PART III ADVANCED TOPICS IN POWER SYSTEM DYNAMICS
10.5.3 Control Based on Local Measurements
The control strategy given by Equation (10.19) is based on the state variablesδandω. As these quantities are not normally available at the shunt element busbar, the practical implementation of the control must be based on other signals that can be measured locally. How exactly such a local
2 3 1
6 7 8 4 5
δ′
Pm P
Figure 10.14 Interpretation of the control strategy using the equal area criterion.
control emulates the state-variable control depends on the choice of the measured quantities and the structure of the controller.
10.5.3.1 Dynamic Properties of Local Measurements
LetqGandqBbe some quantities used as input signals to the shunt element controller. In the control strategy given by Equation (10.19) the shunt admittance depends on the rotor speed deviationω.
If the magnitude of the transient emf is assumed constant (classical model), the derivative with respect to time of any electric quantityqGcan be expressed as
dqG
dt = ∂qG
∂δ ω+αGG
dGsh
dt +αGB
dBsh
dt , (10.26)
where the coefficients
αGG= ∂qG
∂Gsh
, αGB= ∂qG
∂Bsh
, (10.27)
determine the sensitivity of qG to a change in the control variablesGsh(t) andBsh(t). Equation (10.26) gives
ω∂qG
∂δ = dqG
dt −αGG
dGsh
dt −αGB
dBsh
dt . (10.28)
If the sensitivity coefficients αGG andαGB are known, the right hand side of Equation (10.28) can be computed in real time and used to determine a signal proportional to the rotor speed deviation necessary for the control of Gsh(t). Comparing the right hand side of the first of the equations in (10.19) with the left hand side of Equation (10.28) shows that the signal obtained from Equation (10.28) is the same as the state-variable control signal if
∂qG
∂δ =
b(ξ+cosδ)
XSHC. (10.29)
Substitution of the right hand side of the first equation in (10.19) by Equation (10.28) gives the following control principle:
Gsh(t)=K dqG
dt −αGG
dGsh
dt −αGB
dBsh
dt
. (10.30)
This means that if a measured quantityqG satisfies the condition in Equation (10.29) then the modulation controller need simply differentiateqGwith respect to time and subtract from the result values proportional to the rate of change of the controlled variablesBsh(t) andGsh(t).
The control principle for the shunt susceptance can be obtained in a similar way as Bsh(t)=K
dqB
dt −αBG
dGsh
dt −αBB
dBsh
dt
, (10.31)
where the coefficients
αBG= ∂qB
∂Gsh
, αBB= ∂qB
∂Bsh
, (10.32)
determine the sensitivity ofqBto changes in the control variablesBsh(t) andGsh(t). Comparison with the second of the equations in (10.19) shows that the input quantityqB should satisfy the following condition:
∂qB
∂δ = bsinδ
XSHC. (10.33)
It now remains to determine what locally measurable quantities qB and qG will satisfy the conditions defined in Equations (10.29) and (10.33).
10.5.3.2 Voltage-Based Quantities
The current flowing from the network to the shunt element in Figure 10.11 is given by Ish=VshYsh= Eg−Vsh
jXg
+ Vs−Vsh jXs
, (10.34)
whereYsh=Gsh(t)+jBsh(t) andXgandXsare the equivalent reactances denoted in Figure 10.11.
Multiplying the current by the short-circuit reactanceXSHCand movingVshto the left hand side gives
Vsh{[XSHCGsh(t)]+j [1−XSHCBsh(t)]} = EgXs+VsXg
j Xg+Xs
. (10.35)
Substituting for the complex voltages
Eg= Eg(cosδ+j sinδ) and Vs=Vs, (10.36) and multiplying the resulting equation by its conjugate gives, after a little algebra,
Vsh2 = b XSHC
ξ+1ξ+2 cosδ
[XSHCGsh(t)]2+[1−XSHCBsh(t)]2, (10.37) whereξ is the coefficient defined in Equation (10.15). When deriving Equation (10.37), it is also possible to find the phase angleθof the shunt element voltage measured with respect to the infinite bus:
tanθ= 1
ξ +cosδ
XSHCGsh(t)+sinδ[1−XSHCBsh(t)]
sinδXSHCGsh(t)+
1
ξ +cosδ
[1−XSHCBsh(t)]
. (10.38)
The inequalities in Equation (10.5) allow Equations (10.37) and (10.38) to be simplified to Vsh2 ∼=b
ξ+1
ξ +2 cosδ
XSHC; tanθ∼= sinδ
1
ξ+cosδ. (10.39)
Using the first of Equations (10.39), and after differentiating with respect toδ, gives
∂Vsh2
∂δ = −2 bsinδ
XSHC. (10.40)
Calculation of the derivative∂θ/∂δis slightly more difficult. The second of Equations (10.39) may be written as f(θ, δ)=0. Henceθis an implicit function ofδ. The derivative of that function can be calculated from
∂θ
∂δ = −∂
f
∂δ
∂f
∂θ
. (10.41)
Using this equation gives
∂θ
∂δ = ξ+cosδ
ξ+1ξ +2 cosδ. (10.42)
The expression in the denominator of Equation (10.42) is the same as the expression in brackets in the first of the equations in (10.39). Substitution gives
Vsh2∂θ
∂δ =
b(ξ+cosδ)
XSHC. (10.43)
Equations (10.40) and (10.43) show that local measurements of the squared magnitude of the shunt element voltage and its phase angle can give good signals for controlling the shunt ele- ment. Comparing Equations (10.40) and (10.33) shows that the signalVsh2satisfies the condition in Equation (10.33) for the control strategy of the shunt susceptance. Similarly, comparing Equations (10.43) and (10.29) shows thatθsatisfies the condition in (10.29) for the required control of the shunt conductance provided that the derivative is multiplied byVsh2.
A sensitivity analysis of the effect of changes in Vsh2 andθ on the changes in the controlled variablesBsh(t) andGsh(t) can be conducted by evaluating the derivatives in Equations (10.27) and (10.32) using Equations (10.37) and (10.38). This involves a lot of simple, but arduous, algebraic and trigonometric transformations which finally lead to the following simplified formulae:
αGG= ∂qG
∂Gsh
∼= −XSHC, αGB= ∂qG
∂Bsh
∼=0
(10.44) αBG= ∂qB
∂Gsh
∼=0, αBB= ∂qB
∂Bsh
∼= −Vsh2XSHC.
Zero values of αGB andαBG signify that changes in the shunt susceptance/conductance have a negligibly small effect on the given quantity.
10.5.3.3 Control Schemes
Substituting Equations (10.44) into Equations (10.30) and (10.31) and taking the squared voltage magnitude and the voltage phase angle as control signals yields
Gsh(t)=K Vsh2 dθ
dt +XSHC
dGsh
dt
, (45a)
Bsh(t)= K
−1 2
d Vsh2
dt +XSHCVsh2
dBsh
dt
. (45b)
– (b)
(a) θ f
K
K
Bsh(t) Gsh(t)
Vsh2 Vsh2
XSHC Ts
Ts 1
Ts Ts 1 Ts
Ts 1
1 2
Figure 10.15 Modulation controller employing the frequency–voltage control scheme for: (a) Gsh(t); (b)Bsh(t).
The rotor angle δ and the voltage phase angleθ are measured with respect to the infinite busbar voltage or the synchronous reference frame. As the derivative ofθwith time is equal to the deviation of the local frequency, that is dθ/dt=2πf, control (10.45) is referred to asfrequency- and voltage-based control.
Figure 10.15 shows the block diagram of the appropriate control circuits. Differentiation has been replaced by a real differentiating element with a small time constantT. The shunt susceptance controller is nonlinear because the output signal in the main feedback loop is multiplied by the main input signal. The shunt conductance controller is linear but its effective gain is modulated by the squared voltage magnitude which is also the main input signal for the shunt susceptance controller. The short-circuit reactanceXSHCplays only a corrective role and its value may be set with a large error. For practical applications its value can be assessed offline and set as a constant parameter.
It is worth noting that the time derivative of the voltage angle dθ/dtis equal to the deviation of the local frequencyf. Thus the proposed shunt element controller is a frequency- and voltage- orientated controller. The input signals for the control system may be measured using digital techniques described by Phadke, Thorap and Adamiak (1983) or Kamwa and Grondin (1992).