Population Variance Harmony Search Algorithm to Solve Optimal Power Flow with Non-Smooth Cost Function

Transcription

1 Populaton Varance Harmony Search Algorthm to Solve Optmal Power Flow wth Non-Smooth Cost Functon B.K. Pangrah 1, V. Ravkumar Pand, Swagatam Das2, and Ajth Abraham3 Abstract. Ths chapter presents a novel Harmony Search (HS) algorthm used to solve securty constraned optmal power flow (OPF) wth varous generator fuel cost characterstcs. HS s a recently developed dervatve-free, meta-heurstc optmzaton algorthm, whch draws nspraton from the muscal process of searchng for a perfect state of harmony. Ths chapter analyses the evoluton of the populatonvarance over successve generatons n HS and thereby draws some mportant attenton regardng the exploratve power of HS. Ths novel methodology of modfed populaton varance harmony search algorthm (PVHS) easly takes care of solvng optmal power flow problem even wth non-smooth and pecewse cost functons. Ths PVHS algorthm was tested on the IEEE30 bus system wth three dfferent types of cost characterstcs and compared wth other reported results. 1 Introducton Optmal power flow s the man tool used for plannng an economc operaton of power system [1]. In the recent attenton n OPF shows the mportance of the electrc utltes to fnd the optmal secure operatng pont correspondng to the each loadng condton. The problem of solvng OPF nvolves estmatng the optmal soluton of control varables lke generator real power, generator voltage magntude and transformer tap settngs correspondng to the best objectve functon. The B.K. Pangrah and V. Ravkumar Pand Department of Electrcal Engneerng, Indan Insttute of Technology, Delh, Hauz Khas, New Delh, Inda E-mal: bkpangrah@ee.td.ac.n, ravkumarpand@gmal.com Swagatam Das Department of Electroncs and Telecommuncaton Engneerng, Jadavpur Unversty, Kolkata , Inda E-mal: swagatamdas19@yahoo.co.n Ajth Abraham Machne Intellgence Research Labs (MIR Labs), USA E-mal: ajth.abraham@eee.org Z.W. Geem: Recent Advances n Harmony Search Algorthm, SCI 270, pp sprngerlnk.com Sprnger-Verlag Berln Hedelberg 2010

2 66 B.K. Pangrah et al. dependent varable ncludes load bus voltage magntude, generator reactve power generaton, transmsson lne thermal loadng. In general OPF s large scale hghly non lnear and constraned problem of mnmzng the fuel cost. OPF problem has been solved usng many tradtonal technques such as non lnear programmng, quadratc programmng, mxed nteger programmng and nteror pont method. The lterature revew on these methods s gven n Momoh et al. [2, 3]. The dsadvantage of these tradtonal methods s t cannot be applcable n case of the prohbted operatng regons and multple fuels. It also has hgher senstvty to ntal soluton, so t may trap nto local optma. The dffcultes n mplementng OPF can be overcome by modern stochastc algorthms such as evolutonary programmng (EP) [4], tabu search (TS) [5], mproved evolutonary programmng (IEP) [6], modfed dfferental evoluton (MDE) [7], partcle swarm optmzaton (PSO) [9], genetc algorthm (GA) [10] and smulated annealng (SA) [11]. In 2001, Geem et al. proposed Harmony Search (HS) [13], a dervatve-free, meta-heurstc algorthm, mmckng the mprovsaton process of musc players. Snce ts ncepton, HS has found several applcatons n a wde varety of practcal optmzaton problems lke ppe-network desgn [14], structural optmzaton [15], vehcle routng problem [16], water dstrbuton networks [17, 23], combned heat and power economc dspatch problem [18], Dam Schedulng [19] and numercal optmzaton [20]. The applcablty of harmony search algorthm for dscrete varable problem s gven n [22]. The hybrd verson of harmony search wth partcle swarm optmzaton appled to water network desgn s proposed n [24]. In the PVHS [21], the control parameter known as dstance bandwdth (bw) has been made equal to the standard devaton of the current populaton. In ths chapter we have used ths PVHS algorthm to solve optmal power flow problem havng varous cost characterstcs. The algorthm s appled to IEEE 30 bus test system effectvely to show the approprateness of the method. The smulaton results wth three dfferent cost characterstcs are comparable wth the recently reported results. 2 OPF Problem Formulaton The objectve of OPF problem s to mnmze the total fuel cost of generators whle satsfyng several power system steady state securty constrants. If x s the vector of state varables consstng of slack bus real power P g1, load bus voltages V Lk, generator reactve power outputs Q gj, and transmsson lne thermal loadng S l, x can be expressed as: x T = [P g1, V l1,, V lnl, Q g1,, Q g, S l1,., S lnb ] (1) where NL, and NB are the number of load buses, the number of generators and the number of transmsson lnes, respectvely. u s the vector of control varables consstng of real power outputs P g except at the slack bus, generator voltages V g, transformer tap settngs T. Hence, u can be expressed as:

3 Populaton Varance Harmony Search Algorthm to Solve Optmal Power Flow 67 u T = [P g2,.,p g,v g1,.,v g,t 1,,T NT ] (2) where NT s the number of regulatng transformers. The objectve of OPF problem can be expressed as Mnmze F = F ( P g ) j= 1 (3) where F s the total generator fuel cost and F s the fuel cost of generator connected to th bus. The system equalty constrants g(x,u) s descrbed by the followng power balance equaton N P g P d = V V j Y j cos θ j δ + δ j = 1,..., N (4) j = 1 N ( δ + δ ) = 1 N Qg Qd = V V j Yj sn θ j j,..., (5) j= 1 where P g s the total real power generaton at th bus, P d s the total real power demand at th bus, Q g s the total reactve power generaton at th bus, Q d s the total reactve power demand at th bus, V s the voltage magntude at th bus, V j s the voltage magntude at j th bus, Y j s the magntude of the j th element of Ybus, θ j angle of the j th element of Ybus, δ voltage angle at th bus and δ j s the voltage angle at j th bus. The system nequalty constrants h(x, u) s consst of the followng 1. Generator constrants: The generator real power outputs, reactve power outputs and voltages are bounded to ts lower and upper lmt. mn Pg Pg Pg = 1,..., (6) mn Qg Qg Qg = 1,..., (7) mn Vg Vg Vg = 1,..., (8) mn where P g and P g are the mnmum and mum real power generaton at th generator bus, mn Q g and Q g are the mnmum and mum reactve power generaton at th mn generator bus, V g and V g are the mnmum and mum voltage magntude at th generator bus. 2. Transformer constrants: Transformer tap settngs are bounded by mnmum and mum lmts as follows T mn T T = 1,..., NT (9)

4 68 B.K. Pangrah et al. mn where T and transformer. T are the mnmum and mum tap settng lmt of th 3. Securty constrants: It ncludes the lmts n the voltage magntude of load buses and thermal loadng lmts of all transmsson lnes as follows. where mn V L and at th load bus, mn VL VL VL = 1,..., NL (10) Sl Sl = 1,..., NB (11) V L are the mnmum and mum voltage magntude S l s the thermal lmt of th transmsson lne. Constrants Handlng: The problem of handlng these constrants n the state varables are accommodated n the algorthm by ncludng the constrants volaton as quadratc penalty terms n the objectve functon tself. F corr = F + K p K lm 2 lm 2 ( Pg 1 Pg 1 ) + K v ( V L VL ) + lm 2 lm ( Q Q ) + K ( S S ) 2 Q g g where K p, K v, K Q and K S are the penalty factors correspondng to slack bus real power generaton, load bus voltage magntude, generator reactve power and transmsson lne thermal loadngs, respectvely. In the equaton (12) the x lm s equals to x mn f x s lesser than the mnmum lmt and x f x s greater than mum lmt. S l l (12) 3 Harmony Search algorthm 3.1 Classcal Harmony Search Algorthm In the harmony search algorthm muscan mprovses the ptches of hs/her nstrument to obtan a better state of harmony. The dfferent steps of the classcal HS algorthm are descrbed below: Step 1: The 1 st step s to specfy the problem and ntalze the parameter values. The optmzaton problem s defned as mnmze (or mze) f (x) such that mn x x x, where f (x) s the objectve functon, x s a soluton vector consstng of N decson varables ( x ) and x mn and x are the lower and upper bounds of each decson varable, respectvely. Other algorthm parameters, such as harmony memory sze (HMS), or the number of soluton vectors n the harmony memory; harmony memory consderng rate (HMCR); ptch adjustng rate (PAR); and the number of mprovsatons (NI) or stoppng crteron are also specfed n ths step.

5 Populaton Varance Harmony Search Algorthm to Solve Optmal Power Flow 69 Step 2: The 2 nd step s to ntalze the Harmony Memory. The ntal harmony memory s generated from a unform dstrbuton n the ranges [ xmn, x ], as ( ) j mn mn x = x + r x x (13) where = 1,2,.., N, j = 1,2,3..., HMS, and r ~ U(0,1). Step 3: The thrd step s known as the mprovsaton step. The New Harmony vector y = ( y1, y2,, yn ) s generated by usng memory consderaton, ptch adjustment, and random selecton. The procedure works as follows: Pseudo-code of mprovsaton n HS for each [ 1, N ] do f U ( 0,1) HMCR then /*memory consderaton*/ j y = x, where j ~ U ( 1,2,, HMS ). U 0,1 then /* Ptch adjustment */ f ( ) PAR Y = Y + r bw, where ~ U ( 0,1) r (14) else /* random selecton */ y = x + r x x (15) endf done ( ) mn mn Step 4: In ths step the harmony memory s updated. The generated harmony vector y = ( y1, y2,, yn ) replaces the worst harmony n the HM (harmony memory) only f ts ftness s better than the worst harmony. Step 5: The stoppng crteron (generally the number of teratons) s checked. If t s satsfed, computaton s termnated. Otherwse, Steps 3 and 4 are repeated. 3.2 Modfed Populaton Varance Harmony Search (PVHS) Algorthm In [21] Mukhopadhyay et al. analyze the exploratve power n HS as follows: Theorem 1. Let x = { x1, x2,, xn } be the current populaton, Y = { Y1, Y2,, YN } the ntermedate populaton obtaned after harmony memory consderaton and ptch adjustment. If HMCR be the harmony memory consderaton probablty, PAR the ptch-adjustment probablty, bw the arbtrary dstance bandwdth and f we consder the allowable range for the decson varables (x ) to be {x mn, x } where x = a, x mn = a, then

6 70 B.K. Pangrah et al. ( ( )) ( m 1) ( ) ( ) E var Y = [ HMCR var x + HMCR 1 HMCR x m _ + HMCR ( 1 HMCR) PAR bw x HMCR PAR a + HMCR PAR bw + ( 1 HMCR) ] If HMCR s chosen to be very hgh (.e. very near to 1) and the dstance bandwdth parameter ( bw ) s chosen to be the standard devaton of the current populaton, then populaton varance (wthout selecton) wll grow almost exponentally over generatons. Now, Neglectng the terms contanng ( 1 HMCR), and choos- bw = σ x = var x the expresson (16) becomes: ng ( ) ( ) 2 (16) ( ( )) ( m 1) 1 HMCR PAR var Y = HMCR + HMCR PAR var ( x) (17) E m 3 4 From equaton (17) t s seen that f we do not nclude selecton n the algorthm, then the expected varance of the g th populaton ( X g ) becomes: E ( ( ) ( m 1) var X g 1 HMCR PAR = HMCR + HMCR PAR var( X 0 ) (18) m 3 4 In equaton (10) f we choose the values of the parameters HMCR, PAR n such a way that the term wthn the second brackets becomes greater than unty, then we can expect an exponental growth of populaton varance. Ths growth of expected populaton varance over generatons gves the algorthm a strong exploratve power. In modfed HS the bw s changed dynamcally as ( x) var( x) σ = (19) We also took HMCR = 0.98 and PAR = 0.67 to equp the algorthm wth more exploratve power after performng a seres of experments. 4 Implementaton for OPF The optmal power flow problem s mplemented usng the PVHS algorthm by takng the control varables u n the each harmony memory and equaton (12) as objectve. The algorthm stops when the current generaton exceeds the total number of generaton. The parameters are selected as: total number of generaton = 100, harmony memory sze (HMS) = 50, harmony memory consderng rate (HMCR) = 0.98, ptch adjustng rate (PAR) = The detaled mplementaton methodology s descrbed as follows: Step 1: Intalze harmony memory sze (HMS), harmony memory consderng rate (HMCR), ptch adjustng rate (PAR) Step 2: Intalze harmonc memory and evaluate objectve functon after runnng load flow g

7 Populaton Varance Harmony Search Algorthm to Solve Optmal Power Flow 71 Step 3: Improvsaton of harmony memory by ptch adjustment Step 4: Run load flow and evaluate the objectve functon Step 5: Update the harmony memory wth ths mprovsed soluton f t s better than worst soluton n memory Step 6: If stoppng crtera s met then prnt the OPF result and stop, otherwse go to step3. 5 Results and Dscusson The PVHS algorthm was tested on IEEE30 bus system conssts of 6 generatng unts, 41 transmsson lnes and 4 tap-changng transformers [8]. The lower and upper lmts on ndependent varables are shown n Table 1. In all the cases bus 1 s consdered as swng bus. The smulaton was done by takng a quadratc cost curve n case 1, a pecewse quadratc cost curve n case 2, and quadratc cost curve wth valve pont loadng n case 3. The result of the PVHS algorthm s compared wth NLP [8], EP [4], TS [5], PSO [9], IEP [6] and MDE [7]. The algorthm s coded on Intel Pentum IV 2.3 GHz processor and 2 GB RAM memory usng Matlab 7.4 [12] programmng language. Table 1 Cost Results of PVHS Algorthm Parameter Lower Upper Lmt Lmt Case1 Case2 Case3 P g P g P g P g P g P g V g V g V g V g V g V g T T T T P loss Sum(P g ) Penalty Best Cost

8 72 B.K. Pangrah et al. 5.1 Case 1 In ths case the fuel cost characterstcs of all the 6 generatng unts are gven by quadratc cost functon as f = = 1 F ( Pg ) = = 1 a + b P g 2 g + c P where a, b and c are the cost coeffcents of the th generator. The generator cost coeffcents are found n [6] and the optmzed parameters correspondng to mnmum cost s gven n Table 1. The results of the PVHS algorthm s compared n Table 2 wth other reported results. The statstcal results of 50 trals are also reported n Table 2. The algorthm converges quckly and the results are better than others. The convergence characterstc of PVHS algorthm for ths case s shown n Fgure 1. Table 2 Cost Comparson wth Other Methods for Case 1 Parameter NLP[8] EP[4] IEP[6] MDE [7] PVHS Best Cost Worst cost Avg cost Std cost (20) 5.2 Case 2 Fg. 1 Convergence of PVHS Algorthm for Case 1 In ths case the fuel cost characterstcs of the generatng unts connected at bus 1 and bus 2 are havng pecewse quadratc cost curve [6] to model dfferent fuels. 2 mn 1 a1 + b1pg + c1pg, Pg Pg Pg a2 + b2 Pg + c2pg, Pg < Pg Pg F ( Pg ) = (21)... 2 k 1 ak + bk Pg + ck Pg, Pg < Pg Pg

9 Populaton Varance Harmony Search Algorthm to Solve Optmal Power Flow 73 where a k, b k and c k are the cost coeffcents of the th generator at the k th nterval. The other 4 generators are havng same quadratc cost curve coeffcents as mentoned n Case 1. The generator cost coeffcents are found n [6] and the optmzed parameters correspondng to mnmum cost s gven n Table 1. The results of the PVHS algorthm are compared n Table 3 wth other reported results usng modfed dfferental evoluton algorthm (MDE) [7]. The algorthm converges quckly and the results are better than the other. 5.3 Case 3 Table 3 Cost Comparson wth Other Methods for Case2 Parameter MDE[7] PVHS Best Cost Worst cost Avg cost Std cost In ths case the fuel cost characterstcs of the generatng unts connected at bus 1 and bus 2 are also havng a sne component to model the valve pont loadng effect of the generators as ( ) 2 mn ( ) sn ( ) F P = a + bp + cp + d e P P (22) g g g g g where a, b, c, d and e are the cost coeffcents of the th generatng unt. The other 4 generators are havng same quadratc cost curve coeffcents as mentoned n Case 1. The generator cost coeffcents are found n [6] and the optmzed parameters correspondng to mnmum cost s gven n Table 1. The results of the PVHS algorthm are compared n Table 4 wth other reported results usng mproved evolutonary programmng (IEP) [6], and modfed dfferental evoluton algorthm (MDE) [7]. The algorthm converges quckly and the results are better than the others. Table 4 Cost Comparson wth Other Methods for Case 3 Parameter IEP[6] MDE[7] PVHS Best Cost Worst cost Avg cost Std cost Summary and Conclusons In ths chapter, detaled dscusson s carred out about the applcaton of populaton varance harmony search algorthm to solve the optmal power flow problem

10 74 B.K. Pangrah et al. n the presence of securty constrants. The algorthm explores the search space quckly wth the help of populaton varance parameter. The PVHS algorthm was tested wth IEEE 30 bus test system havng three dfferent types of cost characterstcs. The comparson of obtaned results wth other prevously reported results shows the effectveness of the algorthm. References 1. Wood, A.J., Wollenberg, B.F.: Power Generaton, Operaton and Control. John Wley & Sons, New York (1984) 2. Momoh, J.A., El-Hawary, M.E., Adapa, R.: A revew of selected optmal power flow lterature to 1993 Part I: Nonlnear and quadratc programmng approaches. IEEE Trans. on Power Systems 14, (1999) 3. Momoh, J.A., El-Hawary, M.E., Adapa, R.: A revew of selected optmal power flow lterature to 1993 Part II: Newton, lnear programmng and nteror pont methods. IEEE Trans. on Power Systems 14, (1999) 4. Yuryevch, J., Wong, K.P.: Evolutonary programmng based optmal power flow algorthm. IEEE Trans. on Power Systems 14, (1999) 5. Abdo, M.A.: Optmal power flow usng tabu search algorthm. Electrc Power Components and Systems 30, (2002) 6. Ongsakul, W., Tantmaporn, T.: Optmal power flow by mproved evolutonary programmng. Electrc Power Components and Systems 34, (2006) 7. sayah, S., Zehar, K.: Modfed dfferental evoluton algorthm for optmal power flow wth non-smooth cost functons. Energy Converson and Management 49, (2008) 8. Alsac, O., Stott, B.: Optmal load flow wth steady-state securty. IEEE Trans. on Power Apparatus Systems 93, (1974) 9. Abdo, M.A.: Optmal power flow usng partcle swarm optmzaton. Electrc Power Energy Systems 24, (2002) 10. Bakrtzs, A.G., Bskas, P.N., Zoumas, C.E., Petrds, V.: Optmal power flow by enhanced genetc algorthm. IEEE Trans. on Power Systems 17, (2002) 11. Roa-Sepulveda, C.A., Pavez-Lazo, B.J.: A soluton to the optmal power flow usng smulated annealng. Electrc Power Energy Systems 25, (2003) 12. Matlab (2007), Geem, Z.W., Km, J.H., Loganathan, G.V.: A new heurstc optmzaton algorthm: harmony search. Smulaton 76, (2001) 14. Geem, Z.W., Km, J.H., Loganathan, G.V.: Harmony search optmzaton: applcaton to ppe network desgn. Int. J. Model. Smulaton 22, (2002) 15. Lee, K.S., Geem, Z.W.: A new structural optmzaton method based on the harmony search algorthm. Comput. Struct. 82, (2004) 16. Geem, Z.W., Lee, K.S., Park, Y.: Applcaton of Harmony Search to Vehcle Routng. Amercan Journal of Appled Scences 2, (2005) 17. Geem, Z.W.: Optmal Cost Desgn of Water Dstrbuton Networks Usng Harmony Search. Engneerng Optmzaton 38, (2006) 18. Vaseb, A., Fesanghary, M., Bathaeea, S.M.T.: Combned heat and power economc dspatch by Harmony Search Algorthm. Internatonal Journal of Electrcal Power and Energy Systems 29, (2007)

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