r ~~~~~~~~~andflow at any one section of a pipe toynamicesponse o a y roe ectric ~~these variables at any other. He is

indebted to Professor Henry M. Paynter,ant ~~~~~~~~~Massachusetts Institute of Technology,

for this formulation. With these ordinarydifferential equations the dynamic re-

R. OLDENBURGER J. DONELSON, JR. ~sponse of a hydraulic turbine and con-

MEMBER AIEE ASSOCIATE MEMBER AIEEnetdpyiassemcnbdsrbdby a set of ordinary differential equations.The validity of such sets of equations was

Summary: Frequency-response tests have of the method. The results of this paper verified by frequency response runs atbeen made on the turbines at the Apalachia are needed to design computer and other the Apalachia powerhouse of the TVA.Hydroelectric Plant, of the Tennessee onrlfrhyoectipatscetf- This paper reports on the results of theseValley Authority, to check the validity of conrl o yreeti lnssinii tests. Part of the theoretical work waspreviously derived mathematical expres- cally. done by Professor Yasundo Takahashi ofsions. These tests confirmed in general For the scientific design of speed con- the University of California. The basicthat the mathematical expressions could be trols of hydraulic turbines in power eqain wredivdb Pofsrused to describe the performance under all sytmanotecotositineear eqtoswredivdb Pofsrmajor operating conditions and for the sytoaem andilother coferntrolsts neessaryon Paynter. The tests were started indirect design of nonlinear power system tohv vial ifrnileutos December 1954, and completed soon after.controls. Hydraulically, this is the most describing the system sufficiently ac- The theoretical and experimental resultscomplicated installation east of the Rockies. curately and at the same time simply of this paper are required for the optimum

enough to be easily manipulated indsg facmue,gvmr n te

mathematical design and analysis com- designl fofa ydcomuer,geri orp andtohe

THE RESULTS OF frequency-response putations. These equations must de- coTrose ohyderboeleticpleatrintests. oUtests are presented on the hydraulic scribe hydraulic components of power sta- Thenarhypferbolic loeruatiorsi therisetsoturbine units at the Apalachia power- tions such as penstocks and surge tanks, oriat dplcifferyenta requationbysimverifehouse of the Tennessee Valley Authority as well as mechanical and electrical ele- oeatorApalachiatmayhe reweplacedobysimle(TVA). These tests were suggested by ments. Adequate techniques for treat- opereatorsls thnpuattedne eqatinponssicanthe Woodward Governor Company with ing the nonhydraulic aspects of power bereapddeiy o

ma nipulated,maipmngpossblthe co-operation of TVA, and were per- systems have been available for a long raidegnocntleqpm tfrformed under the supervision of the au- time. This has not been true of hydraulic hydroelectric installations.thors while R. Oldenburger was director lines. Previously, ASEA made frequency-of research of the Woodward Governor It is well known that in a first aprx- response runs on the effect of variationsCompany. The gates of one of the mation the flow of fluids in hydraulic pipes in input power on the frequency of theturbines were oscillated over wide fre- may be described by partial differential Seihpwrsse. h plci

underallajoroperting equaIon'o h aetp stoe tests were not concerned with the responsequency ranges une l ao prtn qain'o h aetp stoe of the entire TVA system, but only of theconditions and the resulting speed rpm employed for electrical transmission lines.2 units in the Apalachia powerhouse.(revolutions per minute), electrical phase In general, enormous mathematical diffi- The Apalachia tests were made underangle, and power swings recorded. The culties are involved in the solution of all major operating conditions of the twopartial differential equations for fluid flow system control problems when the partial turbines in the powerhouse in relation toin pipes are too complicated mathemati- differential equations are used with- teuisbigo rofteln ncally for direct use in the design of power out approximation. To overcome this, oprtheduisbingly nortogether Traces weresystem controls. Resort has been made Gaden' introduced a lumped first order mpeatdeoftrinelorp undeter. sTraeaystaterto the lumped-parameter approach Of approximation to a unit composed of the conditionsFurbieqrmuency-respneteaystsaonGaden, but this is not accurate enough turbine and the hydraulic line (such as turbneduitios. werequnyresporsetedsby A.for some studies. The partial differential penstock, scroll case, and draft tube). turbpfeunitseerinprtdbyAequations may be replaced by ordinary Evangelisti used the Laplace trans- Klpesindifferential equations in hyperbolic opera- formns of the partial differential equations A description of the hydraulic systemi attors and these operators may be approxi- of pipe flow, where these transforms in- the Apalachia project was published bymated by polynomiials, so that the result- volve hyperbolic functions.' It is well George R. Rich.9 This is a system with a

ing simplified equations can be easily em- known that the relation between' the com- long tunnel, differential surge tank, twoployed for control analysis and synthesis. plex currents and voltages at two sections pntcsspligtoFacstpThe frequency-response runs were made of a transmission line may, under rather turbines, and draft tubes.to verify the validity of these equations general conditions, be expressed by twoand approximations. Good agreement equations involving such functions.5 Old- The Apalachia Projectwas obtained between theory and experi- enburgert published the analogous ordi-

connected to an infinite bus for light load:1.25 cps

Apalachia Powerhouse is normallyISIS.VLJU>#ER00US F 7t t S connected by a 14-mile 161-kv trans-

s ISTrRRtSIJFD,Y I mission line to the Hiwassee switchyard.ITIANKS R ~ 3 - E <> NT ffi>(PENSTA E; I One line comes into this yard from Ocoee

TUNNEL | number 3 which has 27,000-kw generation.lP 7i' | |Blue Ridge Hydro and Ocoee number 2,

SITE PLAN g 6 with 20,000- and 21,000-kw generationSCALE 4000 0 4000 F ET respectively, are connected to the Ocoee

number 3 switchyard. Hiwassee is con-r l/jXlooo ° °° °; o R0;>P\ OACC| nected by a 51-mile 161-kv line to Alcoa.

l 2- ° =-= N eI One 47-mile 161-kv line connects Apa-TRANFORMER TRACK lachia to Chickamauga via East Cleve-

ALA//TRANSFORMER TRACKX I}n , , land. Reactances of all lines were about0.8 ohm per mile. The impedances oftransformers, machines, and transmission

VAVE HOUSE VALVES, STA. 437+20. SURGE TANK lines, and system equivalents are shownVUNITS S F S) S F Nt 1 N\ f in Fig. 2. Connections are indicated in

STA. T0 UCTLINi I A/ J On-line runs were made with the440+ 51.0 POWERHOUSEI. Lilt l2 UNITS40,000 KW (OO// 433+ 5376/ / / | Apalachia units as normally connected toI \ 111 tL«IN ) I C/ / / s / | the TVA system. On-line runs were also

made with one Apalachia unit isolated toChickamauga, i.e., disconnected from

PLAN Hiwassee (except for station supply),BlueRidge Hydro, Ocoee numbers2and 3.

1400 l | This unit was connected to Chickamauga,£E4ASSLO13 through the 51-mile transmission line,

S00 d ,= z ELl35BOpassing through East Cleveland, but notE C. VALVE OUCT LINE 1 16.0 ID \ | connected to Oglethorpe.

1200 -S__ : X__URGETANK The reactances were taken as follows,(some of these are not needed for the

oo10EL 11250

W 1._ ID differential equations, but are included2-11O'D POWERHOUSE for completeness):|PENSTOGIKS =- f COrNE I F

1000 & I IIi---_I I Value of Apalachia transformer reactance on(8POWERHDUSEw ICOD TUNNELS Q lSTA 433+ 53.6 I a 100,000-kva base with Apalachia normally43Ts+110 W_ connected to TVA system, used in computa-

tions: 12%4DISTR--k Transmission line reactance, on same base6EL UNITS SECTIONAL ELEVATION I for same connections, taken to be: 4%

SURGE TANK AND POWERHOUSE Transmission line reactance for on-line runsl______-_____-_____ _______________SURGETANK_AND_POWERHOUSE with unit isolated to Chickamauga takenFig. 1. VA Apalahi. instllationto be: 7%Fig. 1. TVA Apalachis installation Apalachia 161-kv bus impedances on a

100,000-kva base with:Downstream from the surge tank the Gross head, used for computations (about (a) Both generators off line: 8.38%tunnel splits into two 12-foot-diameter half of drop occurs between the forebay (b) The two generators on line: 7.2%tunnels with steel liners followed by 11 and penstocks and the rest in the pen- (c) One generator on line: 7.83%foot-diameter penstoc.ks. Details follow: stocks): 428 feet System (external) reactance for Apalachiafootdiaetepestoks,Detilsfolow: Rated head losses in tujnnel up to surge units directly connected to TVA system:

tank for 1,600-cfs (cubic feet per second) 16%Tunnel length: 43,200 feet flow rate: 100 feet System (external) reactance for ApalachiaDiameter of 41,200-foot concrete-lined Rated head losses in rest of hydraulic lIne units connected to TVA system throughportion of tunnel: 18 feet (penstock, turbine, draft tube, etc.) for trnsmission line to Chickamauga: 19.2%Diameter of 2,000-foot steel-lined portion 1,600-cfs flow rate: 25 feetof tunnel: 16 feet Turbines, 2 Baldwin Southwark units, hp The turbine constants used in the differ-Diameter of surge tank: 66 feet (horsepower) rating is for 360-foot grossDiameter of riser: 16 feet head, i.e. headwater minus tailwater, and entialequationswereobtanedfrommodeDistance from surge tank to Y (concrete- 225 rpm: 5,3,500 hp curves. The efficiency of the prototypelined tunnel): 100 feet Rated turbine speed: 225 rpm was assumed to be 3% greater than thatLength of water column from surge tank to Generators (Westinghouse) rated: 40,000 for themodel. Fig. 4showsacomparsondraft tube discharge: 841 feet kva (kilovolt-amlperes)(100-foot tunnel to Y, 260-foot-long 12- Rated power factor: 0.9 between the prototype and model horse-foot-diameter steel-lined tunnels leading Phase: 3 power versus gate-openling curves. Thefrom Y to valve house; 2 such tunels, Cycles, cps (c:ycles per second): 60 results of TVA index tests for a single340-foot penstocks from valve house to Terminal voltage: 13,800 unit and both units running are shown inscroll cases, 40 feet from start of scroll Computed natural frequency of generators Fig5 Thscuvsho otptmcase to tulrbine inlet, 8 feet through turbine connected to an infinite bus for full-load .5 hs uvsso uptmrunner, 44-foot drft tubes, and miscel- excitation: 1.5-1.6 cps versus gate where the percentage gatelaneous sections) Computed nlatural frequency of generators was read at the turbine.

404 Oldenurger, Donelson-Dynamic Response ofHydro Plant OCrOBER 1962

Instrumentation referred to as cases I-V. A description of mauga, and the other unit shut down.these cases follows: Case V. Both units on line under normal

An "electronic phase comparator" for C I O operation.recodingeletricl pase ngl beteen

Case I. One unit off linle, other unit shutrecording electrical phase angle between down. Let constants K, and B, be defined asthe unit being oscillated and the line was Case II. One unit off line being oscillated follows:developed for the tests of this paper by and other on line. K8 = per-unit synchronizing torque perStuart~~~~~~~~~~~~~A.pe-ulMonfortlmandquEuenrCStuart A. Monfort and Eugene C. Case III. Oscillated unit connected to electrical radianWhitney of Westinghouse Electric Cor- TVA system at Hiwassee, and other unit B8 =per-unit damping torque per electricalporation. This comparator was designed shut down. radian per secondspecifically for use in determining the Case IV. Oscillated unit connected to the The values of K, and B. employed inangular displacement between the rotor TVA system through a line to Chicka- computing the theoretical curves wereof a salient pole synchronous machine,and some fixed reference. In the on-lineruns a beam of light was placed in the Fig. 2 (right). APALACHIA ALCOA SW. STA.generator housing directed to a photoelec- TVA impedances 40 MVA 161 KVV. 161 KV.

13. 8 Ky.tric cell. Each passage of a pole piece of t 42.%SYSTEMthe generator intercepted the light beam, J 2 4.

4 EQUIVALENT

yielding a flash for each half wave of the - J2. 1 %generator output. The mechanical an- J30%gular position of the generator rotor rela-tive to the voltage taken at the machineterminals, or a reference bus, was deter-mined by the phase comparator. Thus, APALACHIA OGLETHORPEtraces of electrical phase angle of the 161 KV. 161 KV.unit with respect to the line were obtained. 40 M VA1389KVA 3-phase alternator was mounted on t SYSTEM

generator shaft. The output of this J20.6 % EQUIVALENT

alternator was rectified and filtered to J49% J4.72%yield a voltage proportional to turbine J30%

rpm. This voltage was used for studiesof "noise" in the rpm signal, which studiesare not reported here, as well as the rpmcurves, which are given here. APALACHIA CHICKAMAUGAGate position was picked up by a trans-

40 MVA 161 K V. 161 KV.ducer mounted at a turbine servomotor 13. KV.and attached to the servomotor piston rod. SY 480- YSTEM

Current and voltage from the current J 5 44. EOUIVALEN28and voltage transformers at the power- J30%house board were employed to yield traces Fig. 3 (beIo) ALL IMPEDANCE VALUESof line current and voltage. Instantane- Connections oFON MVA ASEous current and voltage from these Apalachia to ITVAtransformers were multiplied on an [system TVA IMPEDANCESanalog computer to obtain instantaneouspower output.The mechanical oscillator used as the SYSTEM n

input for the frequency-response runs EQUIVALENT _ lKVwas described elsewhere.'0 With one HIWASSEE |SSTEMexception all of the frequency-responseruns were made with the oscillator outputconnected rigidly to the gate limit shaftof the governor. The governor speed EAST CLEVELANDsetting was raised to a point where theunit was on gate limit control. The fly-weights and overspeed trip were able tooverride the action of the oscillator.

Lags in instrulments were taken intol.account in interpreting experimental data. | v l1The excellence of the instrumentation was oA }) |due largely to the efforts of Karl HansonIof the Woodward Governor Company.c LAPACA

Frequency Responses ( \ C 6SYSTE

FreqXuency-response runs were made at ' fSYSTEM SYSTEMApalachiaunder five operating conditions, EQOUIVAL-ENT EQUIVALENT>

OCTOBER 196i2 Oldenburger, Donelson-Dynamic Response of Hydro Plant 405

60 a fixed load of 35,000 kw to the line,

whereas in Fig. 10 one unit was shut down._____ _____ 11

The slopes of these curves may be com-a pared with the experimental entries in< I I I I \ 'S/%MODPRTOYP Table II obtained from frequency-re-040 __ ____M sponse runs.040x [ ;/ l l At the time of the runs the TVA sys-

z tem had a system droop of 162/3%, i.e.,a: 30 _ -_l _frequency dropped 0.1 cps when load

0 DIA.- 103 was increased 1%. For the ASEA fre-I. HEAD 360' quency-response runs on the Swedishcr 20 toMODL 62 power system the corresponding figureI 'doo, TEST a 664 SERIES was 14%.w 0 x.665 Fig. 11 shows the results' of a linearityM0 check. Here the gates were oscillated

about the base position (8-inch gate corre-_____ _____ _____l_l_ l_ l | sponding to 40-mw output) through

0 10 20 30 40 50 60 70 80 90 100 amplitudes shown, and oscillations withperiods of 1 and 120 seconds. These tests

WICKET GATE OPENING IN PERCENT show that in a first rough approximationthe assumption of linearity between gateposition and power output is justified, butis not valid for accurate analyses.

54 Most runs were made with gate oscilla-PEAK tion amplitudes (total about 10%, i.e., 0.8

FOR UNITS AT SAME GATE: __ X 105% inch). Amplitudes of as low as 0.16%46 _ HEADWATER 1279. 6 FT. - - - and as high as 60o were employed.42 TAILWATER 840 FT.

FOR SINGLE UNITS: _ -HEADWATER 1279.5 FT.38 TAILWATER 839.4 FT. - - - - - Basic Hydraulic Equations

34 _

30 -[If I Consider a horizontal cylindrical pipe26 - 1___ 1__ [..TTTI 1....i - DOUBLE UNIT of uniform cross section as shown in Fig.31 _UNIT I - -.12. Associated constants and variables22 SIN LE U N ITA>1 I I I I I I I I I I 1are defined as follows:

r=internal radius of pipe14 _ ; ; .=1 1 1 _ = | l _ _ <f=thickness of pipe wall

x = longitudinal pipe co-ordinate10 - A .- l- - - l xt=x-co-ordinate at section i for i=I, II

IH=head at section i for i=I, II6 - H=head at arbitrary section

+2 - II. U =velocity (average) of fluid at section0 i for i=I, II-2 LCI II II.II . . U=velocity at arbitrary section0 1 0 20 30 40 50 60 70 80 90 100 Q1=volumetric rate of flow (average) of

% GATE AT TURBINE fluid at section i for i=I, IIg=acceleration of gravity

Fig. 5. Generator output verus gate, from index tests p=density of fluidK= bulk modulus of elasticity of fluidE=Young's modulus for pipe

supplied by Mr. Monfort. Typical values runs. The results of measuring a23 during a =pgF+lare shown in Table I. Curves for K, and the steady state and during frequency- KBo are given in Figs. 6 and 7. response runs at low frequencies such as a = J- wave velocityThe turbine gain a28 is the partial 1/5 cycle per minute were not completely

derivative of per-unit turbine driving consistent. The experimental values ob- L=distance between sections I and IItorque (torque applied by the water to tained for various gate swings at low corresponding to x=xi and x =xiIrotate the turbine runner) with respect frequencies, and the theoretical values TA L/a elastic time

. . . . . ' . . . ~~~~~~~A-internal cross-sectional area of pipeto per-unit gate position. The driving used in the differential equations, are (storque and turbine gate position are on shown in Table II. The theoretical ZO= a/gA impedanceper-unit bases, the torque being on a 40- values were obtained from the model h =deviation in head Hi from steady statemw (megawatt) base and gate on curves supplied by the turbine manufac- q =deviation in volumetric rate Qt froman 8-inch base. The gain a2a proved to turers. steady statebe a critical constant in the differential A calibration of rpm versus turbine D=deiavewtrspctoimequations for the Apalachia runs. The gate for off-line runs is shown in Fig. 8. It is shown in Appendix I that theconstant mustbeknown precisely in order Corresponding results for on-line tests standard equations of flow1' for fluid into obtain close agreement between theore- and mw versus gate are shown in Figs. 9 a uniform pipe with friction neglectedtical and experimental frequency-response and 10. In Fig. 9 one unit was supplying may be put in the form

406 Okien burger, Donelson-Dynamic Response of Hydro Plant OCTOBER 1962

Table I. Samples of K, and B, Values Used HII= (sech T,D)HI-Z0 (tanhi T,D)QuiI

(frequency. 0 ...0.016 ..0.16 . 1Q = (cosh TeDQ1+- (sinh TgD)HuI (1)Case II Ks.....1.7 .. .1.7 .1.85 .2.0 z~Bs.....0.122.. .0.119 .0.106 ..0.037

~frequency .0 ...0.096 ..0.16 ..0.32 ..0.5 ..1 ..3 ..5 ..7 where xi =O0,xi,=L. The same equati'onsCase III' Ks.....1.7 ...1.8 ..1.85 .1.9 ..1.95 .2 ..2.25 .2.45 .2.5 hoditehasH,HIIndfwrtsQlBs.....0.124..0.106 ..0.095 ..0.084 ..0.063..0.037 ..0.0159 ..0.0093hodfteeasH,1adflwas

~frequency.0 ...0.096 .0.16 .0.32 .0.5 .1 .3 . .5 .7 Qri are replaced by the deviations hi, hilCase IV1Ks. 1.0.I ...1.7 .1.75 ..1.8 ..1.85 ..1.9 ..2.15 ..2.3 ..2.5 and qi,qi, respectively.

B......0.124.. .0.119 .0.106 . .0.086 ..0.061.0 .0358. .0.0172 . .0.0106~frequency. 0 ....0.0001. .0.00019. .0.00048. .0.001..0.0011. .0.00118. .0.00119..0.00121 Theoretical and experimental investiga-

Case V Ks.....1.70 ...1.70 ..1.70 ..1.70 ..1.70 ..1.70 -.1.70 .1.70 ..1.70 tion shows.thatfor the Apalachia runs the~Bs.....0.122... .0.122 .0.122 .0.122 .0.122. .0.122 .0.122 ..0.122 ..0.122 drop in head due to resistance varies inNote: Frequency in cycles per second, steady state as the square of the flow rate.

Following a suggestion of H. Paynter, thehead loss due to friction is approximatedby replacing the first equation in the set 1

4- - ~~~~~~CALCULATED SYNCHRONIZING TORQUE COEFFICIENTSH1=(scTD)-40000 KVA 13.8 KV. 225 RPM 3% SONt H (ehTDH

VERTICAL W.W.GEN. ~~~~~Z0(tanh TD)Qir-kjQ1j1 (2)ui ~~~~~~~~~~~~~~~~~~~~~~fora constant k where I Qii I denotes the

W. absquare of Q1,Tand Qii=QI Ifrh0. absolute value IQ,, of Qii. In terms ofdeviations the first equation in the set l0~~~~~~~~~~

CE~ ~ ~ ~ - - - -isnowreplaeb- - - - - = ~~~~~~~~~~~hll=(sechTeD)h,-Z0(tanh T.D)qli-

CURVE B. FUL LOAD-I12% ExT. REACTANCE whr h osatk1i ie yki11 =2ku 1 .0 EXT. INFlNITE ~~~CURVE Dc NOL LOAD - 169.% EXT. REACTANCE OI eadesntr

ELECTRICAL RADIAN's 3.56 MECH. DEGREE resistance one may take equation 3 to hold

111.1111111 1 1 1 1 ~ ~~~~~~fora properly chosen value of k1r. The2 3 4 567 8 10 Ii 4 IS 17 ~~~~~~~~~~los'sin head for a steady flow Q in the pipe

WI 3I will be of the formUS

OSCILLATION FREQUENCY -CPS4 kz

Fig. 6. Synchronizing torque coeFficient for"2 a constant ki and 1 <n.<2, whereinthe deviation in the head loss due to pilperesistance is of the formnknQn-1dQ

Wh _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Co132 6 - - -Let Q0 be base flow rate and H,, baseW. I19I- -- head. Per-unit heads and flows are ob-z ~~~~~CALCULATED DAMPING COEFFICIENTS __ tamned by dividing heads and flows by

14.1060 ~~~40000 KVA 13.8KV. 225 RPM~~~~~VERTICAL w. W. GEN. H, and Q0, respectively. In terms of per-- - ~~~~~~~~unitheads hr, hi, and flows qx, qii equa-CURVE At FULL LOAD'-tosIbcm

ZERO EXT. REACTANCE tos1bcm%0796 CURVE S3 FULL LOAD -

-J ~~~~~~~~~~~~~~~6%EXT. REACTANCE Jl (ehTDh-~tn .)i

cr -AILl - ~~~~~~~~~~~~~~~~~~~~~~~~~~qI=(coshT6D)qxI±-(sinh ToD)hi i

00398 -for the normalized impedance Z,, where0

ET.0265 REC.ZOQO H H11 QI Qu0 __ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~n hi h0 0 Q0 Q0..0132~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r 7

Symbols for the variables follow a22 = partial derivative of per-unit turbine Apalachia Case I. Oscillation oftorque for unit 1 with respect to Unit Off Line, Other Shut Down

w= per-unit head for forebay per-unit turbine speedh= per-unit riser head b2= same for unit 2

I hscs n ntwssu onh = per-unit surge taak head an= partial derivative of per-unit turbine I hscs n ntwssu onhi= per-unit turbine head (at bottom end torque of unit 1 with respect to The other was running at speed, no load.

of penstock just ahead of the turbine per-unit turbinie gate position The gates were oscillated and the rpmscroll case of unit 1) bus= same for unit 2 swings recorded. In writing the differ-

h2 = same for unit 2 The partial derivatives are evaluated at ential equations, it will be assumed thatqc= per-unit tunnel rate of flow near surge thestateabout which oscillations aremade. unit 2 is shuit down.

tankq= per-unit riser rate of flow

q = per-unit surge tank rate of flowTal.

enseeVlyAuhrtTssqp1=per-unit rate of flow at the top end TbeI.TneseVle uhrt etof penstock for unit 1

qp2 = SaMe for unit 2 Gate Oscillations Experimental Theoretical Valueq,= per-unit turbine rate of flow for unit 1 Case Around Value* Used in Computationsq2=same for unit 2mi=Per-unit turbine driving torque for Unit 1 off line oscillating ON.........14........15unit 1 1 Unit 2 shut down ......%.1.6 . .

m2 = same for uinit 2 Unit 1 on line.......7%.........1.42.........1.5ni= per-unit turbine speed for unit 1 Unit 2 off line oscillating)

n2=sameforunit 2 ~ ~ ~ ~ ~ {~Uni 1 shut dowrn53714 . 2.n:g =sameforunit2 Unit 2 on line oscillating ......5%........146........125zi= per-unit gate position for unit 1 Unit 1 shut down 53% .........1.4 .........1.252= same for unit 2 Unit 2 on line oscillatingi.....

UnitIon line osiltn) 63%.1........I-08.........1.25G L(ni) Unit ~~~~~~~~~~~~2on line oclaig .....

L(zi) M Per-unit driving torque deviation

transfer function relating ni and zi where Z Per-unit gate deviationL(ni) and L(z1) are the Laplace transforms where M Is on 40-mw base

of n, and zi,respectively ~~Zis on 8-inch (80%) basecrmx*Taen qua todeviation of per-unit output mw (from micrmx

For frequency-response curves we setpe-ntgedviios=jcoin Gforthe Laplacevariables, .radian frequency Wendj=V 1. The e s9..symbol Gis alsoused for thetransfer20operator n1/z1. 25C-

--I40,=friction coefficient for tunnel ----0.,= friction coefficient for unit 1 penstock

04,2=friction coefficient for unit 2 penstock 2cr -- `.-Q0,=base flow rate (per-unit flow rate is 190 --

flow rate divided by Q0,)H0,=base head (per-unit head is head 7 - ---.

divided by H,) SC_ --.Z0c =tunnel impedance 31C .... .Z0s=impedance of unit 1 penstock 2.5' 4 5 6 7 a 9 10 IIZ2= impedance of unit 2 penstock

ze = normalized tunnel impedance% GATE AT TURBINE (10' BASE)

0o Fig. S. Turbine speed versus gate=Z,=QH, normalized impedance of unit 1

ZP,2 = normalized impedance of unit 2 - NTN.IO IN0T3.0Ho ~~~~~~~~~~~~~~~~~UNITNO. 2 ON LINEAT300KW

penstock45 UINO2ONLE

IS 1/2 MEGAVARS ON UNIT 2- - - - - - -T.,= elastic time for tunnel SC AV. HEADWATER 1277.27 FT.Tg surge tank-riser time 40 AV. TAILWATER 539. 76 FT. - - - - - - -

0 GROSS HEAD 431.51 FT.T.=tutrbine starting time - -T,~= elastic time for penstocksan =partial derivative of per-unit turbine 13 ..-

flow rate of unit 1 with respect to c

per-unit turbine head U.2.

b1s=same for unit 2

0 UNIT SHUT DOWN - - _ - - - - qc=(qt+qr)+qpi (7)UNIT 2 ON LINE

45 AV. HEADWATER 1277.81' where flows in the penstock for unit 2 areAV.TAILWATER 839.04'

40 GROSS HEAD 438 .t7'? neglected. The rate of flow into the surge- - - -tank nser and the surge-tank-riser water

*35o X 3 g X X g X X 1 level are related by the equation0

IX30 ~~~~~~~~~~~~~qt+q=TjDhj (8)- -/- PER UNIT GAtE ON 100% (10') BASE for a constant T1. Equations of type 1

-r0 AL SATE READ ON CABINET ACTUATOR for the penstock of unit 1 are_IL, OS ATE AT TURBINE, POINTS OOING

3'-/ 5 DOWN h,=(sech TeD)hj-Zpi(tanh TeD)qI-0piqs3115 - - - - - - - * GATE AT TURBINE, POINTS GOING UPa X GATE READ ON CABINET ACTUATOR

- - - - - MW (0-4OMW) _ = (cosh TD)l\+ (sinh D)hl (9)(40MW BASE) q,=cs ''.)l----klfh e')l13 _ . <GATE (BE a 80*b% ACTUAL 7ATE'6

o L zT I I I I I I I I I I 1where T8 refers to the elastic time of the0 .5 2 3 4 5 6 7 penstock and 0.1 the friction coefficient

(INCHES) for the penstock. For the turbine theGATE AT TURBINE (10 BASE) following equations hold:

Fig. 10. Generator output versus gate, with both units on lineq- aiih, al2nI +a,3zml= a21hl+aun, +anz, (10)

3.1 --

I SEC. PERIOD 7 - Here the a's are constants which can be

3 - -I | T obtained from characteristic curves of*.9 the turbine. Let Tm be the starting time2.8 - - - - - _- of the turbine. It follows that2. 7 - - - - - --

0O UNIT SHUT DOWN TmDnls =fmi (2.6 - - I I I - UNIT 2 OSCILLATED AT 40MWONLINE Fo - t ta

2.5 -'IS%I GATE/

BASE (PER UNIT)From equations &8 it folows that

z2. /-e - - - - - 8-HEAWATER 1274FsPE "T..4 I s | 40MW * BASE (PER UNlT) hi - (122.4 1? l l l l | ~~~HEADWIATER 127T 4 FT. t h =F, (12)

w ~~~~~~~~~TAILWATER B39. 4B FT.CL 2.3 - - - - GROSSHEAD 43 .00 FT. qpi

-M2.2 - - - - - - - - for an operator F, given by

.1 --L -I1L-__ F1= *c+Zc(tanh TCDD) (13)2.0 1120 SEC. PERIOD 1 | +Tg4cD+Z Tg(tanh TscD)

|.9 | A A a |l_ _ | > t 1 1 m l l l l ~It wil be noted tht the relationship be-tween the surge tank head ht and the pen-

1.7 .1 .2 -3 .4 ,5 , .7 .0 .9 stock rate of flow qp depends only on thetunnel and surge-tank-riser characteristics,

GATE (INCHES) and not on the part of the system followingFig. 11. Linearity check on power versus gate the tank. The hydraulic system up to the

penstock is, for our purposes, completelyIt is clear that when the gates are oscil- where T,, is the elastic time and 4c the described by equation 12. This equation

lated at sufficiently low frequencies the friction coefficient for the tunnel. Since shows how the heat h varies with the flowlevels in the riser and surge tank will be the reservoir is large, we may take rate qpl.practically the same.' It is to be ex- Combining equations 9 and 12 yieldspected that when the frequency is high hw (6) q,enough the water levels in the surge tank The flow rates at the surge tank are related h1, F, (14)and riser will not change, for practicalpurposes. These and other consider-ations lead us to assume that the levels in Ut iH1 U. H Ul Huthe surge tank and riser are identical for allcases in frequency-response runs. There- ffore: !

and the equations are simplified accord- I//////////////ingly. The experimental runs verify theIvalidity of this assumption. I- L-L ----- .4

Equation 3 applied to the tunnel yields 1. ' I l=-(sech TecD)hw"Zc(tanh T.cD)qc-'1qc 0 Xr xxz

(5) Fig. 12. Pipe section CONOUIT

OCTOBER 1962 Oldenbufrger, Donelson-Dynamic Response of Hydro Plant 409

where the operator F3 is given by The transfer operator G relating turbine For the case at hand the constants in

Ft gate position z1 to turbine rpm n1 is given equations 5-9 have the following values:1+ (tanh T.D) (15) Tc = 13 seconds 4c =0.009

Fs = ~~~~~~~~~~~~~~~~~T,=0.25 second otpiO0.001'pI+ F1 +Zvl(tanh T6D) G = a23F3 +alla23-a2la13 (16) T1 =900 seconds Zc=Z1=04Thus the relationship between the turbine Tm(F3+al)D - a22(F3 +,11) +a,2a21 Tm=8 secondsrate of flow q, and the turbine head hi so that In putting the variables on per-unitdepends only on the conduits and the surge n bases for the experimental curves andtank riser, and not on the properties of -=G (17) equations 10 and 11 the following werethe turbine as given by the characteristic used.curves and wr' (inertia) of the turbine. The use of the operators F1 and F,3 notThis hydraulic system is thus described only sheds light on the relationship be- Variable as2rby the single equation 14 relating turbine tween important physical variables, but Gatque 8 inches (80% of strokeflow to turbine head. simplifies the mathematical calculations. at servometer piston)

Speed 225 rpmHead 428 feet (gross, that is,

10 headwater minus tail-7 ITwater)5:X St4: | mWSHiii--Hg Flow rate 1,600 cfs3 E ]Electrical phase 23 degrees

z 4.267 ~~~~~~~~~~~~~~~~~~angleW The basic dimensions are feet, seconds,

I ~~~~~~~~~~~andpounds.With the variables on per-unit basesz 5

the constants in equations 10 are3wCL 2 ON.EDUNITOFFLINE , SWIlN GIING GATE AT ttSt t tS an=-0.57 a2I=1.18

a12 =-0.13 a,=-0.35w 0.1SEDNLODa,,,= 1. 10 a,-= 1.5

W OTHER UNIT SHUT DOWN) 5 BASE FOR GATE80%(8) ACTUAL GATI With the numerical values above the

BASE MW z-40 MW33 transfer operators F1, Fs, and G become

z 3 m BASE RPM * 225 FOR RIGID COLUMN2 UNIT NO. SHUT DOWN 20D +0.009

w0. *UNIT NO. 2 SHUT DOWN F1-

0.01 -THEORETICAL CURVE FOR RIGID 18,OOODI+8==D+:o COLUMN TUNNEL CASE D

W5 1lilil 1 1lilill 1 | Llilill {Ilkilt i I 1111111+l/4F,tanh-4

w 3 ] 1l 1 1 1 1 1 lil 1 1 1 1 lil ___ __ Fs D (18)D 2 4 : :0.001+F1+4 tanh -

5 1il ll1-ii||jl-=_(8Fg+4.6)D+0.35F3+0.0461I. 4

2 ; || 1llill 1 1ilil 1 11lilil 1 1liill 11 1llill Hydraulic resistance (here p qi) hasI I I I 11 1 1 11 ll ll ll ll been included in the mathematical models

0.001 0.01 0. !.0 10 terms are dropped the efect on the fre-

Fig. 13. Magnitude-ratio .1equency-response curve for off-line casequnyrsoecrvsinglib.

+ 30 ______________ ~~~~~~~~Tunnel Neglected-o STADY STATE°\-0 lI 0|1l] 1lXI ll ITIF

-30 - t l l1lll 1111ljl lll 11Xg It is now assumed that the surge tank- 6 1 1 1 111\-14S4LLl ll l l l l l isolates the tunnel from the rest of the

-90-o§ | | | | | | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 hydraulic system. The equations 5 andi-I2O - zo | || | 1]11 1 1| | ||1 14 | || | 111 1 be11 aremodified by dropping equation5s

-001 this a otherandsetting

-,801 - 1111 11111 11111 11111 1111 ge =0 (19)

i-2IO| |||||| ||||| ||||1l iii Equation 19 states that the water flow-240 - il 11111ll il 11111 1 rate in the tunnel does not vary. Essen-

-340 - ii I111 111 111IL IIIII The operator F1 now becomes0.001 0.01 0.1 1.0 10 1

FREQUENCY - CYCLES PER SECOND T:D (0

Fig. 14. Phaseit ngle-frequency-response curve for off-line case For the range of frequences employed in

410 Oldenburger, Donehson-Dynamic Response of Hydro Plant OCTOBER 1962

IO0 IITe 111 E E_Ei1 iItte IFR:iF=F=-+the frequency-response runs some of the5==:lt I 1111 terms occurring in the operator F3 of5

wifmt 8 t_t 3t $ t tt_ t_ X equation 15 are negligible so that onesz~~~~~To l S11 L M- i3=---- 2)1w may take

I-~~~~ 1.021-eE7_- Z;1 t > _= r rm Zp,tanh TeD1t < t 2 t I S S mg---T-mSSubstitution for F3 in equation 16 yieldsthe transfer operator G employed for this

EXACT TRANSFER FUNCTONAl case. It will be seen from the experi-0.1 NEGLECTING TUNNEL ~~~~~~~~~mental results that the surge tank does2 7 - CLASSICAL FORMULA isolate the tunnel from the penstocks, so

* fIRST APPROXIMATIONI_-----that the mathematical simplificationsLd2t s SECOND APPROXIMATION m Wil X" made here are justified.

2 -RIGID COLUMN IN TUNNEL_____ ; i t0°.01 * ELASTIC FLUID THEORY First Approximation4 7 APPLIED TO TUNNEL IIT

w 5 NOTE ALL CURVES FOLLOW THE Replacing the operator tanh T.D by03 SOLID LINE EXCEPT AT THE TeD in formula 15 for Fs yields the first-

z 2 POINTS WHICH ARE PLOTTED order approximation formula for G,namely,

0.001 _7 -= t t llXl t1 | n __ t ---i I -tH G= a23+Z. Te(alla23- a2lal3)D5 (T- a22)(1+ZpiTea1iD) +Zpi Ta1a2iD(22)

For the numerical case at hand

(A) 0.001 0.01 0.1I 1.0 10 G= 0.46D+1.5234.6D2+8D+0.35 (23)

FREQUENCY - CYCLES PER SECONDSecond Approximation

+ LetSTEADY STATEMI

30-- 30 TQ 7 l ll F,-- 60L III ll! II 11 1 1 1 11 _1 11 l ZcTsD

- 90 - 1+z- tanh TeD-120

Fs 4'(4)-150 - I4 ,+Zpltanh T0D(

ISO~~~~~~~~~~~~~~~~~tn .~TD (TeD)' 2(TD)'-240

I-27L Substituting in G of formula 16yidsthe--300 } 4 | lilil 1 l 1 1]lill lilill 1lillA 1 1 1 lilil second approximation to G. This approxi-,,,-330 llt -A { i i1l1mationto G is a polynomial of degree 5

cr. || il killIII IlilM II IHI \ I1 1111 in D divided by one of degree 6.

Rigid-Column Theory0-45z-4510 | |||| ||t > 1 1111il 41il If the water in the tunnel is assumed

>--RIGID COLUMN IN TUNNEL to be rigid, equation 5 is replaced by4-540 -II OUM NTNE-EXACT TRANSFER FUNCTION NEGLECTING TUNNEL hw-TcDq¢+c (25)Q-570 N~Lh cg cc(5& CLASSICAL FORMULA 11 11111111-600 H FIRST APPROXIMATION 1 1 where TC=20 seconds for the Apalachia-630 x SECOND APPROXIMATION case. The operator F1 now becomes

-660 1 * ELASTIC FLUID THEORY APPLIED TO TUNNELNOTE ALL CURVES FOLLOW THE SOLID LINE I l i Fl- TcD±++(26-690 tlEXCEPT AT THE POINTS WHICH ARE 11 l TcTgD'+4+cTgD+l 26

lassicalorm ula(8) 0.001 0.0 1 0. 1 1.0 10

FREQUENCY - CYCLES PER SECOND The classical water-hamer formulabased on a lumped system yields'Fig. 15. Frequency r.sponse curves K,( -TsD+1)A vMognitude-r8tio-frequency-re3poflse curve for rigid-column tunnel case m-- D+1F7B-PFhdsecngle.-frequency response curve for rigid-column tunnel case 2D+

OCTOBER 1962 Oldenburger, Donelson-Dynarnic Response of flydro Pkant 411

for turbine torque m, gate position z, water nominators corresponds to turbine damp- function C relating turbine speed to gatestarting time T.. and constant (gain) K1. ing. position. The solid curves without marksFor Apalachia case I this yields on them are plots of Cl and L Gas given

G= 0.18( - T.+1) (28) Frequency-Response Curves for by equation 16 with F8 of equation 21.Apalachia Case I. Isolated Unit This is the "exact" formula for G withOff Line the tunnel neglected. Superimposed are(.- D+1)(D+O.044)

plots of the magnitude-ratio and phasewhere G is the transfer operator relating In Figs. 13 and 14 are plotted the theo- curves for C for the cases discussed above.turbine rpm to gates as before, T0 = 1 retical magnitude-ratio and phase fre- The reader will note tunnel resonance dipssecond, and the term 0.044 in the de- quency-response curves for the transfer in the curves, obtained by treating the

0000000000000O NOt 0 .DN0t .

t. 00 .- .- 0 a. . 0

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u..

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UC

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o.

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0

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412 Oldenburger, Donelson-Dynarnic Response of Hydro Plant OCTOBER 1962

water in the tunnel as elastic and by the tiznnel resonance dip did not occuir in significantly from the other curves, in-considering the water in the tunnel as practice. A satisfactory theoretical ex- dicating the validity of this approach.rigid. As is to be expected, these dips planation is stillforthcoming. The phase curves show that the classicaldo not occur when the tunnel is neglected, It will be noted that, except for the formula, the "exact" transfer functionas for the solid curve, the first and second classical formulas, the magnitude-ratio with the tunnel neglected, as well as theapproximations to G when the tunnel is curves were identical up to penstock first and second approximation to thisneglected and for the case of the classical resonance, i.e., 2 cps. It will also be ob- function, are practically identical up towater-hammer formula. A remarkable served that the magnitude-ratio curve 1 cps. This frequency is near penstockresult of the experimental ruins was that for the classical formula does not deviate resonance. Except at tunnel resonance,

0 0 0 0 0 0 0 0 0 Q0O D N 0 wZ q a 0 ID N

a o

_I l 1 / T I -] tI I l

: f = =: : : j _ H 11 1 1 1 E 1 1 1 1 WB_1_1 -I II

I =00, 01z oI

* U

Ijsts3 ° 00 ° o ww~~~~~~ 5.I~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~W C

o - - - N N N n - C

_' O

0 r 0 iCEK tEt2 2I: E :~~~~~~~~~~Uop.

soO-¢050S-<ololulil0 n - -1 N N N 1010S3LV9 111111S413.L13 1O 1V 3QflI~ 1 11 1 1I

31t 9 / M W * 011vS wanilNsvw((M n-B) s33893U 3SVHd

OCrOBER1962 Oldenburger, Donelson-Dynamic Response of Hydro Plant 413

N~~ ~ ~ ~ ~~

z3C 0~~~~0C

cn 0~~~~~~~~~~~~~~~~~~~~~~~

o o 2 0 0 CL 10.I0

0

0. n1 y0 -I nC1- *i nci f)C10 10 oON (00- t0 10 00000NtI -0 r- 01-0 - - -N fNN 101010 ~ 0 0011

3±V9 / M b'4 * OIJ.V~~~~ 3OflhIN9VktI '''''' I 9 ) S33t1030 3SVHd

OCTOBE 192OdnIgr,Dnlo-yamcRsos f yr ln 1

the same is true of the phase curve for at least up to 2 cps. Careful experimental borhood of the frequency at which it would

the case when the water in the tunnel is runs show that the expected 360-degree shift occur.

taken to be a rigid column. In the gen- does not occur. Good agreement exists between theo-

eral elastic-fluid theory applied to the Fig. 15 shows the plotted experimental retical curves and experimental curves ex-

tunnel the phase angle changes by 360 frequency-respoilse curves and the theo- cept at tunnel resonance. The frequency

degrees as onie goes through tunnel res- retical curves for the rigid-column tunnel range covered is 0.000566 to 0.75 cps,

onance. Except for this shift, the elastic case. The reader will note that tunnel a range of 1,327 to 1. The frequencies of

fluid case yields the same phase curve as resonance could not be detected in spite concern for governing and system control

the "exact" case with the tunnel neglected, of several experimental runs in the neigh- problem fall in this range above tunnel

0

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4IIt1s _1_ _ _ __ __ -.-tI-1-f-1---p10~ ~~~~L 0 o4=1 I2I

Z ~~~~~CoW 0~~~~o

w1 WZ OD 3: ZR V tQ-

0~~~~~~~~~~~~~~~~~~~

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C)~~~~~~)

rU) POQN 01- In) N 00mnF)CUJ 00000000000 0 0 o00000 010 O-tt0-t.. W _ _ o _N 0 0-CIF _ rL

31V1111lV13131_ _=Nooo Y w_l_ 4m 0 -__ to11 111111111/MNI.o±8abNvdo!u=l iIII II LLc1 1 ' a o = =3,fvo m VI) CI9 )3rVH_IIIIIITI 1 !1 I D11 T 1t- j0tf e 1 1 1 I T I If f

0afXbzD e t e CQ - O o 0 F @ 0 tn00 O

w~~~~~~~~~~ U

31tD / 0~~~n4 11 v al9t°, , ,,, , ,,,,,-~

~~~ ~ ~ ~ ~ ~ ~~ 0

( 31vO - M W)-s33893Q 3- Hr

a _j ~~~~~~~~~~~~~0 (1

z~~~~~~~ <voe@o

1-tO VCN Or-U) )N 01U)~N - N -: N0 0000IoI NI-c1-0

i 1 1IIIIIII

W.~~~~~~~~~~~2t'-j

e-~~~~~~~~~~10If UOe nl I-enO r<Not )f Q-°O°°°°°°°°°°

3iVOI/MV4-OI±V 3QflhNDd 0 (MW-I S383O0S Nd

414 Oldenburger, Donelson-Dynamic Response of Hydro Plant OCTOBER 1962.

resonance, so that the transfer function nG for all of the mathematical models ____covered above is adequate for control 3UNIT MW UNIT ON LINE AT 40 MW - GAtE BLOCKEDstutdies. 2 BASE FOR GATE - 80% (81 ACTUAL GATE

1.0Apalachia Case II. Oscillation of ' 7

Unit Off Line, Other On Line3

The equations for this case are given0

STEADY STATE HIIin Appendix II. The experimental re- , 0.1 * 0.141sults are plotted in Fig. 16 as well as the

La 5EXPERIMENTALztheoretical curve based on the equations ~ 3 THEORETICAL:in Appendix II. The hydraulic coupling z 2between units is taken into account here. a 01Comparison of Figs. 14, 15, and 16 show 7 I L

that this coupling is negligible, and that it IM IImakes little difference whether or not 3

2the unit being oscillated is on the line orshut down. The electrical coupling is 0.001clearly weak in this case. 5

it

Apalachia Cases III and IV. OneUnlt On Line Carrying 35 Mw 0.001 0.01 0.1 o o(7/8 Load) With Gate Oscillated, III IIOther Unit Shut Down 8 I-TERTCLI Il IIl

The line reactances were as follows: EXPERIMENTALZ9@ ECase III: 16%54Case IV: 19.2% 630The equations for these cases are given in 1Appendix III. The following response 900 -curves for the per-unit electrical phase 9-so -angle (base is taken to be 23 degrees) of I- 080--Ili .-the unit relative to the line are shown in Ii70 - - 00i1000l 10 I1Fig. 17. Such curves for mw output areshown in Fig. 18. Similar curves for FREOUENCY - CYCLES PER SECONDcase IV are shown in Figs. 19 and 20.There is littleodifference.between the Fig. 22. Frequency-response curves for nonoscillated unit with bothThere is little difference between the units on linecurves for cases III and IV, which showsthat the external reactance of the system In Fig. 22 the theoretical and experi- accuracy that is adequate for control ap-is not critical. mental curves for the mw swings of unit 1 plications. Agreement was often within

experimental error. For the frequencydue to the oscillation of the gates of unit 2 range of 1 cycle in 2 hours to 2 cps, treatedApalachia Case V. Bothnnits are plotted. The magnitude-ratio curves in the runs, significant deviation occurredOn Line agree reasonably well up to 0.02 cps, but only at resonance points, at high frequencies,

deviate thereafter due to unknown factors ' and in the gain of the unit.In this case both units are on the line not taken into account in the differential 2. For the frequency range of 0.002 to

with the TVA system connected in the equations. 0.5 cps the runs with the unit under oscilla-customary way. The gates of one unit The runs for cases III and IV were re- tion off the line agree with theory to experi-were oscillated and the phase angles of peated with the nonoscillated units carry- mental accuracy.both units relative to the line were re- ing 35 mw, and the oscillated units also 3. Some of the theoretical magnitude-ratiocorded. The equations used to describe carrying this average load. The fre- curves were not identical with the corre-this case are given in Appendix IV. The quency-response curves were virtually spondig expenmental curves, but weregates of the nonoscillated unit were identical, showing that the effect of elec- ted downti. Unit 2 gain used for theblocked. For the oscillated unit the trical coupling between the two units in the theoretical curves was clearly too low fortricl cublnvabeteen he wo nit inthe theset cases.,q The- theonretical eliectrical plhaseptheoretical and experimental curves are same powerhouse is negligible, angle curves were low in each case. Thisgiven in Fig. 21. These curves are plots occurred in spite of the fact that the corn-of mw versus gate. The experimental P l Results of Tests putations and experimental work on themagnitude-ratio curve was higher than p)al e so essrelation between electrical phase angleand power output were done and repeatedthe difeoreniceay,butoeexpained byhape The Apalachia frequency response and with considerable care. In spite of theThe Hernce ay e eplaied y a other runls demonstrate the following discrepancy in gain the agreement wasdiscrepancy between theoretical and ex- points: good enough for control purposes.

perimetal vlues f the ain cnstan a23. 1. The diferential equations employed in 4. UJp to 0.5 cps the magnitude ratio forCareful theoretical and experimental this paper expkain the effect of gate movement the on-line runs (oscillated unit was on line)checks failed to explain the discrepancy. on turbine speed and power output to an was practically constant. The phase lag

OCTOBER 1962 Oldenburger, Donelson-Dynamic Response of Hydro Plant 415

increased with frequency in such a way Appendix 1. Basic Hydraulic To simplify this relationship setthat the power output of the unit could beassumed to be proportional to the gate input Equations xl = 0, XII =L (37)except for a dead time (pure delay) of about for the distance L between sections I and1.0 second. With the notations listed above equation II Thus5. The gate servomotors operated to 1 1 the partial differential equations of flowcps as if there was a lag of less than 1/60 through a uniform pipe with friction neg- C1= UI*, C2 - V/jHi* (38)second (time constant) between governor relay lected are given by andvalve and gate position. This lag is negligible u aH bu oH (29for normal control studies. Above 1 cps --= 9,--g -(29)the relation between relay valve and gate a1x Ota X Un *- UI* cosh TCs- VagHi sinh Tesbroke down. The gate drifted, for reasons for time t. (39)unexplained, in a random fashion. Let U* and H* be the Laplace transforms

6. Transmission lines had a small effect of U and H, respectively. Transforming for the elastc time Te, whereTi =Laa.for on-line runs. How the unit being equations 29 by taking the Laplace trans- It follows from equation 38 and theoscillated was connected to the TVA system forms of the left and right sides we obtain second equation 34 thatmade little difference in the results. UI *Whether or not the unit was connected dU

a[sH-H(x* 0H+A i sinh Tes+HI*cosh Tes (40)by a short or long line to the TVA system dx Vaghad little effect. (30) since7. For a range of gate swings from 0.37% sU*- U(X, 0+)= -g d Qi*=AUi* i=III (41)to 10% the response of the unit was essen- dtially linear. Indications are that this for the complex variable s and initial values equations 39 and 40 can be written aslinearity holds over a much wider range, so H(x, 0+), U(x, 0 ) of Hand U, respectively,that the assumption of linearity appears at t -0+ Assume that nh Tsjustified unless the gate swings are very Qll*=Qt* cosh TS-- Hi silarge, for instance 100%, in which event H(x, 0 )= U(x, 0+)xO (31) o (42)the nonlinear nature of conduit friction HII* ZQ * sinh Tes+Hi cosh TScomes into the picture. so that the initial values of H and U vanish.8. The rpm signal from a turbine is such Equation 31 now becomes where we have introduced impedance ZO

that it can be differentiated twice to obtain dU* dH* for Zo= 1/AV/ag.a modified signal that can be used for -=-asH*, sU*=--d-- (32) By definition the delay operator e-TeDcontrol purposes. dx dx applied to a function f(t) of time t yields

9. Distortion of the input waveform from Solutions of equations 32 yields e T Df(t)=f(t-_Te) (43)sinusoidal caused no noticeable difficulty 5 5 Similarly, the prediction operator eTDin the frequency-response runs, even though U*-= C cosh - x+C2 sinh - x yields, by definition,the distortion was at times appreciable. a a (33)

10. The runs showed that the surge tank CI s s eTDf(t)=f(t+T) (44)essentially isolates the downstream portion H*4 _ sinh -x- cosh - xof the system from the upstream part for Vcg a ag a The operator sinh TeD is by definitionfrequency-response analysis, so that one for functions C1 and C2 of s, and hyperbolic given bycan assume that the tunnel flow is constant. functions cosh s/a x and sinh s/a x. Writing eTD _e-T,DOn the other hand, as is well known, this equation 33 for sections I and II of Fig. 11 sinh TD = 2 (45)is not true for violent conditions, as, for yieldsinstance, when full load is suddenly rejected so that11. The classical operator in equation 28 U,= C1 cosh - xi+C2 sinh - X+ trelating turbine torque m and gate position a a (sinh TeD)f(t) =J(t+ 2e)-f(t-Te) (46)z is a good one. Only when very accurate _ 5 2studies are to be made is it necessary to X/ag Hi* = C1 sinh - Xi+C2 cosh - xigo to hyperbolic operators or their rational a a Similarly,approximations. for i=I,II (34) eT,D+e TOD12. Electrical coupling between machines in Remembering that COSh TeD 2 (47)a powerhouse may be neglected for fre-quency-response analysis except in deter- cosh2 x-sinh2 X= 1 so thatmining the response of a machine on the linewhose gates are stationary, but where the SOlUtiOn Of theSe eqUatiOnS yields f(t+Te) +f(t - TO)response is due to the rapid movement of the (COSh T6D)f(t)= 2 (48)gates of another machine in the powerhouse. C1= UI* cosh - xI+/g I* sinh - xI13. The response of a machine at Apalachia a a Consider the differential equations(35)due to the movement of its gates is little s s 1affected by whether or not the other machine C2 = -/Vg h1 * cosh - xI -- UI * sinh - XI Ql = (cosh T6D)Q2t - - (sinh T.D)His on or off the line. a a z0 (49)14. The hydraulic system above a surge It follows from equation 34 for i=II and HiI= -Z(sinh TgD)Qi +(cosh T.D)Htkink for a single-unit installation can be equations 35 for Cl and C2 thatdescribed by a single transfer function relatingsurge tank head to penstock flow rate at the UII*= UI* COSh - XI COSh -- XII+ SinCe fOr T6>0, and f(t)=0 When t<0, theupper end of the penstock. a aI Laplace transform Lff(t-T6)] of f(t-T)15. For the single-unit case the hydraulic _s s is given by"system (tunnel, surge tank and riser penstock, Vcg HI* cosh - xl sinh - xx- L(-6]=e fLf](0scroll case, draft tube) can be described by asingle transfer function relating turbine _/ 1 sih x csh-X-Silay,frT> an f()=0we

flow rate to turbine head. a a ~~~~~t<T6 the Laplace transform L!f(t+ F.)]16. Hydraulic resistance can be neglecteds sof(+cisgvnbfor frequency-response analysis of power UI* sinh -X1 sinh -XII (36) of f( 6 sgvnbsystem units, a a Lf(t+Tc)=e eL[f(t+Te)I (51)

416 Otdenburger, Donelson-Dynamic Response of Hydro Plant OCTOBER 1962

It follows that if TC>0 and f(t) =0 for enough one can use analogous to equation 11 applies. Thet<T, the Laplace transform L[(sinh TCD) system of equations is now complete. Thet(t)] satisfies the relation 1- (Te)+(T,D) values of the constants for Apalachia

2! case II follow.L [(sinh T.D)f(L)I= F(s) sinh T7s (52)

for aletnoms ffin place of eTSD TC= 20 seconds 4c = 0.12

for the Laplace transform F(s) of f(t). Similarly, the hyperbolic operators (sech Tt= 900 seconds pl-= 0.03Similarly, under the same conditions T,D), etc., occurring in equations 1 may be T, = 0.25 second p2=0

L[(cosh TeD)f(t)I = F(s) cosh Tes (53) expanded in power series involving powers Tm 8 seconds ZP1=ZP2=4of D. Thus, if terms up to and includingIt follows that equation 49 can be trans- the first order are kept, the second equation a1 =0.58 a2i = 1.40formed into equation 42, in fact the sets l simply becomes a12= -0.16 an = -0.80of equations 42 and 49 are equivalent for 1 a13- 1.10 a,3= 1.50

Qr(O, t)unH,(O, t)-O (54) Qi=Qii+- Hii' (65) bil=0.57 b2l=1.18zob12=-0.13 b22=-0.35when tTe.< where HII' is the derivative dHI/dt. In bia= 1.10 b23= 1.50

Making use of identities between hyper- this way one can get rid of the hyperbolicbolic functions, equation 42 can be put in operators. The values of K, and B, were obtainedthe form from Figs. 6 and 7.

HIl * = (sech Tes)Ht*-ZO,(tanh T,s)Qli*

Q'~~~~(coshF Appendix 11. Equations for Appendix Ill. Equations forz(sinhT,S)Hll Apalachia Case II. Oscillated Apalachia Cases Ill and IV.(55) Unit Off Line, Other On Line Oscillated Unit On Line, Other

fAssuming that the water in the tunnel Shut DownQn1(L, I)=Hii(L. t)O (56) is rigid, equations 6 and 26 apply to the

tunnel. The tank equation is now Assuming the rigid column theory forfor t< Te equation 1 is completely equivalent the tunnel, equations 6 and 26 again apply.to the second equation 55. TtDht = qc- (qpl +qp2) (66) The tank equations are now given by

equations 7 and 8 as for Apalachia case l,By definition the operator sech TcD Equation 9 applies to the penstock for it being assumed that unit 2 is shut down,

satisfies the relation unit 1. It is assumed that unit 1 is on the and that qp,2nO. Equations 9 and 10 for

(cosh TeD)(sech T,D)f(t)=f(t) (57)line with blocked gate, therefore the penstock and turbine of unit 1 agains= 0 (67) apply. Since unit 1 is on the line, equations

and the operator tanh T,D is such that - 69-71 hold. At rated speed, that is, 225Equation 10 applies to unit 1 with z1=0, rpm, the per-unit speed n1 of unit 1 is 1.

(cosh TeD)(tanh T,D)f(t)= therefore they become At this speed the electrical output of the(sinh TeD)f(t) (58) generator is 60 cps, that is, 1207r electrical

q, = ai1h, +a2nl6 radians per second. The electrical phaseIt follows that the first expression of (68) angle 8, of unit 1 relative to the line isequation 1 can be put in the form lr,= a21k1+a2,nl thus related to n, by the equation

(cosh TeD)HI =H- Z,(sinh T,D)Q,l (59) Equation 11 is now changed because ds,there is a load torque mr, on unit 1. This - = 120xrn, (75)

which can be transformed into mav be understood to be a synchronizing d!

(cosh Tes)HII*=Hl*-Z,(sinh T,s)QuI* torque, since the damping torque due to The constants for Apalachia cases III(60) the generator being on the line may be and IV are as follows:taken care of by the value of the coefficient

Multiplying both sides by sech Ts yields a22. Equation 11 is replaced by T,= 20 seconds 4c=0.10the first equation 55. Thus, equations 1 are TmDni = ml-mai (69) T-= 900 seconds Opt = 0.03equivalent to equations 55, and hence to Tg =0.25 second Zvi=4equations 49. Here, for system speed n,, Tm= 8 seconds

Since under rather general conditions onf(t) one can expand f(t-T,) as follows: nsi=1=O2rL-+BsJ(n1-nS) (70) ai,=-0.15 a2-1.3

______ ~~~~~~~~Dai3= 0150 a22=1-.73f(t- Te) =f(t) -Tef'(t) +

T f ( t) (61) Since ns is relatively constant we take an = l.00 a23 = 1.252. The values of K, and B, are obtained

one may write n38=0 (71) from the curves of Figs. 6 and 7 for theseFor the penstock of unit 2 the equations quantities at 16% external reactance for

(TeD)' case III, and 19.2% external reactancee- eD _= 1-(TeD)+ +... (62) 1 for case IV. These curves are interpolated

2! qp2=(cosh TgD)q2+- (sinh T,D)h_ to obtain the values of K, and Bs at 7/8Zp2 load.

derivingr ITD7-) -Zp2(tanh T,D)q2+ (72

e~rDJt)=1-TCD'+ l. f(t) (sech TeD)hg-cfP2q2g 72L1 ) 2! J Appendix IV. Equations forTC2 ~~hold for a friction coefficient fp2. For Apalachia Case V. Both Units

=f(t)- Tcf'(t)+ f(t)-... (63) the turbine of unit 2 the relations On Ln2!vn Lnq2 = b,1h, +b11n2 +b13z2

If the series converges fast enough one rn ns+u,+n,(73) Th)ii ounteoyi gisue

can replace e6T.Df(I) by to hold for the tunnel so that equationsanalogous to equations 10 hold, whlere 6 and 26 again apply. For the tank,

(1-T,D)f(t) (64) the b's are constants. An equation equation 66 holds. Equations 9 and 10are valid for the penstock and turbine of

If this approximation is not accurate TmDn,=rns (74) unit 1. Equations 69, 70, and 71 hold.

OCTOBER 1962 Oldenburger, Donelson-Dynamic Response of Hydro Plant 417

Equations 72 and 73 are valid for the bu= 0.58 b2= 1.4 7. FREQUENCY-RESPONSE METHOD APPLIED TOpenstock and turbine of unit 2. For unit b1n= -0.16 b22= -(.8 THE STUDY OF TURBINE REGULATION IN THE

2 the fllowin relatinsarevalid. = 1.00 b23= 1.25SWEDISH POWIER SYSTEM, V. Oja. "Frequency2 the following relations are valid. 3= 1.00 b13=1.25 Response," American Society of Mechanical

Engineers, New York, N. Y., 1956, pp. 109-17.TmDn,= mj-m82 (76)

8. RESPONSE OsF STEAM AND HYDROLBCTRXCKs D f GENERATING PLANTS TO GENERATION CONTROL

192,2=120 +B11(n1-n8) (77) *elerences TESTS, A. Kiopfenstein. AAtEE Transactons, pt._D J III-B (Power Apparatus and Systems), vol. 78, Dec.

where m 2 iS the synchronizing torque for 1. MATHRMATICAL ENGINEERING ANALYSIS 1959, pp. 1371-81.w r

2.i (book), Rufus Oldenburger. The Macmillan 9 HYDRAuLIC TRANSIENTS (book), George R.

unit 2.A system was operating for the

Company, New York, N. Y., 1950, pp. 367-75. Rich. McGraw-Hill Book Company, [nc., 1951,The TVA system was operating for the 2. TRANSIENT ELECTRIC CURRENTS (book), H. H. pp. 134-66.

tests of case V normally, so that the system Skiiling. McGraw-Hill Book Company, Inc., 10. SINE WAvE GENERATORS, D. W. St. Clair,reactance was 16%; K. and B. were obtained New York, N. Y., 1937, p. 269. LW.Srth L.V LGilesie. W.eSt. Refrom the curves of Figs. 6 and 7 for this 3, CONSIDERATIONS SUR LB PROBLikME DR LA Lponse," American Society of Mechanical Engineers,case. STABILITA (book), Daniel Gaden. Editions la Con- 1956, pp. 70-77.The constants are as follows: corde, Paris, France, 1945, p. 116.

4. LA REGOLAZIONE DBLLB TURBINE IDRAULICHE 1 1. See reference 1, p. 374.TC = 20 seconds 4,c 0.24 (boook), Guisoppe Evangellsti. Nicola Zanichelli 12. FLUID MECHANICS (book), R. A. Dodge, M. J.Tt = 900 seconds PI=p2 = 0.03 Editore, Bologna, Italy, 1947, p. 107. Thompson. McGraw-Hill Book Company, Inc.,Te = 0.25 second ZPl=ZP2=4 5. COMMUNICATION ENGINEERING (bOOk), W. L. 1937, pp. 201-26.Tm =8 seconds Everitt, G. E. Anner. McGraw-Hill Book Com- 13. See reference 8, p. 155.

an =0.58 a2l = 1.4pany, Inc., 1956, p. 159.

a12= -0.58 a21= -1.4 6. MATHEMATICAL ENGINEERING ANALYSTS 14. OPERATIONAL MATHEMATICS, Ruel V.Gi2= -0.16 aG2--08 (book), Rufus Oldenburger. Corona Press, Tokyo, Churchill. McGraw-Hill Book Company, Inc.,

als = 0 a23e=0 Japan, Japanese edition, 1956, p. 436. 1958, p. 15.

Discussion elastic time for the tunnel only 13 seconds Now, change in turbine power assuming no

compared to the surge tank riser time of change in efficiency is900 seconds. AP= V10Ai1 ±o V,

T. M. Stout (TRW Computers Company, There are some points which requireCanoga Park, Calif.): This paper is a further clarification, and these concern AG V0O V0onotable combination of careful analysis the results of Figs. 13 and 14 with reference Po- --s(L1+L2)+. L2Cs2-and experimental work. The close agree- to tunnel resonance. [LL2CSment between the theoretical work and In viewing the phenomena involved, _-__ _ _ _ _ _ _the test data should be a source of great the writer finds it useful to draw a lumped- r VO s(Li+L2) VO 1satisfaction to the authors, as well as to parameter electrical analog of the hydraulic +- +±. L2Cs'+LiL?CsIthe other people who had a part in the svstem as described in the Appendix and 'to 2 lOstudy. discussion of reference 1. AG F Li +±2 LIL2An interesting part of the paper is the Fig. 23 shows this analog where water is Po I 1-s +L2Cs1- Cs3

application of hyperbolic functions, stand- considered incompressible. The flow rate Go L Ro R4.ard for analysis of electrical transmission in the tunnel is i2, in the penstock il. The F L,+L2 LLL2Clines, to relate flows and pressures in pipes. inertia of the water columns in the tunnel lR1+s +LsCs+ 2R CsWhile they affect considerable simplification and penstock are described by the induct-in stating the relatioilships, these functions ance terms L2 and LI, respectively. Neg- This expression shows no magnitude dip.are somewhat cumbersome to use. The lecting the effects of water-wheel speed on As a matter of fact, the numerator is greaterauthors can also take credit for working out flow, the turbine can be simulated as an than the denominator, which would explainapproximations that facilitate calculations admittance termination G, and the change the resonant peaking in the experimentalwithout sacrificing accuracy. in gate by a change in the gate admittance results.

Being careful workers, the authors are AG. When the surge tank capacity is verydoubtlessly disturbed by the failure of the Writing the linearized change equations large, C--c and the above expressionexperimental work to turn up the tunnel about a steady state operating point for reduces to the familiar water-hammerresonance dips predicted in Figs. 13 and this analog, one obtains: formula14, and by the differences between thetheoretical and experimental curves shown [s(L +L2) 1 LL C2S2 PoG- [A1-GLRs] PO-[1-Tws]in Fig. 22. Their views on these dis- s(LL+L2 Go [ Gocrepancies would be a useful addition to A V=-il 1 ±LC2s2 AP=- G , ___________the paper. AL (78) +-s] +T.eiS]

(78) _ 2RO_

F. P. deMello (General Electric Company, Also When the inductance (inertia) of thetunnel is considered very large L2- o,

Schenectady, N. Y.): The authors are to i A V, AGbe complimented for a substantial analytical i1 = -_ +- (79) the expression becomesand experimental contribution which adds 2 VO o AGf 1 Lto the store of knowledge in the area of (from the square root relationship POG LR+ CS2Jdynamics of hydroelectric prime mover AP_systems. ii=G <Ji analogous to Q=GVh) F1 +C L1C 12The analysis and results in this paper _2R CsRoS

seem to confirm that for the range of From equations 78 and 79frequencies of interest in control studies, The treatment of the tunnel and penstockthe lumped-parameter approach yields AG as distributed parameter lines does notsufficiently accurate results. This should 2 - V0 significantly affect the nature of the resultsbe expected when one looks at orders of aj= Go obtained by lumped-parameter analysis,magnitudes of wave travel times. -/ L1L2 \ especially for a limited range of frequencies.

tion time is only 0.25 second comparedl to a 2 .---+ _\ L+ 2)1 and experimental results showrn in Fig. 22

water starting time of 1 second, and the so 1+L2C2s2 j seems to deserve further discussion. Could

418 Oldenburger, Donelson -Dynamic Response of Hydro Plant OCTOBER 1962