Governor Control for Kakabeka Falls Generating Station

Robert E. Doan @ September 1999

A Thesis submitted in Ailfillment of the requinments of the Msc. Eng. Degree in

Control Engineering Faculty of E n g i n d g Lakeheaâ University Thunder Bay, Omûio

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Abstract

This thesis examines a specific problem of govemor control on a set of hydroelectric airbines that suffer fkom severe wicket gate leakage. Linear mdeling and the steady state behavior of each turbine is investigated aad used to detennine the best means of s p d conml. Findings show that a secondary water control device may be used with the wicket gates to alleviate the effects of leakage while the generator is disconnecteci from the transmission system. The control method results in two distinct modes of turbine operation, with each hving significantly diffmnt dynamic characteristics.

A digital speed droop govemor scheme is wnsidered in a gain scheduling arrangement to amunt for the two operating modes. The stability boundaries of the govemor parameters are investigated, and an output fdback cost minimiration algorithm is used for obtaining controller gains- An outline of the software used to implement the govemor on an industriai programmable controller is presented, and the experimental performance for the turbines is compareci to the theoretically developed model.

Table of Contents

Abtract .................. ... ........................................................................................ i . . Tabk of Contents ................................................................................................... II I . ....................................................................................... Liit of F i y m & TiMa iii

CBAPTER 1 . STATION OVERVIEW ....e...ee..........eee..........~...........e........ee......e....... 1

CHAPTER 2 O LINEAR MODELINC OF W R O GENERATORS ...................... 9

CHAPTER 3 O SPEED DROOP GOVERNING ..e~..........................~........................ 26

CBAPTER O - FUTURE: WORK .............................................................................. SS

List of Figures & Tables

Figure 1 O 1 Kakabeka GS Electncal Ovewiew ................................................................. 3 Figure 1-2 Water Conveyance S ystem ............................................................................. 4 Figure 1-3 Turbine Assembly ......................................................................................... 7 Figure 2- 1 Wicket Gate actuator Dynamics ................................................................. 15 Figure 2-2 Steaây State Turbine Characteristics ........................................................... 21 Figure 3- 1 Speed Droop Govemor Control ................................................................... -27 Figure 3-2 Govemor gstem Bode Plot O OOnline Mode .................................................. 29 Figure 3-3 Govemor System Bode Plat O ûffline Mode ............................................... 30 Figure 3-4 Governor Stability Regions ..................................................................... 34 Figure 3-5 Govemor Coatrol System ............................................................................. 42 Figure 3-0 Generator 1 =Line Disturbance Response ................................................ 49 Figure 3-7 Generator 1 Simulated On-Line Disturbance Response ................................ 50 Figure 3-8 Generator 1 W icket Gate Setpoint Response .............................................. 5 1 Figure 3-9 Generator 1 Startup Sequence ...................................................................... 53 Figure 3-10 Generator 1 Shutdown Sequence .............................................................. 54

Table 2- 1 Generating Unit Constants ..................................................................... 2 2 Table 2-2 Mode1 Plvameters for Steady State Operating Points ..................................... 23 Table 3- 1 Sumrnary of Computed Governor Settings .................................................... -48

Chapter 1

Station Ovewiew

1.1 Introduction Kakabeka Falls GS is a small hydro-electxic generating station situated on the Kaministiquia river approximately 20km west of Thunder Bay, Ontario. The plant is owneâ and operateci by Ontario Power Generation, and is one of the oldest stations of its type in the province. Construction of the plant began in the late 1800's and was cumplete by 1904, during the early stages of Ontario's electrification.

Although the plant is nearing its 100' year of srnice, fiequent capital reinvestment has allowed the facility to keep pace with modem electrical generating stations. In 1998 a new plant automation system was instaîled, to facilitate remote controi and monitoring. Before this time, many of the critical plant operations were dom m u d l y b y a plant operator. W i i the newly instaîled equipment, Ontario Power Generation hoped to perform the following plant opetatio~s fkom its central operating & maintenance facility:

Automtic unit start-up and shutdown of the genmors. A u t o d c genemor synchronisation to the power system. Automafitc contrd of ekr ica l power pmQuaion and wata usage.

The automation system i n d l e d at the plant is b d on the Modiwn "Quantum" fmily of programmable logic controllas (PLC). A system of this type was chosen becsuse it is

I

adaptable to many industrial applications, it can handle a large numba of field wired I/0 signalis, and it aîlows the end user to tailor fit the software logic to specific design requirement S.

This thesis deals primarily with one challenging aspect of the station automation: the design a digital govemor control algorithm tOr implemedrtion on the PLC. The rniccess of the entire project hhged on the abüity of the control system to provide adquate speed regulation of the g e n d g uni& Ahhough this is a serightfhmd task a typid hydro

generating stations, problems unique to Kakabeka quired fùrthcr review of the fmibility and design of this control bction.

in this chapter an ovaview of the plant equipment and opaation is given, dong with the conditions that d e the task of s p a d govdng at Kakabeka unique. ui Chapter 2, a linear madel is formulateci for each of the plant generating units. S t d y state turbine behavior is investigated and used to determine the best means of operation. The chosen feedback control structure used for speed goveming is set out in Chapter 3. Tuning panmeten are derived based upon the stability boundaries of the closed loop system, and the govemor control aigorithm and experimeatrl data are presenteâ, to support the theorrticat development.

1.2 Piant EIectrical System Kskpbeka Generating Station contains four horizontdly mounted synchronous generators (see Figure 1-1) each co~lccted to a Francis water-wheel turbine. Generators 1 through 3 are equivalent in size and rating, while generator 4 is slightly larger in capacity. Each of these units has a power transformer ~ ~ t ! C t e d duectly to the stator windings. The transfomers are configured in a Delta Y-Ground arrangement and provide voltage step- up Born 4kV at the generators, to the 2S.OkV distriiution bus. Power tiom the plant is transfemd to the ûntario's bulk electrical system via two 25.OkV feeders.

A number of oil-filled cucuit bteakers provide switching and fault isolation for the genentors and distribution feeders. The generator breakers are useâ primarily for synchronizing and removal of individual generating units Eorn the power system. However, they also pmvide isolation in the event of an electrical fault. The line breakers are useâ to isolate the plant fiom the power system in the m n t of a feeder or station bus fault, and also serve as isolation points of electrical equipment whea routine maintenance is paforrned. Fwlt detection on the various pieces of electricd apparatus is achieved with high-speed protective relaying schemes located within the plmt.

Gt -G3- 6350 KVA O.= PF ao KV 3 4 mfz r17RpM

Cl - C3 Tmi- 6000 KVA

4iaaœmw 9375 KVA 0.80 PF 4.0 KV 34 60Hz UTlZPM

ClTrrrlkwr 10,000 KVA

Figure 1-1 Kalubeka CS Electriul Overview

1.3 Water Conveyance System Kakabeka generating station is a "run of the rivet' type hydro plant; A d m located in a section of the Kaministiquia River forms a s d l head pond to facilitate plant operation. The term "run of the river" is us4 because the plant water consumption must closely match river flows at dl times, in order to maintain sufficient water level for operation.

The major element in the water wnveyance system is a 2hn long aqueduct, which is useû to bridge most of the gap ôetween the dam, and the powerhouse (See Figure 1-2). This single burieci concrete pipe is 5 metm in diameter and can aransport up to 54 cubic m a e n ofwater a second âom the dam to the surge tank. The surBe tuik acts as a relief point in the water system, absorbing any pressure fluctuations resulting tiom changes in water flow. Four penstock cerry water from the surge tank to the water-wheel turbines located in the powerhouse. The penstocks are 00nstNcted ofriveteci steel and are supported by a number of concrete ddles at ground level. The fourth pstock is somewhat iarger in diameta than the nnt thra in order to m a t the wata requirements ofthe Iargest genenting unit. The turbines are duai impeller Fnncis type units, al1 with horizontal drive shtts and hotsepower tatings to match the coupled synchroaous

machines. The turbines are physicaily mounted on top of the macrete foundatioa and have large exposed cast iron scroll cases that sit above the power house floor.

nine elements help to control the flow of water through the plant: The headgates located at the dam intake, the butterfly valves located in each penstock, and the turbine wicket gates. Both the headgates and the buttdy valves are operated with the aid of elearic motors, while the wicket gates move with the aid of a high-pressure hydrwlic oil system. During n o d plant opemîion the butterfly valves remain fully open on dl functioning generators; the only time these devices change position is during generator stsmip and shut down. The wicket @es are useâ to regulate generated power, and are used as the primay control element for speed regdation. At typical hydro plants headgates are n o d l y used for water system isolation following a plant shut down; othenivise they remain fully open at all times. At Kakabeka, the head gates are used continuously to thmttle or &ct the amount of water entering the Aqueduct. This operation is necessary because the maximum pennissible water level in the surge tank is less thui the nominal head pond level. Without the water losses introduced by the head gates, the plant surge tank can overfill and the excess water must be sent to the riverbed through an emergency spillway .

A bypass valve on each unit is used to divert water fiom the base of the penstock to the drift tube. The purpose of these devices is to pmtect the turbine scroll case and penstock ftom damaging water hammer effects that can occur with rapid turbine shutdown. On ~eneratin~ units 1.2 and 3 these valves are mecbanically wupled to the wicket gates and are f o r d open when the wicka gates approach there tiilly closed position. On generator 4 the bypass valve is co~eeted to the wicket gates through a dashpot and spnng mechanism. The bypass valve opens whenever the wicket gates move rapidly towards closure.

1.4 Plant Control and the need for Governing Govemor control systems installeci on hydro electric generating units have three primary fiinctions:

- 1. Speed regulation of the generator for synchronizing when the unit is

discomected fiom a power system. 2. Stable Lod regulation via the wicket gates, while the generator is connectecl to

the power system. 3. Frequency stabilization during power system disturbances.

Electromechuiical governor systems are wmmon in older stations, while newer plants or refùrbished plants are now using digitai govemor systems for better performance and reduced maintenance. In both cases- these systems use wicket gate position and speed sensing féedback signds to drive a wicket gate actuator, in order to maintain unit spaad and load at some desid level. The actuators are typically higbpressue hydraulic systems utilizing electncally controlled pilot valves, however servomoton are also utilized on smaller units.

While synchrocüzing, the Bovemor must be capable of matching the generator's spad to thPt ofthe power system so that the slip-rate between the two systems is minimized. With minimai slip, the timing for the generator breaker to close at a zero phase angle crossing bewmes less critical, and the power that is transmitted or taken fiom the system when the rotor is pulled into synchronism is rninimized. When the generator is connecteci to the power system, the govemor nomally provides a means of wntrolling the wicket gates so dut the amount of generated power may be regulateâ. However, during power system dishirbances the govemor will readjust gate position in an effort to correct frequency error.

The dynarnic m d i n g of hydro plants and design of governor systems bas ban studied a geat deai, and for a l o n ~ time. Mimy technical publications may be found which discuss the various aspects of this subject. The most notable work on modeling may be found in refmence (21. This paper gives a very detailed summary of the dynamic equations that may be used to mdel a hydro plant, and @mental test chta is provideci to assess the suitability of the models presented. More recently, IEEE cornmittee reports given in references [3] rad [4] Sunmuue the generaîly accepted techniques for bah modehg and govemor conmi of hydroelectric generation. Likewise, muiy methods of comrol have ben deviscd Speed dmop and PID type govemor systems are commonly

r

used in hydro plants and are prevalent in literature [4,5, IO]. Mon recent publications site methods such as gain scheduling [l 11, adaptive control[12, 131 as well as H, control techniques 1141 for ~ovnning hydro turbines.

Standard speed govemiag procedures are not directly applicable to the Kakebeka plant under al1 operating conditions because of severe wicket gate lealcage. When the Kskrbeka generating units are ~ 0 ~ e d e d to the grid system, theu behavior is not unlike units at other hydro plants, however, when they are disco~ected fiom the power system in an ofMine mode of operation, speed governing is hampered. The residual Bow of water that occurs with the gates are fully closed, causes each unit to spin above rated speed ifunsynchronwd fiom the power system. The ieakage probiem is partialiy due to the design of the turbines and partially because of the age of the plant.

Each of the turbines is composed of twin scroll cases and rumen with a common drail tube. The scroll cases are divided into two parts dong the horizontai centerline. The bottom section of the scroll case is f i e d to the concrete foundotion of the plant atop of the penstock opening. The top section of the scroll case is joind to the bottom by means of large machine bolts. A T-shapeû draft tube sits between each of the scroll cases, and is independently af'fïxed to the concrete foundation. A set of head covers encloses the runners, and provides a seal between each of the scroll cases and the drsft tuôe flanges.

The head wvas pdaily support the wicka gates, and dso house some components of the wicket gate assembly. Each gate is supporteci at one end by a stub shaft that passes through the head cover and at the other by a "blind end" bushing in the back ofthe scroll case. Each of the stub sh&s is wnnected to a common actuator ring by a number of gate anns. The gates are set in motion by way of a hyâraulic cylinder, which rotates the actuator ring (see Figure 1-3).

The relative position between the draft tube and the bottom of the scroll cases has shiRed over time due to movements in the concrete foundation. As such, the alignment of the head covers between the dr& tube and the bottom of each scroll case is pax. Some of the gates are "tippeâ" in relation to the blind end bushing of the d l case, and the heel and t a clearances of the wicket gates are affectecl. Efforts made to improve alignment during turbine maintenance have proved fnitless, as it has ban found that the scroll -actually move sli@tly betw-een the pressurized and non-pressurized state. Any adjustment made to the turbine assembly while in the de-watked state, change whei the turbine is placed ôack in service.

To compound the aügnment problem, the gate amis are not desi~ned with any adjustable linkye or rhear mechanism; the stub shfts are simply keyed to the gîte ana Quite o f h the gates ue forced closed on debris nom the rivabed thrit has fouad its way into the penstock. Force fiom the hydraulic cylinder causes the stub shafts to twist slightly, and over tirne repeated stress on the gate s M s results in poor gate &ut off The Iack of aâjustrble gate Iùikye harnpers MY effort to maice @te djustments, except during major ovahauls when re-keying of the gate wbs is possible.

Figure 1 3 Turbine Assembly

The original mechanicd govemors installed during the plant's inception were removed from service in the 1980's when the leakage problem made off-line speed goveming of the genemtors impossible. At the rime, Ontario Power Generation (then Ontario Hydro) decided that it would be prohibitively expensive to refùrbish and al ip bore ail of the wicket gates as well as change the linkage design. From then on, a somewhat unconventionai operating procedure was adopted.

The penstock butterfly valve upstream of each turbine was used as a throttling device during unit startup. A plant o m o r would rnanually adjust both the buttertly valve position and wicket gate position to acbieve synchronous speed. With the aid of a synchroscope, the plant operator would close the unit b d e r snd then cornpletely open the butterfly valve. The butterfly valve thottling effectively redud the availuble head et the turbine rio that Iowa unit sws were achievable. Aithougb motocid, the buttedy valve could not cornpletely replace the fÛncti*on of the wicket gates h s e of its slow openting speed.

Under normal operatiou, the large power system to which Kakabeka was comected would regulate each turb'ie's speed while online: The wicket gates were used strictly for

the control of generated power. The risk however was that a system disturbance might create an islanding condition, leaving Kakabeka C O M ~ C & ~ to an isolated customer load. The lack of speed goveming would result in a dangerous over-speed condition and possible customer equipment damage. To circumvent this problem additional over and under speed protection was installed at the plant, so thit the generaton would be shut down and isolated fiom the power system during minor tiequency disturbances. Ahhou& the scheme provided adequate speeâ protection and it allowed the generators to fiinction in the same manner as other plants, the units were not able to contribute to power system stability during lod disturbances. Furthemore, the plant ofken required time consuming generstor -arts when these events occurted.

1.3 Probkm Statement At preseng the airbines are et the halfway point in a planned major maintenance cycle of 30 y-. It is likely that that the issue of wicket gate refiirbishment will be revisited, however this will not be for some time to corne. The custs associated with turbine refurbishment are also far beyond the budgeted douars for the plant automation project.

The challenge at hand is to utilize the uewly installed plant programmable controller to achieve automatic startup and synchronization of the genenting units. The "automatic" control megy implemented will have to be similar to the one previously used because of the gate leakage; both the wicket gates and buttedly valve will have to be operated to accomplish off-line speed regulation. Startups m u a be fast so thet extra generation may be called upon in a minimal amount of time, and the off-line speed regulation must be good so that synchronjzation is smooth and non-impactive to the grnerator or the power system. Any closed loop control that is used must be simple and robust so that technical support stafFcan comprehend and maintain the system, but will not be unduly burdened with muent "tuning".

Further improvements tht are to be considered in the design are a limited degree of on- line speed regulation. Aithough the gate leakage maices it impossible to achieve speed conml in the event of a fiil1 load rejection, and the over speed protection described above will still be requireâ, less severe power system disturbances should be matched with an appropriate amount of prime mover power. In this way the plant as a whole will contribute positively to the overall power system stability.

Chapter 2

Linear Modeling of Hydro Generators

2.1 Introduction In this chapter a linear model will be formulated for the Kakaôeka Generating Units. The mode1 will be based on the genedly accepteci techniques found in various references [2, 3,4,5,7]. Any assumptions mde in fomulating or simplifying the model an stated, dong with the ciraimstances under which the asswnptions are valid.

The steady--te turbine behavior will be used to assess how the wicket @es and the butterfly valve should best be used for speed regdation off-line. Stable operating points wül be seleaed for the off-line and on-line modes of operation to determine the values of the parameters in the linear mdel.

Unless othenivise statcd, upper case variables will represent quantities in engineering units, and lower case variables will reprcscnt deviations fiom a steady state operating point in per unit fom. Variables that define a steady state opecating point will be denoted with a subscript 'a', and per-unit base quantities will be denoted with a wbscript 'o'.

2.2 Penstock Water Modeüng The modeling of any hydro gencfating system begins with the study of water flow in the pstock, and the basis of a witer model stems nom a set of partial diffmntiai equations derived in many elementary fluid mechanics textbooks [l].

With the variables defined as:

V - fluid velocity (mis) H - Pressun h d (m of water) p - Fluid density (ICg/rn3) g - Gravitationai constant (9.8 1 NKg) D - Inside pipe diameter (m) e- Pipe wall thickness (m) E - Young's rnodulus for the pipe material (N/rn2) K - Buk modulus of elasticity of the fluid (N/rn2) x- linear displacement along the pipe axis (m) t- Time (s)

These equations d e m i e the flow of a compressible liquid through an elastic pipe, with head loss due to fiction neglected. in [1] a numerical method is given for performing time domain simulations of fluid flow.

In (21 it is shown that Equations (2.1) and (2.2) may be opproximated by:

with rssumptions on the initial conditions, a d some partial derivatives taken as fiil1 derivat ives.

The subscripts on Q and H represtmt some k e d distance along a pipe axis. However, in the case of a penstock where the point of interest is the turbine, HI and Qz may be taken as the h d and flow at the turbine, while HI and QI may be taken as the head and flow at the surge tank or water reservoir. L is the axial distance along the penstock between the tuhime and the water source anû A is the intemai cross section area of the penstock T, is defined as the pipe elastic tirne and Z is defined as the pipe impedrnce. s is the Laplace transfomi operator.

In [2] it is also shown that Equation (2.3) and (2.4) may be d t t e n in variable sets of

1 q, (s) = cosh(T,s) q, (s) + -- Cs) 4 (s)

2 0

, I

1 Equation (2.5) also incorporates a head loss tenn & that rnay be used to uxount for minor water losses in the penstock. The loss is assumed to be of the fonn H I = C Q - I Q 1 . which is

1

standard for turbulent fluid flow in a pipe. The proportiodity constant c may be wmputed using the Darcy- Weisbach equation [Il, or detennined experimentall y.

l

: if the hesd in the surge tank or water resefvou is assumed to remain relatively constant ; (hl=@ then Equation (2.5) may be rewritten as:

This assumption is vdid for most hydro plants with an appropriately shed surge tank, or a penstock that is directly connected to a large water resmoù. The transfer fùnction of

; (2.7) encompasses the efkcts of pipe elasticity and water anapressibility, which are significant with a long penstock ancilor a high head plant. References to Equation (2.7) are cornon, and may be found in [2,4,5,6].

The hyperbolic term in (2.7) rnay be eliminated using a number of approximations so that the transfcr ttnction is of l inw fonn for applications requinng contml design [2,5,6]. The complenity of the approximation depends on the degree to which pipe elasticity and

: water compressibility are considerrd significant. A simple first order approximation suggested in (21 and [SI is

which, when substi~ed into (2.7), pduces an interesting remlt. This has been noted in [SI but is inciudtd here for cl*.

2.3 Turbine Modeiing i

/ The behavior of a hydraulic turbine is nonlinear in nature, with water head h, spesd w,

i and wicket gate position g, al1 acting on the pnmary variables of flow q and mechanical torque m.

The linearized small signal model most o b used to cbacterize turbine behavior for 1 governr design is [2,5] :

i i The partial derivatives aii and a2i are dependent on the turbine operating point, and may be extracteci fiom manufadunr's performance curves if such information is available.

I When not available, it is customary to linearize the equations for an ideai turbine to fit the srnail signal model of (2.14) and (2.15). No turbine information for the Kakabeka units was available and so the traditional ideal turbine model was adopteâ:

Equation (2.16) essentiaJly States that the wbine converts water powerpgH,-Q to mechanical powerM W with an efficiency of y. HI is the turbine water heaâ, Q is volume of water flow, Mis the turbine torque, W the rotational speed, p is the water density and g is the gravitational constant. Al1 variables are show in upper case to denote tbat they are in engineering quatities, and not per-unit quantities.

Equation (2.17) is not wilike (2.12) in that it descriim the relationship between water flow Q and turbine h d H, for a given wicket gate position. Shce the wicket gatee

j operate in a manner that is consistent with a linear vaive, it is reisonable to assume t h this relationship may be c o n s i d d vdid over most of the operating range of the gates.

Under the assumption that the turbine is opentiag at rated s p d aad that small changes in rotational s p d h v e üttk e E i on wrta flow, the partiai derivatives ofEquafions (2.14) and (2.15) mry be obtained by i i n e d g (2.16) and (2.17) about a steedy state opefating point [1 SI.

a,, = 0

The numerical values of these constants will be disaissed in Section 2.6. However, it is important to note here that when the operating point and the per-unit base quantities are both chosen the sarne, the turbine constants of ail =O.S. al2=O, a13=l, az~1.5, a2=-l and a23=l result. In many instances in the literature the assumption is also made that deviations in speed fiom the steady state operating point tend to be small, so that the interaction between torque and speed is negligible ( U ~ ~ = O ) . This is valid principdly when the genemtor is connecteci to a power grid, since generator protection systems ofien incoprate under and over s p a d tripping elementq which only allow the turôine to operate within a narrow speed range (typidly t IO??). It is also worth noting that with these turbine parameters, and the penstock water dynamics of Equation (2. IO), one may derive the classical water hammer formula otten applied in hydro goveming mdies [2.3,4,5].

2.4 Wicket Gate Actuator Modeling Considerable force L required to move the Rubine wicket gates, and so each turbine has a hydroulic cylinder, pump and remvoir to provide gate movement. The cylinder is mechanidly coupled to the wicket gates, and forces the gates open or closed with ünear shaft motion, The rate of flow of hydnulic fluid to and fiom the cylinder detennines the rate at which the gates move and i s regulated by an electridly controllcd proportional vaive. The proportional d v e accepts a 4-2ûmA signal, and regulrtes the oil flow rate to

the cylinder based upon the value of the input. Although the proportional valve does exhibit some dynemic properties of its own, the dominant dywnics occur at the hydraulic cylinder where fluid flow produces a rate of change in the observai wicket gate position.

Figure 2-1 Wicket Gate actuator Dynamia

The force-mass dynamics of the proportional valve spool movement is low pass in nature but has a time constant that is insignifiant when compared to the time constant of the hydraulic cylinder: For this rerw>n it has ban neglected ftom the actuator model. Likewise, the pump dynunics have been aeglected as testing hrs show the supply side pressure at the proportional valve only deviates a small amount when the wicket gates are operated in an segnssive rnanner.

The deadbanci pnsent in the proportional valve spool at the neutmi position was found to k significant, so to cancel its aâvene e f f i s a nonlinear element was introduced into the PLC software. in software, a high gain was applied to the valve-input signal within the region ofthe deaâband, while an e f f i v e gain of unity was applied to the signal elsewhere.

The dynamic model shown in Figure 2.1 w u derived Born the following equations, taking into account wicket gate luluge.

1 g', (s) = -*

Tg - S u, (4

O ag, s1.0

O Sg,, 6 1.0

O s gr's 1.0

- 1 . 0 ' ; ~ ~ c,+1.0

Equation (2.18) is the transfer function from the gate conml signal u, through to the observed wicket gate position gt '. 7'' is the time for the gates to move from the fiil1 y closed position gr '=O.O. to the fully open position g, '=1 .O, with a cuntrol signal ofunity applied. Equation (2.19) describes the actual turbine gate position, and is the sum of the wicket gate leakage g~ and the obsaved gate position, scaled to adhere to the limits given above. '

The smdl signal model for the above, given a per-unit base ofunity and assuming operation about a stable operating point is:

2.5 Cenerator Modeling A simple mechanical model is usually suficient to account for gegeaerator dynamics when considering governor design [2,4,9]. This is because the most severe of speed disturbances often corne in the wake of a gaiaator breaker trip, when the electrical system is separated from the genercitor stator. Al1 power once king tr~nsferred into the connected electrical system is now directed towards d e r a t i n g the generator rotor, and the govemor is called upon to d u c e turbine power kfote dangerous over-speeds occur. So tw one may argue that load changes made to a single generating unit ~ 0 ~ c c t e d to an electrical system with large amounts of generation and load will have littîe eRect on the system fiequency. For these rwsons, a grnerator model like that of Equation (2.21) which negleas the dynamics of the interconnecteci electricd system is sufficient.

Mis the turbine torque, M, is an opposing torque acting on the machine rotor, as a result of the electricil power king produceâ, and R is the generiltor mtatiod losses cuised by windage, bwing fnction and field excitation. The rotationai losses are assumeci to be proportional to the turbine shaft speed. W is the rotationai speed ofthe grnerator shatt, and J is the i n e h of th system.

Assuming that the opposing electrical torque is constant, Equation (2.21) may be rewritten in smdl signal per unit form, for changes about a steady state operating point.

The transfonktion from the time domain to the Laplace domain has been performed assuming zero initial conditions. Furthet simplification is possible in most practical applications because the effects of rotational losses have little effect on shaft speed, due to the enonnous amount inertia contained in the systern, so it is customary to assume rd) [2,4,9l.

2.6 System Tnnsfer Function The penstock wata mode1 given by Equation (2.10) may be combined with the turbine equations of (2.14) and (2.15) to formulate a system the transfer function. The development begins by using Equation (2.10) to eüminate q, in Equation (2.14), making note that the turbine constant a12 = 0.

Now, the above equation may be substituted into (2.15) to find an expression for turbine torque.

The expression above for Nbine torque lbon may be combine with the torque-speed transfct fundon of Eqiution (2.22) to find a single rdationship for generator speeâ as a function ofwicket gate position.

- a23 + (a+, , - %,a,, X V + k) i s ) -

(r,s + r - uP Hl+ a, (s,s + k)) gr (s)

Finally, the gate actuator dynamics may be augment4 to form a transfer fùnction nlating gate control input to generator speed using Equation (2.20).

Equation (2.24) may be fwther simplified to Equation (2.25) if the rotational losses are considered negligible, and the penstock butterfly valve is in the fiilly open position, so that water losses are also negligible.

2.7 Turbine Chanctenstics As stated previously, the beliavior of the Kakabeka units in an online mode of opention is the sarne as other hyQo turbines; incremental increases in gate opening create a propntionate amount a shaft torque. Under normal operating conditions the large interconnected power system maintains the unit speed so that increases in shaft torque rerailt in a net increase in power output. The only adverse effect that the gate leakage bas is a meamrable but small positive power output when the wicket gates are Uly closeû. Only in an abnomul islanding condition, whae the Kakabeka units may k leff connected to a s d l isolated 1 0 4 would the wicket gate leakage cause a possible speed- goveming problem. Thus it is sufficient to use stanâard speed governin~ techniques on the units while online, with the provision that over-sped detection and tripping is present*

It is the off-line mode of operation that requires ttrther investigation, since speed governing is impossible without the added use of the buttdy valve. To determine the best means of providing speeâ control with the wicket gotes and the buttdy vaive, one must invmCgate the stedy-state relationship between unit speed, butterfly valve position, and wicket gate position. Simple fluid flow relationships may be used for the but tdy vaive, wicket @es, and the turbine. in the following analysis, the tubine characteristics d l be assumeci ideal over the fiill operathg range and the bypass valve negîected. Also, the assumption that the wata is incompressible and the penstock inelastic will be used so

that Euler's equation for fluid flow in a conduit may be applied to account for the penstock behavior.

Unda steady state conditions the diffemitial elements with respect to time in Equations (2.10) and (2.21) may be set to zero so that:

To derive Equation (2.26). the variables of (2.10) expressed as per-unit deviations fkom a steady-state opaaing point have been substituted with engineering units, and the buttedy valve head loss assumed proportionai to kq(9 has been replaced with the actual head loss Hb, expressed in meters of water. In Equation (2.21) the opposing el&cal torque actingon the rotor has been taken as M,=O, since we are considering the off-line mode of operation only.

Equation (2.26) may be substituted into (2.12) and then combined with Equation (2.17) to express the turbine head as a function of gatdvalve position and total available head:

This may be combined again with Equation (2.17) to express turbine flow in the same variable set.

Equation (2.27) may be combineâ with Equation (2.16) to fonn a nlationship for generator spad.

!

Now, Equation (2.28), (2.29) and (2.30) may k combined together as follows:

Although it is not entinly obvious, W in Equation (2.30) and (2.3 1) exhibits a maximum for a given wicket gate and buttdy valve position. Ifthe turbine is to be controlled by f h g the butterfly valve at a partjal opening, and the wicket gates are used to regulate speeâ, then the maximum speed achievable may be d e t d e d by differentiating W with respect to x.

The maximum turbine speed is achieved when x= % .c or in more prsctical terms, when the turbine head is twice that of the buttedy valve head loss (k?,=2&). in tenns of gate position, the maximum speed occurs at:

Proofthrit this point is indeeâ a maximum and not a minimum, may be shown with the seeond derivative of W with respect to x.

The second derivative is less then zero at x=%c indicating that the point is a maximum.

Figure 2-2 shows the measured and theoretical steady-state turbine speed m e s of the KakPbeh G1 unit. Note that the turbine exhibits a perik openting speeâ for a given set of

! wicket gate and butterfly valve openings: Similar rklts were obtained for the-other turbines. Since no means of mersuring pstock flow or turôine torque was available at the plant, the experiment data show in Figure 2-2, and thrit coUected fiom the other units, was used to ampute the various unit constants as described klow. These wnstants are sumrnarized in Table 2-1,

Figure 2-2 Steady State Turbine Chrrictedstics

The full loaâ penstock flow Q, the turbine power in megawatts, and efficiency y, given in Table 2-1 were taken fkom reférence [17]. The turbine flow constant Cr was computed using Equation (2.1 7) assuming full load flow, a normal operating head, and a fully open buttertly vaive. The butterfly valve coefficients were computed corn Equation (2.32) based upon the peak operating points observed in the experimentol àata. As well, the rotational loss W o r R was extracted €rom the operating data at the observed peak using Equation (2.30). The i n d a J was determined experimentally by obsewing the time required for the turbine to reach 36.8% of nted speed following a unit shutdomi. The measured time was taken to be Ji'R as per Equation (2.21). The wicket gate leaicage was estimated at 10% for ail nubines baseâ upon the miduil power obsecved while the generator was connected to the power grid and the wicket gates were fully closeci.

Discrepancies W e e n the curves obtoined experimentally and those defineci by Equation (2.3 1) are thought to be atüibuted mody to non-ideal turbine behavior, and smail i n w i e s in the cornputeci constants. The throttling action of die butterfly valve can significaatly rduce the effective airbine head and dimpt the water flow patterns at the turbine inlet; both of which impact aciency of the conversion h m watefpower to mechanid power, and make the turbine behave in a non-ideai muum.

Table 2-1 Gcntnting Unit Conrtanb

2.8 Openting Point Selcetion Determining an on-line operating point is straightforward sina considention need not be given to the effects of wicket gate leakage. in the off-line mode, the steady state openting point is selected based upon the desird operating speed of 6ûHz. It is evident &om Figure 2.2 that two vaiid steady-state opaating points will exist on the speed curves that intemect the 6ûHz grid line. Curves falling short of this value hold no valid operating points. Likewise curves fonned by Iarger buttedy valve opming will also not contain vaiid operating points. The wicket gate opening at the point of intersection for the theoretical curves rnay be detennined by solving the following 3" order polynomial deriveci 60m Equation (2.3 1):

Given the rated turbine speed W. specified in rad/sec of shafk speed, and the desired butterfly valve opening such that c = the steady state wicket gate opening gr# may be found by solving for x in Equation (2.33). From these three vaiuq one rmy cornpute all the 0 t h pertinent steady--te opaating parameters.

With the on-line and off-line opedng points known, ail of transfer fiinction parameters contained in Equation (2.24) may be computed. Table 2-2 summuizes the steady nate operathg points end the transfer hnction parameters for the four Kakabeka generating units. For these vaiues, the fi111 lord openting point of each airbine has been selected as both the per-unit base, and the on-line operating point.

1 Each of the turbines b u two distinctive off-line operatin* points (see Figure 2-2). The ! fust operating point has a characteristic typical of hydro turbines; increased wicket gate I i opening provides increased flow, positive torque, and increased speeû. The second

operating point is very distinctive fkom typical hydro turbine operation; increased wicket i gate opening d a s produce increased flow, however this flow also rernilts in significant ' head loss at the butterfly valve and diminished water power at the turbine. The loss in

turbine head is more significant than the increased flow, thus resulting in lower shaft torque and reduced speed.

Though unusual this "inverse characteristic" bas some significant advantages over the other opaating point. First, rdationship between gate position and turbine speed is quite linear and the operating range of the wicket gates and resuitant sped is large. This is important since fluctuations in surge tank water levels or small mors in butterfly valve positioning may adversely effect any efforts made to regdate speed. With the expanded operating range of the inverse characteristic, any speed govening strategy utilized will be better able to tolerate such deviations.

Secondly, the wicket gates never need to opetate within the bottom range of wicket gate movement, thus eliminating any il1 effects that might be caused by the bypass valve. Expimentai atternpts to speed govem turbines 1-3 on the more traditional 'Torward characteristic" w u hamperd by movements in the bypass valve. This also explains why the plant operators used to find the task of manually synchronizing the Kakabeka units tedious and time wnsuming; they were always operating in the normal turbine characteristic region ratha than the inverse characteristic region. Prior to the findings of this thesis, the inverse mode of operation was never revealed or utilized for speed control, and the bypass valves would adverscly affect rnmud control efforts.

From the arguments above it is clear that the inverse turbine characteristic must be chosen to provide speed regdation in an off-line mode of operation. It will be necesssry for an automatic speed govemor to fint position the buttdy valve at the appropriate fixed partîaî opening, and then use wicket gates within the inverse operathg range of the turbine to attempt to regdate unit speed for generator s t m p & synchronization. Once the unit is successfiilly comected to the power system, the speeâ govemor must filly open the butterfly valve to extract the maximum possible power fiom the wbine. At this point the wicket gates will rgiin o p t e in a manna typical of hydro turbines, and may be aâjusted so as to maintain fiequency and power output at a desirad level. This behavior is captund it the transfer funaion of (2.24); The steady state gain of the transfer fiinaion for the on-line panmeters @en in Table 2-2 is positive, while the off-line paramaers result in a negative steaây state gain

2.9 Summry The linear mode1 formed by Equstion (2.24) and the parameters for Table 2-2 describes the dyaimics of Kakabeka generating units, and may be used to proceeû 6 t h govemor design.

Each turbine exhibits two distinctive modes of opedon with differing dynBMics: The first is temed the "on-line" mode and occm when the generating unit is connecteci to the power system. The w n d "off-line" mode ocairs when the genemtor is disco~ected from the power system.

ui the on-lins mode, turbine behavior is normal and standard speeâ goveming techniques may be applied. In the off-line mode, seven wicket gate leakage requires that both the wicket gates and a penstock buttafly valve be used to provide adquate speed governing. The aubine charaaeristics are the opposite of what one might e q a t : Under steady state conditions, inaeased gate opening causes genemtor fiequency to Ml not nse. Special consideration will have to be ~iven to govemor design due to the characteristics exhibitecl by the turbines in the off-line mode

Chapter 3

Speed Droop Governing

3.1 Introduction In this chapter a digital speed droop governor scheme is considered for control of the Kakabeka Generators. A gain scheduling arrangement is employed to account for the differing dynamic characteristics resulting âom the on-line and off-line o m n g modes. For each mode of operation, the stability boundaries of the govemor parameters are investigated, and an output fadback algorithm is used to assist in finding the unknown controller gains. An outline of the software used io implement the g o v m r on an industrial programmable controller is presented, and the experimental pertonnsnce of the turbines is compareci to the theoreticaily developed mode1 of Chepter 2.

3.2 Sprcd Droop Goveming Spesd drwp govemor systems are common to low and medium head hydro generating stations. Neariy al1 of Ontario Power Generation's hydroelectric facilities in Northwest Ontario utilize a drwp govemor scheme of one fom or another. And since the opaating aad maintenina staff ofthese facilities are fàmiliar with its operation, this scheme becornes a logical choice for the Kakabeka Plant.

Figure 3-1 shows a block dicigram for a speed droop govemor. The wicket gates are forced open or closed based upon e m r Aw beniveen the desired speed wo and measureâ speeci W. The extemal input si@ d is normally provideci for gate wntrd or fnquency adjustment under steady state conditions.

The drwp compensation consists of two pudlel control branches, each providing fccdback h m the wicket gate position to the wicket gate -or. The ont bmch uses a proportional gain 4 commonly r e f d to as the permanent droop setting. The second branch consists of a hquency limiteû daivative element. The parameter RI is tcnown as the transient droop d n g , and rr is refemd to as the reset time.

The permanent d m p determines the amount of gate opening (or dosing) that will result with a fixed fnquency error of Aw. under steady state conditions. The permanent droop is nonnaily specified as the percent ofrated fiequency required to drive the wicket gates to the fùliy open position. Because permanent droop oontrols the amount of prime rnover energy addeû or removed fiom an interconnecteci transmission system, its value is usurlly specified based on system stability considerations and not on machine chnracteristics. Ontario Power Generation requins that a permanent droop setting of 4% be used on al1 hycûo turbines in Northwest Ontario [Ml.

The transient draop and met time together essemially provide derivative feedback on the gate position under transient conditions. It is the transient droop characteristic that is set to maintain govemor stability, and it does this by providing improved gain and phase margins amwd gain crossover fiequency.

In considdon of speed governing for Kakabeka, two points are worth noting; First the aror signai Aw wili be taken as fnqwncy error between the power systern and the generatot measured dusctly in Hz. This essentially provides an aââitional gain on the gate rauitor signal and improves o v d l govemor pediormanœ [4]. This extra gain may be acwunteâ for in the mode1 of Chrpter 2 by multiplying Eqwtion (2.20) by a -or of 60. in gened, [4] makes note of the hct tbat an o v d l gain of 5-10 in the gate actuator dynunics is desinble. Secondly, the typical assipent of a (+) pduity on the fiequency error signal bas ban replaced with a sign change (*) to signify thiit a polarity reversai d l k aecessary to accomt for the inverse m i n e characteristics while openting in the off-line d e .

The effects of perment and transient droop rnay be studid fbriher by investigating the open loop tiequency response of the govemor system show in Figure 3-1. The penstock, turbine, and generator dyaunics may been takm as:

based upon Equation (2.24) and the values in Table 2-2.

The gate actuator dynamics including gain compensation for direct frspuency feedback is:

And the permanent and transient droop compensation denoted as D(s), rnay be show as part of the foward loop characteristic by:

so that the closed loop system in Figure 3-1 may be represented as unity feadback around the open loop t r m f ~ function K(s)G(s)H(s) .

For pemunent droop compensation alone,

and for transient droop done,

Figures 3-2 anâ 3-3 provide Bode plots of the on-line and off-line transfer function m d s derived in Chpter 2. In each Figure the independent effects of permanent and m i e n t droop K(s), have been plotteci dong side the open loop d y ~ m i c s S(s)H(s)). In both aues the teset tirne r, has b n taken as 5rw so that the transient droop provides phise and gain compensation around the gain crossover 6equency. The sign applied to the open loop system is dependent upon the openting mode.

... - - - - - - - - -

Govemor Systm Bode Plot - Online Mode

Frequency (Radlsoc)

Figure 32 Govemor syrtem Bode Plot - Online Mode

From Figure 3-2 one can see thot open loop system without compensation is unstable. This is due to the in part to the phase shiA caused by the gate and generator dynamics but also ôecause of the water hamrner effm in the penstock. Unda transient conditions, an incremental gate opening causes the turbine water pressure and speed to decrease for a short period of time, until the m s s flow of water in the pemtock can accelerate to a new steaây-state value. Thus, the efféas of water hammer dso create p b lag in the open loop system, and reduce staôility.

The permanent droop limits steady-state open loop gain by canceling the Unegntor in the gate dynamics, and the tnasient droop prondes a positive phase shift at the crossova frequency. Since low values of permanent droop are typicrlly required to obtain tight fiequency regdation of the htefco~fcted power system, transient droop compensation must be used to stabilize the closed loop system.

The inverse characteristic exhibited by the turbine in the off-line mode of operation, rmhs in a reduced phase SM of the open loop t r a n s k fiinction, d closed loop

staôility without compensation. This is depicted in Figun 3-3. The reduced phase shift is a result the of inverse airbine behavior coupled with the effects of water hammer. An incremental gate opening to slow the gemrator speed is aided by the trwient response of the mas flow ofwater in the penstock.

Although the system is closed loop stable without any govemor compensation, the phase and gain margins are poor and instability may result if unmodeled dynamics are excited. From the Bode plot in Figure 3-3, it appears that pmnanent droop alom may provide adequate compensation, while fùrther improvements in phase and gain margins are possible with the addition of transient droop.

Govemor Systetn Bode Plot - On Line Mode

Figure 3-3 Covernor System Bode Plot - -Lime M d e

3.3 Stabiiity Regioa of Covernor Settings

In [9] Hovey showed that the stability of the speed droop govenor for the turbine and water dyniimics of Equation (3. l), and the generator dynamics of Equation (3.2), is dependent on the pole locations of the Laplace domain polynomial given by Equation (3.3).

1 ~ ( s ) = - (m(s) - me (s))

rH2

Equation (3.3) is the simplified characteristic polynomial of the transfer function relating electncai toque disturbance input m. to the generator speed output, taken from the block diagram show in Figure 3-1. The governor will be stable to step changes in me if al1 the poles of(3.3) lie in the lefi half plane.

By way of a simple Routh-Hunuiîz test, Hovey showed that a closed stability region aristed for the droop governor based on two parameters I l and At [9]. The stability region is given as:

With I r and Li being finctions of the water starting time r , mechanical time constant s,, transient droop R and met time r,.

P

By searching out an aesthetically pleasing time response within the region of staôility, : Hovey was able to develop the following Nning procedure[9]:

This is an important result and one thrt is particularly usefil for govenwx system auiing.

To assess the staôility region for the two operaîing modes defined in Chapter 2, Hovey's work may be npeated in a more generalized ûamework. To account for the changes in the turbine and water model, the msfer fùnction of Equation (3.1) may be restated with coefficient elements nl. t12 and dl.

The sign reversal that occurs in the govemor control loop between the on-line and off- Iine modes, cm be accounted for by modifying the polynomial equation of(3.3) as:

Using a ~o~th-~urwiitz Table [IS], one may determine the conditions (if any) under which al1 of the poles of Equation (3.7) are in the left half plane:

t l d ~ r r r ) s t t n r r *(nA - Vw)

For the on-line mode of operaion, assuming small deviations in speed so that ail=O, the turbine and water model may be taken as:

- 'a(a23ali -u21a13 Prs Ms) - g(s) (1 + a,, rws)

CompPnng this to Equation (3.6) and substituting the turbine parameters fkom Table 2-2, the coefficients ni. n2 and dl evaluate to:

Equation (3.7) will be stable with ail poles in the left haIf plane if the elements on the right of the Routh-Hurwitz table have the same sign (+ in this case). The polynomial of Equation (3.7) is stable ifthe following conditions are met:

Of the 4 conditions, 3 is the most restrictive and is equivalent to that derived in [9]. Condition 1 requires only that Rr is positive. Condition 2 is less restrictive than Condition 3 given the turbine constants, and the values of ?,kW that are of interest. The tuibine model alone satisfies condition 4. The relationship defined by Equation (3.8) is shown graphically in Figure 34 .

For the off-line mode, with the head losses due to the buttdly vaive taken into account, and assuming srnaIl deviations in speed so that aJ2=0, the turbine and water model rnay be taken as:

M s ) = a, + (a& 1 - +t,a,3 X7,s + 4 g(s)

(1 + a,, (v + 4)

Again, cornparkg this to Equation (3.6) and substituting the turbine parameters from Table 2-2, the coefficients n,, n2 and dl evaluate to:

Based on the Routh-Hurwitz table fiom above, the conditions for stability for the off-line mode of operation are:

Condition 1 requires that & be positive. Condition 2 rad Condition 3 are less restrictive since they requin tht R be appmximately pater tban -m. And condition 4 is satisfied by the M m e parameters alone. Thus Equation (3.10) becornes the necessary condition for stabüity.

It is interesting to note that without the sign change in the govemor loop for the off-line mode no stabüity region exists. The Routh-Hu~witz test confirms the need to change the wntrol structure bssed upon the inverse characteristic of the turbine in the off-line operating mode.

Figure 3-4 shows graphically the stability regions for each of the operating modes. Note that Hovey's govemor tuning rule falls within a cornmon stability region, dlowing the sarne values of transient droop and reset time to be used for ôoth operathg modes. Also, the computed stability regions agree closely with the Bode plots presented in the previous section.

Goverwr Stabiiity Regioas

Figure 3 4 Covernor Stabiiity Regions

In the development aôove and in that of [9] a aumber of assumptions have been made to derive the polynomiaîs ofEquations (3.3) and (3.6). First the permanent droop setting b i s b a n assumd negiigible unda transient conditions, The changes in per unit speed are assumeci to k s d l so tht the cffécts ofthe turbine constant au may be negleacd The

rotational losses or "load self regdation factor" is assumed to be negligible, and the gate actuator time constant is ne@&.

A nurnber of authors have expanded upon Hovey's work by nmowing the numbei of assumptions and redefining the stability region shown above. Moditied tuning niles are summarized in [4] while good treatment of the modified stability regions is given in [IO]. In [l O] consideration is given to both permanent drmp and rotational losses. In both cases the stability boundaries fint defineâ by Hovey are expanded providing an extra degree of robustness when considering the auiing d e s of Equation (3 S).

3.4 Digital Speed Droop Govemor

Figure 3-5 is a biock diagram of the digital govemor conad system implemented at Kakabeka GS. The govemor logic consists of3 compoaents: A speed drwp govemor block, a fiepuency tracking control block and a wicket gate sapoint control block.

The droop governor block shown at the bottom of the figure is the discrete time equivalent ofthat present in Figure 3-1. It consists of permanent and transient droop feedback elements. The discnte transient droop wmponent has been derived fiom its continuous time wunterpart by suitable 2-domah transformation with a zero orda hold element 11 51.

The Wuency-tracking block is a discnte proportional + integral (PI control) element enableâ whenever the generator is openting in an off-line mode. With w , set to the power system fhquency, this control element nulls any fiequency error between the power system and the gmaator under steady state conditions. The discnte integntor mimics thu of a continuous time int-tor by way ofa trapemdal summing rule. The present integral of =or io is the sum of the past integral of mor il, and the area sum of the control anu ea el over the iast sampling pend T assuming 1- interpolation between adjacent error samples.

This may k written as the Zdornain transfer function:

The speed matching between the generator and the system is required to provide generator synchronization to the power system Although this scheme d a s not gumtee that the electtical phase angle between the power system and the generator wiil be zero, small hydraulic disturbances in the system normally produce enough phase crossings to facilitate synchronizing. OtheMRse, the extemal govenor input 6 may be used disturb the generator speed a s d l arnount to reduce or advance any phase disparity ailet spded matching is achieved.

The gate setpoint control block functions during the online mode of operation, and provides a means of regulating generated power through wicket gate positioning. Like the fiequency-tracking block, this block also bas a discnte PI structure and uses the same transfer fiinction. Under normal power system conditions, the block is enabled and provides wicket setpoint control fùnctionality to a plant operator. However, during power system disturbances when thne is amr b a n the desued fkquency w , = 60 Hz and the actual system fkequency w, the block is disabled so that the speeâ droop governor may react to the fkequency error. Disabling this PI loop insuns that the gate control will not fight the actions of the speed govemîng, and that the generator will react in a manner that helps to sustain power system stability.

Ramp limiters are used on both the Frquency setpoint signal and the gate setpoint signal. For the frrquency setpoint, the limiter prevents gate aauator saairation during generator startup, whiîe the gate setpoint limiter reduces the efléctive rate at which the gates move when a setpoint change takes place. This allows the changes in water consumption by the turbine to ocw slowly and in a linear fishion so that a plant o-or can respond to surge tank water requirements.

The govemor conml is coordinated with the operation of the butterfly valve and the generator breaker, with programmable logic controller. On generator stuiup the buttedîy valve is opeaed to the throttld position gb = ZOO/, prior to the governor block king enabled in the off-line operating mode. When the generator achieves synchronization with the powa system anl the break closes, the govemor algorithm automatidly switches to the o d i e operating mode, and the butterfiy valve is for& to the Mly open position.

A shutdown sequence is similar: When the plant operator unloaâs a generatot and requests that it is taken off-line, the buttdy valve is tirst c l o d to the throttled position. The unit breakcf is then tripped and the govanor algorithm is switched fkom the on-line to the off-line mode.

A gain scheduiing arrangement is used when the mode of the governor is changed. While off-line the generator dywnics are stabilized by a set of gains applied to the dmop govemor md the fisquency-tracking PI blocks. In the on-line mode, an independent set of gains is applied to the h p and gûe setpint blocks to account for the diffaiag dynimic characteristics.

The governor control algorithm is coded in a basic like language made available by the programming software for the PLC. The algorithm rnakes use of measund field data collected by way of transducers and an ND converter. The wmplete set of algorithm instructions nui e v q scan cycle of the PLC, and upon compktion retum the gate actuator control signal via a DIA converter. The measurement and conversion of these signais is depicted in Figure 3-5 by the indicated sampling points. Setpoint information entas into the algorithm by way of digital communications from a plant supewisory control and data acquisition system (SCADA).

The govemor algorithm accepts a number of inputs defined as:

Auto: BOOL; ( O Govcrnor enable Auto (1) 1 Han (8) Hode: BOOL; (O Operating !!ode On-Line (1) / Off -Cine (8) 1 Gsp: REAL; ( O Gate Setpoint 8.8-188.8 X G: REAL; (O Gate Posit ion 8.8 180.8 X 1 Grmp: BOûL; ( O Gate Setpofnt Ramp L fmi t lng Enable (1) 1 Disable (8) Grate: REAL: (O Gate Ramp Limi t l n g Rate X / sec. 1 Fsp: REAL; ( O Cenerator Frequency Setpoint Hz 1 F: REAL; ( * Generator Frequency Hz ) Frmp: BOOL; (* Frequency Setpoint Ranip L imi t ing EnabLe (1) / Disable (0)') Frate: REAL: ( * Frequency Ramp L lm i t i ng Rate Hzlsec 1 Umax: REAL; ( * Haxinrua Control Signal 8.8 <= Umax c= 1.0 1 Umin: REAL; (* iîfninuiil Control Signal -1,W= Umfn *= 8.8 " 1 Uman: REAL; (. Manual mode Control Signal -1.8 <= Uman <= 1.0 a 1 Fm: Hgov; (* Off-Line Operating Parameters 1 Gm: Hgov: ( O On-Llne Operating Parameters 1

And the algorithm outputs are:

U: REAL: ( O Control Slgnal 1 Gset: BOOL: (* Gate Ramp L imf t fng Status On(B)/Off(l) ) Fstt : BOOL: ( * Frequency Ramp L lm i t ing Status On(B)lOff(l) 1

Two data structures Fm and Gm are used to store the off-line and on-line gains for the wntrol algorithm. The formet ofeach structure is the same, and each contains the

Hgov : STRUCT

T: REAL ; (. Reset Time Constant *) Rd: REAL; (* Governor Droop 5e t t l ng *) Rt : REAL: ( O Governor Transjcnt Oroop Sett tng O) P: REAL ; ( O Proportional Loop Sett lng * ) 1 : REAL ; ( O In tegra l Loop Set t fng * ) Gmax: REAL; ( * îlaxinum Gate Posl t i o n * ) Gnin: REAL; (* Piininun Gate Posi t ion .)

END-STRUCT;

The values of T, Rd and Rt are used for the droop govemor contrd block, while P and I are used for the gate or fkquency tracking control blocks. Gmar and Gmin allow wicket gate limits to be assigned to eech of the operating modes. The elements of the structure are accessed using the language syntax '. ' dot operator.

The Auto and Made inputs determine how the govemor algorithm is to operate. With Auto=O govemor operaiion is disabled anâ the eonml signal is set to the value of Uïwm unconditionally. With Auto=I, the governor aigorithm is active and operates in either the on-üne or off-line opefathg modes depending upon the value of the Mode input. Made

follows the position of the generator breaker: Marie = I when the breaker is closed and M&=O when the brdcer is open.

For the off-line mode of operation Equations (3.1 1) and (3.12) are implemented within the PLC usine the following conml logic:

( O Conpute con t ro l le r States O) dgû:=dglo€KPREAL(-Ts/Fm.T)+(g&gl); ifû:=(Ts/2.8)*(fcû+fel);

The transient droop signal d ' is wmputed baseci upon the past value dgl, the sampling rate Ts, reset t h e Fm. Tand the present and past gate positions gO and gl derived ôrorn the input G. The integral ofthe fiequency error in the Iast sampling p&od r@ is a function ofthe sampling rate Ts, and the present and past frequency crrorfi0 andjél derived fiom the inputs Fsp and F.

Before the control action is computed a number limit checks are paformed. First are checks to see if the gate position is greater than the maximum allowable gate opening Fm. Gnmc. If it iq then the integral action is blocked unless it is building in a negative direction. Likewise, if gate achiator conml signal uf is positive calling for more gate opening then it too is set to zero. Also if the gates have gone beyond the maximum pennissible o@ng tban ufis augmented with a negatively directed value to attempt to bring the gates within operating range. The computation of uf includes an extemal speed sipal Spd that is quivalent to 6 in the block diagram, the PI proportional gain and integral gains Fm.P and Fm.& md the permanent and transient droop Fnr.Rd and Fm.Rt. A similar check is also perfonned on the lower gate limit Fm.Gmzn. Limiting the gate travel between d m u m and minimum values is important particularly in the off-line mode of operation where instability may result if the gates are allowed to close too far.

(O check maximum gate l i n t i t agalnst the gate pos i t i on O )

If (68 > Fa.Gmax) THEN ( * Block I n t t g ra to r * ) I F ( i fe*Fn.I) > 0.0 THEN

i fû:=8.0; END-IF ;

(* Colpute con t ro l act ion and check d i rec t ion O )

uf:=Spd + fe0 + fe0eFm.P + ( i f l+ i fû)°Fm.I - gB°Fn.Rd - dgB°Fa.Rt; I F u f > 8.8 THEN

u f :=8.8; ENDJ F ;

( O Force gate i n negative d i r ec t i on *) I F NOT(Fn.Gnax >= 188 -0) THEN

uf:=uf+UminO(g0-Fm.Gmax)/(l88.B-Fm.Gmax): END-f F ;

(O check the n i n i r u a gate I l n i t against the gate pos i t i on O) ELSIF (60 c Fa.Gmin) THEH

( O Block In tcg ra to r * ) I F ( i fB8F~.1 ) < 0.8 THEN

if8:=8,0; END-IF :

( O Corputc con t r o l act ion and check d i r ec t i on uf:=Spd + fa0 + fe0Vn.P + ( i f l+f fe)OFn.I - gB0Fa.Rd - dgB8Fn.Rt; I F u f < 0.0 THEW

( O Force gate i n post t ive d i rect lon * ) I F MOT(Fm.GMin <= 8.8) THEW

uf:=uf+Unax~(Fm.Gnin-gB)I(Fm.Gmin); END-IF ;

(* gates are w i th ln operating range * ) ELSE

(* gates are w i t h j n the speclf ied L lmi ts so coapute * ) (* the appropriate control act ion 1 uf:=Spd + fe8 + feû0Fn.P + ( i f l+ i f0 ) *Fm. I - geaFm.Rd - dge0Fm.Rt;

ENOI F :

Limit checking is also p d o m e d on the control action, and üke the gate limits, the integral action on the frequency error is blockd if the control signai i s saturatecl uf> Umar or 6 Umin , and the integral action is building in the same direction

(' Check the con t ro l act ion agalnst Umax/Umin * ) I F uf > Umax THEM

u f : =Unax ; I F ( ifVFm.1) > 0.8 THEN

If0:=8.8: END-IF ;

ELSIF u f < Unin THEN u f :=Umln: I F (ifû*Fm.I) q 8.8 THEM

if0:=8.0: END-1 F ;

END-f F ;

The algorithm states are initializeû whenever the on-line mode of operation is enabled, under the assumption o f steady state conditions. Likewise the integral state is precharged with a value thrt results in a bumpless transfer as long as the gain Fm.1 is not zero.

I F (Fn.1 = 0.8) THEN ifI:=B.B:

ELSE ifl:=(gQ.Fn.Rd - te0 - Spd - feû°Fm,P)/Fm,I;

EMû-1 F :

At the end of the algorithm execution, the states are updated in the following mannec (. Update the states f o r the n t x t i t e ra t i on O)

fl:=f0: gi:=ge; fel:=fe@; ge l : =ge0; l f l : = i f l + i f e ; dg1 :=dg8 ; U=uf :

Logic for the online mode of operation is much the same aceptfi0 andfil are nplaced withgeû and gel for the gate position aror, and the feedôack gains becorne Gm.P and

Gm.1 for the gate PI control, Gm.Rd and GmRt for the permanent and transient droop smings, and Grn. T is the reset time.

Variable initializat ion:

I F (Gm.1 = 8.8) THEN igl:=8.8;

ELSE igl:=(gBoGm.Rd - fe8 - Spd - ge8*Gm.P)/Gm.I;

END-1 F :

Compute updates for the transient drwp state and integral of the gate aror:

Check maximum and minimum gate limits: (* check maximum gate l i t n i t a ~ a i n s t the gate pos i t ion * ) I F (go > Gn.Gmax) THEN

( O Bkock Integrator @ ) I F (igûoGni.I) > 0.8 THEN

lg8:=8.0; END-I F :

( O Conpute contro l act ion and check d i rec t ion *) ug:=Spd + feB + geBoGni.P + (lgl+igû)*Gnr.I - gû8Gm.Rd - dgBoGiii.Rt: I F ug > 0.8 THEN

ug:=8.8: END-I F ;

( O Force gate i n negative d i rec t ion O )

I F IOT(Gm.Gmax >= 109.8) THEN ug:=ug + Unino(g8-Gm.Gmax)/(l88.8-Gm.Gmax);

END-IF :

(* check the ninimun gate l i m f t against the gate pos i t lon '1 ELSIF (g8 c Gn.Gnin) THEN

(* Block In tegrator O) I F (ig8°Gm,I) c 0.8 THEN

ig8:=8.8: END-1 F :

( O Coapute contra1 act ion and check d i rec t ion @) ug:=Spd + fe0 + ge0Wn.P + (fgl+igû)oGa.I - gBoGn.Rd - dgB*Gm,Rt: IF U6 c 8.8 THEN

ug:=9.8: END-1 F ;

(O Force gate i n p o s i t l v t direction *) I F NOT(Ga.GPlin <= 0.8) THEU

ug:=ug + Unax~(Gm.Gmin-g8)f(GmIG~~~in): END-IF :

(* gates are w i th in operating range O )

ELSE

gates are w i th in the sp tc i f i ed l i n f t s so corpute *) (8 the appropriate con t ro l act ion ) ug:=Spd + fe8 + ge8°Gn,P + (fgl+igB)*Gm-f - g8°Gir-Rd - dgB*Gn.Rt:

END-IF ;

Check for oontrol signal saturation: ( O Check the contro l act ion against U~axlUmin *) I F ug > U m x THEM

ug :=Wax : I F (l@'Gn.I) > 0.0 THEW

1g8:=8.8: END-1 F ;

Update the algorithm states for the next scan cycle:

( * Update the s t a t c s for the next Sterat ion * ) fl:=f0; gl : =ge ; fet:=feû; gel : =ge0 ; lgl:=f gl+i@: dg1 : =dg0 : V=ug :

The disabling ofthe gate setpoint control loop is done extemal to the governor control algorithm, and is accomplished by disabling the gate setpoint nmp limita and replacing the gate setpoint with the present gate position. In doing this, the bop is taken off line in a bumpless rnanner since the integral state holds its present value, and the operator's last setpoint is replad with the present gate position when the loop is reestablished.

The algorithm code was found to mn at a sampling rate of 0.025 seconds on average, under wont cue conditions. However, to m m that fiinire code additions do not unduly affect the algorithm operation, and to maice the program code more portable to other CPU's, a fixed fisquency wunter pmvided by the CPU was used to cwtinuously track the sampling rate and update the variable Ts every 1 0 scan cycles.

For this specific application with the sampling rate given, the effects of aliasing occur a! a âequency that is much higher than the generator and turbine dpamics. And because the dismte time Equations (3.1 1) and (3.12) have been chosen to behave as continuous time dynamics given in Figure 3-1, one may apply either discrete or continuous time design principles when selectiag feedback gains.

The govemor system of Figure 3-5 requires 7 operating parameters. The droop control block uses the permanent droop Rp, the transient droop & and the reset tirne rr. while the ftequency tracking and gate conml PI blocks use the gains K' &and Km, K, respective1 y.

The tasic of selecting each ofthese values c m be broken down into three control problems:

1. Select values for Rn Rt, rr and Kpfi &to provide stable speeâ regulation and steady state tracking of the powef system fiequency while off-line.

2. Select values for Rp Rt, r, for the online mode of opemeon to provide stable speed regulation of the grnerator, and the appropriate amount of prime mover energy dispatch when ûequency mon aist.

3. Based upon the seleaed values of R , R , rr in 2 sbove, and the dynamics of the gate actuator alone, select vaiues for Km, and K, to provide wicket gate positioning with zero steady m e emor.

With the stability analysis of Section 3.2, some of the complexity in parametet selection may be reduceû since the off-line and on-line modes of operation share a cornmon stability re@on. The panmeter values for Rt, and r, in steps 1 and 2 above may be chosen the same provided that the values Ml within this region, and that Kpf is used to reverse the sign on hv while opetating in the off-line mode. Likewise, because the panwent drwp parameter is dictated by the power system utility 1161 and will have no il1 effects on stability betwcen on-line and off-line operating modes, it too rnay be chosen the same for steps 1 and 2.

It is necessuy to select 4,=4.0% baseci upon the Ontario Hydra's specitications as outîined in [16], then choose R,, and rr based upon the tuning methods of [9,4]. Sening the parameter KM= -2.0 Mectively reverses the sign on dw, and stabilizes the off-line operating mode. Ali that is required is to complete steps 1 and 2 above, is to find a suitably small value for Kfthat achieves steady sute fnquency tracking within a reasonable time.

Step 3 is relatively straightfotward: Given the values of R, Rt, and r, in 2, and the gate ' -or dynamics, choose values for Km, and Kg thu provide stable gate position

eontroi. Althougû this loop operates while the generator is on-line, the dyndcs of the ' genemtor and interwnnected power system a d not be considered since the muency ' aror is assumed to be zero r al1 times. When this is not the cra the loop is intentionally disabled.

An aigorithm that is partiailady usehl for twiing al1 of the unknown parameters of the govemor is presented in Reference [19]. A uses a gradient search to mi~mize the quadratic cost bction:

for a sampled data system defined by the linear matrix equations:

and the static oatput feedbadc law:

Q and R an weighting matrices that may be arbiucvily assiped on the system inputs and outputs to achieve a "pleasing" closed loop system response. Q must be positive semi- definite and R must k positive definite.

The cost fiinction of Equuion (3.13) is minimid if values for the matrices P. k and L CM be found that satisfies the following set oflinear matrix equations:

A numerical procedure for doing this follows:

1. Chwse a starting k that staôilizes the process (A, B, C, D) and compute k P = ~ ) - l & from (3.19).

2. Solve the discrete Lyapunov equation (3.16) for the matrix P. 3. Solve the discrete Lyapunov equation (3.17) for the d x L. 4. From Equation (3.18) compute:

5. Compute a new k' baseâ upon k'=&'+ adk ' for O <&1.0. Note tbat the value of l

1 a mu4 be chosen so that (A+BkBC) remains stable and the trace of the matnx P is ceduced at evety iteraîion, ' 1 6. Repcat seps 2 through 5 until dk' diminishes to an insignificantly sxnall v a k .

I ! 7. Cornpute the fdback gain &=Ç'(I+D& Erom (3.19).

To guarantee a unique solution to the Lyapunov equations, no two eigenvaiues of (A +Bk 'C) miy have the nlotionship At = UA, This condition will always be m a if k ' is chosen at each iteration to make @+Bk %) asymptotically stable, or stable with at most one eigenvalue lying on the unit circle . Unda these conditions both P and L will be positive definite.

Static output feedback algorithrns of this type are particularly useful for sdving a number of g e n d control problems involving incomplete state variable informotion. [20] provides a survey of published work on the output feeâback problem, including the algorithm of [19]. The algorithm of [19] and others presented in [20], if convergent, only guarantee a iocal minimum for Jthat is dependent on the initial choice of gain vector k.

To pro& with the task at hand, the system transfer function of Equation (2.24) may be represented iir state spaa fonn as:

With the system mattices defined as:

And the state vector:

Matrices A and B form a coupled system of first order diffmntial equations, with x' rqresenting the first derivative of the state vector x with respect to tirne. The factor of 60 introduccd into the B maaVt is to accout for the direct frrquency input discussed in S d o n 3.2. The supdpt 'Tg on matrices and vectors represents the tnnspose operator.

The continuous time repmntation of E q d o n (3.20) may k converted to a dismete tirne format given by Equation (3.21) using the standard zaoorder-hold tmnsforrrmtion [15]:

With suitable choices of matrices Ml and Mz, the state space system of Equation (3.21) may be augmentcd to include the dynamics of the transient drwp compensation and one of the integrators fiom eitber PI bop:

With:

The state d,, represents the output fiom the transient droop compensation befon it is scaled by Rt, and the state i, is one of the integrator States befote application of the gain kf or kig.

The augmented system can be used in the cost rninimization algorithm using the following state spaœ npresemation:

The vector &I contains the values of known govmor system control variables, while k2 contains the control gains to be obtained by cost rninimitcrtioa The matrices CI and C? are made up of constants that fom the structure of the output feedback law.

For ProbIem 1 above, with Rp=J% and KM =-2 the following ma& selections dlow the computation of R and ki/.

Or alternately, the tu- niles provided by [9] may be used to obîain Rl d k d y so that only the variable krneeds to be computed.

To solve Problem 2 above, the augmentation of an integrator state is not necessary and the matrices FI and GI may be simplifiecl somewhat.

G, =[G M , G ~

5. =[% 4 P

M, = [0.9 O O]

h f~ cos minimiation may be eliminated compietely if the tuning d e s of [9] are applied with R,=.J%, otherwise the panmeter&may be obtained by choosing k1C1 and k-C2 as follows:

To solve Problem 3, the values for MI, MA k1C1 and k g 2 are:

The effects offiequenq error are neglected by choice of CI, while Mz augments the matrix FI with the gate integrator dynamics.

The govemor parameters used for each of the generaton are swnmarized in Table 3-1. In al1 cases 2, was chosen as 4t* as per the recornmendations of [9]. For the off-Line mode of operation Rt was selected based upon the tuning rules of[9], since the cost minimization resulted in slightly longer M i n g times* For the on-line mode, Rt was selected basecî upon the results of the wst minimization because it offered the sarne senling time with less oscillation. KPf has been selected as -2 in the off-line opaating mode for al1 generators to provide a sign reversal necessary to achieve stability.

TiMe 3 1 Summa y oCComputed Govtrnor Setting

Figure 3-6 shows a fiequency disturbance response for Generator 1 operating in the Off- Line mode. The disturbance was created by injecting a signa! of 1 .O at the 6 input shown in Figure 3-5. The traces shown in r d are the a d responses measured at the PLC inputs, the trace shown in green is the predicted generator response based on the mode1 presented in Chapter 2 and the applied govemor settings. The remaining bhie trace is the predicted response of the generator had the transient droop been tuned with the cost minimization algorithm. AB can be san fiom the figure, the satling time is achieved in approximately 100 seconds, and the PI frequency-tracking block provides good steady state tracking.

On-Line Mode: Govsmor Frequency Response 1 I 1 f I 1 I

I

Tlme (seconds)

On-Une Mode: Govemot Widcet Gate Rssponse 60 ! I

l I I

1 l

-. - - ~ 1 l

30 j i I I

1 * d l y Vdve Position , 20 t

I I I 1 I

'O0 i

50 100 150 200 250 300 Time (seconds)

Figure 3-7 shows the simulated disturbance nsponse of Generator 1 operating in the on- line mode. The simulation was dom with a signai of 1 .O was injecteci at the 6 input show in F i p e 3-5, and the mode1 developed in Chapter 2 without additional power system dynamics. The mi trace is for governor settings baseci upon the h n g rules of Reference [9], wMe the green traa is for R computd with the cost minimization algorithm of the pmious sedion. The second set of settings were applied to the genemton given that the setting times are approximately the same, but with less oscillation resulting from the cost minimization methoci.

No adwl responses for the genentor are available, since power system distuhances occur infiequently and cannot be cceated artificiaily.

i Figure 3-7 Cenerator 1 Simulitcd On-Lime Diaturbance Reipoase

l

l ' Figure 3-8 shows the actual and predicted wicket gate response for Geaerator 1, with the : ramp limiter of Figure 3-5 disabled. Steady state tracking of the desired wicket gate

setpoint is achieved with the PI controller. The overshoot did not aeate any practical problems, and waa masked for the most part by the ramp limiter that aaed on the setpoint.

F i y n 36 Gentrator 1 Wicket Gate Setpoint Rcrponw

Wcket Gate Setpoint Response 40 . 1 1 1 1

-

Figures 3-9 and 3-10 show a complete sumip and shutdown sequence for generator 1. At startup the butterfly valve is opened to the throttled position - 20%, in preparation for the govemor cuntrol to be initiated in the off-line mode of operation. With the partial butterfiy valve openhg and the wicket gate leaicage alone, the genentor shaft receives enough torque to begin accelenting toward synchronous speed. Shortly &er the butterfly

% 25

20

15

10

has «>me to rest, the govemor is enabled and the wicket gates immediately tnvel hto the stable operaîing range of the inverse turbine characteristic gr>20%. The governor's frequency setpoint is set to tht of the power system with ramp limiting of 0.25 Hz/ sec. In approximately 300 seconds the govemor bas locked on the power system frrquency, and the phase angie slip-nte is slow enough allow syncbronizing8 When the unit b n i k a closes the butterfly valve is forceci to the fùlly open position, and the govemor algorithm

O 100 200 300 400 500 sec

- - - , , . - - .. . - -

1 1

i I 1 I

immediately switches to the on-line mode of operation. Likewise, the gate control PI loop is enabled with an initial setpoint l0%, ununtil comanded to do otherwise by the plant operator.

The shutdown sequence is similar. The butterfly valve is closed to the thronled position =20%, in preparation for a bnaker open signal. Upon a breaker trip, the govemor algorithm switches to the off-line mode ofoperation, and the wicket gates immediately travel into the stable operating range of the inverse hubine characteristic. The generator frsquency deviates a s d l arnount as the gates adjust to compensate for the disturbance that is aeated when the breaker is opened. The genaator remains at rated speed tracking the pown system fhquency until commandeci to stop by a plant operator. When the stop signal occurs, both the wicket gates and the butterfly vaive are forced to the closed position and the govemor control is disabled.

Similar results to those presented above were obtained for the other units. Due to the increased mechanical time-constant and wicket gate tirneanstant of Genmtor 4, the sdtüng times for the various opersiihg modes were found to be slightly greater than that of Cienenton 1,2 and 3.

Chapter 4

Future Work

1 The modeling and governor design presented in this thesis meets al1 of the design 1

l objectives set out in Chapter 1 and his been successfully implemented at the plant. At the I time of this writing, the aigorithm has been in successtiil operation for a pend of eight

months. The govemor bas semeci the plant fw better than the previous means of control by providing quick generator startup, synchronization, and relieble operation.

The cost minimitaion algorithm presented in chapter 3 converged well for the problem at hand and providecl inherently stable and robust solutions. However, the n a d to supply an initial stabilizing gain vector, and lack ofcorrelation between the Q and R matrices and any practical design objectives begs one to seek out betta methods of tuning.

Futwe efforts rnight be best directed towards expanding on the work of Hovey [9] to develop a generalized stability region and tuning parameters ôased not only on the generator time constansts, but also on a fiil1 set of nubine constants a~i-clu. The fiequency tracking PI block giwn proved to be very usehl for synchronhtion, and so its conml @ns might also be considnd in any development of a generalized set ofhining niles.

Stteeter, Victor L., Wylie, E. Benjamin, "Fluid Mechanics", McGraw-Hill Book Company, New York, 1979.

Oldenburger, R, Donelson, Jr. J., "Dynamic Response of a Hydroelectric Plant", AIEE Tmsactions, Vol. 8 1, Pt pp. 40348,1962.

IEEE Cornmittee Report, "Dynmic Models for Steam and Hydro Turbines in Power System Studies", IEEE Transactions Vol. PAS-92, pp. 1904-19 1 5, 1973.

Working Croup on Prime Mover and Energy Supply Models for System Dynami*c Performance Snidies, "Hydraulic Turbine and Turbine Control Models for System Dyii~mic Saidies", IEEE Transactions on Power systems, Vol. 7, No. 1, 1992.

Woodward, J. L. "Hydreulic Turbine Transfer Function for use in Govemor Studies", IEEProc., Vol. 115, No. 3, 1968.

Trudnowski, D. J., Agee J. C., "ldentifying a Hydraulic Turbine Mode1 from Measured Field Data", E E Transactions on Energy Conversion, Vol. 10, No. 4, 1995.

Vournas, C. D., Papaioannou, G., "Modeling and Stability ofa Hydro Plant with Two Surge Tanks". EEE Transactions on Energy Conversion, Vol. 10, No. 2, 1995.

Shinskey, F.G. "Rocess Contrd Systems - Application, Design, and Tuning", Third Edition, McGraw-Hill Book Company, New York 1988.

Hovey, L. M. "Optimum Adjustment of Hydro Govemors on Manatoba Hydro System", AIEE T m . Vol 8 1, Part iiI, Dec. 1962.

10. Hagihara, S., Yokota, K., Goda, K., Isobe, K., "Stability of a hydraulic turbine genersting unit controlled by PID govemor", IEEE Trans. Power Apparatus and Systems, Vol. PAS-98, No. 6, 1979.

11. ûrelind, G., Womiak, L., Medanic, J., Whittemore, T., "Optimal Ptû gain schedule for hyhydrenerators - design and applications" IEEE Trans. on Energy Conversion, Vol. 4, No. 3, Sept. 1989.

12. Findlay, D., Davie, H., Foord, T. R, Marshall, AG., Winning, D.J., "Micropcocessor based adaptive water turbine governof', IEE Proc. Vol. 127, Part C, No. 6, Nov. 1980.

13. Malik, O. P., Zen& Y., ''Design of a robust aâaptive controiler h r a water turbine govenllng system" IEEE Tram. on Energy Conversion, Vol. 10, No. 2, June 1995.

14. Jiang, J. "Design of an optimal roôust govemor for hydraulic turbine genenting units" EEE Trans. on Energy Conversion, Vol. 1 O, No. 1, Mar& 1995.

15. &langer, P. R "Coatrol Engineering: A modern approich", Saunders Collqe Publishing, 1995.

16. Bayne, JP. "Govemor Settings in the Northwest Region" Ontario Hydro Research Division Report 77-1 18-H, hlrrch 1977.

17. Ontario Hydro, "Operathg Hydraulic Data for Kakabeka Falls GS", Regioaal Trades aad Operating Information System Technicd Directive HO-9 1 1-R1, September 1983.

18. Hostetter, G.H., Savant, C.J., Stefiuii, RT., 'Design of Feedback Control Systems", Holt, Rinebart and Wilson, New York, 1982.

19. Marder, D., Calise, A., "Convergence of a Numencal Algorithm for Cdculating Optimai Output Feedback Gains". IEEE Transactions on Automatic Control, Vol. AC30, No. 9, September 1985.

20. Synnoq VL., Abdallah, C., Dorato, P., "Static Output feedback: A Survey", IEEE Proceedings of the 33" Confaena on Decision and Control, Lake Buena Florida, December 1994.