Operation and control of a water supply system

ISA TRANSACTIONS® ISA Transactions 42 共2003兲 461–473

Operation and control of a water supply system I˙lyas Eker,* Tolgay Kara† Department of Electrical and Electronic Engineering, Division of Control Systems, University of Gaziantep, 27310 Gaziantep, Turkey

共Received 4 April 2002; accepted 22 September 2002兲

Abstract The control of water supply systems is becoming more important, since there are increasing requirements to improve operation. A need exists to model and simulate water supply systems so that their behavior can be fully understood and the total process optimized. This paper describes the simulation and control of a water supply system consisting of a sequence of pumping stations that deliver water through pipelines to intermediate storage reservoirs. The system is represented by dominant system variables that represent active and passive dynamical elements. The hydraulic models include the nonlinear coupling between flow rates and reservoir heads. The bisection numerical solution approach is used to obtain a roughness dependent friction coefficient. The whole system is simulated and the results are presented and compared with the real-time measured data. A water level controller using the robust polynomial H ⬁ optimization method by manipulating pump speed is obtained. The stochastic nature of the disturbance and loads is considered for controller design. The parametrized dynamic weighting functions of the design theory are selected to achieve the required control functions and robustness. © 2003 ISA—The Instrumentation, Systems, and Automation Society. Keywords: Water supply systems; Modeling; Feedback control; Friction

1. Introduction Water supply systems are of great importance among hydro systems, being built and operated to provide enough water to consistently meet the demand 关1–3兴. Global demand for water is continuously increasing due to population growth, industrial developments, and improvements of economic conditions, while accessible sources of water keep decreasing in number and capacity. Due to financial and environmental reasons, control of water supply systems is needed to provide continuous and stable operation in meeting this demand 关2– 4兴. Thus it is necessary to carefully manage water transfer 关5兴. This requires modeling *Corresponding author. Tel: ⫹90 342 360 12 00 Ext: 2135; fax: ⫹90 342 360 11 00; E-mail address: [email protected] † E-mail address: [email protected]

and simulation of water supply systems as a first step, so that their behavior can be fully understood and the total process optimized 关4 –7兴. Understanding the technical side of the water supply systems such as knowledge about how the system behaves if a disturbance occurs at any point of the system is a crucial problem 关8兴. Crucial targets also include the continuity of operation and improvement of supply systems 关2兴. These many challenges involve different technical disciplines 关9兴. Water supply systems are generally composed of a large number of interconnected pipes, reservoirs, pumps, valves, and other hydraulic elements which carry water from retention to demand areas 关5,10,11兴. The hydraulic elements in a supply system may be classified into two categories: active and passive. The active elements are those which can be operated to alter the flow rate of water in

0019-0578/2003/$ - see front matter © 2003 ISA—The Instrumentation, Systems, and Automation Society.

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specific parts of the system, such as pumps and valves. The pipes and reservoirs are passive elements, insofar as they receive the effects of the active elements. These elements in the supply systems play important roles in dynamic behavior of the water supply systems 关12,13兴. Simulations of the water supply systems have been an indispensable work to understand their behavior to produce a feasible control solution as well as modeling 关5,6,8,14兴. The simulations can thus be used to generate ideas in order to develop flexible management and design schemes 关4,5,7兴. Consequently, this process may facilitate a better exchange of ideas among representatives of different professions. It also combines technical and financial viewpoints 关4,15兴. The first step in simulation and control is to establish a mathematical model for the plant to be controlled. Furthermore, an adequate model is an important step in determining the behavior and producing a control algorithm 关7兴. A successful model should be able to take advantage of system features that lead to simpler mathematical formulations and the proper choice of solution method 关2兴. Hydraulic systems generally require complex models. Derivation of control strategies on the basis of the complex models is difficult. For these reasons, the plant model should be chosen to be simple with a minimum number of dominant variables, which, nevertheless, adequately reflect the dynamics of the plant 关7兴. The plant can be described by the parameters that characterize its functioning such as the pumps discharges, water heads in the reservoirs, and flow rates through the system 关5,12兴. Thus the simulation of the model that represents a water supply system may prove an efficient measure to contribute to the correct transfer of water and to reduce operational cost, as well as to improve the operation 关5兴. This paper discusses simulation and control of a water supply system. The active and passive elements are represented by dominant system variables. Nonlinear friction coefficient equation is solved using the bisection numerical solution method. The main objectives are to ensure the proper operation of a water supply system and to regulate the water flow rates and heads by manipulating the water pumps. The polynomial H ⬁ optimization algorithm is used to design a controller that regulates flow rates and heads along the water supply system. The application to the supply sys-

tem of Gaziantep in Turkey is presented. The measured quantities and simulation results show the significance of the study. 2. Hydraulic models Water supply systems are composed of active and passive hydraulic elements such as pumps, pipes, valves and reservoirs that play important roles in dynamic behavior of the supply systems 关5,10,11,14兴. Water is assumed incompressible and the individual system components are stationary. 2.1. Pumps Head developed by n variable-speed pumps running in parallel varies nonlinearly with their speed N rpm and output water flow rate Q p ( t ) m3 s⫺1 关11,14,16兴:

h p 共 N,Q p 兲 ⫽A o N 2 ⫹

Bo Co NQ p ⫺ 2 Q 2p , n n

共1兲

where A o , B o , C o are the constants for a particular pump depending on component characteristics. These constants can also be calculated using appropriate manufacturer’s specifications 关5,11兴. 2.2. Pipes Consider a pipe section with length l p ( m ) and cross-sectional area A p ( m 2 ) . If the head difference ⌬h between two ends of the pipe section is considered, the following differential equation is obtained 关11,14,16兴:

dQ 共 t 兲 gA p ⫽ 关 ⌬h⫺h loss 共 t 兲兴 , dt lp

共2兲

where h loss denotes the total head loss along the piping section and g denotes the acceleration of gravity. The flow rate and head loss may be given as o h loss 共 t 兲 ⫽h loss ⫹⌬h loss 共 t 兲 ,

共3兲

Q 共 t 兲 ⫽Q ⫹⌬Q 共 t 兲 , o

where ( • ) o denotes steady-state value and ⌬h loss ( t ) designates the variable head loss caused by the variable water flow rate ⌬Q ( t ) .

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2.3. Head loss

2.4. Water reservoirs

Friction losses and local losses are two different types of losses in hydraulic systems 关9,10,16兴. There are several approaches obtained from theoretical considerations and experimental data for the friction loss in pipes 关17,18兴. The total loss in a pipe section may be given as 关11,14兴

When a reservoir discharges under its own head without external pressure, the continuity equation 关20兴 simplifies to

h loss 共 t 兲 ⫽h loss⫺ f p 共 t 兲 ⫹h loss⫺l 共 t 兲 ,

where ␳, ␳ i , ␳ o represent the water densities inside the reservoir, water inflow, and outflow, respectively, and these are assumed equal ( ␳ ⫽ ␳ i ⫽ ␳ o ) . Q i ( t ) m3 s⫺1 and Q o ( t ) m3 s⫺1 denote reservoir input and output water flow rates, respectively, and V ( t ) m3 is the volume of the reservoir.

共4兲

where h loss⫺ f p denotes friction loss, h loss⫺l denotes local losses. Hazen-Williams’s method and Darcy-Weisbach’s method have been frequently used in calculation of head loss in piping systems 关5,18兴. The Hazen-Williams method has nonlinear characteristics with the water flow rate Q ( t ) as 关18兴

h loss⫺ f p 共 t 兲 ⫽

冉

冊

10.78l p Q 1.852共 t 兲 , c 1.852D 4.87

共5兲

where the constant c is the pipe coefficient which depends on age and type of material used, D is the inner diameter of the pipe in meters. The DarcyWeisbach’s formula depends on the flow rate and it changes with the flow rate nonlinearly as 关19兴

冉

冊

f pl p h loss⫺ f p 共 t 兲 ⫽ Q 2共 t 兲 , DA 2p 2g

共6兲

冉 冊

Km Q 2共 t 兲 , 2gA 2p

共7兲

where K m is the local loss coefficient that depends on the physical characteristics of the system such as the curvature radius, pipe diameter, central angle, and roughness dependent friction coefficient „f p ( ␧ ) ….

d 关 V 共 t 兲兴 ⫽ ␳ iQ i共 t 兲 ⫺ ␳ oQ o共 t 兲 , dt

共8兲

3. Friction coefficient for pipes There are several methods to obtain friction coefficient f p ( ␧ ) 关11,14,16,17,21兴. Use of the friction coefficient which depends on surface roughness under rough conditions can give accurate predictions of flow rates 关22兴. Colebrook and White’s empirical correlation seems the best for this purpose as recommended in several references 关5,11,14兴:

1

where f p denotes friction coefficient. Local losses resulting from rapid changes in the direction or magnitude of the velocity of water can be neglected for long pipes of which length is 100 times the pipe diameter 关11,13,18兴. But, the local losses are quite important for small pipe sections 关16,17兴. Expansions, contractions, bends, valves, and flow rates at reservoir entrances and flow rates at reservoir exits may cause local losses that can be expressed as a multiple of the square of flow rate in the pipe configuration 关11,14兴:

h loss⫺l 共 t 兲 ⫽

␳

冑f p

⫽⫺4 log10

冉

冊

2.51 ␧/D ⫹ , 3.71 2 冑2 f p N R

共9兲

where ␧ is the roughness of the pipe and N R denotes Reynolds number. This nonlinear equation may be solved using graphical methods or using a numerical method and a computer program. The ‘‘bisection’’ numerical solution method 关23兴 can be used to solve the nonlinear equation, Eq. 共9兲. The algorithm needs two values initially, the upper and the lower values of friction coefficient. The developed algorithm is based on an iteration process by narrowing the upper ( f pu ) and lower ( f pl ) limit values to obtain desired friction coefficient with the defined error tolerance. Iteration is proceeded until the defined error tolerance is achieved. 4. System studied The water is taken from Kartalkaya dam, which is 53 km away from the city of Gaziantep. Fig. 1 illustrates the diagram of the water supply system along which there are three pumping stations

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Fig. 1. Diagram of the city of Gaziantep water supply system.

共PST-1, PST-2, PST-3兲 and three reservoirs 共RS-1, RS-2, RS-3兲. The position of the first pumping station is taken as the reference point. h ti ( t ) , h si , l pi denote the variable heads ( m ) in the reservoirs, static heads ( m ) , and the lengths of pipes ( m ) , respectively. The variables Q a ( t ) , Q b ( t ) , and Q c ( t ) designate the water flow rates through pipes. The pipelines are buried underground and are assumed to be free of chemical reaction, biochemical effects, and significant thermal disturbances. Check valves are fitted on the discharge side of each pump to maintain forward flow rates in the main and to prevent backflow 关11,12,14兴. The system does not include cavitations. All relief valves and pressure regulating valves are fitted on the piping system. The flow rates, reservoir heads, and pump speeds exhibit deviations from the nominal steady-state operating values to obtain behavior of the overall system. Uniform flow rates are assumed for the water supply system and the variations around nominal operating values do not deteriorate these assumptions 关7,9–11,22兴. Distributed flow is not taken into account because this model considers simple piping systems with low velocities 关17兴. The flow rate disturbances in water supply systems are common 关7,9,14兴 and should be taken into account. Pumps are stable because they handle incompressible fluids 关24兴. Three pumps work in parallel in each pump station with the nominal speed of 985 rpm. The pipes are concrete type with an inner diameter of 1.4 m, a crosssectional area of 1.5394 m2 and 15 years old. The reservoirs have a cross-sectional area ( A t ) of 475 m2 . Bending curvatures of the pipes were measured to be larger than the pipe diameter ( D ) . In this case the coefficient K m in Eq. 共7兲 is taken as 0.08 and the loss coefficient of the check valves

are taken as K m ⫽3 关25兴. The numerical data about the water supply system of the city of Gaziantep are given in Table 1. The output water flow rate was measured at 1-h intervals in a day, so 24 measurements were taken using a flow meter installed on the real system. Using the data obtained, the average water flow rate is about Q o ⫽2.83 m3 s⫺1 ( 10 188 m3 /h) , and it changes between 10 175 m3 /h and 10 200 m3 /h. The system was linearized using the Taylor series expansion method around a steady-state operating point ( N so ⫽985 rpm; Q so ⫽2.83 m3 s⫺1 ). The pump characteristics were obtained from the pump’s manufacturer. Head developed by the pump was calculated around the operating point using the characteristic curve as

h p 共 N,Q p 兲 ⫽0.000 143 3N 2 ⫹0.005 015NQ p ⫺3.98Q 2p . A single-input-single-output 共SISO兲 model is developed here with the single input as the first pump speed N rpm and the single output as the output flow rate from the third reservoir Q o ( t ) m3 s⫺1 关14兴. The block diagram of the system is illustrated in Fig. 2. The system can be Table 1 Numerical data of the supply system. lp1 ⫽669.27 m lp2 ⫽13805.04 m lp3 ⫽20094.69 m

A pi⫽1.5394 m2 h s1 ⫽113.4 m h s2 ⫽210.4 m

lp4 ⫽4689.04 m A t⫽475 m2

h s3 ⫽283.4 m h s4 ⫽279.7 m

D⫽1.4 m g⫽9.81 m/s2 ␳ i⫽ ␳ ⫽ ␳ o ⫽1 g/cm3 N so⫽985 rpm Q so⬵2.83 m3 s⫺1

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Fig. 2. Block diagram of the water supply system.

represented in state space matrix form such that the reservoir heads and flow rates can be considered as states 关11,12,14兴. The canonical state space form is

x˙ 共 t 兲 ⫽Ax 共 t 兲 ⫹Bu 共 t 兲 , y 共 t 兲 ⫽Cx 共 t 兲 ,

and y ( t ) is the system output. The state matrix x ( t ) , input u ( t ) , and calculated constant matrices A, B, C, are

x 共 t 兲 ⫽ 关 Q o h t3 Q c h t2 Q b h t1 Q a 兴 T ,

共10兲

where x ( t ) is the state matrix, A, B, C are the constant system matrices, u ( t ) is the system input,

A⫽

冤

u 共 t 兲 ⫽N,

共12兲

⫺0.0253

0.0032

0

0

0

0

0

⫺0.0021

0

0.0021

0

0

0

0

0.0008

0

0

0

0

0.0021

0

0

0.0011

0

0

0.0021

0

⫺0.0008 ⫺0.0398

0

0

⫺0.0021

0

0

0

0

0

0

0

⫺0.0021

0

0

0

0

0

B⫽ 关 0 0 0 0 0 0 0.0067兴 T , C⫽ 关 1 0 0 0 0 0 0 兴 . The system is seventh order. The zeros and

⫺0.0011 ⫺0.0465

共11兲

⫺0.0226 ⫺0.4553

冥

,

poles of the supply system were calculated and given in Table 2, and the system is minimum phase and stable. The last three poles are the closest to the origin with time constants of 0.86, 1.11, and 4.47 h, respectively.

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Table 2 Zeros and poles of the water supply system. Poles (s⫺1 )

Zeros

⫺0.4551564 ⫺0.0396796 ⫺0.0255573 ⫺0.0250177 ⫺0.0003240 ⫺0.0002504 ⫺0.0000621

⫺0.0396805 ⫺0.0255868 ⫺0.0250794 ⫺0.0002201⫾0.0000598i

4.1. Open loop system response The output response 关 Q o ( t ) 兴 is illustrated in Fig. 3 to 985⫾10 rpm square wave speed variations with a frequency of 5⫻10⫺6 Hz in the first pump station. The speed variation was applied at t⫽105 sec ( t⫽27.7 h) after starting. The steadystate output water flow rate was obtained at about 2.835 m3 s⫺1 . The frequency response of the normalized system is illustrated in Fig. 4, the gain and phase margins are 16 dB and 72°, respectively. This implies that the open loop uncompensated system is minimum phase and stable. The gain crossover frequency is 0.0001 rad sec⫺1 and the system bandwidth is about of 6.5 ⫻10⫺5 rad sec⫺1 . The effect of local and friction losses through the pipe section 3 ( l p3 ) is illus-

Fig. 3. Output flow rate Q o (t) for N⫽985⫾10 rpm square wave speed variation.

trated in Fig. 5. The friction loss is larger in magnitude compared with the local losses. 4.2. Friction methods Friction losses for pipe section 3 were obtained from the open loop test as shown in Fig. 6. Approximately 45 m of head loss in steady state was obtained from the Darcy-Weisbach method, which is illustrated by the solid line. The smaller head loss in magnitude 共dashed line兲 was obtained from the Hazen-Williams method that was about 44.5 m in steady-state operating conditions.

Fig. 4. Frequency response of the open loop uncompensated system.

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Fig. 5. Friction 共major兲 and local 共minor兲 losses in pipe section 3 (l p3 ).

4.3. Friction coefficient The nonlinear friction coefficient equations for different methods were solved using the bisection numerical solution algorithm that was imple-

mented to run in the MATLAB environment. The algorithm is very simple providing fast convergence and is easy to use. The calculated values from different methods in literature 关18兴 are given in Table 3. The methods of Davies and White,

Fig. 6. Head losses obtained from the Hazen-Williams method 共dashed兲 and the Darcy-Weisbach method 共solid兲 for the pipe section 3, l p3 .

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Table 3 Calculated friction coefficients for different methods. f p 共␧兲

Methods Davies and White Blasius Prantdl and von Karman Gainguillet and Kutter Nikuradse Colebrook and White Colebrook Moody

0.00873830 0.00862908 0.01055540 0.01754690 0.01810300 0.01833740 0.01834350 0.01901630

Prantdl and von Karman, and Blasius were developed for smooth pipes, and that is why the calculated friction coefficients from these methods were different and smaller than the calculated values from the other methods. The results obtained from the methods for the rough pipes show that all values are very close to each other. Colebrook and White’s method was used in the rest of the simulations. The steady-state water heads were measured and obtained with simulations in the reservoirs as given in Table 4. The water level in RS-1 was about 4.15 m just at the overflow limit 共4.5 m兲. The other reservoirs work satisfactorily, since the water head in RS-2 was about 2.07 m and about 3.15 m in RS-3. Therefore RS-1 needs level control. 5. Robust H ⬁ optimization and level controller design Feedback control is needed to provide more stable operation and to improve the robustness margins. Since the water head is just at the overflow limit in RS-1, a level controller should be designed. The feedback control diagram to regulate the water head in RS-1 is given in Fig. 7. The

Table 4 Water heads in reservoirs. Reservoir no.

Measured head 共m兲

Simulation results 共m兲

1 2 3

4.2 2.15 3.2

4.15 2.07 3.15

Fig. 7. Feedback control diagram of the water supply system.

speed of one of the pumps is manipulated to control the water head in RS-1 because manipulating the speed of the three pumps simultaneously has higher costs than that of the speed of a single pump, and most of the water supply systems work in this manner in Germany 关26兴. This reduces the cost of operation and saves several kW h of electricity per day. N 1 represents the manipulated input variable as the pump speed. The nominal speed is assumed for the other pumps in the same pump station ( N 2 ⫽N 3 ⫽985 rpm) . h t denotes the output variable as the water level at the first reservoir to control, Q t is the total water flow rate pumped by the first pump station, Q di and Q do denote input and output water flow rate disturbances, respectively. h re f is the set point for h t 共4 m兲. The linearized time-domain state space system, given by Eq. 共10兲 using the matrices A, B, and C, is transferred into the s -domain transfer function. C ( s ) denotes the controller, G 1 ( s ) and G 2 ( s ) represent the linearized pumps, and G t ( s ) denotes the linearized reservoir, G p1 ( s ) and G p2 ( s ) are the linearized pipe transfer functions including pipe losses, and h c1 and h c2 are the static heads. The polynomial GH ⬁ controller design method 关27,29兴 can be considered for the controller design, since this method has several advantages and applications 关14,27–31兴 The polynomial solution approach that is based on frequency domain is used to solve the problem. Dynamic weighting functions have direct influences on the controller. The controller is designed to eliminate lowfrequency disturbances and to attenuate highfrequency disturbances and measurement noise.

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For example, the high-frequency roll-off property can be included in the controller to obtain sufficient attenuation of measurement noise. The design algorithm is suitable for on-line adaptive and self-tuning control applications. The GH ⬁ robust control design method has following features in general: • It is developed specifically to tolerate for the modeling errors. • It uses dynamic weighting functions to introduce some controller properties. • The stochastic nature of the system models, such as disturbance and measurement noise models, are available for use in the design. • The technique is easy to use since the algorithms are available in commercial software packages such as MATLAB. • Stability and robustness properties can be predicted from the frequency response analysis. • The technique provides multivariable design for the multi-input multi-output systems. • The stability can be guaranteed for bounded uncertainty. • The closed loop system is guaranteed to be stable even if the open loop system is nonminimum phase or unstable. The method has been applied to several industrial problems such as furnace temperature control problem 关28兴, gas turbine control, beam rolling mill control, submarine motion control, wind turbine power control, flight control systems 关30兴, generator terminal voltage control problem 关31兴. The GH ⬁ cost function to be minimized is

J ⬁ ⫽ 储 ⌽ ␸␸ 共 s 兲 储 ⬁ ,

共13兲

where ⌽ ␸␸ ( s ) is the spectral density of the weighted sum of the error and control signals. In the Laplace transform domain, ␸ ( s ) ⫽ P e e ( s ) ⫹F c u ( s ) . The dynamic weighting functions P e and F c , to achieve good disturbance rejection and attenuation at high frequencies and good tracking, are defined as 关14,28兴

P e⫽

s⫹k c1 P en s⫹k e1 F cn ⫽ , F c⫽ ⫽␤ , P ed k e2 s⫹ ␦ F cd k c21s⫹k c22 共14兲

where k e1 , k e2 , k c1 , k c21 , k c22 , ␤, and ␦ are positive tuning parameters.

469

The optimal controller C ( s ) is

C共 s 兲⫽

F cd G , P ed H

共15兲

where the polynomials G and H are the solution of a couple of diophantine equations in the H ⬁ design algorithm. The H ⬁ controller design algorithm and more information about selection of dynamic weightings and their parameters can be found in Refs. 关14,28兴. The integral action and high-frequency roll-off are required in the controller to eliminate steady-state error, high-frequency disturbances, and measurement noise. The normalized linear models used for the controller design are calculated as

G 1 共 s 兲 ⫽5.151 95,

G 2 共 s 兲 ⫽0.017 59,

G p1 共 s 兲 ⫽44.3225s⫹1.0473, G p2 共 s 兲 ⫽

G t共 s 兲 ⫽

1 , 475s

330.047 . 13 805.04s⫹330.047

The dynamic weighting functions were chosen as

P e共 s 兲 ⫽

1 共 0.01s⫹0.001兲 . F c共 s 兲 ⫽ 1 共 1 000 000s 兲

The optimal H ⬁ controller was calculated as

C共 s 兲 ⫽

0.154s 3 ⫹0.0732s 2 ⫹0.0017s⫹0.18⫻10⫺6 . s 共 s 3 ⫹0.575s 2 ⫹0.0586s⫹0.001兲

The integral control eliminates steady-state error, input, and output disturbances. The highfrequency roll-off eliminates high-frequency noise, measurement disturbance, and unwanted signals 关14,27,28兴. 5.1. Load test The system was simulated using the feedback controller. ⫾5% ( Q do ⫽⫾0.142 m3 s⫺1 ) square wave output flow rate disturbance is applied with a frequency of 0.000 005 Hz at t⫽105 s. ( t ⫽27.7 h) after starting. The applied output disturbance Q do to the compensated system is illustrated in Fig. 8共a兲. Input water flow rates into RS-2 and RS-3 and output water flow rate Q o are illustrated in Fig. 8共b兲. The water input flow rate into RS-2 is

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Fig. 8. 共a兲 Applied water flow rate disturbance Q do (⫽0.145 m3 s⫺1 ) from input of RS-2; 共b兲 water flow rate inputs to RS-2, RS-3, and output water flow rate Q o .

increased for the positive cycle of the flow rate disturbance changes and decreased for the negative cycle of the flow rate disturbance. The output water flow rate Q o and input flow rate into RS-3

has smooth characteristics since the disturbance is applied just from input of RS-2. The steady-state values of all flow rates in Fig. 8共b兲 are equal. This is an expected response. The water heads in the

Fig. 9. Water heads in the reservoir for ⫾5% output load flow rate disturbance Q do (⫽⫾0.1415 m3 s⫺1 ). Solid line ⫽water head in RS-1; dotted line⫽water head in RS-2; dashed line⫽water head in RS-3.

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Fig. 10. Speed change of the variable pump, N 1 for ⫾5% output load flow rate disturbance Q do .

471

Fig. 11. Input disturbance 共solid line兲 and output disturbance 共dotted line兲 sensitivities.

6. Conclusions reservoirs were illustrated in Fig. 9. The output flow rate disturbance caused very small water head changes in RS-1 共solid line兲, since the disturbance was applied at a point after RS-1. However, significant changes occurred in the RS-2 level 共dotted line兲 such that the water head was increased to 3.8 m for the positive cycle of the flow rate disturbance changes and decreased to 0.25 m for the negative cycle of the flow rate disturbance. This is a natural consequence of the fact that the flow rate disturbance was applied just before RS-2. About 0.5 m of peak-to-peak water head change was observed in RS-3. The variation in the speed of the pump in PST-1 that occurred as a result of the square wave output load flow rate disturbance changes is illustrated in Fig. 10. The speed of the variable speed pump was reduced to 972 rpm at steady state when the feedback control was used. The speed increased or decreased according to the applied disturbance cycle. The system performance obtained from the closed loop frequency response was improved by the feedback control so that 38 dB gain margin, 82° phase margin, and 0.0027 rad/sec of operation frequency were obtained. The robustness to disturbances is achieved such that 储 h t /Q do 储 ⬁ ⬍1 and 储 h t /Q di 储 ⬁ ⬍1 should be satisfied 关14,28兴. Fig. 11 illustrates frequency responses of the input and output disturbance sensitivities, 关 h t ( s ) /Q di ( s ) and h t ( s ) /Q do ( s ) ] , respectively. The largest gains of the disturbances were about ⫺26.5 and ⫺26 dB, respectively, that satisfy the disturbance rejection conditions given.

Simulation and control of water supply systems is an efficient means for a water transfer operation to achieve management goals such as flow rate regulation and cost minimization. The study presented here seems to be effective in understanding the behavior of the supply system and to generate a feasible and reasonable solution to improve system operation. A suitable modeling approach is chosen to enhance our understanding of the observed system. This was not primarily to provide definitely accurate representation, but to enrich our understanding of the fundamental behavior of the system to produce a control mechanism. Models of active and passive elements were incorporated in obtaining the behavior of the system. The hydraulic models, in particular, included the nonlinear coupling between flow rates and reservoir heads. The study revealed the dynamics of behavior and interactions among active and passive elements in the water supply system. The type of advanced control design method applied can improve the control of the water supply system by limiting unacceptable variations in flow rate and level. The suggested control structure can be modified to meet possible requirements concerning energy consumption and to handle hard constraints in future works. The case study presented in this paper showed that operation can be improved by using an optimal robust control method to control water transfer operation. The city of Gaziantep water supply system in Turkey was considered. A good approximation for pumps was obtained using appropriate

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manufacturer’s specifications. The whole system was simulated and the results were presented in both time and frequency domains. A robust H ⬁ control approach was used to maintain the flow rates and reservoir heads. This concept has the intrinsic ability to compensate for changes in water disturbance that may occur at any point of the water supply system. The model that was developed allowed us to simulate the different scenerios of the system to compare the results with observed phenomena of the real-time system. The bisection numerical solution method was used to solve nonlinear equations for the friction coefficient numerically using the iterative type of solution.

References 关1兴 Brookshire, D. S. and Whittington, D., Water resources issues in the developing countries. Water Resour. Res. 29, 1883–1888 共1993兲. 关2兴 Mousavi, H. and Ramamurthy, A. S., Optimal design of multi-reservoir systems for water supply. Adv. Water Resour. 23, 613– 624 共2000兲. 关3兴 Lee, Y. W., Bogardi, I., and Kim, J. H., Decision of water supply line under uncertainty. Water Resour. Res. 34, 3371–3379 共2000兲. 关4兴 Gieling, T. H., Janssen, H. J. J., Straten, G. V., and Suurmond, M., Identification and simulated control of greenhouse closed water supply systems. Comput. Electron. Agric. 26, 361–374 共2000兲. 关5兴 Cembrano, G., Wells, G., Quevedo, J., Perez, R., and Argelaguet, R., Optimal control of a water distribution network in a supervisory control system. Control Eng. Pract. 6, 1177–1188 共2000兲. 关6兴 Obradovic, D., Modelling of demand and losses in real-life water distribution systems. Urban Water 2, 131–139 共2000兲. 关7兴 Elbelkacemi, M., Lachhab, S., Limouri, M., Dahhou, B., and Essaid, A., Adaptive control of a water supply system. Control Eng. Pract. 9, 343–349 共2001兲. 关8兴 Tillman, D., Larsen, T. A., Pahl-Wostl, C., and Gujer, W., Modelling the actors in water supply systems. Water Sci. Technol. 39, 203–211 共1999兲. 关9兴 Simons, D. B., Future trends and needs in hydraulics. J. Hydraul. Eng. 118, 1608 –1620 共1992兲. 关10兴 Brdys, M. A. and Ulanicki, B., Operational Control of Water Systems. Prentice-Hall International Ltd., Hertfordshire, U.K., 1994, pp. 142–155. 关11兴 Eker, I˙. and Kara, T., Modelling and simulation of water supply systems for feedback control. 36th Universities Power Engineering Conference 共UPEC’2001兲, Power Utilisation, Part-5C, University of Wales Swansea, 12–14 September, Swansea, U.K., 2001. 关12兴 Ermolin, Y. A., Zats, L. I., and Kajisa, T., Hydraulic reliability index for sewage pumping stations. Urban Water 4, 301–306 共2001兲. 关13兴 Rothe, P. H. and Runstadler, P. W., First order pump surge behaviour. J. Fluids Eng. 100, 459– 465 共1978兲.

关14兴 Eker, I˙. and Kara, T., Control of water supply systems. 36th Universities Power Engineering Conference 共UPEC’2001兲, Power Utilization, Part-8C, University of Wales Swansea, 12–14 September, Swansea, U.K., 2001. 关15兴 Tillman, D., Larsen, T. A., Pahl-Wostl, C., and Gujer, W., Interaction analysis of shakeholders in water supply systems. Water Sci. Technol. 43, 319–326 共2001兲. 关16兴 Fox, J. A., Hydraulic Analysis of Unsteady Flow in Pipe Networks. The MacMillan Press Ltd., London, U.K., 1977, pp. 96 –98. 关17兴 McInnis, D. and Karney, B. W., Transients in distribution networks: Field tests and demand models. J. Hydraul. Eng. 121, 218 –231 共1992兲. 关18兴 Morris, H. M. and Wiggert, J. M., Applied Hydraulics in Engineering. Ronald Press Comp., New York, 1972, pp. 130–180. 关19兴 Araya, W. F. S. and Chaudhry, M. H., Computation of energy dissipation in transient flow. J. Hydraul. Eng. 123, 108 –115 共1977兲. 关20兴 Seborg, D. E., Edgar, T. F., and Mellichamp, D. A., Process Dynamics and Control. Wiley Series in Chemical Engineering. Wiley, New York, 1989, pp. 18 –19. 关21兴 Rosso, M., Schiara, M., and Berlamont, J., Flow stability and friction factor in rough channels. J. Hydraul. Eng. 116, 1109–1118 共1990兲. 关22兴 Douglas, J. F., Gasiorex, J. M., and Swaffield, J. A., Fluid Mechanics. Longman Scientific and Technical-UK Lmt., London, 1985, pp. 92–259. 关23兴 Chapra, S. C. and Canale, R. R., Numerical Methods for Engineers. McGraw-Hill, Inc., Singapore, 1990, pp. 240–246. 关24兴 Dean, R. C. and Young, L. R., The time domain of centrifugal compressor and pump stability and surge. J. Fluids Eng. 99, 53– 63 共1977兲. 关25兴 Giles, R. V., Fluid Mechanics and Hydraulics. Schaum’s Outline Series. McGraw-Hill Inc., Singapore, 1983, pp. 40– 42. 关26兴 Arisoy, A., Design of water systems. Installation Engineering Natural Gas Special Issue. Turkey, 1996, pp. 27–35. 关27兴 Grimble, M. J., H⬁ robust controller for self-tuning control applications, Part 2. Self-tuning and robustness. Int. J. Control 46, 1819–1840 共1987兲. 关28兴 Eker, I˙. and Johnson, M. A., New aspects of cascade and multi-loop process control. Trans. Inst. Chem. Eng., Part A 74, 38 –54 共1996兲. 关29兴 Grimble, M. J., Industrial Control Systems Design. Wiley, Chichester, 2000, pp. 15–50. 关30兴 Grimble, M. J., Robust Industrial Control: Optimal Design Approach for Polynomial Systems. PrenticeHall International 共U.K.兲 Limited, Hertfordshire, 1994, pp. 5– 42, 261. 关31兴 Eker, I˙., Robust multivariable H⬁ automatic voltage regulator design for synchronous generators. 30th Universities Power Engineering Conference 共UPEC’95兲, 5–7 September, The University of Greenwich, London, U.K., pp. 473– 476, 1995.

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Ilyas Eker 共1965兲 received B.Sc. in Electrical and Electronic Engineering 共EEE兲, Middle East Technical University 共METU兲/Turkey in 1988. From 1988 to 1992 he completed his M.Sc. and he worked as research assistant in EEE, Gaziantep University, Turkey. He joined the Industrial Control Centre, University of Strathclyde, Glasgow, U.K. in 1992, where he received his Ph.D. degree in 1995. At present, he currently works as assistant professor in EEE, Gaziantep University/Turkey. His current research interests are self-tuning control, robustness, fuzzy control, linear and nonlinear control, and their applications.

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Tolgay Kara received the B.S. degree in Electrical and Electronics Engineering from the Middle East Technical University at Ankara, Turkey in 1996 and the M.S. degree in Electrical and Electronics Engineering from the University of Gaziantep at Gaziantep, Turkey in 1999. He is currently a Research Assistant in the Department of Electrical and Electronics Engineering, University of Gaziantep, Gaziantep, Turkey. His research interests include system dynamics and modeling, environmental systems, identification and control of nonlinear systems, and adaptive control.