CHAPTER 2 MODELING OF FACTS DEVICES FOR POWER SYSTEM STEADY STATE OPERATIONS

Transcription

1 19 CHAPTER 2 MODELING OF FACTS DEVICES FOR POWER SYSTEM STEADY STATE OPERATIONS 2.1 INTRODUCTION The electricity supply industry is undergoing a profound transformation worldwide. Maret forces, scarcer natural resources and an ever-increasing demand for electricity are some of the drivers responsible for such an unprecedented change. Against this bacground of rapid evolution, the expansion programmes of many utilities are being thwarted by a variety of well-founded, environmental, land use and regulatory pressures that prevent the licensing and building of new transmission lines and electricity generating units. An in-depth analysis of the options available for maximizing existing transmission assets, with high levels of reliability and stability, has pointed in the direction of power electronics. There is general agreement that novel power electronics equipment and techniques are potential substitutes for conventional solutions, which are normally based on electromechanical technologies that have slow response times and high maintenance costs. The power electronics options we have, are HVDC transmission systems and FACTS (Acha et al 2004). Until recently, active and reactive power control in AC transmission networs was exercised by carefully adjusting transmission line impedances, as well as regulating terminal voltages by generator excitation control and by transformer tap changes. At times, series and shunt

2 20 impedances were employed to effectively change line impedances. FACTS technology is most interesting for transmission planners because it opens up new opportunities for controlling power and enhancing the usable capacity of present, as well as new and upgraded, lines. The possibility that current through a line can be controlled at a reasonable cost enables a large potential of increasing the capacity of existing lines with large conductors and use of one of the FACTS controllers, to enable corresponding power to flow through such lines under normal and contingency conditions. By providing added flexibility, FACTS controllers can enable a line to carry power closer to its thermal rating. It must be emphasized that FACTS is an enabling technology, not a one to one substitute for mechanical switches. In its most general expression, the FACTS concept is based on the substantial incorporation of power electronic devices and methods into the high-voltage side of the networ, to mae it electronically controllable. Comprehensive modeling of most popular FACTS devices namely SVC, TCSC, STATCOM and UPFC for power flow studies are carried out. The effectiveness of modeling and convergence is tested on a five bus system without any FACTS devices and further analyzed it with different FACTS devices. The Newton-Raphson method is used to solve the nonlinear power flow equation. The wor is also extended for IEEE 30 bus system. 2.2 POWER FLOW IN A MESHED SYSTEM To understand the free flow of power, consider a very simplified case in which generators at two different sites are sending power to a load centre through a networ consisting of three lines in a meshed connection (Figure 2.1).

3 21 Figure 2.1 Power flow in a mesh networ: (a) system diagram (b) system diagram with Thyristor-controlled series capacitor in line AC (c) system diagram with Thyristorcontrolled series reactor in line BC Let the lines AB, BC and AC have continuous ratings of 1000 MW, 1250 MW and 2000 MW. If one of these generators is generating 2000 MW and the other generates 1000 MW, then a total of 3000 MW would be delivered to the load centre. For the impedances shown, the three lines would carry 600, 1600 and 1400 MW respectively, as shown in Figure 2.1(a). Such a situation would overload line BC and therefore generation would have to be decreased at B and increased at A, in order to meet the load without overloading line BC. Power, in short, flows in accordance with transmission line impedances (which are 90% inductive). If however a capacitor whose reactance is -j5 at the synchronous frequency is inserted in one line (Figure 2.1(b)), it reduces the line impedance from 10 to 5, so that power flow

4 22 through the lines AB, BC and AC will be 250, 1250 and 1750 MW respectively (Hingorani and Gyugyi 2000). A series capacitor in a line may lead to sub-synchronous resonance (typically at 10 to 50 Hz for a 60 Hz system). If such resonance persists it will soon damage the turbine and shaft. If all or part of the series capacitor is thyristor controlled it can be modulated to rapidly damp, any sub synchronous resonance condition and also low frequency oscillations in power flow. This enhances easy operation within steady state conditions thus improving stability of the networ. Similar results can be obtained by increasing the impedance of one of the lines in the same meshed configuration by inserting a +j7 reactor (inductor) in series with line BC (Figure 2.1(c)). 2.3 BENEFITS OF FACTS DEVICES Edris et al (1997) proposed terms and definitions for Flexible AC Transmission System (FACTS) as: Alternating-current transmission systems incorporating power electronics-based and other static controllers to enhance controllability and increase power transfer capability. The main objective of FACTS devices is to replace the existing slow acting mechanical controls required to react to the changing system conditions by rather fast acting electronic controls. These devices can dynamically control line impedance, line voltage, active power flow and reactive power and when storage becomes economically viable; they can supply and absorb active power as well. All this can be done in high speed. The implementation of FACTS devices requires technology for high power electronics with its real time operating control. With the use of fast acting controls, the power system security margins could be enhanced by utilizing the full capacity of transmission lines maintaining the quality and reliability of power supply. Generation patterns that lead to heavy line flows result in

5 23 higher losses leading to reduced security and stability margin are economically undesirable. Further, transmission constraints mae certain combinations of generation and demand unviable, due to the potential line outages. In such situations, FACTS devices may be used to improve system performance by controlling the power flows in transmission lines. By using reliable, high speed power electronic controllers, technology offers utilities the following opportunities for increased efficiency: Greater control of power, so that it flows on the prescribed transmission routes. Loading of transmission line to levels nearer their thermal limits. Greater ability to transfer power between controlled areas so as to have reduced generation reserve margin. Prevention of cascading outages by limiting the effects of faults and equipment failure. Damping of power system oscillations, which could damage equipment failure. Improvement in steady state stability limit and transient stability margin. 2.4 PRINCIPLE OF OPERATION This section describes briefly the principle of operation and control of SVC, TCSC, STATCOM and UPFC (Mohan Mathur and Rajiv Varma 2002).

6 Static Var Compensator The Static Var Compensators are the most widely installed FACTS equipment at this point in time. They mimic the woring principles of a variable shunt susceptance and use fast thyristor controllers with settling times of only a few fundamental frequency periods. From the operational point of view, the SVC adjusts its value automatically in response to changes in the operating conditions of the networ. By suitable control of its equivalent susceptance, it is possible to regulate the voltage magnitude at the SVC point of connection, thus enhancing significantly the performance of the power system. The majority of Static Var Compensators have similar controllable elements. The most common ones are, Thyristor-Controlled Reactor (TCR) Thyristor-Switched Capacitor (TSC) Thyristor-Switched Reactor (TSR) In the case of the TCR a fixed reactor, typically is connected in series with a bidirectional thyristor valve. The fundamental frequency current is varied by phase control of the thyristor valve. A TSC comprises of a capacitor in series with a bidirectional thyristor valve and a damping reactor. The function of the thyristor switch is to connect or disconnect the capacitor for an integral number of half-cycles of the applied voltage. A practical configuration of a TCR-TSC SVC system is shown in Figure 2.2. The equivalent susceptance B eq is determined by the firing angle of the thyristor. The variation of B eq as a function of is as shown in Figure 2.3.

7 25 V HV bus PT Controller Filter Figure 2.2 A practical example of TCR-TSC SVC Beq Bmax unavailable 0 o res 90 o Bmin unavailable Figure 2.3 Variation of B eq as a function of for SVC V-I Characteristics of the SVC The steady-state and dynamic characteristics of SVC describe the variation of SVC bus voltage with SVC current (Figure 2.4).

8 26 V svc Over current limit Steady-State Characteristics Over load range B min V 1 V ref Dynamic Characteristics V 2 B max Linear Range of Control I Cr Capacitive 0 I Lr Inductive I svc Figure 2.4 V-I characteristics of SVC Dynamic Characteristics Reference voltage, V ref : This is the voltage at the terminals of the SVC during the floating condition, that is, when the SVC is neither absorbing nor generating any reactive power. Linear range of SVC control: This is the control range over which SVC terminal voltage varies linearly with the SVC current or reactive power as it is varied over its entire capacitive to inductive range. Slope or current droop: The slope or droop of the V-I characteristics is defined as the ratio of voltage-magnitude change to currentmagnitude change over the linear-controlled range of the compensator. Overload Range: When the SVC traverses outside the linearcontrollable range on the inductive side, the SVC enters the overload zone, where it behaves lie a fixed inductor.

9 27 Over current Limit: To prevent the thyristor valves from being subjected to excessive thermal stress, the maximum inductive current in the overload range is constrained to a constant value by an additional control action. Steady-State Characteristics The steady-state characteristic of the SVC has a dead band. In the absence of this dead band, in the steady state the SVC will tend to drift toward its reactive power limits to provide voltage regulation. It is not desirable to leave the SVC with very little reactive-power margin for future voltage control or stabilization excursions in the event of system disturbance Thyristor Controlled Series Capacitor TCSC is a capacitive reactance compensator, which consists of a series capacitor ban shunted by a thyristor-controlled reactor in order to provide a smoothly variable series capacitive reactance. The basic TCSC module comprises a series capacitor C, in parallel with a thyristor-controlled reactor L s as shown in Figure 2.5. However, a practical TCSC module also includes protective equipment normally installed with series capacitors. An actual TCSC system usually comprises a cascaded combination of many such TCSC modules, together with a fixed-series capacitor C F. This fixed-series capacitor is provided primarily to minimize costs. Figure 2.5 A basic TCSC module

10 Basic Operation of TCSC A TCSC is series-controlled capacitive reactance that can provide continuous control of power on the ac line over a wide range. From the system viewpoint, the principle of variable-series compensation is simply to increase the fundamental frequency voltage across a fixed capacitor (FC) in a series-compensated line through appropriate variation of the firing angle. A simple understanding of TCSC functioning can be obtained by analyzing the behavior of a variable inductor connected in parallel with a FC (Figure 2.6). The impedance of FC is given by -j(1/ C). Figure 2.6 A variable inductor connected in shunt with an FC If C - (1/ L) 0 or in other words, L (1/ C), the reactance of the FC is less than that of the parallel-connected variable reactor and that this combination provides a variable-capacitive reactance are both implied. If C - (1/ L) = 0, a resonance develops that results in an infinite-capacitive impedance an obviously unacceptable condition. If, however, C - (1/ L) 0, the LC combination provides inductance above the value of the fixed inductor. In the variable-capacitance mode of the TCSC, as the inductive reactance of the variable inductor is increased, the equivalent capacitance reactance is gradually decreased.

11 Static Synchronous Compensator A static synchronous generator operated as a shunt-connected Static Var Compensator whose capacitive or inductive output current can be controlled independent of the ac system voltage. Figure 2.7 The STATCOM operating principle diagram (a) power circuit (b) equivalent circuit and (c) power exchange A STATCOM is a controlled reactive-power source. It provides the desired reactive-power generation and absorption entirely by means of electronic processing of voltage and current waveforms in a voltage-source converter (VSC). A single-line STATCOM is shown in Figure 2.7(a), where a VSC is connected to a utility bus through magnetic coupling. In the Figure 2.7(b) STATCOM is seen as an adjustable voltage source behind a reactance meaning that capacitor bans and shunt reactors are not needed for reactive power generation and absorption, thereby giving STATCOM a compact design.

12 30 Figure 2.8 The power exchange between the STATCOM and the AC system The exchange of reactive power between the converter and the ac system can be controlled by varying the amplitude of the 3-phase output voltage, E s of the converter, as illustrated in Figure 2.7(c). If the amplitude of the output voltage is increased above that of the utility bus voltage, E t, then a current flows through the reactance from the converter to the ac system and the converter generates capacitive-reactive power for the ac system. If the amplitude of the output voltage is decreased below the utility bus voltage, then the current flows from the ac system to the converter and the converter absorbs inductive-reactive power from the ac system. If the output voltage equals the ac system voltage, the reactive-power exchange is zero, in which case the STATCOM is said to be in floating state. The reactive and the real-power exchange between the STATCOM and the ac system can be controlled independently of each other. Any combination of real power generation or absorption is achievable if the

13 31 STATCOM is equipped with an energy-storage device of suitable capacity as depicted in Figure 2.8. With this capability, extremely effective control strategies for the modulation of reactive and real-power can be achieved. Furthermore, a STATCOM does the following: 1. It occupies a small footprint, for it replaces passive bans of circuit elements by compact electronic converters; 2. It offers modular, factory-built equipment, thereby reducing site wor and commissioning time; 3. It uses encapsulated electronic converters, thereby minimizing its environmental impact Unified Power Flow Controller A combination of a static synchronous compensator (STATCOM) and a static synchronous series compensator (SSSC) which are coupled via a common dc lin, to allow bi-directional flow of real power between the series output terminals of the SSSC and the shunt operated terminals of STATCOM, and are controlled to provide concurrent real and reactive series line compensation without an external electric energy source. The UPFC, by means of angularly unconstrained series voltage injection, is able to control, concurrently or selectively, the transmission line voltage, impedance and angle or alternatively, the real and reactive power flow in the line. The UPFC may also provide independently controllable shunt-reactive compensation. The UPFC which is one of the most promising device in the FACTS concept, has been researched and put into practical use (Schauder 1998).

14 32 Figure 2.9 The implementation of UPFC with two bac-to-bac VSCs with a common dc-terminal capacitor The UPFC consists of two voltage source converters. The dc voltage for both converters is provided by a common capacitor ban (dc lin) (Figure 2.9). Converter 2 provides the main function of the UPFC by injecting an ac voltage with controllable magnitude V pq, in series with the transmission line via a series transformer which can be varied from 0 to V pqmax and phase angle of V pq can be independently varied from 0 o to 360 o. The basic function of converter 1 is to supply or absorb the real power demand of converter 2 which it derives from the transmission line itself. Although the reactive power is internally generated / absorbed by the series converter, the real power generation /absorption is made feasible by the dcenergy-storage device i.e., the capacitor. It can also generate or absorb controllable reactive power and provide independent shunt reactive compensation for the line. Converter 2 supplies or absorbs locally the required reactive power and exchanges the active power as a result of the series injection voltage. Thus the net real power drawn from the ac system is equal to the loss of the two converters and their coupling transformers. In addition

15 33 the shunt converter functions lie a STATCOM and independently regulates the terminal voltage of the interconnected bus. 2.5 POWER FLOW STUDIES Planning the operation of power systems under existing conditions, its improvement and also its future expansion require the load flow studies, short circuit studies and stability studies. The satisfactory operation of the system depends upon nowing the effects of interconnections, new loads, new generating stations and new transmission lines before they are installed. They also help to determine the best size and favourable locations for the power capacitors both for the improvement of the power factor and also raising the bus voltages of the electrical networ. They help us to determine the best locality as well as optimal capacity of the proposed generating stations, sub stations or new lines. Through the load flow studies we can obtain the voltage magnitudes and angles at each bus in the steady state. This is rather important, as the magnitudes of the bus voltages are required to be held within a specified limit. Once the bus voltage magnitudes and their angles are computed using the load flow, the real and reactive power flow through each line can be computed. Also based on the difference between power flow in the sending and receiving ends, the losses in a particular line can also be computed The Newton Raphson Algorithm A popular approach to assess the steady state operation of a power system is to write equations stipulating that at a given bus, the generation and powers exchanged through the transmission elements connecting to the bus must add up to zero. This applies to both active and reactive power. These

16 34 equations are termed mismatches power equations and at bus, they tae the following form: P = P G P L P cal = P sch - P cal = 0 (2.1) Q = Q G Q L Q cal = Q sch Q cal = 0 (2.2) P, Q - mismatch active and reactive powers P G, Q G - active and reactive powers injected by the generator P L, Q L - active and reactive powers drawn by the load at the bus Under normal circumstances the customer has control over these variables and in the power flow formulation they are assumed to be nown variables. In principle, at least, the generation and the load at bus may be measured by the electric utility and in the parlance of power system engineers, their net values are nown as the scheduled active and reactive powers as given in equations (2.3) and (2.4): P sch = P G - P L (2.3) Q sch = Q G - Q L (2.4) cal The transmitted active and reactive powers, P and Q cal, are functions of nodal voltages and networ impedances and are computed using power flow equations. Provided the nodal voltages throughout the power networ are nown to a good degree of accuracy then the transmitted powers are easily and accurately calculated. In this situation, the corresponding mismatch powers are zero for any practical purposes and the power balance at each bus is satisfied. However, if the nodal voltages are not nown precisely then the calculated transmitted powers will have only approximated values and the corresponding mismatch powers are not zero. The power flow solution taes the approach of successively correcting the calculated nodal

17 35 voltages and hence, the calculated transmitted powers until values accurate enough are arrived at, enabling the mismatch powers to be zero or fairly close to zero. In modern power flow programs, it is normal to all mismatch equations to satisfy a tolerance as tight as 1e-12 before the iterative solution can be considered successful. In order to develop suitable power flow equations, it is necessary to find relationships between injected bus currents and bus voltages. Based on the Figure 2.10 the injected complex current at bus, denoted by I, may be expressed in terms of the complex bus voltages E as follows: I = 1*( E E m ) / Z m = y m * (E E m ) (2.5a) Similarly for bus m, I m = 1*(E m E ) / Z m = y m * (E m E ) (2.5b) Figure 2.10 Equivalent impedance The above equations can be written in matrix form as, I I m Y Y m Y Y m mm E E m (2.6) The bus admittances and voltages can be expressed in more explicit form: Y ij = G ij + jb ij (2.7) E i = V i e = V i * ( cos i + jsin i ) (2.8)

18 36 The complex power injected at bus consists of an active and reactive component and is expressed as a function of the nodal voltage and the injected current at the bus as given in equation (2.9). S = P + jq = E I * (2.9) S = E ( Y E + Y m E m ) * where I * is the complex conjugate of the current injected at the bus. The expressions for P cal and Q cal are as follows: P cal = V 2 G + V V m [G m cos( m ) + B m sin( m )] (2.10) Q cal = -V 2 B + V V m [G m sin( m ) - B m cos( m )] (2.11) For specified levels of power generation and power load at bus and according to equations (2.1) and (2.2), the mismatch equations may be written down as follows: P =P G P L { V 2 G + V V m [G m cos( m ) + B m sin( m )]} = 0 (2.12) Q =Q G Q L {-V 2 B + V V m [G m sin( m ) - B m cos( m )]} = 0 (2.13) Similar equations may be obtained for bus m simply by exchanging subscripts and m in equations (2.12) and (2.13). It should be remared that equations (2.10) and (2.11) represent only the powers injected at bus through the i th transmission element, i.e., i P cal i and Q cal. However a practical power system will consists of many buses and transmission elements. This calls for equations (2.10) and (2.11) in more general terms, with the net power flow injected at bus expressed as the summation of the powers flowing at each one of the transmission elements terminating at this bus.

19 37 The generic active and reactive powers injected at bus are as follows: P cal = P i cal Q cal = Q i cal (2.14) (2.15) where P i cal respectively. and Q i cal are computed using equations (2.10) and (2.11) As an extension, the generic power mismatch equations at bus are as follows: P = P G P L P i cal = 0 (2.16) Q = Q G Q L Q i cal = 0 (2.17) In large-scale power flow studies, the Newton Raphson has proved most successful owing to its strong convergence characteristics. The power flow Newton Raphson algorithm is expressed by the following relationship. P P / P /( v / v) = - Q Q / Q /( v / v) ( v / v) (2.18) It may be pointed out that the correction terms V m are divided by V m to compensate for the fact that jacobian terms ( P m V m )V m and Q m V m )V m are multiplied by V m. It is shown in the directive terms that this artifice yields useful simplifying calculations. Consider the l st element connected between buses and m in Figure 2.10, for which self and mutual Jacobian terms are given below: For m P,l = )] Vm [ Gm sin( m ) Bm cos( m m,l V (2.19)

20 38 V P m, l, l V m, l = V V G cos( ) B sin( )] (2.20) m[ m m m m Q,l P =, l m,l Vm, l Vm, l (2.21) V Q m, l, l V m, l = P,l m,l (2.22) for = m, P,l = cal 2 Q V B,l (2.23) V P, l cal 2 P V, l V, l G (2.24) Q, l cal P V 2, l G (2.25) V Q, l cal 2 Q V, l V, l B (2.26) The mutual elements remain the same whether we have one transmission line or n transmission lines terminating at the bus The Sample Five Bus System First we have considered the five bus system as a case study shown in Figure 2.11 (Stagg and Abiad 1968). The input data for the considered system are given in Table 2.1 for the bus and Table 2.2 for transmission line.

21 39 Figure 2.11 The five-bus networ Table 2.1 Input Bus data (p.u.) for the system under study Bus No. Bus code (-m) Impedance (R+jX) Line charging admittance j j j j j j j j j j j j j j0.05

22 40 Table 2.2 Input Transmission line data (p.u.) for the system under study Bus No. Type Generation Load Voltage P Q P Q v 1 slac P-V P-Q P-Q P-Q Assuming base quantities of 100 MVA and 100 KV. The power flow result for the above system without any FACTS devices is mentioned in Table 2.3. All the nodal voltages are achieved to be within acceptable voltage magnitude limits. Table 2.3 Bus voltage of system under study without FACTS devices Parameter BUS 1 BUS 2 BUS 3 BUS 4 BUS 5 V (p.u) (deg) FACTS CONTROLLERS MODELING This chapter explains the power flow modeling of different FACTS devices we have adopted in our project. The unified approach that combines the state variables describing controllable equipment with those describing the networ in a single frame of reference for iterative solutions using the Newton-Raphson algorithm is followed, which retains its quadratic convergence characteristics (Acha et al 2004).

23 Power Flow Model of SVC Two models are presented in this category, namely, the variable shunt susceptance model and firing-angle model Variable susceptance model In practice the SVC can be seen as an adjustable reactance with either firing angle limits or reactance limits. The equivalent circuit shown below in Figure 2.12 is used to derive the SVC nonlinear power equations and the liberalized equations required by the Newton s method. Figure 2.12 SVC -Variable shunt susceptance With reference to Figure 2.12, the current drawn by the SVC is given by equation (2.27), I SVC = jb SVC V (2.27) The reactive power drawn by the SVC, which is also the reactive power injected at bus, is given by equation (2.28), Q SVC = Q = -V 2 B SVC (2.28)

24 42 At the end of each iteration, the variable shunt susceptance B is updated as given in equation (2.29), B B ( i ) ( i 1) SVC SVC B B SVC SVC ( i) B (2.29) ( i 1) SVC The changing susceptance represents the total SVC susceptance necessary to maintain the nodal voltages at specified value Firing angle model An alternative SVC model, which circumvents the additional iterative process, consists in handling the TCR firing angle as a state variable in the power flow formulation. The positive sequence susceptance of the SVC, is given by equation (2.30). Q V X X C 2 L { X L X C [2( ) sin(2 )]} SVC SVC (2.30) From the above equation, the linearised SVC equation is given as follows: P Q 0 0 (i) = 2 2V 0 [cos(2 SVC ) 1] X L (i) SVC (i) (2.31) At the end of iteration (i), the variable firing angle SVC is updated as given in equation (2.32). SVC (i) = SVC (i-1) + SVC (i) (2.32) Power Flow Model of TCSC Two alternative power flow models to assess the impact of TCSC equipment in networ wide applications are presented in this section. The simpler TCSC model exploits the concept of a variable series reactance. The

25 43 series reactance is adjusted automatically, within limits, to satisfy a specified amount of active power flows through it. The more advanced model uses directly the TCSC reactance-firing angle characteristics, given in the form of a non-linear relation. The TCSC firing angle is chosen to be the state variable in the Newton-Raphson power flow solution Variable series reactance power flow model The TCSC power flow model presented in this section is based on the simple concept of a variable series reactance, the value of which is adjusted automatically to constrain the power flow across the branch to a specified value. The amount of reactance is determined efficiently using Newton s method. The changing reactance X TCSC, shown in Figure 2.13(a) and 2.13(b), represents the equivalent reactance of all the series-connected modules maing up the TCSC, when operating either in the inductive or in the capacitive regions. Figure 2.13 TCSC equivalent circuits (a) Inductor (b) Capacitive operative regions given by The transfer admittance matrix of the variable series compensator is I I m = jb jb m jb jb m mm V V m (2.33)

26 44 For inductive operation, we have B 1 = B mm = - X TCSC (2.34) B m = B m = 1 X TCSC (2.35) For capacitive operation the signs are reversed. The active and reactive power equations at bus are as follows: P V V B sin( ) m (2.36) m m Q V 2 B V V B cos( ) m (2.37) m m For the power equations at bus m, the subscripts and m are exchanged in equations (2.36) and (2.37). The state variable X TCSC of the series controller is updated at the end of each iterative step as given in equation (2.38). ( i) X X X X (2.38) ( i) ( i 1) TCSC ( i 1) TCSC TCSC TCSC X TCSC Firing angle power flow model The model presented above in section uses the concept of an equivalent series reactance to represent the TCSC. Once the value of reactance is determined using Newton s method then the associated firing angle TCSC can be calculated. However, such calculations involve an iterative solution since the TCSC reactance and the firing angle are nonlinearly related. One way to avoid the additional iterative process is to use the alternative TCSC power flow model presented in this section.

27 45 Figure 2.14 TCSC compensator firing angle modules The fundamental frequency equivalent reactance X TCSC of the TCSC module shown in Figure 2.14 is given by equation (2.39). X TCSC X C 2 C 1 {2( ) sin[ 2( )]} C 2 cos ( ){ tan[ ( )] tan( )} (2.39) where X C X LC C 1 (2.40) C 2 4X X L 2 LC (2.41) X X C L X LC (2.42) X C X L X X C L 1 / 2 (2.43) The TCSC active and reactive power equations at bus are as follows: P V V B sin( ) m (2.44) m m 2 Q V B V V B sin( ) (2.45) m m m

28 46 where B B B (2.46) m TCSC For equations at bus m, exchange subscript and m in equation (2.44) and (2.45) Power Flow Model of STATCOM The Static Synchronous Compensator is represented by a synchronous voltage source with minimum and maximum voltage magnitude limits. The bus at which STATCOM is connected is represented as a PV bus, which may change to a PQ bus in the events of limits being violated. In such case, the generated or absorbed reactive power would correspond to the violated limit. The power flow equations for the STATCOM are derived below from the first principles and assuming the following voltage source representation: Figure 2.15 STATCOM equivalent circuit Based on the shunt connection shown in Figure 2.15, the following equation can be written. EvR VvR (cos vr j sin vr ) (2.47) S V I V Y ( V V ) (2.48) * * * * vr vr vr vr vr vr

29 47 converter at bus : The following are the active and reactive power equations for the P V G V V [ G cos( ) B sin( )] (2.49) 2 vr vr vr vr vr vr vr vr Q V B V V [ G sin( ) B cos( )] (2.50) 2 vr vr vr vr vr vr vr vr P V G V V [ G cos( ) B sin( )] (2.51) 2 vr vr vr vr vr vr Q V B V V [ G sin( ) B cos( )] (2.52) 2 vr vr vr vr vr vr Power Flow Model of UPFC The equivalent circuit consists of two coordinated synchronous voltage sources should represent the UPFC adequately for the purpose of fundamental frequency steady state analysis. Such an equivalent circuit is shown in Figure Figure 2.16 UPFC equivalent circuits

30 48 The UPFC voltage sources are as follows: EvR VvR (cos vr j sin vr ) (2.53) EcR VcR (cos cr j sin cr ) (2.54) where V vr and vr are the controllable magnitude (V vrmin V vr V vrmax ) and phase angle (0 vr ) of the voltage source representing the shunt converter. The magnitude V cr and phase angle cr of the voltage source representing the series converter are controlled between limits (V crmin V cr V crmax ) and (0 cr ), respectively. The phase angle of the series injected voltage determines the mode of power flow control. If cr is in phase with the nodal voltage angle, the UPFC regulates the terminal voltage. If cr is in quadrature with, it controls active power flow, acting as a phase shifter. If cr is in quadrature with line current angle then it controls active power flow, acting as a variable series compensator. At any other value of cr, the UPFC operates as a combination of voltage regulator, variable series compensator and phase shifter. The magnitude of the series injected voltage determines the amount of power flow to be controlled. Based on the equivalent circuit shown in Figure 2.16 and equations (2.53) and (2.54), the active and reactive power equations at bus are as follows: P V G V V [ G cos( ) B sin( )] 2 m m m m m V V [ G cos( ) B sin( )] cr m cr m cr (2.55) V V [ G cos( ) B sin( )] vr vr vr vr vr

31 49 Q V B V V [ G sin( ) B cos( )] 2 m m m m m V V [ G sin( ) B cos( )] cr m cr m cr (2.56) V V [ G sin( ) B cos( )] vr vr vr vr vr At bus m P V G V V [ G cos( ) B sin( )] 2 m m mm m m m m m V V [ G cos( ) B sin( )] m cr mm m cr mm m cr (2.57) Q V B V V [ G sin( ) B cos( )] 2 m m mm m m m m m V V [ G sin( ) B cos( )] m cr mm m cr mm m cr (2.58) Series converter P V G V V [ G cos( ) B sin( )] 2 cr cr mm cr m cr m cr V V [ G cos( ) B sin( )] cr m mm cr m mm cr m (2.59) Q V B V V [ G sin( ) B cos( )] 2 cr cr mm cr m cr m cr V V [ G sin( ) B cos( )] cr m mm cr m mm cr m (2.60) Shunt converter P V G V V [ G cos( ) B sin( )] (2.61) 2 vr vr vr vr vr vr vr vr Q V B V V [ G sin( ) B cos( )] (2.62) 2 vr vr vr vr vr vr vr vr Assuming lossless converter values, the active power supplied to the shunt converter, P vr, equals the active power demanded by the series converter, P cr ; that is, P P 0 (2.63) vr cr

32 50 Further more, if the coupling transformers are assumed to contain no resistance then the active power at bus matches the active power at bus m. Accordingly, P P P P 0 (2.64) vr cr m networ. The UPFC power equations are combined with those of the AC 2.7 CASE STUDIES WITH FACTS CONTROLLERS The five bus networ is modified to include different FACTS devices and examine the voltage control capabilities and power flow control capabilities of the same Power Flow Study with SVC The five bus networ is modified to examine the voltage control capabilities of different SVC models. One SVC is included in the bus 3 (Figure 2.17) to maintain the nodal voltage at 1 p.u. Figure 2.17 Study system with SVC

33 51 The SVC inductive and capacitive reactances are taen to be p.u. and 1.07 p.u. respectively. The SVC firing angle is set initially at 140 0, a value that lies on the capacitive region. Convergence is achieved in 5 iterations, satisfying a prespecified tolerance of 1e-12 for all variables. The result for the voltage magnitude and phase angle obtained for both variable susceptance and firing angle model is shown in Table 2.4. Due to the inclusion of SVC in the third bus, its voltage is maintained at 1 p.u. The SVC susceptance values and firing angle values are shown in Table 2.5 for each step of the iterative process. Table 2.4 Bus voltage of modified networ (with SVC) Type of model Parameter BUS 1 BUS 2 BUS 3 BUS 4 BUS 5 SVC (Varying Susceptance) SVC (Varying Firing Angle) V (p.u) (deg) V (p.u) (deg) Table 2.5 SVC state variables Susceptance model Firing angle model Iteration Bsvc (p.u) Bsvc (p.u) SVC Power Flow Study with TCSC The original five-bus networ is modified to include one TCSC (Figure 2.18) to compensate the transmission line connected between bus 3

34 52 and 4. The TCSC should maintain the real power flow in transmission line 6 as 21 MW. Figure 2.18 Study system with TCSC The starting value of TCSC is set at 50 percent of the value of the transmission inductive reactance i.e p.u. for varying reactance model. Convergence is achieved in 6 iteration to power mismatch tolerance of 1e-12. The TCSC upholds the target value of 21 MW, which is achieved with 70 percent series capacitive compensation. The initial value of firing angle is set at Convergence is obtained in 6 iterations, with final value of at , with TCSC upholding the target value of 21 MW in line 6. The result for the voltage magnitude and phase angle obtained for the above system for both variable reactance and firing angle model is shown in Table 2.6.

35 53 Table 2.6 Bus voltage of modified networ (with TCSC) Type of model Parameter BUS 1 BUS 2 BUS 3 BUS 4 BUS 5 TCSC (Varying Reactance) TCSC (Varying Firing Angle) V (p.u) (deg) V (p.u) (deg) Power Flow Study with STATCOM The STATCOM is included in bus 3 (Figure 2.19) of the sample system to maintain the nodal voltage at 1 p.u. Figure 2.19 Study system with STATCOM The power flow result indicates that the STATCOM generates 20.5 MVAR in order to eep the voltage magnitude at 1 p.u. at bus3. The source

36 54 impedance is Z vr = 0.1 p.u., the STATCOM parameters associated with this amount of reactive power generation are V vr = p.u. and vr = Use of STATCOM results in improved networ voltage profile as shown in Table 2.7. Table 2.7 Bus voltages of STATCOM upgraded networ Type of model Parameter BUS 1 BUS 2 BUS 3 BUS 4 BUS 5 V (p.u) STATCOM (deg) Power Flow Study with UPFC The original five-bus networ is modified to include one UPFC to compensate the transmission line lining bus 3 and bus 4 (Figure 2.20). UPFC should maintain real and reactive power flowing towards bus 4 at 40 MW and 2 MVAR, respectively. The UPFC shunt converter is set to regulate the nodal voltage magnitude at bus 3 at 1 p.u. Figure 2.20 Study system with UPFC

37 55 The starting values of the UPFC voltage sources are taen to be V cr =0.04 p.u., cr = , V vr = 1 p.u. and vr = 0 0. The source impedances have values of Z cr = Z vr = 0.1 p.u.. Convergence is obtained in five iterations to a power mismatch tolerance of 1e-12. The UPFC upheld its target values and the bus voltage are given in Table 2.8. Table 2.8 Bus voltages of UPFC upgraded networ Type of model Parameter BUS 1 BUS 2 BUS 3 BUS 4 BUS 5 V (p.u) UPFC (deg) RESULTS AND DISCUSSION The power flow for the five bus system was analyzed without and with FACTS devices performing the Newton-Raphson Method. The consolidated comparative bus voltage results have been represented in Table 2.9. The largest power flow taes place in the transmission line connecting the two generator buses: 89.3 MW and MVAR leave bus 1 and 86.8 MW and 72.9 MVAR arrive at bus2. The operating conditions demand a large amount of reactive power generation by the generator connected at bus1 (i.e., MVAR). This amount is well in excess of the reactive power drawn by the system loads (i.e., 40 MVAR). The generator at bus 2 draws the excess of reactive power in the networ (i.e., MVAR). This amount includes the net reactive power produced by several transmission lines, which is addressed by different FACTS devices. Thus SVC upholds its target value and as expected identical power flows and bus voltages are obtained for both Shunt variable susceptance model and Firing angle power flow models.

38 56

39 57 The TCSC Variable series compensator model is used to maintain active power flowing from the extra fictious bus 6 towards bus 3 at 21 MW. The starting value of the TCSC is set at 50% of the value of transmission line inductive reactance. The TCSC upholds the target value of 21 MW, which is achieved with 70% series compensation of the transmission line 6. In the case of the firing angle model the initial value of firing angle is set at 145º and the TCSC upholds the target value of 21 MW. The power flow result indicates that the STATCOM generates 20.5 MVAR in order to eep the voltage magnitude at 1 p.u at bus 3. Use of STATCOM results in an improved networ voltage profile, except at bus 5, which is too far away from bus 3 to benefit from the influence of STATCOM. The original five-bus networ is modified to include one UPFC to compensate the transmission line lining bus 3 and bus 4. The UPFC is used to maintain active and reactive powers leaving UPFC, towards bus 4, at 40 MW and 2 MVAR, respectively. Moreover the UPFC shunt converter is set to regulate the nodal voltage magnitude at bus 3 at 1 p.u. There is a 32% increase of active power flowing towards bus 3. The increase is in response to the large amount of active power demanded by the UPFC series converter. Thus from the above analysis we find that within the framewor of traditional power transmission concepts, the UPFC is able to control, simultaneously or selectively, all the parameters affecting power flow in the transmission line (voltage, impedance and phase angle) and this unique capability is signified by the adjective Unified in its name. The power flow study is then extended for IEEE 30 bus system whose single line circuit diagram is shown in Figure Voltage at bus 12 for the base case (without any FACTS devices) is found to be V = & = The SVC is connected at bus 12, which bring the bus voltage to 1 p.u. and convergence is obtained within 4 iterations. The impact of different

40 58

41 59 FACTS devices on the neighbouring buses is given in Table The power flow in the line between bus 13-bus 12 is found to be 16.9 MW and 38.4 MVAR. The SVC is able to control the reactive power flow of the line to the specified control reference 0.0 p.u., while the active power flow is almost unchanged. By driving the reactive power flow on the line to zero using SVC, the un-used (available) transmission line capacity can be increased. It can be seen that the base case reactive power of line is 38.4 MVAR, so the reactive power control by the SVC is significant. Similar result is also obtained when SVC is replaced by STATCOM at bus 12. Desired real power flow is also obtained when TCSC is connected in the line connecting bus 12 and bus 15. The UPFC is installed between bus 12 and the sending end of the transmission line connecting bus 12 bus 15. In the simulation, the bus voltage control reference is V 12 = 1 p.u. Convergence is achieved within 6 iterations with specified real power and reactive power in line connecting bus 12 bus 15. The detail results show the validity of different types of model proposed for FACTS devices with the model proposed by Zhang et al (2006) Figure 2.21 IEEE 30 Bus system

42 SUMMARY The non-linear power flow equations of the various FACTS controllers have been linearised and included in Newton Raphson power flow algorithm. The state variables corresponding to the controllable devices have been combined simultaneously with the state variables of the networ in a single frame of reference for unified, iterative solutions. The effectiveness of modeling and convergence is tested and results are analyzed. The next chapter discusses about dynamic modeling of Unified Power Flow Controller (UPFC) and proposed suitable controller, for damping the electromechanical oscillations in a power system.