35 CHAPER-3 MODELING OF INERCONNECED AC-DC POWER SYSEMS 3. INRODUCION he modeling of power system is essential before entering into the design of Load Frequency Control (LFC). In this chapter, a model of two-area and three area reheat thermal power system are developed. he load frequency control is concerned with the evaluation of response of the system for small changes (Allen 984). In order to derive the transfer function of the interconnected power system, it is necessary to develop a linear model. A linear model can be developed by making the suitable approximations and assumptions. he two most common mathematical models are transfer function model and state space model. he mathematical equations of the interconnected power system are described below. 3.2 MODELING OF POWER GENERAING SYSEM 3.2. Governor Model Governors are the units that are used in power systems to sense the frequency bias caused by the load change and cancel it by varying the inputs of the turbines. he schematic diagram of a speed governing unit is shown in Figure 3.. If without load reference, when the load change occurs, part of the change will be compensated by the valve/gate adjustment while the rest of the change is represented in the form of frequency deviation (Prabha undur
36 2008). he goal of LFC is to regulate frequency deviation in the presence of varying active power load. he measured rotor speed r is compared with reference speed 0. he error signal or f is amplified and integrated to produce a control signal X e which actuates the main steam supply valve. Where R is the speed regulation, g is the time constant of the governor and g is the governor gain. hus, the load reference set point can be used to adjust the valve/gate positions so that the load change is cancelled by the power generation rather than resulting in a frequency deviation. Steam/ Water Valve/Gate urbine P d P g - + Generator X e s g r - 0 + Ref. set point U - + R Figure 3. Schematic diagram of speed governing unit he reduced form of Figure 3. is shown in Figure3.2.he Laplace transform representation of the block diagram in Figure 3.2 is given by (Prabha undur 2008) g X e(s) U(s) F(s) s g R (3.)
37 F(s) R U(s) + - g +s g X e (s) Figure 3.2 Reduced block diagram of speed governing unit 3.2.2 urbine Model A turbine unit in power systems is used to transform the natural energy, such as the energy from steam or water, into mechanical power (P m ) that is supplied to the generator. In LFC model, there are two kinds of commonly used turbines: non-reheat and reheat turbines, all of which can be modelled by transfer functions (Elgerd 2005). he first stage of turbine is named as non-reheat turbine. From a response point of view this turbine is quite simple. Upon opening the control valve the steam flow will not reach the turbine cylinder instantaneously. A time delay (denoted by t ) occurs between switching the valve and producing the turbine torque. he transfer function of the non-reheat turbine is represented as where G NR P(s) t t (s) X (s) s e t X e is the incremental change in valve/gate position. P t is the incremental change turbine power. t is the gain of turbine. t is the turbine time constant. (3.2)
38 he representation of non-reheat turbine model is shown in Figure 3.3. he turbine time constant t assumes values in the typical range 0. 0.5 seconds (Elgerd 2005). X e (s) t s t P t (s) Figure 3.3 Block diagram representation of Non-reheat turbine model Consider the two stage turbine shown in Figure 3.4. Assume that the High Pressure (HP) and Low Pressure (LP) stages have the per-unit megawatt ratings r and - r respectively. Steam Valve X e HP Stage LP Stage P t Reheater Figure 3.4 wo stage steam turbine Upon changing the control valve with the increment X e the HP turbine contributes the power component t P,HP r X e MW st (3.3) he reheater represents a delay r and the LP stage thus contributes t P,LP ( r) X e MW ( s t) ( s r) (3.4)
39 he total power is obtained by adding the Equation 3.3 and 3.4. One thus obtain the overall turbine transfer function. G P,HP P,LP t s r r X s s e t r (3.5) he time constant r assumes values in the typical range 4 0 seconds making the reheat turbine dynamically very slow (Elgerd 2005). he representation of reheat turbine model is shown in Figure 3.5. X e (s) t s t s r r s r P t (s) Figure 3.5 Block diagram representation of reheat turbine model 3.2.3 Generator Load Model A generator unit in power systems converts the mechanical power received from the turbine into electrical power. But for LFC, one focus on the rotor speed output (frequency of the power systems) of the generator instead of the energy transformation. Since electrical power is hard to store in large amounts, the balance has to be maintained between the generated power and the load demand (Nagrath 2005). he representation of the generator load model is shown in Figure 3.6. he system is originally running in its normal state with complete power balance is P G = P D. he frequency is at normal value f. By connecting additional load to the system, the load demand is increased by P D. he generator increases its output P G to match the new load that is given by
40 P G = P D (3.6) where P G = P, incremental turbine power output (assuming generator incremental loss to be negligible) and P D is the load increment. From the power balance equation (Nagrath 2005), one have 2HPr d P G P D f D f f dt (3.7) Dividing throughout by P r and rearranging, one get 2Hd P G(pu) P D(pu) f D(pu) f f dt (3.8) where D = P D / f = Damping coefficient in MW/Hz. aking the Laplace transformation, one can write F(S) as F(s) P (s) P (s) G D D (3.9) 2H f s PS F(s) P G(s) P D(s) sps (3.0) PS 2H Df Power system time constant PS D Power system gain
4 P D (s) P g (s) + - PS F(s) s PS Figure 3.6 Block diagram representation of generator load model 3.2.4 ie-line Model An extended power system can be divided into a number of load frequency control areas interconnected by means of tie lines. Without loss of generality we shall consider a two area case connected by a single tie line as shown in Figure 3.7. Control Area- ie line Control Area-2 Figure 3.7 ie line interconnected control area he objective of an interconnected power system is to regulate the frequency deviation between the control areas. Based on the demand, the power interchange between the areas takes place. he power out of area is given by V V P sin ( - ) (3.) 2 tie, 2 X2 where, 2 = Power angles of equivalent machines of the two areas.
42 can be expressed as For incremental changes in and 2, the incremental tie line power Ptie, 2 ( - 2 ) (3.2) where V V cos ( - ) Synchronizing power coefficient 2 2 2 PX r 2 Since the incremental frequency f is related to the phase angle of deviation, 2 f (3.3) f d 2 2 dt (3.4) i.e., f d 2 dt (3.5) 2f dt (3.6) Substitute the Equation 3.6 in Equation 3.2, one gets (3.7) P 2 ( f dt - f dt ) tie, 2 2 where f and f 2 are incremental frequency changes of areas and 2 respectively. aking the Laplace transformation of the above equations, the signal P tie (s) is obtained as 22 P tie, (s) F (s) - F 2(s) (3.8) s
43 In an interconnected power system, different areas are connected with each other via tie-lines. he tie-line connections can be modelled as shown in Figure 3.8. Considering i th control area, the change in tie line power can be expressed as 2ij P tie,ij (s) F i(s) - F j(s) s (3.9) F i - + 2 ij s P tie,ij F j Figure 3.8 Block diagram of the tie line model between the two areas 3.3 MODELING OF WO AREA AC-DC REHEA HERMAL POWER SYSEM 3.3. ransfer Function Model he transfer function model of a two-area thermal reheat power system using simplified model is shown in Figure 3.9. A two-area system may be represented for the load frequency control in terms of its components like governor system, turbine, generator, load and tie line between two areas. It is convenient to obtain the dynamic model in state variable form from the transfer function model (Nagrath 2005, Issarachai 2002).
44 Figure 3.9 Modelling of HVDC power modulator in the two area thermal reheat with parallel AC-DC power system 3.3.2 State Space Model he state space model can be developed from the transfer function model shown in Figure 3.9. he following equations can be written by inspection. F (s) [ P (s) P (s) P (s) ] PS g d tie sps (3.20) s [ P (s)] r r P g(s) r sr (3.2)
45 P (s) t r st [ X (s)] e (3.22) X (s) [u (s) F (s) ] g sg R (3.23) u (s) [ F (s) P tie(s)] (3.24) s 2 2 P AC(s) [ F (s) F 2(s)] (3.25) s P (s) [ (s) (s)] DC DC F PAC sdc (3.26) F (s) [ P (s) P (s) P (s) ] PS2 2 g2 d2 tie2 sps2 (3.27) s [ P (s)] r2 r2 P g2(s) r2 sr2 (3.28) t2 P r2(s) [ X e2(s)] st2 (3.29) X (s) [u (s) F (s) ] g2 2 2 2 sg2 R2 (3.30) u 2(s) [ 2F 2(s) P tie2(s)] (3.3) s
46 aking inverse Laplace transforms for the above equations F [ PS! Pg F PSPd PS P tie ] (3.32) PS (3.33) r t r Pg Pg Pr Pr- Xe r r t t (3.34) t Pr Xe Pr t t Xe u F X (3.35) g g e g Rg g u F P (3.36) tie P 2 F 2 F (3.37) AC 2 2 2 (3.38) F PAC PDC DC DC PDC DC DC DC F 2 [PS2Pg2 F2 PS2Pd2 PS2 P tie2 ] (3.39) PS2 (3.40) r2 t2 r2 Pg2 Pg2 Pr2 Pr2 - Xe2 r2 r2 t2 t2 (3.4) t2 Pr2 Xe2 Pr2 t2 t2
47 X u F X (3.42) g2 g2 e2 2 2 e2 g2 R2g2 g2 u F P (3.43) 2 2 2 tie2 he above equations are put in the matrix form x A x B u E d (3.44) where the system state vector [x] is x F P g P r X e u P AC P DC F 2 P g2 P r2 X e2 u2 (3.45) System control input [u]is u u u 2 (3.46) System disturbance input vector is [d] d P P d d2 (3.47)
he system matrix A is written as ps 0 0 0 0 0 0 0 0 0 0 ps ps r t r 0 0 0 0 0 0 0 0 0 r r t r t 0 0 0 0 0 0 0 0 0 0 t t g - g 0 0 0 0 0 0 0 0 0 Rg g g 0 0 0 0 0 0 0 0 0 0 0 22 0 0 0 0 0 0-22 0 0 0 0 A - 0 0 0 0 0 0 0 0 0 0 0 DC ps2 0 0 0 0 0 0 0 0 0 0 ps2 ps2 r2 t2r2 0 0 0 0 0 0 0 0 0 r2 r2 t2 r2 t2 0 0 0 0 0 0 0 0 0 0 t2 t2 g2 - g2 0 0 0 0 0 0 0 0 0 R2g2 g2 g2 0 0 0 0 0 0 0 2 0 0 0 0 48
49 0 0 - Ps 0 Ps 0 0 0 0 0 0 0 0 g 0 g 0 0 0 0 0 0 0 0 0 0 B = E = 0 0 0 0 0 - Ps2 0 0 Ps2 0 0 0 0 0 0 0 0 0 0 0 g2 g2 0 0 0 0 where, A - System matrix (n x n) B - Input matrix (n x m) E - Disturbance matrix (n x p) x - State vector (n) u - Control vector (m) d - Disturbance vector (p)
50 3.4 MODELING OF HREE AREA AC-DC REHEA HERMAL POWER SYSEMS 3.4. ransfer Function Model Modern control theory is applied to design an optimal load frequency controller for a three area interconnected AC-DC reheat thermal power systems. he transfer function model of a three area AC-DC interconnected thermal reheat power system using simplified model is shown in Figure 3.0. It is conveniently assumed that each control area can be represented by an equivalent governor system, turbine, generator, load and tie line between areas to obtain the dynamic model in state variable form from the transfer function model. 3.4.2 State Space Model he state space model can be developed from the transfer function model shown in Figure 3.0. he following equations can be written by inspection. F (s) [ P (s) P (s) P (s) ] PS g d tie sps (3.48) s [ P (s)] r r P g(s) r sr (3.49)
5 Figure 3.0 Modelling of three area interconnected thermal reheat with parallel AC-DC power system P (s) t r st [ X (s)] e (3.50) X (s) [u (s) F (s) ] g sg R (3.5)
52 u (s) [ F (s) P tie2(s)] (3.52) s 2 2 P AC(s) [ F (s) F 2(s)] (3.53) s P (s) [ (s) (s)] DC DC F PAC sdc (3.54) F (s) [ P (s) P (s) P (s) ] PS2 2 g2 d2 tie2 sps2 (3.55) s [ P (s)] r2 r2 P g2(s) r2 sr2 (3.56) t2 P r2(s) [ X e2(s)] st2 (3.57) X (s) [u (s) F (s) ] g2 2 2 2 sg2 R2 (3.58) u 2(s) [ 2F 2(s) P tie2(s)] (3.59) s 2 23 P AC2(s) [ F 2(s) F 3(s)] (3.60) s F (s) [ P (s) P (s) P (s) ] PS3 3 g3 d3 tie32 sps3 (3.6) s [ P (s)] r3 r3 P g3(s) r3 sr3 (3.62)
53 P (s) t3 r3 st3 [ X (s)] e3 (3.63) X (s) [u (s) F (s) ] g3 3 3 3 sg3 R3 (3.64) u 3(s) [ 3 F 3(s) P tie32(s)] (3.65) s aking inverse Laplace transforms for the above equations F [ PS Pg F PSPd PS P tie2 ] (3.66) PS (3.67) r t r Pg Pg Pr Pr- Xe r r t t (3.68) t Pr Xe Pr t t X u F X (3.69) g g e e g Rg g u F P (3.70) tie2 P 2 F 2 F (3.7) AC 2 2 2 (3.72) F PAC PDC DC DC PDC DC DC DC
54 F 2 [PS2Pg2 F2 PS2Pd2 PS2 P tie23 ] (3.73) PS2 (3.74) r2 t2 r2 Pg2 Pg2 Pr2 Pr2 - Xe2 r2 r2 t2 t2 (3.75) t2 Pr2 Xe2 Pr2 t2 t2 X u F X (3.76) g2 g2 e2 2 2 e2 g2 R2g2 g2 u F P (3.77) 2 2 2 tie2 P 2 F 2 F (3.78) AC2 23 2 23 3 F 3 [ PS 3Pg3 F3 PS 3Pd3 PS3 P tie 32 ] (3.79) PS3 (3.80) r3 t3 r3 Pg2 Pg3 Pr3 Pr3 - Xe3 r3 r3 t3 t3 (3.8) t3 Pr3 Xe3 Pr3 t3 t3 X u F X (3.82) g3 g3 e3 3 3 e3 g3 R3g3 g3 u F P (3.83) 3 3 3 tie32
55 In the optimal control scheme the control inputs u, u 2 and u 3 are generated by means of feedbacks from all the states with feedback constants to be determined in accordance with an optimality criterion. matrix form he above equations can be organized in the following vector x A x B u E d (3.84) where the system state vector [x] is [x] = [F P g P X e u P AC P DC F 2 P g2 P r2 X e2 u 2 P AC2 F 3 P g3 P r3 X e3 u 3 ] (3.85) System control input is u [u] u 2 u 3 (3.86) System disturbance input vector is P [d] P P d d2 d3 (3.87)
A ps ps ps r r t r t t g - R 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 r t r 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 t 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g g g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0-22 0 0 0 0 0 0 0 0 0 0-0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 DC ps2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ps2 ps2 r2 t2 r2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 t2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 t2 2 g2 g2 g2 2 r2 r2 t2 r2 g2 - g2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 R 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0-2 0 0 0 0 0 0 0 0 0 0 23 23 t2 ps3 0 0 0 0 0 0 0 0 0 0 r3 t3 r3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 t3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ps3 ps3 t3 3 g3 g3 g3 3 r3 r3 t3 r3 - R 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 t3 g3 56
57 0 0 0 - Ps 0 0 Ps 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B = E = 0 0 0 0 0 0 0 - Ps2 0 0 0 0 Ps2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g2 0 g2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - Ps3 Ps3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g3 g3 0 0 0 0 0 0 where, A - System matrix (n x n) B - Input matrix (n x m) E - Disturbance matrix (n x p)
58 x - State vector (n) u - Control vector (m) d - Disturbance vector (p) he prime objective is to minimize the ACE which stabilizes the system frequency for a sudden load disturbance. he objective function of the load frequency controller (Nagrath 2005, Vaibhav Donde 200) is given by, ACE i Fi Ptie, i (3.88) where, i - Number of areas, F - Change in frequency, P tie, i - Change in tie-line power - Biasing factor (<= ) 3.5 SUMMARY he problem of controlling the power output of the generators of a closely knit electric area is considered so as to maintain the scheduled frequency. he boundaries of a control area will generally coincide with that of an individual electricity board company. he load frequency control is one of the mechanisms to maintain or restore the frequency within the specified limit. o understand the load frequency control problem, the modelling of governor, turbine, generator load and tie-line were designed. Based on the concept, the modelling of two area interconnected AC-DC thermal reheat power system was developed and it will be extended to the three area interconnected AC-DC thermal reheat power systems.
59 Load frequency control with integral controller achieves zero steady state frequency error and a fast dynamic response, but it exercises no control over the relative loadings of the AC-DC tie line interconnected multi area power systems.