20 CHAPTER 2 MATHEMATICAL MODELLING OF AN ISOLATED HYBRID POWER SYSTEM FOR LFC AND BPC 2.1 INTRODUCTION The technology of the hybrid power system is at an exciting stage of development. Much research effort is in progress on the controls of the system such that the frequency and the voltage of the generated power are maintained within the specified limits. For having high efficiency and minimum wear and tear of the consumer s equipment, these parameters should be maintained at the rated values. The wind turbine generators are the main constituents of the hybrid power systems and are designed to operate in parallel with micro- hydro turbine and the diesel based local grids. The reason is to achieve economic advantages of no fuel consumption by wind turbines, and increase in power capacity to fulfil the growing demand with the locally available, sustainable, clean source of energy. These parameters, frequency and voltage, may get affected considerably because of the fluctuating nature of wind and frequent change in load demands. Therefore, a proper control strategy is essential to maintain these parameters at the desired level. It has been recognised that the study of the response of the power system to disturbances and operational changes is greatly assisted by the use of mathematical models and computer simulations (Freris 1990, Hunter and Elliot 1994. These models are not only of great importance in studying the behaviour, but also in the design and adjustment of controllers
21 (Anderson and Bose 1983. The design and adjustment of controller are based on dynamic stability analysis concerning system oscillations due to small perturbations in load or generation (Tsistsovits and Freris 1983a, 1983b, Tripathy et al. 1993. Elgerd and Fosha (1970, Hadi Saadat (1999 and Kundur (2008 discussed the state space modelling of load frequency control and voltage control loops. Generally it is assumed that the load frequency control and excitation control are non-interactive, as small change in frequency is due to small change in active power while small change in voltage is because of the small change in reactive power. Furthermore, excitation control is fast acting major time constant contributed by the generator field winding, while power frequency control is slow acting due to large inertia of the prime mover and of the generator. The later time constant is much larger than that of the generator field. Therefore the load frequency control loop and the excitation voltage control loop are assumed to be decoupled (Elgerd and Fosha 1970. In a hybrid power system, IG is generally used to produce the electrical power and connected to a bus which is linked with the diesel genset terminals. The mathematical model is developed for isolated power hybrid system by considering the parallel operation wind, and diesel and micro-hydro power generation unit. Wind and micro-hydro use induction generator, but diesel uses synchronous generators for electric power conversion (Bansal 2003c. The configuration of isolated wind-micro hydrodiesel-hybrid system is shown in Figure 2.1. To describe the dynamic behaviour of a system, these different constituent of the hybrid system may require detailed modelling. Simple subsystem mathematical models are incorporated in the state space modelling of the hybrid system in the present work.
22 Figure 2.1 Configuration of an isolated wind-micro hydro-diesel hybrid system 2.2 STATE SPACE MODELLING OF HYBRID SYSTEM The function of the LFC is to eliminate a mismatch created either by small real power-load or input wind power changes. The input power to the wind power generating unit is not controllable in the sense of generation control, but a supplementary controller known as LFC can control the generation of the diesel unit and thereby of the system. The small real power mismatch causes perturbations about the nominal operating point and the system dynamics may therefore be described by a set of linear differential equations (Mohadjer and Johnson 1984, Kamva 1990, Tripathy et al. 1984, 1992, Kariniotakis and Stavrakakis 1995. The transfer function block diagram of the system consists of parallel operation of a wind power machine, diesel electric power generating unit and a micro hydro unit. The diesel power generation system is equipped with mechanical speed governor to regulate the speed of the diesel engine
23 (Bhatti et al. 1995. The wind power generation system is equipped with blade pitch control mechanism to regulate the power output of wind turbine if it exceeds the maximum power set point (Hinrichsen and Nolan 1982, Kamva 1990, Kariniotakis and Stavrakakis 1995. A micro hydro generating unit is added in parallel with the hybrid system, where water streams are abundantly available and the plus point is that the flow in these streams is fairly constant throughout the year. Therefore, input power variations are not aggravated by the micro hydro addition. 2.2.1 Diesel Generation System Modelling An exact forecast of real power demand is impossible due to random changes in the load and therefore an imbalance occurs between the real power generation and the load demand (plus losses. This causes kinetic energy of rotation to be either added to or taken from the generating units (generator shaft either speed up or slow down and the frequency of the system varies as a result (Elgerd and Fosha 1970. Whenever the real power demand changes, a frequency change occurs. This frequency error is amplified, mixed and changed to a command signal which is sent to turbine governor. The governor operates to restore the balance between the input and output by changing the turbine output. The Load Frequency Controller (supplementary controller of the diesel generating unit sends the command signal to the speed-gear changer of the diesel engine in response to the frequency error signal FS. It uses a system frequency deviation of the power system as a feedback input. For mathematical modelling, the transfer function block diagram of the diesel generating unit with LFC is shown in Figure 2.2.
24 Figure 2.2 Transfer function model of diesel generating unit The speed-gear changer must not act too fast, as it will cause wear and tear of the engine and, also should not act too slow, as it will deteriorate the system performance. Therefore, an optimum load frequency controller is required for satisfactory operation of the system. Conventional PI controller is used for LFC and its transfer function block is included in the diesel generating unit transfer function model. Optimum selection of the gains of the controller is obtained using Integral Square Error (ISE technique. This conventional PI controller is simulated and compared with the proposed intelligent controllers in further chapters for investigating the hybrid system for various load disturbances and wind input power disturbances. The transfer function of the mechanical speed-governing system in diesel unit can be written in partial fraction form as K D(1 ST d1 K1 K2 (1 ST (1 ST (1 ST (1 ST d2 d3 d2 d3 (2.1 where K 1 K D(1 ST d1 K D(Td2 T d1 (1 ST (T T d3 1 S d2 d3 T d 2 (2.2
25 K 2 K D(1 ST d1 K D (Td3 T d1 (1 ST (T T d2 1 S d3 d2 T d3 (2.3 T, T and T are the time constants of the speed governing mechanism, K is the part of power supplied by diesel power generation to the load. Equation (2.1 can be written in terms of the canonical state variables X ED11 and X ED21, ( P CD (s- F S (s]= X ED11 (s+ X ED21 (s (2.4 where R is the speed regulation due to the governor speed action. From Equation (2.1 and Equation (2.4, we get X ED11 (s = [ P CD(s - F S (s] (2.5 and X ED21 (s = [ P CD(s - F S (s] (2.6 Therefore, the state differential equations of the mechanical speed governing mechanism are X ED11 = - X ED11 - F S + P CD (2.7 X ED21 = - X ED21 - F S + P CD (2.8 The transfer function equation for the change in diesel power generation P GD, can be written in terms of the state variables as P GD (s = [ X ED11(s + X ED21 (s] (2.9
26 where T d4 is the time constant of the diesel power generation system in sec. From Equation (2.9, the state differential equation can be written as P GD = - P GD + X ED11 + X ED21 (2.10 2.2.2 Wind Turbine Power Generation System Modelling The intermittent wind power may affect the power quality of the hybrid system. The deviations in generating power and frequency fluctuations are eliminated by blade pitch control mechanism, which continuously monitors the wind turbine speed and acts accordingly in an active feedback control system added to the turbine. The transfer function block diagram of the wind-turbine generation system with blade pitch controller is shown in Figure 2.3. A Supplementary controller, which is shown as PI controller in transfer function block model acts as a Blade pitch controller. Figure 2.3 Transfer function model of wind generating unit
27 The transfer function equation for the wind power generation system is F T (s = [ - P GW(s + P IW (s + P CW (s + K F T (s ] (2.11 and P GW (s = K [ F T (s F S (s ] (2.12 where T is the time constant of the wind-turbine power generation system in sec. K is the part of power supplied by wind-power generation to load and is a function of slip. K co-efficient that depends on the slope of C p, curve of the windturbine (Tsistsovits 1983b. From Equation (2.11 and Equation (2.12 the state differential equation can be written as F T = - F T + F S + P IW + P CW (2.13 The real power load change P L or change in wind power generation P GW experienced by the hybrid system deviates the power generation from a specified level. The power generation of the hybrid system can be maintained by the diesel engine controller by changing its power generation by an amount P GD. The net surplus power P I will be absorbed by the system either by increasing the kinetic energy of the system or by increased load consumption. The surplus power P I = ( P GD + P GW - P L (2.14
28 The total kinetic energy varies as the square of the speed or frequency of the system, therefore, the transfer function equation of the system subject to change in real power load or input wind power can be written as F S = [ P GD (s + P GW (s P L (s ] (2.15 where K = and D = (2.16 T p = (2.17 H = P.U. Inertia constant F = Nominal system frequency D = Damping coefficient The state differential equation is F S = - F S + P GD + F T - P L (2.18 The combined transfer function of different blocks of the blade pitch control mechanism is given by, P CU (s = P CW (s (2.19 where T and T are the time constants of the hydraulic blade pitch actuator in sec. T is the time constant of the data fit pitch response unit K and K are gain constants of the hydraulic pitch actuator.
29 K is the gain constant of the data fit pitch response unit K is the blade characteristic constant Equation (2.19 can be rewritten as K T + P CU (s = P CW (s (2.20 In terms of intermediate state variables, Equation (2.20 can be expressed as P CW (s = [K P C1 (s + K T P C2 (S ] (2.21 P C1 (s = P C2 (S (2.22 P C2 (s = P CU (s (2.23 The state differential equations for the transfer function Equation (2.21, Equation (2.22 and Equation (2.23 are given by P CW = - P CW + P C1 + P C2 (2.24 P C1 = - P C1 + (1- T P C2 (2.25 P C2 = - P C2 + P CU (2.26 2.2.3 Micro Hydro Turbine Generation System Modelling A micro hydro power unit is added in parallel with the above mentioned wind-diesel system. The transfer function block diaphragm of the micro hydro turbine generating system is shown in Figure 2.4.
30 Figure 2.4 Transfer function model of micro hydro generating unit The transfer function equations for the micro hydro power generation system are X H = - X H + P IH P GH (2.27 P GH = Kgh ( F H F S (2.28 F H = X H 2( P IH P GH (2.29 get Solving Equation (2.27, Equation (2.28 and Equation (2.29, we X H = - X H + F S + P IH (2.30 and P GH = X H - ( F S - P IH (2.31 where Kgh is the part of the power supplied by micro hydro unit to the load and is a function of the slip.
31 Th is the nominal starting time of water in penstock. The Equation (2.18 is modified by the addition of micro hydro unit in parallel and is re-written as F S = - + K + ( F S + P GD + K F T + ( X H - P L ( P IH (2.32 2.3 CONVENTIONAL PI CONTROLLER FOR LFC Conventional PI controllers are included as supplementary controller in the transfer function block diagram model of the hybrid system for LFC and BPC. The input power to the renewable sources of power generation is fluctuating, particularly in case of wind by nature. Therefore, there are two reasons that the diesel power generation system has to be equipped with the Load Frequency Controller. One is that the input power to the renewable sources of power generation is uncontrollable in nature; therefore, raising and lowering of input power as per demand change may not be easily possible. The other is that the maximum power available may be utilized to meet the partial load demand and thus achieving saving in fuel. To regulate the system frequency, the diesel engine is equipped with mechanical speed governor. The speed governor regulates the frequency of the system when mismatch between generation and load occurs. To accomplish this we must manipulate the speed changer by Load Frequency Controller in accordance with some suitable control strategy and the system frequency change may be considered as the feedback signal (Jaleeli et al. 1992. The objective of this Thesis is to investigate the problem of frequency control of an isolated hybrid power system using intelligent control techniques and their performance is compared with conventional PI controller
32 designed here. The parameter of the controllers is tuned for optimum performance of the system subjected to step load change and step wind input power change. The continuous time dynamic behaviour of the load frequency control system is modelled by a set of state vector differential equations. X = AX + BU+ P (2.33 where X, U and P are the state, control and disturbance vectors respectively. A, B and are real constant matrices of the appropriate dimensions associated with the above vectors. In this chapter, the Load frequency controller of proportional plus integral type is designed for the hybrid system so as to achieve zero steady state error in frequency and is used for comparing the performance of the hybrid system with intelligent controllers in further chapters. The state vector in Equation (2.33 is to be augmented by two additional state variables X n+1 and X n+2 defined as X n+1 = F dt (2.34 X n+2 = F dt (2.35 Therefore the additional state differential equations are X n+1 = F (2.36 X n+2 = F (2.37 Equation (2.36 and Equation (2.37 can be written in matrix form as X X n 1 n 2 [A ]X 1 (2.38
33 Now the state vector in Equation (2.33 is modified by including the state variables defined in Equations (2.34 and Equation (2.35. The augmented set of differential equations can be written as ˆ A 0 ˆ B A1 0 0 0 1 X X U P 2 3 4 (2.39 where O, O, O and O are null matrices of appropriate dimensions. The control vector U can be expressed in terms of the augmented state vector as U=[H] ˆX (2.40 The final augmented set of differential equations can be written as Xˆ AX ˆ ˆ ˆ P ˆ (2.41 where  = A O A O + B O [H] (2.42 and = O (2.43 2.4 PARAMETER OPTIMIZATION The optimization is a theoretical approach of minimizing or maximizing a performance index by the variation of system variables, under certain constraints. An attempt is made to determine the optimum set of PI controller parameters. The performance index used in the parameter
34 optimization on problem of frequency control of the hybrid power system is the Integral Square Error (ISE criterion (Elgerd 1982, which is PI = ( F 2 dt (2.44 By focussing on the square of the error function, the performance index penalizes both positive and negative values of the error. Therefore, minimization of PI with respect to known parameters will provide the best frequency performance of the system. The PI controller gain parameters, Kpp and Kpi of blade pitch controller, and Kdp and Kdi of Load Frequency controller, respectively are optimized for wind-micro hydro- diesel hybrid power system. The values of Kpp and Kpi are optimized when the system is subjected to change in wind input power, the reason being that the blade pitch controller becomes active when input wind power exceeds the maximum power set point. The values of Kdp and Kdi are optimized when the system is subjected to change in load as the generation can be raised or lowered to meet the demand. For a particular value of Kdi, the value of PI is evaluated over a range of values of Kdp until a minimum value occurs for PI. The value of Kdp obtained corresponding to minimum PI is the first optimum value. Using the optimum value of Kdp again, the value of PI is evaluated over a range of values of Kdi until a minimum occurs for PI. These optimum values of parameters Kdp and Kdi of Load frequency controller are used for finding the optimum values of parameters Kpp and Kpi of the blade pitch controller. This procedure is repeated until the variation in optimum value of the parameters is negligible. Figure 2.5 shows the transfer function block diagram of wind-micro hydro-diesel hybrid power system for LFC and BPC.
35 Figure 2.5 Transfer function model of an isolated wind-micro hydrodiesel hybrid power system as The state vector without supplementary PI controller can be written X T = [ F P X X F P P P X ] (2.45 The control and disturbance vectors are given by U T = [ P P ] (2.46 P T = [ P P P ] (2.47
36 The matrices [A], [B] and [ ] for this system are given in Appendix 2 and the system data chosen for computation is given in Appendix 1. The matrix [H] required for the augmented set of differential equations defined in Equation (2.40 is given by Kdp 0 0 0 0 0 0 0 0 Kdi 0 T [H] KigKpp 0 0 0 KigKpp 0 0 0 0 KigKpi KigKpi (2.48 The parameters Kpp, Kpi, Kdp and Kdi of the controllers are optimized and optimum values are given in Table 2.1. Using the optimum gain parameters, the transfer function model of the hybrid system for LFC is simulated with conventional PI controller. The performance of the hybrid system with conventional PI controller is used for analysis and comparison in this thesis work. Table 2.1 Optimum values of PI controller parameters for LFC and BPC Kdp Kdi Kpp Kpi 282.8 19.5 81.5 1.75 The proposed work concentrates on the design of intelligent controllers to damp out the frequency oscillations and to maintain the power generation of the hybrid system subjected to step load changes and step wind input power disturbances. The control objectives is not only to maintain the scheduled frequency following a disturbance but also have a minimum settling time and minimum deviations of the swings.
37 2.5 SUMMARY In this chapter, the system configuration of the wind-micro hydrodiesel hybrid system has been presented. The mathematical model for LFC and BPC of an isolated wind-micro hydro-diesel hybrid power system has been presented. State space equations have been derived for wind- micro hydro- diesel hybrid system. The optimum values of the gain parameters of conventional PI controller have been evaluated using the ISE technique.