**5. Numerical illustration**

170 Induction Motors – Modelling and Control

MSC at location i under contingency k,

the SVC at location i under contingency k,

Bimin\_MSC: minimum size of the MSC at location i, Bimax\_MSC: maximum size of the MSC at location i,

Bimin\_svc: minimum size of the SVC at location i, and

*4.2.2. Updated successive mixed integer programming* 

Bimax\_svc: maximum size of the SVC at location i.

programming problem.

Mr: required voltage stability margin,

*(k)\_MSC*: size of the MSC to be switched at location i under contingency k,

M(k): voltage stability margin under contingency k and without controls,

τdip,n,r: maximum allowable time duration of voltage recovery at bus n,

Note from (13) that SVCs can also be used to increase the voltage stability margin.

The output of the mixed integer-programming problem in section 4.2.1 is the combined reactive compensation locations and amounts for all concerned contingencies. Now the network configuration is updated by including the identified reactive power support under each contingency. After that, the voltage stability margin is recalculated using CPF to check if sufficient margin is achieved for each concerned contingency. Also, time domain simulations are carried out to check whether the requirement of the transient voltage recovery performance is met. This step is necessary because the power system model is inherently nonlinear, and the mixed integer programming algorithm uses linear sensitivities to estimate the effect of variations of reactive support levels on the voltage stability margin and post-fault voltage recovery. So if need be, the reactive compensation locations and/or amounts can be further refined by re-computing sensitivities (with updated network configuration) under each concerned contingency, and solving a second-stage mixed integer

This successive MIP problem based on updated sensitivity and system performance information is again formulated to minimize the total installation cost of MSCs and SVCs, subject to the constraints of the requirements of voltage stability margin and voltage

*<sup>M</sup> <sup>i</sup> S* : sensitivity of the voltage stability margin with respect to the shunt susceptance of

: sensitivity of the voltage recovery time duration at bus n with respect to the size of

τdip,n(k): time duration of voltage recovery at bus n under contingency k and without controls,

*<sup>k</sup> Bi* : size of the SVC at location i under contingency k,

Ω: union of ΩMSC and Ωsvc,

*Bi*

( ) \_ svc

( ) , *k*

( ) , , *k <sup>n</sup> <sup>i</sup> S*

The control planning method described in this paper was applied to a particular portion of US Eastern Interconnection system consisting of about 16173 buses. This subsystem belonging to a particular utility's control area, henceforth will be referred to as the "study area." The study area within this large system consisted of 2069 buses with 30065.2 MW of loading and 239 generators producing 37946.7 MW.

The contingencies considered for the study are the more probable ones, i.e., N-1 and N-G-T. For N-1 and N-G-T contingencies, according to the WECC/NERC performance table, minimum steady state performance criteria is to have a post-contingency voltage stability margin of at least 5% of the sub-system's base load. For the slow voltage recovery problem, the minimum performance should be such that it avoids the induction motor tripping. The trip relay timer of the trip induction motor 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. Therefore, the objective of the coordinated RPP is to identify a minimum cost mix of static and dynamic Var resources that results in satisfactory voltage stability and transient voltage recovery performance for all considered contingencies.

The study area was grouped into 6 Market Zones (MZ), representing the 6 different load increase (stress) directions required to perform CPF analysis. For a particular stress direction, the sink is characterized by the set of loads inside a MZ, and the source is characterized by generators outside of that MZ, but within the study area.

For the transient study, dynamic models for generators, exciter, governor systems, and appropriate load and SVC models are used. Loads in the focus area were partitioned as 50% induction motor load (dynamic) and 50% ZIP load (static). Induction motor was modeled using CIM5 model in PSS/E, which is sensitive to changing voltage and frequency. SVCs were modeled using CSVGN1 and CSVGN3 family of SVC models. The PSS/E manual presents the block diagram, parameters and detailed description for each of these models. The dynamic models of induction motor loads were further split equally into three different kinds, i.e., large, small, and trip motors. Table 2 shows some of the important parameters of each of these motor loads as defined in section 3.1.2.1. The ZIP load is modeled as 50% constant impedance and 50% constant current for real power load and 100% constant impedance for reactive power load.

The steady state contingency analysis using CPF is performed in Matlab, while the dynamic voltage stability analysis using time domain simulation is performed in PTI PSS/E [32]. As part of this study, which required Matlab using the input files in PSS/E's "raw" data format, a data conversion module was built that converted the system raw data to a format that was understandable by Matlab. The conversion module includes tasks such as careful modeling

of 3-winding transformer data, zero-impedance lines, switched shunt data, checking system topology, and checking for any islanding, so that the utility's base case is transported without any errors into Matlab.

Role of Induction Motors in Voltage Instability and Coordinated Reactive Power Planning 173

the overall planning methodology. This in effect means we consider only constraints (13), (15), and (17) in original MIP of section 4.2.1, and also in updated successive MIP of section 4.2.2. We also analyze the impact of induction motors on the transient voltage recovery performance, and demonstrate the ineffectiveness of steady state planning approach to counter the ill-dynamics caused by induction motors in voltage stability phenomena. This strongly motivates the need for considering the short-term voltage performance and fast-

Steady state voltage stability analysis using CPF-based contingency screening was performed for all credible N-1 (2100 branches (T) and 168 generators (G)) and N-G-T (all possible combinations of G and T) contingencies under 6 different stress directions. The results indicated that only the stress direction corresponding to the MZ1 region was found to have post contingency steady state voltage stability problems, as indicated by the 56 critical N-G-T contingencies (28 line outages under 2 N-G base case, where G = {14067, 14068}) in Table 4. The total base load in MZ1 zone (sink) is 2073 MW. The collapse point for basecase was at 2393 MW, which is equivalent to a stability margin of 15.44%. All the 56 contingencies in Table 4 have stability margin lesser than 5%, with the top 14 facing instability issues. Table 4 also shows the final optimal static VAR solution that just satisfies the minimum performance criteria under every critical contingency. The final optimal investments will be the maximum amount of MSCs required at each location. The total investment cost of MSCs to solve all the steady state voltage stability problems is 2.665 M\$.

> **Stability Index (Margin in %)**

14079 14083 138 Unstable Unstable 1.5 0 1.5 0.25 14094 14326 230 Unstable Unstable 1.5 1.5 0.75 0 14175 14319 230 Unstable Unstable 1.2 1.5 0 0.85 14296 14322 500 Unstable Unstable 1 1.2 1.25 0 14319 14321 230 Unstable Unstable 1.5 1.5 0.5 0 14319 14326 230 Unstable Unstable 1.5 1.5 0.9 0 10983 14129 345 Unstable Unstable 1.5 1.5 1.5 1 14077 14080 138 1.495417 1.784853 0.3 0 0 0 14130 14142 138 2.411963 2.556681 0.65 0 0 0 14129 14162 345 2.556681 2.701399 0.85 0 0 0 14294 14319 230 2.990835 3.135552 0.68 0 0 0 14295 14302 138 2.990835 3.135552 0.65 0 0 0 14071 14079 138 3.28027 3.424988 0.57 0 0 0 14126 14142 138 3.28027 3.424988 0.55 0 0 0 14109 14363 138 3.617945 3.617945 0.52 0 0 0 14148 14232 138 3.617945 3.762663 0.45 0 0 0

**Gen#2-**

**14067** 

**Capacitor Allocation (p.u Mvar)** 

**<sup>14068</sup>14073 14071 14080 14160** 

responding dynamic reactive resource within a coordinated planning approach.

**Cont. No** 

**Transmission** 

**Line Base**

**KV** 

**From To Gen#1–**


**Table 2.** Induction Motor Dynamic Model Parameters

Based on the pre-processing performed using these computational tools, respective sensitivity information of steady state and dynamic performance measures with respect to the reactive control device are computed, as explained in section 3.3. All the necessary input including candidate locations and control device cost information are fed as input to the coordinated planning algorithm, which is coded in Matlab and executed using CPLEX. Since candidate control locations and linear sensitivity information are used, the MIP optimization is faster even for a very large scale system. PSS/E's ability to store the results in \*.csv format is utilized in post-processing the simulation results using MS Excel.

The cost is modeled as two components [24]: fixed cost and variable cost. For SVCs the fixed and variable costs were taken to be 1.5 M\$ and 5 M\$/100Mvar respectively. The maximum capacity limit for SVCs was fixed at 300 MVar. Table 3 shows the MSC cost information and the default maximum capacity constraints at every feasible location under different voltage levels. The maximum MSC limit at various voltage levels ensures avoiding overdeployment of MSCs at those voltage levels, which could degrade the voltage magnitude performance. At every solution validation step, if any bus has post-contingency voltage exceeding 1.06 p.u, a new maximum MSC (MVar) constraint is developed, and the optimization is re-run with the new constraint included.


**Table 3.** MSC Cost Information and Maximum Compensation

## **5.1. Steady state reactive power planning and impact of induction motors**

In this section we present only the steady state RPP, i.e., plan to satisfy post-contingency voltage stability margin only using MSCs, without including dynamic problem and SVCs in the overall planning methodology. This in effect means we consider only constraints (13), (15), and (17) in original MIP of section 4.2.1, and also in updated successive MIP of section 4.2.2. We also analyze the impact of induction motors on the transient voltage recovery performance, and demonstrate the ineffectiveness of steady state planning approach to counter the ill-dynamics caused by induction motors in voltage stability phenomena. This strongly motivates the need for considering the short-term voltage performance and fastresponding dynamic reactive resource within a coordinated planning approach.

172 Induction Motors – Modelling and Control

without any errors into Matlab.

**Table 2.** Induction Motor Dynamic Model Parameters

is utilized in post-processing the simulation results using MS Excel.

optimization is re-run with the new constraint included.

**Table 3.** MSC Cost Information and Maximum Compensation

**Fixed Cost (Million \$)** 

**Base KV** 

of 3-winding transformer data, zero-impedance lines, switched shunt data, checking system topology, and checking for any islanding, so that the utility's base case is transported

**Large Motor** 1.5 0 20 2 1 **Small Motor** 0.5 0 20 2 1 **Trip Motor** 0.5 0.7 20 2 1

Based on the pre-processing performed using these computational tools, respective sensitivity information of steady state and dynamic performance measures with respect to the reactive control device are computed, as explained in section 3.3. All the necessary input including candidate locations and control device cost information are fed as input to the coordinated planning algorithm, which is coded in Matlab and executed using CPLEX. Since candidate control locations and linear sensitivity information are used, the MIP optimization is faster even for a very large scale system. PSS/E's ability to store the results in \*.csv format

The cost is modeled as two components [24]: fixed cost and variable cost. For SVCs the fixed and variable costs were taken to be 1.5 M\$ and 5 M\$/100Mvar respectively. The maximum capacity limit for SVCs was fixed at 300 MVar. Table 3 shows the MSC cost information and the default maximum capacity constraints at every feasible location under different voltage levels. The maximum MSC limit at various voltage levels ensures avoiding overdeployment of MSCs at those voltage levels, which could degrade the voltage magnitude performance. At every solution validation step, if any bus has post-contingency voltage exceeding 1.06 p.u, a new maximum MSC (MVar) constraint is developed, and the

0.025 0.41 30 0.05 0.41 75 0.07 0.41 120 0.1 0.41 150 0.28 0.41 200 0.62 0.41 300 1.3 0.41 300

**5.1. Steady state reactive power planning and impact of induction motors** 

In this section we present only the steady state RPP, i.e., plan to satisfy post-contingency voltage stability margin only using MSCs, without including dynamic problem and SVCs in

**Variable Cost (M \$/100MVar)**  **Maximum MSC (MVar)** 

*H* **(p.u. motor base)** *VT* **(p.u)** *CT* **(cycles)** *D Tnom* **(p.u)** 

Steady state voltage stability analysis using CPF-based contingency screening was performed for all credible N-1 (2100 branches (T) and 168 generators (G)) and N-G-T (all possible combinations of G and T) contingencies under 6 different stress directions. The results indicated that only the stress direction corresponding to the MZ1 region was found to have post contingency steady state voltage stability problems, as indicated by the 56 critical N-G-T contingencies (28 line outages under 2 N-G base case, where G = {14067, 14068}) in Table 4. The total base load in MZ1 zone (sink) is 2073 MW. The collapse point for basecase was at 2393 MW, which is equivalent to a stability margin of 15.44%. All the 56 contingencies in Table 4 have stability margin lesser than 5%, with the top 14 facing instability issues. Table 4 also shows the final optimal static VAR solution that just satisfies the minimum performance criteria under every critical contingency. The final optimal investments will be the maximum amount of MSCs required at each location. The total investment cost of MSCs to solve all the steady state voltage stability problems is 2.665 M\$.



Role of Induction Motors in Voltage Instability and Coordinated Reactive Power Planning 175

Recovery time Cycles Recovery time Cycles

With appropriate modeling of induction motor dynamics, time domain simulations were run for the top 7 contingencies by applying a 3-phase fault at t=0 at one end of the transmission circuit and then clearing the fault and the circuit at 6 cycles (t = 0.1s). All the contingencies lead to a slow voltage recovery due to the presence of induction motor loads that ultimately tripped. A list of all the buses having slow voltage recovery problem under each severe contingency was made. Table 5 shows the time domain simulation results under few of the contingencies of Table 4, wherein we notice the buses at which the minimum criteria for transient voltage recovery (more than 20 cycles below 0.7p.u) is violated. The induction motor loads connected to these buses also get tripped, in a similar manner as was shown in Figure 7. The transient voltage response under other contingencies was also very poor. For instance, under contingencies 2, 4, 5, and 6, about 76, 18, 72, 64 buses respectively

**Bus Number Contingency 1 Contingency 3** 

**Table 5.** Buses resulting in Slow Voltage Recovery and Induction Motor Tripping

**Table 6.** Contingency Ranking in terms of Worst-case Recovery Times

**5.2. Coordinated reactive power planning** 

according to the following criteria:

0.841 50.46 0.344 20.64 0.771 46.26 0.36 21.6 0.694 41.64 0.36 21.6 0.614 36.84 0.353 21.18

Table 6 ranks the top 7 contingencies (under both gen. outages) shown in Table 4, based on their severity, which is quantified in terms of worst-case recovery times. It can be expected

**Contingency No. Bus Numbers kV Rank From To**

The proposed coordinated planning, as discussed in section 4.2, was performed to mitigate both transient as well as steady state problems under these contingencies. Sensitivities of post-contingency voltage stability margin and transient voltage recovery times are used as one of the inputs to find the optimal solution. The candidate locations were identified

14079 14083 138 **2**  14094 14326 230 **4**  14175 14319 230 **6**  14296 14322 500 **7**  14319 14321 230 **3**  14319 14326 230 **1**  10983 14129 345 **5** 

that the most severe contingencies will drive the amount of dynamic VARs needed.

violated the minimum post-fault voltage recovery criteria.


Even though the system is planned against steady state voltage instability using MSCs, nevertheless the effect of induction motors in transient phenomena has not been investigated.

Figure 7 shows the voltage recovery phenomenon at a certain bus with and without dynamic modeling of induction motor. The figure shows the significance of modeling induction motor properly, which has important role in ascertaining post-fault-clearance short-term voltage stability of the system. Once this phenomenon is captured by appropriate modeling, we can counter it by the proposed coordinated RPP.

**Figure 7.** Voltage recovery phenomenon with and without Induction Motor modeling

With appropriate modeling of induction motor dynamics, time domain simulations were run for the top 7 contingencies by applying a 3-phase fault at t=0 at one end of the transmission circuit and then clearing the fault and the circuit at 6 cycles (t = 0.1s). All the contingencies lead to a slow voltage recovery due to the presence of induction motor loads that ultimately tripped. A list of all the buses having slow voltage recovery problem under each severe contingency was made. Table 5 shows the time domain simulation results under few of the contingencies of Table 4, wherein we notice the buses at which the minimum criteria for transient voltage recovery (more than 20 cycles below 0.7p.u) is violated. The induction motor loads connected to these buses also get tripped, in a similar manner as was shown in Figure 7. The transient voltage response under other contingencies was also very poor. For instance, under contingencies 2, 4, 5, and 6, about 76, 18, 72, 64 buses respectively violated the minimum post-fault voltage recovery criteria.


**Table 5.** Buses resulting in Slow Voltage Recovery and Induction Motor Tripping

Table 6 ranks the top 7 contingencies (under both gen. outages) shown in Table 4, based on their severity, which is quantified in terms of worst-case recovery times. It can be expected that the most severe contingencies will drive the amount of dynamic VARs needed.


**Table 6.** Contingency Ranking in terms of Worst-case Recovery Times

## **5.2. Coordinated reactive power planning**

174 Induction Motors – Modelling and Control

investigated.

14302 14363 138 3.617945 3.907381 0.52 0 0 0 14124 14126 138 3.907381 3.907381 0.5 0 0 0 14148 14149 138 3.907381 3.907381 0.4 0 0 0 14232 14328 138 3.907381 4.052098 0.5 0 0 0 14237 14328 138 3.907381 4.052098 0.5 0 0 0 14322 14494 500 3.907381 4.052098 0.5 0 0 0 14106 14109 138 3.95562 4.052098 0.5 0 0 0 14297 14311 138 4.196816 4.341534 0.47 0 0 0 14134 14290 138 4.245055 4.486252 0.35 0 0 0 14103 14130 138 4.486252 4.63097 0.4 0 0 0 14106 14110 138 4.486252 4.63097 0.4 0 0 0 14238 14297 138 4.63097 4.823927 0.4 0 0 0 **Table 4.** Contingency List and Optimal Allocation of MSCs by Steady State Reactive Power Planning

Even though the system is planned against steady state voltage instability using MSCs, nevertheless the effect of induction motors in transient phenomena has not been

Figure 7 shows the voltage recovery phenomenon at a certain bus with and without dynamic modeling of induction motor. The figure shows the significance of modeling induction motor properly, which has important role in ascertaining post-fault-clearance short-term voltage stability of the system. Once this phenomenon is captured by appropriate

> **Slow voltage recovery phenomenon and**

modeling, we can counter it by the proposed coordinated RPP.

**Figure 7.** Voltage recovery phenomenon with and without Induction Motor modeling

The proposed coordinated planning, as discussed in section 4.2, was performed to mitigate both transient as well as steady state problems under these contingencies. Sensitivities of post-contingency voltage stability margin and transient voltage recovery times are used as one of the inputs to find the optimal solution. The candidate locations were identified according to the following criteria:

	- the bus being among the top 5 worst voltage dips and
	- the bus has induction motor load that trips

Role of Induction Motors in Voltage Instability and Coordinated Reactive Power Planning 177

**Figure 8.** Voltage profiles of worst-hit buses under cont. 2 without SVC

**Figure 9.** Voltage profiles of worst-hit buses under cont. 2 with SVC from stage 1 MIP result

**Bus Number** 

**From To** 

**Table 8.** Final Solution of Coordinated Optimal Planning

**No.** 

**N-G-T Contingencies under both the N-G SVC (p.u MVAR)** 

14079 14083 138 3 2.65 14094 14326 230 3 2.7 14175 14319 230 3 0.7 14296 14322 500 1.7 0 14319 14321 230 3 2.7 14319 14326 230 3 2.85 10983 14129 345 3 2.1

**KV 14071 14084** 

Table 7 shows the stage 1 MIP result, which selects two SVC locations to solve both static as well as dynamic problems. Investigation indicates the reason for this is that the transient voltage problems are so severe that the amount of SVC required to solve them is also sufficient to mitigate the steady state voltage stability problems.


When stage I MIP solution was validated for steady state voltage stability problems for the top 7 contingencies (under both gen. outages) of Table 4, it was found that they not only attained equilibrium, but also had post contingency voltage stability margin more than the minimum requirement of 5% as shown in Table 7. The remaining contingencies in Table 4 were also validated with this stage I MIP solution and were found to have sufficient post contingency voltage stability margin.

When time domain simulations were done to validate stage 1 MIP result, it was found that contingencies 1, 2, 5, 6, and 7 still had buses that violated the minimum recovery time requirement, resulting in tripping of some induction motors. Figures 8 and 9 show the voltage profiles at the most severely-affected buses under contingency 2 before and after stage 1 MIP result implementation, respectively.

Figure 9 also shows that the SVC placed after stage 1 MIP at buses 14071 and 14084, which are just sufficient to provide voltage recover within 20 cycles at some buses. For the contingencies having slow voltage recovery problems at some buses even after stage 1 MIP solution, a successive MIP is performed with updated sensitivity information using stage 1 SVC solution.

Table 8 shows the final operational solution of the coordinated planning problem, which chooses only SVCs due to the nature of problems under contingencies.

**Figure 8.** Voltage profiles of worst-hit buses under cont. 2 without SVC

1. Buses for which one or more contingencies result in:

the bus has induction motor load that trips

also efficiently address the static problems.

**Bus Number**

**From To** 

contingency voltage stability margin.

SVC solution.

stage 1 MIP result implementation, respectively.

the bus being among the top 5 worst voltage dips and

sufficient to mitigate the steady state voltage stability problems.

2. Buses must have high steady state voltage stability margin sensitivity so that they can

Table 7 shows the stage 1 MIP result, which selects two SVC locations to solve both static as well as dynamic problems. Investigation indicates the reason for this is that the transient voltage problems are so severe that the amount of SVC required to solve them is also

**(%) No.** 

14079 14083 138 3 0.85 9.64 14094 14326 230 3 0.8 9.16 14175 14319 230 3 0.7 9.4 14296 14322 500 1.7 0 8.68 14319 14321 230 3 1.5 9.4 14319 14326 230 3 1.53 8.92 10983 14129 345 3 0.95 5.1

When stage I MIP solution was validated for steady state voltage stability problems for the top 7 contingencies (under both gen. outages) of Table 4, it was found that they not only attained equilibrium, but also had post contingency voltage stability margin more than the minimum requirement of 5% as shown in Table 7. The remaining contingencies in Table 4 were also validated with this stage I MIP solution and were found to have sufficient post

When time domain simulations were done to validate stage 1 MIP result, it was found that contingencies 1, 2, 5, 6, and 7 still had buses that violated the minimum recovery time requirement, resulting in tripping of some induction motors. Figures 8 and 9 show the voltage profiles at the most severely-affected buses under contingency 2 before and after

Figure 9 also shows that the SVC placed after stage 1 MIP at buses 14071 and 14084, which are just sufficient to provide voltage recover within 20 cycles at some buses. For the contingencies having slow voltage recovery problems at some buses even after stage 1 MIP solution, a successive MIP is performed with updated sensitivity information using stage 1

Table 8 shows the final operational solution of the coordinated planning problem, which

chooses only SVCs due to the nature of problems under contingencies.

**Table 7.** Stage 1 MIP Result of Coordinated RPP and Steady State Stability Validation

**N-G-T Contingencies SVC (p.u MVAR) Steady State** 

**KV 14071 14084** 

**Stability Margin** 

**Figure 9.** Voltage profiles of worst-hit buses under cont. 2 with SVC from stage 1 MIP result


**Table 8.** Final Solution of Coordinated Optimal Planning

Figure 10 shows the improved voltage profile at Bus 14071 under contingency 1 after implementing the final SVC solution. Figure 10 also shows the output of the two SVCs. The SVC peak output at bus 14071 is 290 MVar, and that of bus 14084 is 270 MVar.

Role of Induction Motors in Voltage Instability and Coordinated Reactive Power Planning 179

many critical contingencies simultaneously and averts steady-state and dynamic voltage

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**Author details** 

**7. References** 

*Iowa State University, USA* 

**Figure 10.** Bus 14071 voltage profile under cont. 1 with final allocation

The final investment solution for the coordinated planning problem includes SVC placements at buses 14071 and 14084 of maximum capacity 3.0 MVar and 2.85 MVar respectively1. The total cost is 32.25 M\$ under the cost assumptions used for this study.
