Electric Power Systems

Linear Motion Alternators (LMAs)

noreply@blogger.com (Sajib Barua) — Tue, 03 Jun 2014 17:50:00 +0000

Transverse Flux and Flux Reversal Permanent Magnet Generator Systems

noreply@blogger.com (Sajib Barua) — Tue, 03 Jun 2014 17:45:00 +0000

Permanent Magnet Synchronous Generator Systems

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Switched Reluctance Generators and Their Control

noreply@blogger.com (Sajib Barua) — Tue, 03 Jun 2014 17:30:00 +0000

Permanent-Magnet- Assisted Reluctance Synchronous Starter/ Alternators for Electric Hybrid Vehicles

noreply@blogger.com (Sajib Barua) — Tue, 03 Jun 2014 17:25:00 +0000

Induction Starter/ Alternators (ISAs) for Electric Hybrid Vehicles (EHVs)

noreply@blogger.com (Sajib Barua) — Tue, 03 Jun 2014 17:18:00 +0000

Stator Converter Controlled Induction Generators (SCIGs)

noreply@blogger.com (Sajib Barua) — Tue, 03 Jun 2014 17:04:00 +0000

Stator Converter Controlled Induction Generators (SCIGs)

noreply@blogger.com (Sajib Barua) — Tue, 03 Jun 2014 16:38:00 +0000

Self-Excited Induction Generators

noreply@blogger.com (Sajib Barua) — Tue, 03 Jun 2014 07:31:00 +0000

Wound Rotor Induction Generators (WRIGs): Design and Testing

noreply@blogger.com (Sajib Barua) — Sun, 01 Jun 2014 16:41:00 +0000

Wound Rotor Induction Generators: Transients and Control

noreply@blogger.com (Sajib Barua) — Sat, 31 May 2014 12:35:00 +0000

Variable Speed Generators-Ion Boldea

noreply@blogger.com (Sajib Barua) — Sat, 31 May 2014 12:24:00 +0000

Power flow and measurement

noreply@blogger.com (Sajib Barua) — Sun, 24 Nov 2013 08:26:00 +0000

Single-phase

Suppose we have a single-phase load as in Figure 2.7 supplied with a sinusoidal voltage whose instantaneous value is . The RMS value is and the phasor value is V. If the load is linear (i.e. its impedance is constant and does not depend on the current or voltage), the current will be sinusoidal too. It leads or lags the voltage by a phase angle , depending on whether the load is capacitive or inductive. With a lagging (inductive) load, ; see Figure 2.29.

The instantaneous power is given by , so

(2.30)

This expression has a constant term and a second term that oscillates at double frequency. The constant term represents the average power : we can write this as

(2.31)

is equal to the product of the rms voltage , the RMS current , and the power factor . The amplitude of the oscillatory term is fixed: i.e. it does not depend on the power factor. It shows that the instantaneous power varies from to and back to twice every cycle. Since the average power is VmIm/2, this represents a peak-peak fluctuation 200% of the mean power, at double frequency. The oscillation of power in single-phase circuits con- tributes to lamp flicker and causes vibration in motors and transformers, producing undesirable acoustic noise.

Fig. 2.29 Instantaneous current, voltage and power in a single-phase AC circuit.

Two-phase

Suppose we have a two-phase load with phases a and b, with υ_a = V_m cos ωt, i_a = I_m cos (ωt – ϕ) and υ_b = V_m sin ωt, i_b = I_m sin (ωt - ϕ). This system is said to be balanced, because the voltages and currents have the same RMS (and peak) values in both phases, and their phase angles are orthogonal. The total instantaneous power is now given by

The oscillatory term has vanished altogether, which means that the power flow is constant, with no fluctuation, and the average power P is therefore equal to the instantaneous power p. Note that if the phases become unbalanced, an oscillatory term reappears.

Three-phase
Suppose we have a three-phase load as in Figures 2.20 and 2.22, with phases a, b and c, with

This system is said to be balanced, because the voltages and currents have the same RMS (and peak) values in all three phases, and their phase angles are equi-spaced (i.e. with a 120^o symmetrical phase displacement). The total instantaneous power is now given by

As in the two-phase system, the oscillatory term has vanished. The power flow is constant, with no fluctuation, and the average power P is equal to the instantaneous power p. If the phases become unbalanced, an oscillatory term reappears.

The voltages and currents in equation (2.34) are phase quantities. In terms of line quantities, for a wye connection we have and I_L = I_ph, whereas for a delta connection we have and V_LL = V_ph. In both cases, therefore,

where ϕ is the angle between the phasors V_ph and I_ph.
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Three-phase systems

noreply@blogger.com (Sajib Barua) — Sat, 22 Jun 2013 17:36:00 +0000

Most power systems (from 415 V upwards) are three‐phase systems. When the phases are balanced, the phasor diagrams and equations of one phase represent all three phases.

Why three‐phase? The main reasons for having more than one phase are as follows:

(a) better utilization of materials such as copper, iron, and insulation in lines, cables, transformers, generators and motors

(b) constant power flow

(d) ‘natural rotation’, permitting the widespread use of AC induction motors.

Development of three‐phase systems

To achieve ‘diversity’ —that is, the ability to supply different loads from different circuits so that a failure in one circuit would not affect the others —we can use separate circuits or ‘phases’ as shown in Figure 2.18.

The power in each phase is where and are the RMS voltage and current as shown in Figure 2.18. The total power is 3. Assuming that the cable works at a certain current‐density determined by its allowable temperature rise, the total cross‐section area of conductor is 6A.

Suppose that the three phase currents are shifted in time phase by 120° from one another as shown in Figure 2.19. The RMS currents are unchanged, as is the power in each phase and the total power. The sum of the three currents is zero, and we can express this in terms of instantaneous or phasor values

and (2.23)

Fig. 2.18 Three single-phase cable.

Fig. 2.19 Three‐phase instantaneous and phasor currents.

This suggests that the circuit could be equally well served by the three‐phase connection shown in Figure 2.20, which has only half the number of conductors compared with Figure 2.18. Figure 2.20 also shows the cross‐section of a three‐phase cable capable of carrying the required current. The total cross‐section of conductor is 3, that is a saving of 50%.

In the voltage phasor diagram in Figure 20, the voltage across each phase of the load remains the same as in Figure 2.20, but the voltage between lines is increased: evidently from the geometry of the triangles

(2.24)
where V is the reference phase voltage taken as , measured between line A and the ‘star point’ of the load or the ‘neutral point’ of the supply. Likewise the phase voltage is measured between line and the star‐point of the load, and between line and the star point. The line‐line voltages are times the phase voltages. In Figure 2.20 both the load and the generator are ‘wye connected’ and in terms of the RMS values only, we have
and (2.25)

Fig. 2.20 Three‐phase connection with wye‐connected load and phasor diagram.

where is the line current and is the phase current. With wye connection they are one and the same current. The dotted line in Figure 2.20 shows the possibility of a connection between the neutral point of the supply and the star point of the load. This connection may be used to stabilize the potential of the star point where there is an excess of triplen harmonics in the current or voltage waveforms of the load.

The current and voltage in Figures 2.19 and 2.20 are displaced in phase by the power factor angle . Figure 2.21 shows a complete phasor diagram for a balanced wye‐connected load with a lagging power factor.

An alternative connection of the three phases is the delta connection shown in Figure 2.22 together with the construction of the phasor diagram under balanced conditions with a lagging power factor.

For the delta connection,

and (2.26)

The delta connection is used to provide a path for triplen harmonic currents. For example, when transformers operate at higher than normal voltage the magnetizing

Fig. 2.21 Phasor diagram for balanced wye‐connected load.

Fig. 2.22 Three‐phase connection with delta‐connected load and phasor diagram.

Fig. 2.23 Wye‐delta transformation.

current in each phase tends to become distorted, and the triplen harmonics are allowed to flow locally in a delta‐connected winding without entering the external circuit. In electric motors a delta winding permits the use of a larger number of turns of smaller‐gauge wire, because the phase voltage is increased while the phase current is decreased, compared with the wye connection.

The wye-delta transformation

A wye‐connected load can be represented by a virtual load connected in delta, and vice‐versa, Figure 2.23. To transform the delta connection into a wye connection,

; ;

where . To transform the wye connection into a delta connection,

; ; (2.28)

where

Balancing an unbalanced load

It can be shown by means of a series of diagrams, that an unbalanced linear ungrounded three‐phase load can be transformed into a balanced, real three‐phase load without changing the power exchange between source and load, by connecting an ideal reactive compensating network in parallel with it. Assume that the load is delta‐connected with admittances , as shown in Figure 2.24.

The power factor of each phase can be corrected to unity by connecting compensating admittances in parallel, as shown, where , and . The resulting network is real, Figure 2.25.

Fig. 2.24 Unbalanced delta‐connected load with power‐factor correction admittances.

Fig. 2.25 Unbalanced load corrected with unity power factor in each phase.

If we now take one of the resistive admittances, , we can connect this in a so‐called Steinmetz network with an inductor and a capacitor to produce balanced line currents as shown in Figure 2.26. The phasor diagram in Figure shows how the balanced line currents are achieved, and the resulting equivalent circuit in Figure 2.28 is real, balanced, and wye‐connected.

Fig. 2.26 Steinmetz network with balanced line currents.

Fig. 2.27 Phasor diagram for Steinmetz network.

Fig. 2.28 Balanced network resulting from compensation of with the Steinmetz network.

The resulting compensating admittances are given in equation (2.29).

(2.29)

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Relationships between power, reactive power, voltage levels and load angle

noreply@blogger.com (Sajib Barua) — Tue, 11 Jun 2013 16:20:00 +0000

The phasor diagram for the system in Figure 2.16 is shown in Figure 2.17, assuming that the load has a lagging power factor angle ϕ. The line or cable is represented by its impedance , and is again neglected (being usually much smaller than . The voltage drop across the transmission line is j, which leads the phasor I by . The angle between and is the load angle, δ and

and (2.17)
Also
(2.18)
From this we get the power flow equation
(2.19)

and the reactive power equation for the receiving end
(2.20)
Evidently as long as the transmission losses are negligible. At the sending end,
(2.21)

Fig. 2.17 Phasor diagram for Figure 2.16.

from which it can be shown that

(2.22)

Note the symmetry between this expression and the one for in equation (2.20).
Example 1

Suppose p.u. and p.u.(The per‐unit system is explained in Section 2.13. If you aren’t familiar with it, try to read these examples as practice in the use of normalized (per‐unit) values. In effect, they make it possible to forget about the units of volts, amps, etc.) The transmission system has . and is negligible. We have so and . Then p.u. and p.u. Thus the receiving end is generating reactive power and so is the sending end. The power factor is lagging at the sending end and leading at the receiving end. The phasor diagram is shown in Figure 2. (not to scale).

Example 2

Suppose p.u. while the receiving‐end voltage is reduced to p.u., with . and . Now and δ, slightly larger than in example 1 because is reduced by 5%. Also . and p.u. Since is positive, the sending‐end generator is generating VArs. is also positive, meaning that the receiving‐end load is absorbing VArs. The phasor diagram is similar to that shown in Figure (but not to the same scale).

Notice that a 5% reduction in voltage at one end of the line causes a massive change in the reactive power flow. Conversely, a change in the power factor at either end tends to cause a change in the voltage. A 5% voltage swing is, of course, a very large one. Changes in the power tend to produce much smaller changes in voltage; instead, the load angle δ changes almost in proportion to the power as long as δ is fairly small then .

The transmission system has an inductive impedance and therefore we would expect it to absorb VArs. If we regard as another impedance in series with the load impedance, we can treat it the same way. The current is obtained from p.u., and I p.u. Therefore p.u. p.u. (with as reference phasor). The voltage drop across is and the reactive power is p.u. Note that this equals the difference between and .

We could have made a similar calculation in example 1, where . Again this is . In Example 1 also , which means that each end of the line is supplying half the reactive VArs absorbed in .

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Control of Power and frequency

noreply@blogger.com (Sajib Barua) — Tue, 11 Jun 2013 15:13:00 +0000

In power systems it is essential to keep the frequency and the voltage close to their rated values. The frequency is controlled by controlling the balance between the power supplied to the system and the power taken from it. Figure 2.16 shows a transmission system with a prime mover driving a generator, and a motor driving a mechanical load. Table 2.5 gives examples of prime movers and loads.

The power P_in supplied to the system is determined by the prime mover(s). In a steam‐turbine generator, the steam valves arc the main means of control. Of course, if the valves arc opened wide, the boiler must be able to provide sufficient steam (at the correct pressure and temperature) to develop the required power. This means that the boiler control must be coordinated with the steam valves. Similarly, in a wind turbine the power transmitted to the generator is determined by the wind speed and the blade pitch, which can be varied to control the power to the required level.

The power P_out taken from the system is determined by the mechanical and electrical loads. For example, consider a direct‐connected induction motor driving a pump. The motor rotates at a speed determined by the intersection of its torque/ speed characteristic with the pump’s torque/speed characteristic. Since the motor torque/speed characteristic is very steep near synchronous speed, the motor tends to run near synchronous speed and the torque is then determined by the requirements of the pump (depending on the pressure head and the flow rate). So the power is jointly determined by the pump and the motor. With passive electrical loads (such as lighting and heating), the power supplied to the load depends on the voltage and the load impedance.

Fig. 2.16 Transmission system with prime mover, generator, motor and load. The voltages at both ends of the transmission system are assumed to be controlled, so the symbol E is used instead of V. At the sending end, the voltage is E_s; at the receiving end, E_r.

Table 2.5 Examples of prime movers and mechanical loads

Examples of prime movers	Examples of mechanical loads
Steam turbine‐generator (coil, oil, gas, nuclear, etc. - i.e. ‘thermal’) Hydro‐electric turbine generator Wind turbine generator Diesel engine	Pumps (water, sewage, process fluids, foods, etc.) Fans and blowers (air-moving) Compressors Machinery, hoists, conveyors, elevators

It is evident that and are determined quite independently. Yet in the steady state they must be essentially equal, otherwise energy would be accumulating some‐ where in the transmission system (Losses in the transmission system are assumed to be negligible for the purposes of this discussion). The power system operator can control but s/he has no control over , since customers can connect and disconnect loads at will. The power system operator does not even have any practical means of measuring for the entire system, and in any case, even if this parameter was available, there may be several generating stations in the system, so it appears to be somewhat arbitrary as to what contributions should be supplied by the individual generating stations at any instant.

In the short term (i.e. over a period of a fraction of a second), it is the frequency control that ensures that , and this control is effected by maintaining the speed of the generators extremely close to the nominal value. Suppose the power system is in a steady state and . Suppose that the load increases so that more power is taken from the system, tending to make . The prime mover and the generator will tend to slow down. Therefore the prime mover has a governor (i.e. a valve controller) that increases when the frequency is below the rated value, and decreases when the frequency is above the rated value.

In an isolated power system with only one generator, the governor has a relatively simple job to do, to maintain the speed of the generator at the correct synchronous speed to hold the frequency constant. But what happens in a power system with multiple generators? In this case usually there is a mixture of power stations. The large ones which produce the most economical power are usually best operated at constant power for long periods, without varying their contribution to . Apart from the economics, one reason for this is that if the power is varied, the temperature distribution in the turbine, boiler, and generator will be affected, and ‘thermal cycling’ is considered undesirable in these very large machines. So these generators have a relatively steep or insensitive governor characteristic, such that the frequency would have to change by quite a large amount to change the contribution to (Quite a large amount’ might mean only a fraction of 1 Hz). Elsewhere in the power system, or sometimes in the same power station, there are special generators assigned to the task of frequency control. These generators have very flat governor characteristics such that a tiny change in frequency will cause a large swing in power. They are usually gas turbine powered, up to 20 MW or so, but very large rapid‐response generators are sometimes built into hydro‐electric pumped‐storage schemes. For example, the Dinorwic power station in North Wales has a rating of 1800 MW and can change from zero to maximum power in a few tens of seconds.

The rapid‐response generators in a large interconnected power system (such as the United Kingdom system) are used for frequency control in the short term (over a few minutes or hours). They provide a time buffer to allow the larger power stations to vary their contribution. As the total system load changes during the day, the frequency is maintained almost constant, within 0.1 Hz. Averaged over 24 hours, the frequency is kept virtually dead accurate.

(In fossil‐fuel power stations two‐pole generators predominate, and the speed is in a 50‐Hz system or 3600 in a 60‐Hz system. In nuclear power stations, four‐pole generators are more common, running at 1500 (1800 at 60 Hz). In hydro plants, the generators have larger numbers of poles with speeds in the range .)

Some of the generators in a large system may be operated at light load in a state of readiness or ‘spinning reserve’, in case the system load increases suddenly by a large amount. This can happen, for example, at the end of television transmissions when the number of viewers is exceptionally high.

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Compensation and voltage control

noreply@blogger.com (Sajib Barua) — Mon, 11 Mar 2013 11:56:00 +0000

Figure 2.12 shows a one-line diagram of an AC power system, which could represent either a single-phase system, or one phase of a three-phase system. Figure 2.13 shows the phasor diagram for an inductive load.

When the load draws current from the supply, the terminal voltage V falls below the open-circuit value E. The relationship between V and the load current I is called the system load line, Figure 2.14.

The 'load' can be measured by its current I, but in power systems parlance it is the reactive volt-amperes Q of the load that is held chiefly responsible for the voltage drop. From Figures 2.12 and 2.13,

(2.10)

where I is the load current. The complex power of the load (per phase) is defined by equation (2.2), so

(2.11)

and if V = V + j0 is taken as the reference phasor we can write
(2.12)

The voltage drop ∆V has a component ∆V_R in phase with V and a component ∆V_X in quadrature with V; Figure 2.13. Both the magnitude and phase of V, relative to the open-circuit voltage E, are functions of the magnitude and phase of the load current, and of the supply impedance R_s + jX_S. Thus ∆V depends on both the real and reactive power of the load.

Fig. 2.12 Equivalent circuit of supply ant load.

Fig. 2.12 Phasor diagram (uncompensated).

Fig. 2.14 System load line.

By adding a compensating impedance or 'compensator' in parallel with the load, it is possible to maintain |V| = |E|. In Figure 2.15 this is accomplished with a purely reactive compensator. The load reactive power is replaced by the sum Q_S = Q + Q_γ, and Qγ (the compensator reactive power) is adjusted in such a way as to rotate the phasor ∆V until |V| = |E|. From equations (2.10) and (2.12),

(2.13)

The value of Qγ required to achieve this 'constant voltage' condition is found by solving equation (2.13) for Q_s with V = |E|; then Q_γ = Q_s - Q. In practice the value can be determined automatically by a closed-loop control that maintains constant

Fig. 2.15 Phasor diagram, compensated for constant voltage.

voltage V. Equation (2.13) always has a solution for Q_S, implying that: A purely reactive compensator can eliminate voltage variations caused by changes in both the real and the reactive power of the load.

Provided that the reactive power of the compensator Q_γ can be controlled smoothly over a sufficiently wide range (both lagging and leading), and at an adequate rate, the compensator can perform as an ideal voltage regulator.

We have seen that a compensator can be used for power-factor correction. For example, if the power factor is corrected to unity, Q_s = 0 and Q_γ = Q. Then

(2.14)

which is independent of Q and therefore not under the control of the compensator. Thus: A purely reactive compensator cannot maintain both constant voltage and unity power factor at the same time.

The only exception is when P = 0, but this is not of practical interest.

System load line

In high-voltage power systems R_s is often much smaller than X_s and is ignored.

Instead of using the system impedance, it is more usual to talk about the system short- circuit level S = E²/X_S. Moreover, when voltage-drop is being considered, ∆V_x is ignored because it tends to produce only a phase change between V and E. Then

and (2.15)

and

(2.16)

This relationship is a straight line, as shown in Figure 2.14. It is called the system load line.

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Power factor correction

noreply@blogger.com (Sajib Barua) — Sun, 10 Mar 2013 13:22:00 +0000

The load in Figure 2.7 can be expressed as an admittance Y = G + jB supplied from a voltage V, where Y = 1/Z. G is the conductance, i.e. the real part of the admittance Y, and B is the susceptance, i.e. the reactive or imaginary part of the admittance Y. The load current is I and for an inductive load the reactive component is negative (equation (2.3)) so we can write

(2.8)

Both V and I are phasers, and equation (2.8) is represented in the phaser diagram (Figure 2.11) in which V is the reference phaser. The voltage V and current I are in common with Figure 2.9, but Figure 2.11 shows the components of the current I and omits E and the voltage drop across the supply impedance. The load current has a 'resistive' or 'real' component I_R in phase with V, and a 'reactive' or 'imaginary' component, I_X = VB in quadrature with V. The angle between V and I is ϕ, the power-factor angle. The apparent power supplied to the load is given by equation (2 .2) with P = V²G and Q = V²B. For a capacitive load I_X is positive and Q = V²B, which is negative.

The real power P is usefully converted into heat, mechanical work, light, or other forms of energy. The reactive volt-amperes Q cannot be converted into useful forms of energy but is nevertheless an inherent requirement of the load. For example, in AC induction motors it is associated with production of flux and is often called the 'magnetizing reactive power'.

The supply current exceeds the real component by the factor 1/cos ϕ, where cos ϕ is the power factor: that is, the ratio between the real power P and the apparent power S. The power factor is that fraction of the apparent power which can be usefully converted into other forms of energy. The Joule losses in the supply cables are increased by the factor 1/ cos²ϕ. Cable ratings must be increased accordingly, and the losses must be paid for by the consumer.

The principle of power-factor correction is to compensate for the reactive power; that is, to provide it locally by connecting in parallel with the load a compensator having a purely reactive admittance of opposite sign to that of the reactive component of the load admittance. An inductive load is compensated by a capacitive admittance +jB_γ and a capacitive load by an inductive admittance -jB_γ. If the compensating admittance is equal to the reactive part of the load admittance, then for an inductive load the supply current becomes

(2.9)

Fig. 2.11 Phasor diagram for power factor correction.

which is in phase with V, making the overall power-factor unity. Figure 2.11 shows the phaser diagram.

With 100% compensation the supply current I_s now has the smallest value capable of supplying full power P at the voltage V, and all the reactive power required by the load is supplied locally by the compensator. The reactive power rating of the compensator is related to the rated power P of the load by Q_γ = P tan ϕ. The compensator current Q_γ/V equals the reactive current of the load at rated voltage. Relieved of the reactive requirements of the load, the supply now has excess capacity which is available for supplying other loads. The load may also be partially compensated (i.e. |Q_γ| < |Q|).

A fixed-admittance compensator cannot follow variations in the reactive power requirement of the load. In practice a compensator such as a bank of capacitors can be divided into parallel sections, each switched separately, so that discrete changes in the compensating reactive power may be made, according to the requirements of the load. More sophisticated compensators (e.g. synchronous condensers or static compensators) are capable of continuous variation of their reactive power.

The foregoing analysis has taken no account of the effect of supply voltage variations on the effectiveness of the compensator in maintaining an overall power factor of unity. In general the reactive power of a fixed-reactance compensator will not vary in sympathy with that of the load as the supply voltage varies, and a compensation 'error' will arise. Later the effects of voltage variations are examined, and we will find out what extra features the ideal compensator must have to perform satisfactorily when both the load and the supply system parameters can vary.

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Leading and lagging loads

noreply@blogger.com (Sajib Barua) — Sat, 09 Mar 2013 13:58:00 +0000

Figure 2.7 shows a circuit with a supply system whose open-circuit voltage is E and short-circuit impedance is , where . The load impedance is

Z = 1Ω but the power factor can be unity, 0.8 lagging, or 0.8 leading. For each of these three cases, the supply voltage E can be adjusted to keep the terminal voltage V = 100 V. For each case we will determine the value of E, the power-factor angle ----, the load angle φ, the power P, the reactive power Q, and the volt-amperes S.

Unity power factor. In Figure 2.8, we have E cos δ = V = 100 and E sin δ = X_sI =0.1 × 100/1 = 10V. Therefore E = 100 +j10 = 100.5e^j5.71⁰ V. The power-factor angle is φ = cos^-1(1) = 0, δ = 5.71⁰, and S = P + jQ = VI* = 100 × 100e^j0 = 10 kVA, with P = 10kW and Q = 0.

Lagging power factor. In Figure 2.9, the current is rotated negatively (i.e. clockwise) to a phase angle of ϕ = cos^-1(0.8) = -36.87°. Although I = 100 A and X_sI is still 10 V, its new orientation 'stretches' the phasor E to a larger magnitude: E = V + jX_SI = (100 +j0) + j0.1 × 100e^-j36.87° = 106.3e^j4.32° V. When the power-factor is lagging a higher supply voltage E is needed for the same load voltage. The load angle is δ = 4.32° and S = VI* = 100 × 100e^+j36.87 = 8000 +j6000VA. Thus S = 10 kVA, P = 8 kW and Q = +6 kVAr (absorbed).

Fig. 2.7 AC supply ant load circuit.

Fig, 2,8 Phasor diagram, resistive load.

Fig, 2.9 Phasor diagrams inductive load.

Leading power factor. The leading power factor angle causes a reduction in the value of E required to keep V constant: E = 100 + j0.1 x 100e^+j36.87°= 94.3e^j4.86°V.

The load angle is δ = 4.86°, and S = 10000e^-j36.87°= 8000 - j6000; i.e. P = 8 kW and Q = 6 kVAr (generated).

We have seen that when the load power and current are kept the same, the inductive load with its lagging power factor requires a higher source voltage E, and the capacitive load with its leading power factor requires a lower source voltage. Conversely, if the source voltage E were kept constant, then the inductive load would have a lower terminal voltage V and the capacitive load would have a higher terminal voltage. As an exercise, repeat the calculations for E = 100 V and determine V in each case, assuming that Z = 1Ω with each of the three different power factors.

Fig. 2.19 Phasor diagram, capacitive boat.

We can see from this that power-factor correction capacitors (connected in parallel with an inductive load) will not only raise the power factor but will also increase the voltage. On the other hand, if the voltage is too high, it can be reduced by connecting inductors in parallel. In modern high-voltage power systems it is possible to control the voltage by varying the amount of inductive or capacitive current drawn from the system at the point where the voltage needs to be adjusted. This is called reactive compensation or static VAR control. In small, isolated power systems (such as an automotive or aircraft power system supplied from one or two generators) this is not generally necessary because the open-circuit voltage of the generator E can be varied by field control, using a voltage regulator.

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Complex power, apparent power, real and reactive power

noreply@blogger.com (Sajib Barua) — Fri, 08 Mar 2013 05:31:00 +0000

Consider a simple load R+jX with a current I and voltage V, Figure 2.6. The complex power S is deﬁned as
(2.2)
S can be expressed graphically as the complex number P + jQ, as shown in Figure 2.6, where

P is the real power in W, kW or MW, averaged over one cycle
Q is the reactive power in VAr, kVAr, or MVAr, also averaged over one cycle
S=|S| is the apparent power or 'volt amperes', in VA, kVA or MVA

Let V be the reference phasor, and suppose that the load is inductive. Then
(2.3)

where . The negative phase rotation means that the current lags behind the voltage. When we take the conjugate I* and multiply by V we get

(2.4)

Evidently P is positive and so is Q. A load that has positive reactive power is said to (absorb' VArs. Inductive loads absorb VArs. Conversely, a capacitive load would have

(2.5)

In this case the current leads the voltage. P is still positive, but when we take the conjugate I* we get negative Q. We say that a capacitive load generates or supplies VArs.

Fig. 2.6 Development of the complex power triangle.

There is a distinction between the receiving end and the sending end. The expression is correctly interpreted as power absorbed by the load at the receiving end. But at the sending end the generated power P is supplied to the system, not absorbed from it. The distinction is that the sending end is a source of power, while the receiving end is a sink. In Figure 2.5, for example, both and are positive, supplied to the system at the sending end and taken from it at the receiving end.
A similar distinction arises with reactive power. The receiving end in Figure 2.5 evidently has a lagging power factor and is absorbing VArs. The sending end has a leading power factor and is absorbing VArs. In Figure 2.9, the power factor is lagging at both the generator and the load, but the load is absorbing VArs while the generator is generating VARS. These conventions and interpretations are summarized in Table 2.4.
Note that

and (2.6)

where is the power factor.
Remember that phasors apply only when the voltage and currents are purely sinusoidal, and this expression for power factor is meaningless if either the voltage or current waveform is non-sinusoidal. A more general expression for power factor with non-sinusoidal current and waveforms is
(2.7)
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Basic source/load relationships

noreply@blogger.com (Sajib Barua) — Fri, 08 Mar 2013 05:22:00 +0000

Fault level and circuit-breaker ratings

The fault level (sometimes called short-circuit level) is a term used to describe the 'strength' of a power supply: that is, its ability to provide both current and voltage. It is defined as

Fault level = Open-circuit voltage × Short-circuit current [VA/phase]

The fault level provides a single number that can be used to select the size of circuit-breaker needed at a particular point in a power system. Circuit-breakers must interrupt fault currents (i.e. the current that flows if there is a short-circuit fault). When the contacts of the circuit-breaker are separating, there is an arc which must be extinguished (for example, by a blast of compressed air). The difficulty of extinguishing the arc depends on both the current and the system voltage. So it is convenient to take the product of these as a measure of the size or 'power' of the circuit-breaker that is needed. The fault level is used for this. The rating of a circuit-breaker should always exceed the fault level at the point where the circuit-breaker is connected –otherwise the circuit-breaker might not be capable of interrupting the fault current. This would be very dangerous: high-voltage circuit-breakers are often the final means of protection, and if they fail to isolate faults the damage can be extreme – it world like having a lightning strike that did not switch itself off.

Fig. 2.1 Thevenin equivalent circuit of one phase of a supply system (neglecting resistance).

Thevenin equivalent circuit model of a power system

At any point where a load is connected to a power system, the power system can be represented by a Thevenin equivalent circuit (The Thevenin equivalent circuit is a series equivalent circuit, in which the source is a voltage source and it is in series with the internal impedance. In the Notion equivalent circuit, the source is a current source in parallel with the internal impedance) having an open-circuit voltage E and an internal impedance Z_s = R_s +jX_s (see Figure 2. 1). Usually X_s is much bigger than R_s and Z_s is approximately equal to jX_s (as in the diagrams). The short-circuit current is I_sc ≈ E/X_s and the short-circuit level is EI_sc = E²/X_s in each phase. The short-circuit level is measured in volt-amperes, VA (or kVA or MVA), because E and I_sc are almost in phase quadrature.

Loads and phasor diagrams

A resistive load R on an AC power system draws power and produces a phase angle shift 6 between the terminal voltage V and the open-circuit voltage E. 6 is called the load angle (see Figure 2.2). The voltage drop across the Thevenin equivalent

Fig. 2.2 Resistive load. (a) circuit diagram; and (b) phasor diagram.

Fig. 2.3 Purely inductive load. (a) circuit diagram; and (b) phasor diagram.

Fig. 2.4 Purely capacitive load. (a) circuit diagram; and (b) phasor diagram.

impedance is jX_sI which is orthogonal to the terminal voltage V( = RI). Because of the orthogonality, V does not fall very much below E, even though X_sI might be a sizeable fraction of E. Note that the power factor angle φ is zero for resistive loads; φ is the angle between V and I (It is assumed that the AC voltage and current are sinewaves at fundamental frequency, so+ is the phase angle at this frequency).

A purely inductive load draws no power and produces no phase-angle shift between V and E: i.e. δ = 0 (see Figure 2.3). The terminal voltage V is quite sensitive to the inductive load current because the volt-drop jX_sl is directly in phase with both E and V. You might ask, 'what is the use of a load that draws no power?' One example is that shunt reactors are often used to limit the voltage on transmission and distribution systems, especially in locations remote from tap-changing transformers or generating stations. Because of the shunt capacitance of the line, the voltage tends to rise when the load is light (e.g. at night). By connecting an inductive load (shunt reactor), the voltage can be brought down to its correct value. Since the reactor is not drawing any real power (but only reactive power), there is no energy cost apart from a small amount due to losses in the windings and core.

A purely capacitive load also draws no power and produces no phase-angle shift between V and R i.e. δ = 0. The system volt-drop jX_sl is directly in anti-phase with E and V, and this causes the terminal voltage V to rise above E. Again you might ask 'what is the use of a load that draws no power?' An example is that shunt capacitors are often used to raise the voltage on transmission and distribution systems, especially in locations remote from tap-changing transformers or generating stations. Because of the series inductance of the line, the voltage tends to fall when the load is heavy (e.g. mid-morning), and this is when shunt capacitors would be connected.

Fig. 2.5 Symmetrical system.

Shunt reactors and capacitors are sometimes thyristor-controlled, to provide rapid response. This is sometimes necessary near rapidly-changing loads such as electric arc furnaces or mine hoists. Of course the use of thyristors causes the current to contain harmonics, and these must usually be filtered.

The symmetrical system

The symmetrical system is an important example – indeed the simplest example – of an interconnected power system, Figure 2.5. It comprises two synchronous machines coupled by a transmission line. It might be used, for example, as a simple model of a power system in which the main generating stations are at two locations, separated by a transmission line that is modelled by a simple inductive impedance JX. The loads (induction motors, lighting and heating systems, etc., are connected in parallel with the generators, but in the simplest model they are not even shown, because the power transmission system engineer is mostly concerned with the power flow along the line, and this is controlled by the prime-movers at the generating stations (i.e. the steam turbines, water turbines, gas turbines, wind turbines etc.).

Although the circuit diagram of a symmetrical system just looks like two generators connected by an inductive impedance, power can flow in either direction. The symmetrical system can be used to derive the power flow equation, which is one of the most important basic equations in power system operation. If E_s and E_r are the open-circuit voltages at the two generators, then

(2.1)

where δ is the phase angle between the phasors E_s and E_r. Note that in Figure 2.5 there are two power factor angles: φ_s between E_s and I at the sending end, and φ_r between E, and I at the receiving end.

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Power systems engineering - fundamental concepts

noreply@blogger.com (Sajib Barua) — Fri, 08 Mar 2013 04:40:00 +0000

Reactive power control

In an ideal AC power system the voltage and frequency at every supply point would be constant and free from harmonics, and the power factor would be unity. In particular these parameters would be independent of the size and characteristics of consumers' loads. In an ideal system, each load could be designed for optimum performance at the given supply voltage, rather than for merely adequate performance over an unpredictable range of voltage. Moreover, there could be no interference between different loads as a result of variations in the current taken by each one (Miller, 1982).

In three-phase systems, the phase currents and voltages must also be balanced ( Unbalance causes negative-sequence current which produces a backward-rotating field in rotating AC machines, causing torque fluctuations and power loss with potential overheating). The stability of the system against oscillations and faults must also be assured. All these criteria add up to a notion of power quality. A single numerical definition of power quality does not exist, but it is helpful to use quantities such as the maximum fluctuation in rms supply voltage averaged over a stated period of time, or the total harmonic distortion (THD), or the 'availability' (i.e. the percentage of time, averaged over a period of, say, a year, for which the supply is uninterrupted).

The maintenance of constant frequency requires an exact balance between the overall power supplied by generators and the overall power absorbed by loads, irrespective of the voltage. However, the voltage plays an important role in maintaining the stability of power transmission, as we shall see. Voltage levels are very sensitive to the flow of reactive power and therefore the control of reactive power is important. This is the subject of reactive compensation. Where the focus is on individual loads, we speak of load compensation, and this is the main subject of this chapter along with several related fundamental topics of power systems engineering.

Load compensation is the management of reactive power to improve the quality of supply at a particular load or group of loads. Compensating equipment — such as power-factor correction equipment—is usually installed on or near to the consumer's premises. In load compensation there are three main objectives:

power-factor correction
improvement of voltage regulation ('Regulation' is an old-fashioned term used to denote the variation of voltage when current is drawn from the system)
load balancing.

Power-factor correction and load balancing are desirable even when the supply voltage is 'stiff': that is, even when there is no requirement to improve the voltage regulation. Ideally the reactive power requirements of a load should be provided locally, rather than drawing the reactive component of current from a remote power station. Most industrial loads have lagging power factors; that is, they absorb reactive power. The load current therefore tends to be larger than is required to supply the real power alone. Only the real power is ultimately useful in energy conversion and the excess load current represents a waste to the consumer, who has to pay not only for the excess cable capacity to carry it, but also for the excess Joule loss in the supply cables. When load power factors are low, generators and distribution networks cannot be used at full efficiency or full capacity, and the control of voltage throughout the network can become more difficult. Supply tariffs to industrial customers usually penalize low power-factor loads, encouraging the use of power-factor correction equipment.

In voltage regulation the supply utilities are usually bound by statute to maintain the voltage within defined limits, typically of the order of 15% at low voltage, averaged over a period of a few minutes or hours. Much more stringent constraints are imposed where large, rapidly varying loads could cause voltage dips hazardous to the operation of protective equipment, or flicker annoying to the eye.

The most obvious way to improve voltage regulation would be to 'strengthen' the power system by increasing the size and number of generating units and by making the network more densely interconnected. This approach is costly and severely constrained by environmental planning factors. It also raises the fault level and the required switchgear ratings. It is better to size the transmission and distribution system according to the maximum demand for real power and basic security of supply, and to manage the reactive power by means of compensators and other equipment which can be deployed more flexibly than generating units, without increasing the fault level.

Similar considerations apply in load balancing. Most AC power systems are three-phase, and are designed for balanced operation. Unbalanced operation gives rise to components of current in the wrong phase-sequence (i.e. negative- and zero-sequence components). Such components can have undesirable effects, including additional losses in motors and generating units, oscillating torque in AC machines, increased ripple in rectifiers, malfunction of several types of equipment, saturation of transformers, and excessive triplen harmonics and neutral currents [Triplen (literally triple-n) means harmonies of order 3n, where n is an integer].

The harmonic content in the voltage supply waveform is another important measure in the quality of supply. Harmonics above the fundamental power frequency are usually eliminated by filters. Nevertheless, harmonic problems often arise together with compensation problems and some types of compensator even generate harmonics which must be suppressed internally or filtered.

The ideal compensator would:

supply the exact reactive power requirement of the load;
present a constant-voltage characteristic at its terminals; and
be capable of operating independently in the three phases.

In practice, one of the most important factors in the choice of compensating equipment is the underlying rate of change in the load current, power factor, or impedance. For example, with an induction motor running 24 hours/day driving a constant mechanical load (such as a pump), it will often suffice to have a fixed power-factor correction capacitor. On the other hand, a drive such as a mine hoist has an intermittent load which will vary according to the burden and direction of the car, but will remain constant for periods of one or two minutes during the travel. In such a case, power-factor correction capacitors could be switched in and out as required. An example of a load with extremely rapid variation is an electric arc furnace, where the reactive power requirement varies even within one cycle and, for a short time at the beginning of the melt, it is erratic and unbalanced. In this case a dynamic compensator is required, such as a TCR or a saturated-reactor compensator, to provide sufficiently rapid dynamic response.

Steady-state power-factor correction equipment should be deployed according to economic factors including the supply tariff, the size of the load, and its uncompensated power factor. For loads which cause fluctuations in the supply voltage, the degree of variation is assessed at the 'point of common coupling' (PCC), which is usually the point in the network where the customer's and the supplier's areas of responsibility meet: this might be, for example, the high-voltage side of the distribution transformer supplying a particular factory.

Loads that require compensation include arc furnaces, induction furnaces, arc welders, induction welders, steel rolling mills, mine winders, large motors (particularly those which start and stop frequently), excavators, chip mills, and several others. Non-linear loads such as rectifiers also generate harmonics and may require harmonic filters, most commonly for the 5th and 7th but sometimes for higher orders as well. Triplen harmonics are usually not filtered but eliminated by balancing the load and by trapping them in delta-connected transformer windings.

The power-factor and the voltage regulation can both be improved if some of the drives in a plant are synchronous motors instead of induction motors, because the synchronous motor can be controlled to supply (or absorb) an adjustable amount of reactive power and therefore it can be used as a compensator. Voltage dips caused by motor starts can also be avoided by using a 'soft starter', that is, a phase-controlled thyristor switch in series with the motor, which gradually ramps the motor voltage from a reduced level instead of connecting suddenly at full voltage.

Load	Limits of voltage fluctuation
Large motor starts	1-3% depending on frequency
Mine winders, excavators, large motor drives	1-3% at distribution voltage level 0.5 – 1.5% at transmission voltage level
Welding plant	1/4-2% depending on frequency
Induction furnaces	Up to 1%
Arc furnaces	<1/2%

Table 2.1 Typical voltage fluctuation standards.

Continuous and short-time reactive power requirements.
Rated voltage and limits of voltage variation.
Accuracy of voltage regulation required.
Response time of the compensator for a specified disturbance.
Maximum harmonic distortion with compensator in service.
Performance with unbalanced supply voltages and/or with unbalanced load.
Environmental factors: noise level; indoor/outdoor installation; temperature, humidity, pollution, wind and seismic factors; leakage from transformers, capacitors, cooling systems.
Cabling requirements and layout; access, enclosure, grounding; provision for future expansion; redundancy and maintenance provisions.
protection arrangements for the compensator and coordination with other protection systems, including reactive power limits if necessary.
Energization procedure and precautions.

Table 2.2 Factors to consider in specifying compensating equipment.

Standards for the quality, of supply. One very noticeable effect of supply voltage variations is flicker especially in tungsten filament lamps. Slow variations of up to 3% may be tolerable, but rapid variations within the range of maximal visual sensitivity (between I and 25 Hz) must be limited to 0.25% or less. A serious consequence of undervoltage is the overcurrent that results from the fact that AC motors run at a speed which is essentially determined by the frequency, and if the voltage is low the current must increase in order to maintain the power. On the other hand, overvoltage is damaging to insulation systems.

Table 2.1 gives an idea of the appropriate standards which might be applied in different circumstances, but local statutes and conditions should be studied in each individual case.

Specification of a load compensator. Some of the factors which need to be considered when specifying a load compensator are summarized in Table 2.2.

Conventions used in power engineering

In power engineering it is helpful to have a set of conventions for symbols. Unfortunately many people disregard conventions, and this causes confusion. There is no universal standard, but the simple conventions given in Table 2.3 are widely used, practical, and consistent with most classic textbooks.

Type

What is meant

Examples

Lower-case italic

Instantaneous values

V, i

Upper-case italic

RMS values or DC values

Resistance, reactance, and impedance magnitude

Inductance and capacitance

V, I

R, X Z

L, C

Upper-case boldface roman

Phasors
Impedance

V, I
Z

Table 2.3 Font and symbol conventions

In handwritten work, you can't really use boldface, so use a bar or arrow or tilde — preferably over the symbol, e.g.Subscripts can be roman or italic; it is a matter of style

In three-phase systems, various conventions are used for the subscripts used to denote the three phases. In Europe (particularly Germany): U, V, W. In the UK: R, Y, B (for red, yellow, blue), or a, b, c. In the United States: a, b, c or A, B, C. You will also see I, 2, 3 used: this seems an obvious choice, but if you are working with symmetrical components these subscripts can be confused with the positive, negative, and zero-sequence subscripts I, 2, 0 (sometimes +, -, 0). The best advice is to be very careful!

Never confuse phasor values with scalar values!

	Examples
Typeset	Comment
V=RI	RMS AC; or DC
V=jXI	V and I are phasores X is a scalar (reactance) jX is an impedance (complex)
Z=R+jX	Z is complex (impedance) R is scalar (resistance) X is scalar (reactance)
v=V_mcosωt	v is an instantaneous value V_m is a fixed scalar value

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Random nature of system load

noreply@blogger.com (Sajib Barua) — Fri, 08 Mar 2013 04:15:00 +0000

The system load varies continuously with time in a random fashion. Significant changes occur from hour to hour, day to day, month to month and year to year (Gross and Galiana, 1987). Figure 1.20 shows a typical load measured in a distribution substation for a period of four days.

The random nature of system load may be included in power flow studies and this finds useful applications in planning studies and in the growing 'energy stock market'. Some possible approaches for modelling random loads within a power flow study are:

modelling the load as a distribution function, e.g. normal distribution;
future load is forecast by means of time series analysis based on historic values, then normal power flow studies are performed for each forecast point;
the same procedure as in two but load forecasting is achieved using Neural Networks.

Non-linear loads

Many power plant components have the ability to draw non-sinusoidal currents and, under certain conditions, they distort the sinusoidal voltage waveform in the power network. In general, if a plant component is excited with sinusoidal input and produces non-sinusoidal output, then such a component is termed non-linear, otherwise, it is termed linear (Atha and Madrigal, 2001). Among the non-linear power plant components we have:

Fig. 1.20 A typical load measured at a distribution substation.

power electronics equipment
electric arc furnaces
large concentration of energy saving lamps
saturated transformers
rotating machinery.

Some of the more common adverse effects caused by non-linear equipment are:

the breakdown of sensitive industrial processes
permanent damage to utility and consumer equipment
additional expenditure in compensating and filtering equipment
loss of utility revenue
additional losses in the network
overheating of rotating machinery
electromagnetic compatibility problems in consumer installations
interference in neighboring communication circuits
spurious tripping of protective devices.

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Power Network fault studies

noreply@blogger.com (Sajib Barua) — Thu, 07 Mar 2013 17:45:00 +0000

If it is assumed that the power network is operating in steady state and that a sudden change takes place due to a faulty condition, then the network will enter a dynamic state. Faults have a variable impact over time, with the highest values of current being present during the first few cycles after the disturbance has occurred. This can be appreciated from Figure 1.19, where a three-phase, short-circuit at the terminals of a synchronous generator give rise to currents that clearly show the transient and steady (sustained) states. The figure shows the currents in phases a, b and c, as well as the field current. The source of this oscillogram is (Kimbark, 1995).

Faults are unpredictable events that may occur anywhere in the power network. Given that faults are unforeseen events, strategies for dealing with them must be decided well in advance (Anderson, 1973). Faults can be divided into those involving a single (nodal) point in the network, i.e. shunt faults, and those involving two points in one or more phases in a given plant component, i.e. series faults. Simultaneous faults involve any combination of the above two kinds of faults in one or more locations in the network. The following are examples of shunt faults:

three-phase-to-ground short-circuit
one-phase-to-ground short-circuit
two-phase short-circuit
two-phase-to-ground short-circuit.

The following are examples of series faults:

one-phase conductor open
two-phase conductors open
three-phase conductors open.

In addition to the large currents flowing from the generators to the point in fault following the occurrence of a three-phase short-circuit, the voltage drops to extremely low values for the duration of the fault. The greatest voltage drop takes place at the point in fault, i.e. zero, but neighbouring locations will also be affected to varying degrees. In general, the reduction in root mean square (rms) voltage is determined by the electrical distance to the short-circuit, the type of short-circuit and its duration.

The reduction in runs voltage is termed voltage sag or voltage dip. Incidents of this are quite widespread in power networks and are caused by short-circuit faults, large motors starting and fast circuit breaker reclosures. Voltage sags are responsible for spurious tripping of variable speed motor drives, process control systems and computers. It is reported that large production plants have been brought to a halt by sags of 100 ms duration or less, leading to losses of hundreds of thousands of pounds (McHattie, 1998). These kinds of problems provided the motivation for the development of Custom Power equipment (Hingorani, 1995).

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Snapshot-like power network studies

noreply@blogger.com (Sajib Barua) — Thu, 07 Mar 2013 17:35:00 +0000

Power flow studies

Although in reality the power network is in a continuous dynamic state, it is useful to assume that at one point the transient produced by the last switching operation or topology change has died out and that the network has reached a state of equilibrium, i.e. steady state. This is the limiting case of long-term dynamics and the time frame of such steady state operation would be located at the far right-hand side of Figure 1.16. The analysis tool used to assess the steady state operation of the power system is known as Load Flow or Power Flow (Arrillaga and Watson, 2001), and in its most basic form has the following objectives:

to determine the nodal voltage magnitudes and angles throughout the network;
to determine the active and reactive power flows in all branches of the network;
to determine the active and reactive power contributed by each generator;
to determine active and reactive power losses in each component of the network.

In steady state operation, the plant components of the network are described by their impedances and loads are normally recorded in MW and MVAr. Ohm's law and Kirchhoff's laws are used to model the power network as a single entity where the nodal voltage magnitude and angle are the state variables. The power flow is a non-linear problem because, at a given node, the power injection is related to the load impedance by the square of the nodal voltage, which itself is not known at the beginning of the study. Thus, the solution has to be reached by iteration. The solution of the non-linear set of algebraic equations representing the power flow problem is achieved efficiently using the Newton-Raphson method. The generators are represented as nodal power injections because in the steady state the prime mover is assumed to drive the generator at a constant speed and the AVR is assumed to keep the nodal voltage magnitude at a specified value.

Flexible alternating current transmission systems equipment provides adaptive regulation of one or more network parameters at key locations. In general, these controllers are able to regulate either nodal voltage magnitude or active power within their design limits. The most advanced controller, i.e. the UPFC, is able to exert simultaneous control of nodal voltage magnitude, active power and reactive power. Comprehensive models of FACTS controllers suitable for efficient, large-scale power flow solutions have been developed recently (Fuerte-Esquivel, 1997).

Optimal power flow studies

An optimal power flow is an advanced form of power flow algorithm. Optimal power flow studies are also used to determine the steady state operating conditions of power networks but they incorporate an objective function which is optimized without violating system operational constraints. The choice of the objective function depends on the operating philosophy of each utility company. However, active power generation cost is a widely used objective function. Traditionally, the constraint equations include the network equations, active and reactive power consumed at the load points, limits on active and reactive power generation, stability and thermal limits on transmission lines and transformers. Optimal power flow studies provide an effective tool for reactive power management and for assessing the effectiveness of FACTS equipment from the point of view of steady state operation. Comprehensive models of FACTS controllers suitable for efficient, large-scale optimal power flow solutions have been developed recently (Ambriz-Perez, 1998).

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