THE IMPORTANCE OF INCLUDING ELASTIC PROPERTY OF PENSTOCK IN THE EVALUATION OF STABILITY OF HYDROPOWEWR PLANTS

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1 6 th IAHR Interntionl Meeting of the Workgroup on Cvittion nd Dynmic Problems in Hydrulic Mchinery nd Systems, September 9-, 05, jubljn, Sloveni THE IMPORTANCE OF INCUDING EASTIC PROPERTY OF PENSTOCK IN THE EVAUATION OF STABIITY OF HYDROPOWEWR PANTS Torbjørn K. Nielsen Dep. of Energy nd Process Engineering, Wter Poer bortory, NTNU, Nory ABSTRACT To gin stble opertion of hydro poer plnt, it is mostly mtter of hving the right rtio beteen the time constnt of the rotting msses, T, nd the time constnt for the ter msses, T. If T/T> 6 (or t lest >4), the stbility is normlly not problem. Hoever, for poer plnts ith long penstocks, this criterion is not enough. The elstic property of the penstock becomes n issue. The solution of the ve eqution includes term, hich mthemticlly is defined s tnh (tngents hyperbolicus). This function is notorious unstble. It hs similrity to the tn-function, hich goes from to ± s it pproches ± 90 o. The cross frequency defines the frequency up to hich the governor ill function. Above the cross frequency, ny disturbnce ill go ithout ny interference from the governor. Therefore, the issue is to mke sure tht the elstic frequency is ell bove the cross frequency. KEYWORDS Wter poer, stbility, penstock,elsticity. INTRODUCTION Simultions ith the purpose of controlling system stbility is preferbly done in the frequency domin, becuse then ll the eigen frequencies of the system re identified. If there is stbility problem, the cuse cn esily be detected. This is not the cse if one do the simultions in time domin. The differentil equtions for the system must be linerized round the point of opertion, i.e. t given flo Q 0, hed, H o nd speed of rottion n 0, nd then plce trnsformed to frequency domin. The system is hereby defined by it s trnsfer functions presented in block digrm. This is ell documented method. There re numerous methods to nlyse the result by grphicl representtion, like Bode, Nyquist, Nichols, Root-locus to mention fe. They ll ends up finding the stbility mrgins, i.e. the Phse mrgin nd the Gin mrgin. The trnsfer function of hydro poer plnt ssumed rigid penstock is shon in Fig.. The most importnt time constnts re T nd T. T is the time constnt for the inerti of the ter msses, hich is defined s the time it tkes to ccelerte the ter msses from zero to nominl flo. The ter msses prticipting in the governing process is from the nerest surfce up-strem the turbine to the nerest ter surfce don-strem the turbine. T is the time constnt for the rotting msses, hich is defined s the time it tkes to ccelerte the rotting msses, i.e. minly the genertor, from zero to nominl speed of rottion ith full torque.

2 µ ref (+ Tds)(+ TNs) b Ts t d y Ts + 0.5Ts p h p e p Ts n µ n 0 Figure : Block digrm for hydro poer plnt T cn be derived using Neton s. : T Q H ga 0 () 0 T is highly dependent on the hydrulic design of the poer plnt, hile T is more or less given by the genertor mnufcturer. The trnsfer function beteen the guide vne position, y, nd the hydrulic poer, p h, is in more detil shon in Figure. Figure Block digrm beteen guide vne position y nd poer, p, rigid penstock If one include the elstic property in the penstock, the solution of the Allievi eqution [3] ill include tnh-function; hence, the block digrm ill be s shon in Fig. 3. The constnt, h, is the Allievi constnt, hich is defined s: h Q o () AgH o The norml criterion is tht if h >, preferble good del bigger thn, the elsticity cn be disregrded. Figure 3 Block digrm beteen guide vne position, y, nd hydrulic poer, p h. Elstic penstock.

3 The Allievi constnt is in fct the rtio T /T r, here T r is the reflection time of the elstic ve: T r (3) Where is the length of the penstock n is the pressure propgtion speed or the speed of sound. The criterion h > cn then be interpreted s: T T > or > (4) T T r r The cross frequency is very ner /T, so h > mens tht the frequency for the elstic ves must be higher, or even fr higher thn the cross frequency. In tht cse, the governor ill not rect on the elstic ves, nd the elsticity ill not be stbility issue.. COMPARISON OF THE RIGID AND EASTIC TRANSFERE FUNCTION The to block digrms shon in Fig. nd Fig. 3, gives the to folloing trnsfer functions beteen guide vne position nd poer: Rigid: p y Ts (5) + 0.5Ts Elstic: h tnh( s) p y + h tnh( s) (6) Where s is the complex vrible: s jω. According to methods described in control theory [, ] the mplitude nd the phse ngle cn be solved nd gives n illustrtion of the difference beteen rigid nd elstic representtion. In generl, for trnsfer function: ( + Ts )( + Ts ) As () A( jω) ( + Ts)( + Ts) 3 4 (7) The mplitude is: A( jω) ( + ( ωt) )( + ( ωt ) ) ( + ( ωt ) )( + ( ) ) 3 ωt4 (8)

4 The phse ngle is: A( jω) tn( ωt) + tn( ωt ) tn( ωt ) tn( ωt ) (9) 3 4 For the rigid trnsfer function, the clcultion is rther stright forrd. For the elstic trnsfer function, the complex vrible s in the expression tnh( s ) mkes problem. The complex j hs to jφ be outside the prenthesis. Using Euler eqution e cosφ+ jsinφ, illustrted in Fig. 4, mkes it possible to rerrnge the eqution. Figure 4 Illustrtion of the Euler eqution The Euler eqution for this prticulr cse: j ω e cos( ω) + jsin( ω) (0) Inserted in the eqution for the tnh term, mthemticlly defined s: s s ( e e ) tnh( s) s s ( e + e ) here sjω () cos( ω) + jsin( ω) cos( ω) jsin( ω) tnh( jω) cos( ω) + jsin( ω) + cos( ω) + jsin( ω) () tnh( s) jtn( ) ω (3) The tnh(/s) term hs trnsformed to j tn(/ω) nd the mplitude nd ngle cn be clculted. For the rigid trnsfer function: Amplitude: A( jω) + ( (0.5 ) ( ( ωt ) ) + ωt (4)

the result is shon in Figure 5 nd Figure 6, left Figure rigid nd right Figure elstic property. Figure 5 Rigid (left) nd elstic trnsfer function plotted in Re-Im plne.

5 Phse ngle: A( jω) tn( ωt ) tn(0.5 ωt ) (5) For the elstic trnsfer function: Amplitude: A( jω) ( + (h tn( ω)) A ( + ( h tn( ω)) A (6) Phse ngle: A( j ω ) tn(h tn( )) tn(h tn( )) A ω A ω (7) Solving the equtions for incresing ω, the result is shon in Figure 5 nd Figure 6, left Figure rigid nd right Figure elstic property. Figure 5 Rigid (left) nd elstic trnsfer function plotted in Re-Im plne. The difference is tht the rigid function goes from nd stops t - on the Re-xes, hile the elstic function tkes the hole turn ll the y bck to gin. The intuitive explntion of this is tht the rigid just stops, hile the elstic one bounces bck due to the elsticity. Figure 6 shos the sme performnce in Bode plot (Amplitude nd Phse vs frequency) Figure 6 Bode plot for rigid nd elstic penstock

6 3. CONSEQUENCE FOR THE STABIITY SIMUATION OF A POWER PANT To estblish if poer plnt is stble or not, is question of checking the stbility mrgins, i.e. the mplifying mrgin nd the Phse mrgin. The complete trnsfer function of the poer plnt must be estblished nd the mplitude nd phse ngle clculted s function of the frequency. There re mny ys of plotting the result in order to find conclusion on hether the stbility mrgin is stisfctory. The uthor prefers the Bode plot, hoever to plot the result in Re-Im pln, Nyquist digrm, might give dditionl informtion. The complete trnsfer functions for rigid nd elstic model is: Rigid: (+ T s)(+ T s) ( T s) A(j ω ) b T T s ( 0.5T s) d N t d + (8) Elstic: A(j ω ) btt For the rigid function the Amplitude is: ( h tnh( s)) ( + T s)( + T s) d N t d s + ( h tnh( s)) (9) + ( ω T ) )( + ( ω T ) + ( ωt ) ) A(j ω ) b T T (0.5 T ) ) d N 4 t d ω + ω (0) And the Phse ngle is: A( j ω ) tn( ω T d) + tn( ωt N) tn( ωt ) tn(0.5ωt ) π () Becuse of the negtive sign in the expression ( Ts) in the numertor of eq.8, the phse shift is negtive. This is the key issue regrding stbility nd T. It mkes the phse go tords The -π term in eq. comes in becuse of the to poles t s0. For the elstic function the Amplitude is: + (h tn( ω)) + ( ω T d) )( + ( ωt N) A(j ω ) 4 btt t d ω + (h tn( ω)) () And the phse ngle is: A( j ω ) tn( ω T d) + tn( ω T N) tn(h tn( )) tn(h tn( )) ω ω π (3) Figure 8 shos Bode plots for both rigid n elstic model. Ignoring the elsticity of the penstock, the hydro poer plnt seems to be stble s both Phse mrgin nd Gin mrgin is sufficient. Including the elsticity, the system is on the border of instbility, s shon in Fig. 8, right.

7 Gin mrgin Phse mrgin Figure 8: Bode plots. eft side shos the rigid simultion nd right side the elstic simultion, T 0,7sec, T 6sec, h In the upper Figures, A is open loop, M is closed loop nd N is the sensitivity. The to figures t the bottom shos the sme, plotted in Re-Im plne (Nyquist digrm) The PID prmeters used for the simultions shon in Figure 8 re b t0., T d6sec, T N0.0. It is of course possible to stbilize the system by tuning the PID prmeters, hoever, the cross frequency ill esily be t too lo frequency, hich mens tht the governing ill be to slo nd stnding oscilltions ill occur. Incresing the inerti of the genertor, i.e. incresing T ill hve the sme effect. 4. CONCUSIONS High hed poer plnts hve often long penstocks, hich mkes the reflection time, T r, too big compred to T. It is quite possible to chieve T <, hich is often the design criterion used, nd still get instbility becuse of the elstic property of the penstock. In order to design stble system ith sufficient stbility mrgins, Allievi s constnt must be checked by clculting the rtio T /T r. This rtio, Allievi s constnt, should be t lest bigger thn. T > i.e. > (4) T T T r r Which mens tht the frequency of the elstic ve must be bigger thn the cross frequency, hich is very ner /T.

8 If this criterion is not fulfilled, it might still be possible to mke the system stble by optimizing the governor settings; hoever, the qulity of the governing system ill be lousy. 5. REFERENCES [] Normn S. Nise: Control Systems Engineering,The Benjmin/Cummings Publishing Compny, Inc., ISBN [] Blchen, Fjeld, Solheim: Reguleringsteknikk, TAPIR, ISBN [3] Brekke, H: A Stbility Study on Hydro Poer Plnt Governing Including the Influence from Qusi Nonliner Dmping of Oscilltory Flo nd from the Turbine Chrcteristics, PhD disserttion NTNU, NOMENCATURE T (s) Penstock time constnt (m.s - ) Pressure propgtion speed T (s) Time constnt for rot. msses (m) ength of penstock T d (s) Dsh-pot time constnt h (-) Allievie s constnt T N (s) Derivtive time constnt s (-) Complex vrible T r (s) Reflection time q (-) Dimensionless flo b t (-) Trnsient speed droop h (-) Dimensionless hed y (-) Guide vne position p (-) Dimensionless poer µ (-) Dimensionless speed of rottion

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response