swing equation. The oad is odeed by a sipified induction otor in parae with constant P-Q oad and constant ipedance as described in Chiang. H. D et a. [9]. The induction otor ode specifies the rea and reactive power deands in ters of oad votage and frequency. The oad ode is described by P = P + P1 + δ + pv ( + T ) (1) Q = Q + Q 1 + δ + qv + qv where P, Q are the constant rea and reactive powers of the induction otor and P 1, Q 1 are the constant P-Q oad and, pv,, qv and qv are constants associated with the dynaic oad. The rea and reactive powers suppied to the oad by the network are P = Y sin ( δ + Θ ) Y sin ( δ - δ + Θ ) + Y sin Θ Y sin Θ (3) ( ) + ( Y cosθ Y cosθ ) Q = + (4) Y cos(δ + Θ ) + Y cos(δ-δ Θ ) + The oad bus incudes a capacitor as part of its constant ipedance representation in order to aintain the votage agnitude at a noina and reasonabe vaue. Instead of incuding the capacitor in the circuit, it is convenient to account for the capacitor by adjusting and Y to give Thevenin equivaent of the circuit with the capacitor. Pries are used to indicate Thevenin equivaent circuit vaues. The adjusted vaues are as given in Chiang. H. D et a [9]. = (5) C Y C Y cos Θ 1/ Y ( ) 1 ( 1+ C Y CY cosθ ) 1 / = Y (6) () ISSN : o 6 No 5 Oct-Nov

3 1 CY sin Θ Θ = Θ + tan 1 1 (7) 1 CY cosθ B. Static AR Copensator A sipified first order deay ode of SC, which is tered as conventiona SC in this paper, has been considered to represent the dynaics of its contro action epoyed for deaying the dynaic bifurcations. This is shown in Fig.. REF + Σ SC 1+ ST SC B B MIN B MAX B SC Fig. Bock diagra representation of Static AR copensator Fro Fig.., the equations describing the dynaics of SC are written as: B = T T B SC SC ( ref ) (8) SC where B B B in ax where B is the susceptance, SC is the SC gain, T SC is the tie constant and ref is the reference votage. In order to ode the SC hard iiters, a tanh function isepoyed, such that for a susceptance B with iits ± B LIMIT, the output of the iiter is given by, B B SC = BLIMIT tanh (9) BLIMIT The net effect of the variabe susceptance introduced to the syste by the SC is odeed as an additiona reactive power source, Q = B. SC The equations describing the dynaics of the power syste ode with SC are: δ = ω = Dω + P + Y sin ( δ - δ Θ ) + Mω Y sin Θ (11) δ = qv qv + Q + Q Q SC T pv = qv ( P + P P ) + 1 Q 1 ( ) + ( + Q QSC) qv Equations (1)-(13) aong with (8)-(9) describe the dynaics of the syste shown in Fig.. pv III. Design of SFLC The SFLC is designed for fuzzy ogic controers (FLC) with skew syetric property in the contro rue tabe, i.e., the contro inputs above and beow the switching ine have opposite signs as shown in Fig. 3. A new variabe caed a signed distance, is derived which is the distance to an actua state fro the ain diagona ine (or hyper pane) and is positive or negative according to the position of the actua state. Q Q 1 (1) (1) (13) ISSN : o 6 No 5 Oct-Nov

4 e NS NB ZR e PS PB e + λ e = Fig. 3. Rue Tabe with infinitesia quantization eves The signed distance is used as a soe fuzzy input variabe in SFLC aso caed Sipe FLC. Magnitude u of the contro input is proportiona to the signed (d s ) fro the straight ine caed switching ine S : S : e+ λ e= Let H(,) e e be the intersection point of the switching ine and ine perpendicuar to the switching ine fro an operating point Pe ( 1, e 1) as iustrated in Fig. 4. e H(e, e) d 1 e P(e 1, e 1 ) e + λ e = Fig. 4. Derivation of the signed distance ISSN : o 6 No 5 Oct-Nov

5 The distance d 1 between H(e, e) and P(e 1, e 1 ), can be expressed as e1 +λe 1/ 1 d1= ( e e1) + e e1 = 1+λ (14) Equation (14) can be written in genera for any (e, e ) as: d = e +λ e 1+λ (15) Then, the signed distance d s is defined for a genera point P(e, e ) as foows: d s = sgn( S ) e +λe 1+λ 1, for S > sgn( S ) = 1, for S < The sign of the contro input is negative for S > and positive for S < and its absoutey agnitude is proportiona to the distance fro the ine S =. Therefore u d s Then, a fuzzy rue tabe can be estabished on a 1-D space on d s instead of the -D space of the phase pane for FLCs with skew- syetric rue tabe. That is, the contro action can be deterined by d s ony. Hence the nae SFLC (Singe input FLC). The rue for for the SFLC is given in Tabe I. Tabe I Rue Tabe for SFLC based SC ds NB NS ZR PS PB u PB PS ZR NS NB (16) If d s is NB then u is PB.Where NB-Big negative, NS-Sa negative, NR- Zero, PS-Sa positive, PB-Positive big. The paraeters to be tuned are λ and gain constant for each controer. Hence, the nuber of rues is greaty reduced copared to the case of the conventiona FLCs. and rues can be easiy odified or added for fine contro. A. Ipeentation of SFLC based SC Bock diagra representation of SFLC based SC is shown in Fig 5. The signed distance is derived fro the deviation of oad votage fro its reference vaue. The controer s output is SC susceptance, which is aowed to vary between B in and B ax by using a wind-up iiter. B MAX d s SFLC B B MIN B SC Fig. 5. Bock Diagra representation of SFLC based SC ISSN : o 6 No 5 Oct-Nov

6 I. Siuation Resuts and Discussion Bifurcation diagras of equiibriu as we as periodic branch are obtained fro the coputer ipeentation of the agorith proposed by Pada Subraanian. D. et a. [8]. A. Bifurcation Anaysis with Conventiona SC Fig. 6 shows the bifurcation diagra of the periodic soution branch superiposed on the stationary branch diagra. v R1 (UHB) U PDB ssb usb R SHB ssb usb R3 SNB Q 1 Fig. 6. Bifurcation diagra of oad votage agnitude with conventiona SC S-stabe periodic branch, U- unstabe periodic branch, ssb-stabe stationary branch, usb- unstabe stationary branch. Three critica points R1, R, and R3 are observed. At Q 1 = , i.e., at point R1, there is a UHB with the eergence of an unstabe iit cyce. At Q 1 = , there is a SHB. The oci of Foquet utipiers shown in Fig. 7. confir the occurrence of SHB. The frequency and period of stabe osciation at SHB point are.635 Hz, s respectivey. Fig. 7. Moveent of copex conjugate eigenvaues in the vicinity of SHB. The period doubing route to chaos with conventiona SC (point PDB in Fig. 6.) is deonstrated using threediensiona pots, Fig. 8. ISSN : o 6 No 5 Oct-Nov

7 Fig. 8. Cascade of period doubing with conventiona SCe Period Doubing Bifurcation occurs with the foowing detais: (a) P1 osciation at Q 1 = , (b) P osciation at Q 1 = (c) P4 osciation at Q 1 = 11.51, (d) P8 osciation at Q1 = (e) Chaos at Q 1 = otage coapse point occur at Q 1 = Tie response pot of votage coapse is shown in Fig. 9. Fig. 9. otage coapse at Q 1 = B. Bifurcation Anaysis with SFLC Based SC Singe input fuzzy ogic controer as expained in Section III is epoyed for iproving the perforance of SC. Bifurcation diagras are potted with SFLC based SC. In Fig. 1 bifurcation diagra of the periodic soution branch is superiposed on the stationary branch diagra. ISSN : o 6 No 5 Oct-Nov

8 a HBF U b c SHB d SNB e Q 1 Fig. 1. Bifurcation diagra with SFLC based SC a-b: ssb, b-c: usb, c-d: ssb, d-e: usb, U: unstabe periodic branch ssb-stabe stationary branch, usb-unstabe stationary branch Extensive tie doain anaysis aong with observation of evoution of Foquet utipiers have been carried out with each set of soutions in the periodic branch. It is found that with SFLC based SC, there are no period doubings and hence chaos. Moreover, there is a raarkabe eve of constriction in the periodic soutions branch. Load votage reains constant unti Q 1 is increented upto 11p.u. Tabe II suarizes the various Q 1 vaues at which bifurcations occur in Fig. 6 and 1. TABLE II Surization of Bifurcation Points in Figs. 6 and 1 Type With conventiona SC With SFL based SC Q 1 Q 1 UHB PDB SHB SNB Fro Tabe II, it can be observed that singe input fuzzy ogic controer for SC deays the occreence of HBF, SNB, and hence increases the oadabiyt iit. It aso eiinates dynaic votage coapse. The proposed controer aintais the oad votage at a constant vaue unti Q 1 is increented upto 11p.u.. Concusion A singe input Fuzzy Logic based Static AR Copensator is designed to iprove dynaic perforance of a sape power syste. The SFLC uses ony one input which is the signed distance and has the advantage of reduced nuber of rues.the signed distance is derived fro the deviation of oad votage fro its reference vaue. The proposed controer for SC is effectve in deaying the HBF, SNB, and hence increasing the oadabiity iit. The controer is effective in eiinating the dynaic votage coapse which otherwise occurred with conventiona SC. Moreover, the proposed controer has the advantage of aintaining constant votage at the oad bus for higher oading eve. Tracing of stabe and unstabe steady state equiibriu as we as periodic soution branches is perfored by a nuerica agorith, based on continuation technique. Fro bifurcation diagras, existence of various critica points is identified with the hep of eigen vaue anaysis. Siuation resuts prove the effectiveness of the designed Singe Input Fuzzy ogic Controer based SC in iproving the dynaic perforance of the sape power syste. ISSN : o 6 No 5 Oct-Nov

9 Appendix The paraeters considered for siuation, except those for SC are taken fro Chiang. H. D et a. (199) and are given beow for easy reference. A the paraeter vaues are in p.u. except anges which are in eectrica degrees. Network paraeters: Y =., Θ = 5., = 1., Y = 5., Θ = Generator paraeters: = 1., P = 1., M =.3, D =.5 Load paraeters: Q =.4, = 1.3, P 1 pv =.3, = SC paraeters: SC =., T SC =.1, B LIMIT = 1. SFLC paraeters: λ =, = 4 =.3, qv =.8, References =.1, T = 8.5, =.6, [1] Ji and Z. XueSong (11), Study on ffect of wind power syste paraeters for Hopf Bifurcation based on Continuation Method, Eectrica Power Systes and Coputers LNEE Springer Series, o. 99, pp [] Mithuananthan. N, Canizares. C. A, Reeve. J and Rogers G.J (3), Coparison of PSS, SC and STATCOM Controers for daping Power Syste Osciations, IEEE Transaction on Power Systes, o. 18, Issue, pp [3] Wang. Y, Chen. H, Zhou. R, Studies of votage stabiity via non-inear SC contro (), In Proceedings of IEEE Power Engineering Society Winter Meeting,, pp [4] Gracia asusky. R, Fuerte-Esquive. C. R, Torres- Lucio. D (3), Assessent of SC s effect on noninear instabiities and votage coapse in eectric power systes, In Proceedings of IEEE Power Engineering Society genera Meeting, 4, pp [5] Mohaed. S. Saad, Munther. A. Hassouneh, Eyad. H. Abed, Abde-Aty Edris (5), Deaying instabiity and votage coapse in power systes using SCs with washout fiter aided feedback, In Proceedings of the Aerican Contro Conference, June 6, pp , [6] Byung-Jae Choi, Seong-Woo wak, Byung ook i, Design of a singe input fuzzy ogic controer and its properties (1999), Fuzzy Sets and Systes, 16, pp [7] Londhe P.S, Patre B.M and Tiwari A. P., Design of Singe Input Controer for Spatia Contro of Advanced Heavy Water Reactor (14), IEEE Transactions on Non inear Science, o. 61, No., pp [8] D. Pada Subraanian, R. P. uudini Devi and R. Saravanasevan (11), A New Agorith for anaysis of SC s Ipact on Bifurcation, Chaos and otage Coapse in Power Systes, Internationa Journa of Eectric Power and Energy Systes-Esevier, o. 33, pp [9] Chiang. H. D, Dobson. J, Thoas. R. J, Thorp J. S and Ahed L.F (199), On votage coapse in eectric power systes, IEEE Transactions on Power systes, 5, qv P ISSN : o 6 No 5 Oct-Nov 14 35