System studies
3.5 Stability studies
A power system is at equilibrium when the voltage magnitude and angle at each bus are such that power flows from buses where there is an excess of generation over demand to buses where demand exceeds supply. At each bus, there is thus a balance between the power generated, the power consumed and the power transmitted to and from other buses. Power flow programs calculate this equilibrium. This balance applies also to the generating units: the mechanical power provided by the prime mover is equal to electrical power produced by the generator, if we ignore the losses. In mechanical terms, this implies that the accelerating torque applied to the shaft by the prime mover is equal to the decelerating torque caused by the production of electrical power in the generator. Since the net torque is zero, the shaft rotates at constant speed. The angular position of the rotor is usually measured in a reference frame that rotates at synchronous speed. It is then called the rotor angle. At equilibrium, the rotor angle is a measure of the amount of power injected by the generator into the network.
In practice, a power system is never in the steady state as the loads and Table 3.3 Zero sequence impedances for the network used for unbalanced fault
studies
All values are per unit
System studies 77
generations are constantly changing. In most cases, the voltages adjust themselves naturally to restore the equilibrium and the power system is said to be stable with respect to these small perturbations. On the other hand, a major disturbance such as a fault can induce an instability in the system and trigger a sequence of events leading to the collapse of part of the network.
The purpose of stability studies is therefore to verify that the system is designed in such a way that it can withstand the most severe credible disturbance. Conversely, stability studies are also used to determine the operating limits of an existing system. This section first develops a simple model of the mechanical behaviour of a synchronous generating unit.
This generator is then connected in a quasi-radial manner to a very large power system. It is then shown how the equations of the mechanical and electrical subsystems can be used to study the transient stability of the system. Finally, an example shows how these principles can be applied to the study of an embedded generation scheme.
3.5.1 A simple dynamic model of the mechanical subsystem
The mechanical speed of a generator is described by Newton’s equation for rotating masses,
Jd2θm
dt2 = Tm− Te (3.44)
where J is the combined moment of inertia of the generator and its prime mover
θm is the angular position of the shaft
Tm is the accelerating torque applied to the shaft by the prime mover
Te is the reaction torque applied by the generator to the shaft.
The mechanical power supplied by the prime mover is transformed in electrical power by the generator through the interaction of the stator and rotor magnetic fields. As a result of this transformation, a reaction torque is applied by the generator to the shaft. In the steady state, if we neglect the losses due to friction and windage, this reaction torque is equal and opposite to the prime mover torque and eqn. (3.44) shows that the shaft speed remains constant.
To study the interactions between the mechanical and electrical parts of the system, it is convenient to modify eqn. (3.44) in several ways. First, if we multiply both sides of eqn. (3.44) by the mechanical speed, the right-hand side will be expressed in terms of power rather than torque:
Jωm
d2θm
dt2 = Pm− Pe (3.45)
78 Embedded generation
Secondly, since we are concerned about deviations between the actual speed and the synchronous speed, it is convenient to measure the angular position of the rotor with respect to a synchronously rotating reference frame. We define
Finally, electrical and mechanical angles are related by the number of poles in the generator windings,
δ =N
2δm (3.48)
Replacing in eqn. (3.47), we obtain 2 NJωm
d2δ
dt2= Pm− Pe (3.49)
To normalise this equation, both sides are usually divided by the MVA or kVA rating of the generator SB:
2 The moment of inertia is then expressed in terms of the inertia constant of the generating unit, which is defined as the ratio of the kinetic energy stored at synchronous speed to the generator kVA or MVA rating,
H=–¹
²Jωmsyn2
SB (3.51)
Extracting J from eqn. (3.51) and replacing in eqn. (3.50), we get 2 Normalising the angular frequencies using
ωsyn=N
2 ωmsyn (3.53)
and
ωpu= ωm
ωmsyn (3.54)
we obtain the final version of what is often called the swing equation, 2H
ωsyn
ωpud2δ
dt2= Pmpu− Pepu (3.55) System studies 79
3.5.2 Power transfer in a two-bus system
Consider the simple circuit shown on Figure 3.25. The active and reactive powers injected by the ideal voltage source E into the system are given by:
Pe= Re(S) = Re(E I*) (3.56)
Qe= Im(S) = Im(E I*)
The current I¯ can be found using Kirchhoff’s voltage law, I¯=E− V
jX =E∠ δ − V ∠ 0°
jX =E∠ (δ − 90°) − V ∠ (−90°)
X (3.57)
Substituting eqn. (3.57) into eqn. (3.56), we obtain Pe= Re⎧
Consider now the simple two-bus power system shown in Figure 3.26.
A single generator at bus A is connected to the rest of the power system by two parallel lines. Bus B is represented as connected to a brick wall to suggest that it is part of a very large power system. This system is assumed to have a much larger inertia than the generator connected at bus A. Furthermore, its Thevenin impedance as seen from bus B is assumed to be much smaller than the impedances of the lines connecting A and B and the internal impedance of generator G. Consequently, the magnitude and angle of the voltage at bus B can be assumed to be
Figure 3.25 Two-source network 80 Embedded generation
constant, i.e. unaffected by what happens to the left of bus B. These assumptions are traditionally summarised by calling bus B an infinite bus.
Figure 3.26 also shows the equivalent circuit of this simple network.
As discussed above, the power system beyond bus B can be represented by an ideal voltage source V= V∠0°, which will be taken as the reference for the angles. Generator G is represented by an ideal voltage source E= E∠δ in series with a reactance XS. The two transmission lines are assumed to have purely reactive impedances jXL. Eqn. (3.59) is directly applicable to the system of Figure 3.26 if these various impedances are combined into a single equivalent impedance,
X0= XS+ –¹²XL (3.60)
Figure 3.27 shows how the power injected by the generator into the network varies as a function of the angle δ for given values of E and V.
The most important conclusion to be drawn from this graph is that the amount of power transferred increases with the angle δ until this angle reaches 90 degrees, where it reaches a maximum value
Pmax=EV X0
If we neglect the losses in the generator, the mechanical power supplied Figure 3.26 Two-bus power system for stability studies and its equivalent circuit
System studies 81
by the prime mover is equal (in the steady state) to the electrical power injected by the generator in the network,
P0m= P0e (3.61)
Eqn. (3.59) or Figure 3.27 can thus be used to find the steady-state angle δ° at which the generator operates given the mechanical power.
The reader will undoubtedly have noticed that δ has been used to represent both the mechanical oscillations of the rotor and the angle of the phasor E representing the internal emf of the generator. This choice is deliberate, as these two angles are identical. The internal emf E is induced in the stator windings of the synchronous generator by the rota-tion of the rotor flux created by the field winding. It is therefore tied to the position of this rotor. Changes in electrical and mechanical angles are therefore rigidly linked. Any deviation in the position of the rotor alters the amount of electrical power produced by the generator. Con-versely, any change in the electrical power flows causes mechanical transients.
Figure 3.27 Power transfer curve in the two-bus power system 82 Embedded generation
3.5.3 Electro-mechanical transients following a fault
Consider again the two-bus system shown in Figure 3.26. Suppose that a fault occurs on one of the lines connecting buses A and B, very close to bus A. To analyse the dynamic behaviour of this system, we must consider three periods: before the fault, during the fault and after the fault.
Before the fault, generator G operates in the steady state and injects an electrical power Pe0 into the network. Since the mechanical power Pm0
delivered by the prime mover is equal to the electrical power injected in the network, eqn. (3.55) shows that the rotor angle δ0 is constant.
During the fault, the voltage at bus A is essentially equal to zero. Eqn.
(3.59) shows that the active power injected by the generator into the system is therefore also equal to zero. There is thus no longer a balance between the accelerating power provided by the prime mover and the decelerating power resulting from the injection of electrical power into the network. The rotor of the generator accelerates and eqn. (3.55) shows that the rotor angle δ increases.
After an interval of time called the clearing time, the protection system detects the faults and triggers the opening of the breakers at both ends of the faulted line, clearing the fault. At that point, the voltage at bus A is restored to a non-zero value and the network is again capable of trans-mitting the active power produced by the generator. Since the generator is again transforming mechanical power into electrical power, it applies a decelerating torque to the shaft. If this decelerating torque is applied soon enough, it will succeed in slowing and reversing the increase in the rotor angle δ. Stability will have been maintained. Otherwise, this angle will continue to increase uncontrollably until the protection system of the generator opens the generator breaker to prevent damage to the plant. Stability has been lost.
The purpose of stability studies is thus to determine whether all cred-ible faults will be cleared quickly enough to maintain stability. If the clearing time is given, these studies can be used to determine the max-imum load that the network can handle without causing instability in the event of a fault.
3.5.4 The equal area criterion
The equal area criterion is a simple graphical method for determining whether a one-machine infinite bus system will remain stable. It provides a useful representation of the factors that affect stability. In practical systems, it may also be used to obtain a first approximation of the stability limit.
System studies 83
This criterion is based on the application of the power transfer curves and is illustrated by Figure 3.28. Before the fault, the generator operates along the power transfer curve defined by the generator internal emf E, the infinite bus voltage V and the equivalent impedance X0 given by eqn.
(3.60). The generator angle and the mechanical power input are related by
P0m=EV
X0 sinδ° (3.62)
During the fault, there is no decelerating electrical torque and the rotor accelerates, increasing the angle δ. By the time the fault is cleared at tclear, this angle has reached the value δclear. The kinetic energy stored by the rotor is proportional to the area labelled A1 in Figure 3.28. After the fault clears, the voltage at bus A is restored, and power can again be transmitted between the generator and the infinite bus. However, this transmission takes place on a new power transfer curve because the equivalent impedance between the internal emf and the infinite bus is now
X1= XS+ XL (3.63)
because one of the transmission lines has been taken out of service. Since X1> X0, this postclearing fault is below the prefault curve. Note that for δ = δclear, the electrical power transferred on this new curve is larger than the mechanical power applied by the prime mover. The decelerating
Figure 3.28 Equal area criterion 84 Embedded generation
torque is thus larger than the accelerating torque and the rotor starts to slow down. However, the rotor angle δ continues to increase for a while because of the kinetic energy stored during the fault. The equal area criterion states that δ will increase to a value δmaxsuch that the area A1 is equal to the area A2. This criterion is demonstrated in the Appendix (Section 3.8).
Figure 3.28 shows that, if δ reaches δlimit before the equal area criter-ion is satisfied, the electrical power absorbed by the network becomes smaller than the mechanical power provided by the prime mover. Eqn.
(3.55) shows that the rate of increase in δ will then again be positive and stability will be irretrievably lost. On the other hand, if δmax is less than δlimit,δ starts decreasing and stability is maintained. If the damp-ing caused by the electrical and mechanical losses was taken into account, it could be shown that δ settles, after damped oscillations, at the value δ1.
The equal area criterion suggests that the stability of the system is enhanced if the area A1 is reduced or if the potential area A2 is increased.
This can be achieved in several ways:
1 Reducing δclear decreases the amount of kinetic energy stored during the fault and provides a larger angular margin (δlimit− δclear). A smaller δclear requires a shorter clearing time (i.e. a faster protection system and faster breakers) or a larger generator inertia.
2 Reducing Pm0 also decreases the amount of kinetic energy imparted to the generator during the fault. It also increases the potential energy that the system can absorb after the clearing. Unfortunately, reducing Pm0 implies putting a limit on the amount of electrical power supplied by the generator.
3 Reducing the system impedance decreases the prefault angle δ°, increases the angular margin (δlimit− δclear) and increases the amount of potential energy that the system can absorb after the clearing. A reduction in impedance can be achieved by connecting the generator to the rest of the system through a line operating at a higher nominal voltage.
4 Operating at higher voltages has, in the steady state, the effect of reducing the prefault angle δ°. Boosting the excitation of the generator to increase the internal emf E after the fault clearing also increases the amount of potential energy that the system can absorb.
3.5.5 Stability studies in larger systems
While the analysis presented above provides useful insights into the mechanisms leading to transient instability in power systems, it relies on a series of simplifying assumptions that may not be justified in an actual power system:
1 Modelling the system as a single generator connected to an infinite bus may not be acceptable if several embedded generators are connected to a relatively System studies 85
weak network. It may be necessary to model the dynamic interactions between these generators or between one of these generators and large rotat-ing loads.
2 The generator was modelled as a constant voltage behind a single reactance.
This very simple model does not reflect the complexity of the dynamics of synchronous generators and may give a misleading measure of the system stability. In particular, it neglects the stabilising effect of the generator’s exci-tation system.
3 Since electrical and mechanical losses have been neglected, the system has no damping. This approximation distorts the oscillations taking place after a fault and overstates the risk of instability.
4 Finally, it may be necessary to model the effect of faults at different locations in the network rather than simply at the terminals of the generator.
Removing these limitations requires the use of much more complex models for the generators, for the network and for the controllers.
Unfortunately, the equal area criterion no longer holds under these conditions and another stability assessment method must be imple-mented.
Most commercial-grade stability assessment programs rely on the numerical solution of the differential and algebraic equations describing the power system. The solution of these equations tracks the evolution of the system following a disturbance. If it shows the rotor angle of one or more generators drifting away from the angles of the rest of the generators, the system is deemed unstable. On the other hand, if all variables settle into an acceptable new steady state, the system is con-sidered stable.
The model of each generating unit contains between two and eight non-linear differential equations, not counting those required for model-ling the excitation system. The differential equations of all the generators are coupled through the algebraic equations describing the network. This coupling requires the solution of a power flow at each time step of the solution of the differential equations.
3.5.6 Stability of induction generators
Historically, induction generators have not been significant in large power systems and so they have not been represented explicitly in many power system analysis programs. The behaviour of large induction motors can be important in transient stability studies, particularly of industrial systems, e.g. oil facilities, and so induction motor models are usually included in transient stability programs. These can be used to give a representation of induction generators merely by changing the sign of the applied torque. However, the models used for these gener-ators have often been based on a representation of a voltage source 86 Embedded generation
behind a transient reactance similar to that used for simple transient modelling of synchronous generators [8]. The main simplifying assumption is that stator electrical transients are neglected and there is no provision for saturation of the magnetic circuits to be included.
Such models may not be reliable for operation at elevated voltages or for investigating rapid transients such as those due to faults. More sophisti-cated models of induction generators are available in electromagnetic transient programs or can be found in advanced textbooks [9, 10].
3.5.7 Application to an embedded generation scheme
To illustrate the concepts discussed in the previous sections, we will consider the small system shown in Figure 3.29. Let us first assume that faults at bus D are cleared in 100 ms and that the generator at bus E is producing 20 MW. Figure 3.30 shows the oscillations in the rotor angle of this generator following such a fault at bus D. The generator initially accelerates and the rotor angle reaches a maximum value of approxi-mately 120 degrees 150 ms after the fault. It is clear from the figure that
Figure 3.29 Small system used to illustrate the stability studies
System studies 87
stability is maintained. Let us now suppose that the protection system and the switchgear at bus B are such that faults are cleared after only 200 ms. Figure 3.31 shows that stability would be lost even if the generator was only producing 11.6 MW. 320 ms after the fault, the rotor angle reaches 180 degrees and ‘slips a pole’. At that point, the equipment pro-tecting the generator would immediately take it off-line to protect it from damage. Stability problems could be avoided by further reducing the active power output of the generator. Figure 3.32 illustrates the margin-ally stable case where the generator produces 11.5 MW. The rotor angle reaches a maximum angle of 165 degrees before dropping. Note that this angle will ultimately return to its original steady-state value of about 45 degrees but that this may take some time, as the system is marginally
stability is maintained. Let us now suppose that the protection system and the switchgear at bus B are such that faults are cleared after only 200 ms. Figure 3.31 shows that stability would be lost even if the generator was only producing 11.6 MW. 320 ms after the fault, the rotor angle reaches 180 degrees and ‘slips a pole’. At that point, the equipment pro-tecting the generator would immediately take it off-line to protect it from damage. Stability problems could be avoided by further reducing the active power output of the generator. Figure 3.32 illustrates the margin-ally stable case where the generator produces 11.5 MW. The rotor angle reaches a maximum angle of 165 degrees before dropping. Note that this angle will ultimately return to its original steady-state value of about 45 degrees but that this may take some time, as the system is marginally