**3.2. Post-contingency performance criteria for voltage stability assessment**

In order to effectively plan against steady state and dynamic voltage stability problems under a certain set of contingencies, we need to identify proper performance criteria. Voltage stability of the power system should be assessed based on voltage security criteria of interest to, and accepted by, the utility.

As far as the steady state performance criteria are concerned, there are many criteria such as reactive reserve in different parts of the system, post-contingency voltage, Eigenvalues, etc. that enable to quantify the severity of a contingency with respect to voltage stability. In this work we utilize the most basic and widely accepted criteria, namely, post-contingency voltage stability margin [21, 22] for steady state performance assessments. Voltage stability margin, a steady state performance criterion, is defined as the amount of additional load in a specific pattern of load increase that would cause voltage instability. Contingencies such as unexpected component (generator, transformer, transmission line) outages often reduce the voltage stability margin, and control actions increase it [23, 24], as shown in figure 4 [25].

Role of Induction Motors in Voltage Instability and Coordinated Reactive Power Planning 165

The objective of the reactive power planning problem is to satisfy these minimum performance criteria mentioned in the previous section under various credible contingencies. This is greatly dependent upon the amount, location and type of reactive power sources available in the system, which are all decision variables in the proposed coordinated RPP, as will be dealt in the next section of this chapter. If the reactive power support is far away, or insufficient in size, or too dependent on shunt capacitors (slow acting static device), a relatively normal contingency (such as a line outage or a sudden increase in load) can trigger a large system voltage drop. Hence, in order to properly allocate the reactive power support in terms of optimal type, location and amount, in this work we employ linear sensitivities of performance measure with respect to various control actions. These sensitivities enable obtaining the necessary information, i.e., the sensitivities of performance measure for the type of reactive support, the location and amount of reactive support, which are very useful to perform the proposed coordinated RPP. This section sheds

The sensitivity of voltage stability margin refers to how much the margin changes for a small change in system parameters such as real power and reactive power bus injections, regulated bus voltages, bus shunt capacitance, line series capacitance etc [2, 27]. References

Let the steady state of the power system satisfying a set of equations in the vector form be,

*Fxp* (, , ) 0 

where x is the vector of state variables, p is any parameter in the power system steady state equations such as demand and base generation or the susceptance of shunt capacitors or the reactance of series capacitors, and λ denotes the system load/generation level called the scalar bifurcation parameter. The system reaches a state of voltage collapse, when λ hits its maximum value (the nose point of the system PV curve), and the value of the bifurcation parameter is equal to λ\*. For this reason, the system equation at equilibrium state is

> <sup>0</sup> (1 ) *li lpi li P KP*

<sup>0</sup> (1 ) *Q KQ li lqi li* 

<sup>0</sup> (1 ) *gj gj gj P KP* 

where Pli0 and Qli0 are the initial loading conditions at the base case corresponding to λ=0. Klpi and Klqi are factors characterizing the load increase pattern (stress direction). Pgj0 is the real power generation at bus j at the base case. Kgj represents the generator load pick-up

(2)

(3)

(4)

(5)

[28, 29] first derived these margin sensitivities for different changing parameters.

parameterized by this bifurcation parameter λ as shown below.

**3.3. Sensitivities of post-contingency performance criteria** 

light on these linear sensitivity measures.

*3.3.1. Voltage stability margin sensitivity* 

The disturbance performance table within the NERC (North American Electric Reliability Corporation)/WECC (Western Electricity Coordinating Council) planning standards [26] provides the minimum acceptable performance specifications for post-contingency voltage stability margin under credible events, that it should be atleast,


**Figure 4.** Voltage stability margin

As far as the performance measure for dynamic stability phenomena is concerned, in [11] it is stated that the needs of the industry related to voltage dips/sags for power system stability fall under two main scenarios. One is the traditional transient angle stability where voltage "swing" during electromechanical oscillations is the concern. The other is "short-term" voltage stability generally involving voltage recovery following fault clearing where there is no significant oscillations, for which much greater load modeling detail is required (specifically induction motor loads) with the fault applied in the load area rather than near generation. In [25], it is stated that many planning and operating engineers are insufficiently aware of potential short-term voltage instability, or are unsure on how to analyze the phenomena. In this work we focus on planning for the transient voltage recovery after a fault is cleared. For the transient voltage analysis, the minimum planning criteria is,

Real Power

 **Slow voltage recovery**: The induction motor trip relay timer is actuated when the bus voltage dips below 0.7p.u and trips if voltage doesn't recover to 0.7p.u within next 20 cycles.

### **3.3. Sensitivities of post-contingency performance criteria**

The objective of the reactive power planning problem is to satisfy these minimum performance criteria mentioned in the previous section under various credible contingencies. This is greatly dependent upon the amount, location and type of reactive power sources available in the system, which are all decision variables in the proposed coordinated RPP, as will be dealt in the next section of this chapter. If the reactive power support is far away, or insufficient in size, or too dependent on shunt capacitors (slow acting static device), a relatively normal contingency (such as a line outage or a sudden increase in load) can trigger a large system voltage drop. Hence, in order to properly allocate the reactive power support in terms of optimal type, location and amount, in this work we employ linear sensitivities of performance measure with respect to various control actions. These sensitivities enable obtaining the necessary information, i.e., the sensitivities of performance measure for the type of reactive support, the location and amount of reactive support, which are very useful to perform the proposed coordinated RPP. This section sheds light on these linear sensitivity measures.

### *3.3.1. Voltage stability margin sensitivity*

164 Induction Motors – Modelling and Control

Voltage

**Figure 4.** Voltage stability margin

cycles.

unexpected component (generator, transformer, transmission line) outages often reduce the voltage stability margin, and control actions increase it [23, 24], as shown in figure 4 [25].

The disturbance performance table within the NERC (North American Electric Reliability Corporation)/WECC (Western Electricity Coordinating Council) planning standards [26] provides the minimum acceptable performance specifications for post-contingency voltage

Real Power

As far as the performance measure for dynamic stability phenomena is concerned, in [11] it is stated that the needs of the industry related to voltage dips/sags for power system stability fall under two main scenarios. One is the traditional transient angle stability where voltage "swing" during electromechanical oscillations is the concern. The other is "short-term" voltage stability generally involving voltage recovery following fault clearing where there is no significant oscillations, for which much greater load modeling detail is required (specifically induction motor loads) with the fault applied in the load area rather than near generation. In [25], it is stated that many planning and operating engineers are insufficiently aware of potential short-term voltage instability, or are unsure on how to analyze the phenomena. In this work we focus on planning for the transient voltage recovery after a fault is cleared. For the

 **Slow voltage recovery**: The induction motor trip relay timer is actuated when the bus voltage dips below 0.7p.u and trips if voltage doesn't recover to 0.7p.u within next 20

transient voltage analysis, the minimum planning criteria is,

total system load

M0

M2 M1

M0: normal voltage stability margin M1: reduced voltage stability margin M2: increased voltage stability margin

stability margin under credible events, that it should be atleast,

 greater than 5% for N-1 contingencies, greater than 2.5% for N-2 contingencies, and greater than 0% for N-3 contingencies.

normal

contingency control

> The sensitivity of voltage stability margin refers to how much the margin changes for a small change in system parameters such as real power and reactive power bus injections, regulated bus voltages, bus shunt capacitance, line series capacitance etc [2, 27]. References [28, 29] first derived these margin sensitivities for different changing parameters.

Let the steady state of the power system satisfying a set of equations in the vector form be,

$$F(\mathbf{x}, p, \mathcal{A}) = 0 \tag{2}$$

where x is the vector of state variables, p is any parameter in the power system steady state equations such as demand and base generation or the susceptance of shunt capacitors or the reactance of series capacitors, and λ denotes the system load/generation level called the scalar bifurcation parameter. The system reaches a state of voltage collapse, when λ hits its maximum value (the nose point of the system PV curve), and the value of the bifurcation parameter is equal to λ\*. For this reason, the system equation at equilibrium state is parameterized by this bifurcation parameter λ as shown below.

$$P\_{li} = (1 + K\_{lpi}\mathcal{Z})P\_{li0} \tag{3}$$

$$\mathbf{Q}\_{li} = (\mathbf{1} + \mathbf{K}\_{lqi}\mathcal{A})\mathbf{Q}\_{li0} \tag{4}$$

$$P\_{\mathcal{K}^j} = (1 + K\_{\mathcal{K}} \mathcal{Z}) P\_{\mathcal{K}^j} \tag{5}$$

where Pli0 and Qli0 are the initial loading conditions at the base case corresponding to λ=0. Klpi and Klqi are factors characterizing the load increase pattern (stress direction). Pgj0 is the real power generation at bus j at the base case. Kgj represents the generator load pick-up factor. For a power system model using ordinary algebraic equations, the bifurcation point sensitivity with respect to the control variable pi evaluated at the saddle-node bifurcation point is [27]

$$\frac{\partial \mathcal{X}}{\partial p\_i} = -\frac{w^\* F\_{p\_i}^\*}{w^\* F\_{\lambda}^\*} \tag{6}$$

Role of Induction Motors in Voltage Instability and Coordinated Reactive Power Planning 167

( ) () *BB B*

 (11)

∆Bsvc. This procedure is easy to implement for a large power system using available

svc svc svc

*BB B*

dip dip dip svc svc dip svc

Time

To restore post-disturbance equilibrium and increase post contingency voltage stability margin beyond the minimum criteria

To improve the characteristics of post-fault transient voltage recovery phenomenon satisfying the required minimum criteria,

after a severe contingency that can cause voltage instability

 (2) 

(1) 

simulation tools.

*S*

Bus Voltage Magnitude

**Figure 5.** Post-fault clearance slow voltage recovery

**A (Steady State)** 

> **B (Dynamic)**

**4. Coordinated reactive power planning** 

**Table 1.** Coordinated Reactive Power Planning Objectives

Figure 6 shows the general flow of the RPP procedure developed.

The planning algorithm addresses the problems discussed in Table 1.

**Problem Planning Objective** 

and prevent induction motor tripping

 

 

where w is the left eigenvector corresponding to the zero eigenvalue of the system Jacobian Fx, Fλ is the derivative of F with respect to the bifurcation parameter λ and FPi is the derivative of F with respect to the control variable parameter pi.

The sensitivity of the voltage stability margin with respect to the control variable at location i, Si, is

$$S\_i = \frac{\partial M}{\partial p\_i} = \frac{\partial \mathcal{L}}{\partial p\_i} \sum\_{i=1}^n K\_{lpi} P\_{li0} \tag{7}$$

where M is the voltage stability margin given by

$$M = \sum\_{i=1}^{n} P\_{li}^{\*} - \sum\_{i=1}^{n} P\_{li0} = \mathcal{A}^{\*} \sum\_{i=1}^{n} \mathcal{K}\_{lpi} P\_{li0} \tag{8}$$

### *3.3.2. Post-fault transient voltage recovery sensitivity*

The sensitivity of the voltage dip time duration to the SVC capacitive limit (*Bsvc*) can be defined as the change of the voltage dip time duration (voltage recovery time) for a given change in the SVC capacitive limit. Let τ (1) be the time at which the transient voltage dip begins after a fault is cleared, and τ (2) be the time at which the transient voltage dip ends, as shown in figure 5 [25]. Then the time duration of the transient voltage dip τdip is given by

$$
\tau\_{\rm dip} = \tau^{(2)} - \tau^{(1)} \tag{9}
$$

Thus, the sensitivity of the voltage dip time duration to the capacitive limit of an SVC, Sτ, is

$$S\_{\tau} \equiv \frac{\partial \tau\_{\rm dip}}{\partial B\_{\rm scc}} = \frac{\partial (\tau^{(2)} - \tau^{(1)})}{\partial B\_{\rm scc}} = \frac{\partial \tau^{(2)}}{\partial B\_{\rm scc}} - \frac{\partial \tau^{(1)}}{\partial B\_{\rm scc}} = \tau^{(2)}\_{\, \space ncc} - \tau^{(1)}\_{\, \space B\_{\rm scc}} \tag{10}$$

where svc (1) *B* and svc (2) *B* are calculated based on trajectory sensitivity computations as derived in [25]. For a large power system, this method requires computing integrals of a set of high dimension differential algebraic equations. An alternative to calculate the sensitivities is using numerical approximation [25], as shown by equation (11). This procedure of sensitivity calculation by numerical approximation requires repeated simulations of the system model for the SVC capacitive limits Bsvc and Bsvc+∆Bsvc. The sensitivities are then given by the change of the voltage recovery time divided by the SVC capacitive limit change ∆Bsvc. This procedure is easy to implement for a large power system using available simulation tools.

$$S\_{\tau} = \frac{\partial \tau\_{\rm dip}}{\partial \mathcal{B}\_{\rm scc}} \approx \frac{\Lambda \tau\_{\rm dip}}{\Delta \mathcal{B}\_{\rm scc}} = \frac{\tau\_{\rm dip} (\mathcal{B}\_{\rm scc} + \Delta \mathcal{B}\_{\rm scc}) - \tau\_{\rm dip} (\mathcal{B}\_{\rm scc})}{\Delta \mathcal{B}\_{\rm scc}} \tag{11}$$

**Figure 5.** Post-fault clearance slow voltage recovery

166 Induction Motors – Modelling and Control

point is [27]

i, Si, is

where svc (1) *B* 

factor. For a power system model using ordinary algebraic equations, the bifurcation point sensitivity with respect to the control variable pi evaluated at the saddle-node bifurcation

\* \* \*

*p w F*

where w is the left eigenvector corresponding to the zero eigenvalue of the system Jacobian Fx, Fλ is the derivative of F with respect to the bifurcation parameter λ and FPi is the

The sensitivity of the voltage stability margin with respect to the control variable at location

\*

*i lpi li i i i <sup>M</sup> <sup>S</sup> K P p p* 

\* \*

*nn n*

11 1

The sensitivity of the voltage dip time duration to the SVC capacitive limit (*Bsvc*) can be defined as the change of the voltage dip time duration (voltage recovery time) for a given change in the SVC capacitive limit. Let τ (1) be the time at which the transient voltage dip begins after a fault is cleared, and τ (2) be the time at which the transient voltage dip ends, as shown in figure 5 [25]. Then the time duration of the transient voltage dip τdip is given by

*ii i M P P KP*

1

*li li lpi li*

(2) (1)

 

(2) (1) (2) (1) (2) (1)

 

are calculated based on trajectory sensitivity computations as derived

dip 

( )

 

svc svc svc svc

Thus, the sensitivity of the voltage dip time duration to the capacitive limit of an SVC, Sτ, is

*<sup>B</sup> S B B BB*

in [25]. For a large power system, this method requires computing integrals of a set of high dimension differential algebraic equations. An alternative to calculate the sensitivities is using numerical approximation [25], as shown by equation (11). This procedure of sensitivity calculation by numerical approximation requires repeated simulations of the system model for the SVC capacitive limits Bsvc and Bsvc+∆Bsvc. The sensitivities are then given by the change of the voltage recovery time divided by the SVC capacitive limit change

  *n*

*i*

derivative of F with respect to the control variable parameter pi.

where M is the voltage stability margin given by

*3.3.2. Post-fault transient voltage recovery sensitivity* 

*dip*

 and svc (2) *B* 

\* \* *i p*

0

0 0

(6)

(7)

(8)

(9)

(10)

svc svc

*B*

*w F*

### **4. Coordinated reactive power planning**

The planning algorithm addresses the problems discussed in Table 1.


**Table 1.** Coordinated Reactive Power Planning Objectives

Figure 6 shows the general flow of the RPP procedure developed.

Role of Induction Motors in Voltage Instability and Coordinated Reactive Power Planning 169

The objective of mixed integer program (MIP) is to minimize the total installation cost of MSCs and SVCs while satisfying the requirements of long-term voltage stability margin and

\_M \_ M \_ M \_ M \_ svc \_ svc \_ svc \_ svc [ ] *vi SC i SC fi SC i SC vi i fi i*

*SB B M Mk*

, , \_ svc dip, dip, ,r , , *kk k ni i n n*

 

*S B n k*

( ) 0 , \_M \_M *k*

(12)

(13)

(14)

*i MSC i MSC i MSC i MSC i MSC* min\_ \_ \_ max\_ \_ *B qBB q* (17)

*i i svc i i i* min\_ svc \_ \_ svc max\_ svc \_ svc *B qBB q* (18)

(k)\_svc, Bi\_svc, and qi\_svc.

*i SC i SC BB k* (15)

*i i BB k* (16)

\_ M \_ svc , 0,1 *i SC i q q* (19)

*C B C q CB Cq*

() () () () , \_M \_ <sup>r</sup> [ ], *kk k k*

*M i i SC i svc*

() () ()

( ) \_ svc \_ svc 0 , *<sup>k</sup>*

(k)\_MSC, Bi\_MSC, qi\_MSC, Bi

qi\_MSC=1 if the location i is selected for installing MSCs, otherwise, qi\_MSC=0,

the superscript k represents the contingency causing insufficient voltage stability margin

qi\_svc=1 if the location i is selected for installing SVCs, otherwise, qi\_svc=0,

Cf\_MSC is fixed installation cost and Cv\_MSC is variable cost of MSCs,

Cf\_svc is fixed installation cost and Cv\_svc is variable cost of SVCs,

*4.2.1. Original mixed integer programming* 

*i*

The decision variables are Bi

Bi\_MSC: size of the MSC at location i, Bi\_svc: size of the SVC at location i,

and/or slow voltage recovery problems,

ΩMSC: set of candidate locations to install MSCs, Ωsvc: set of candidate locations to install SVCs,

Minimize

Subject to

short-term post-fault transient voltage characteristics.

*i*

*i*

svc

**Figure 6.** Flowchart of RPP procedure with successive MIP
