**4.2. Formulation of coordinated control planning algorithm**

This section presents the formulation of optimization developed to address the coordinated VAR planning problem for all the contingencies that have either one or both the planning problems shown in Table 1. The information required for the optimization algorithm are reactive device cost and maximum capacity limit at various voltage levels, performance measures and their sensitivities with respect to MSCs and SVCs under each critical contingencies, and an initial set of candidate locations for MSCs and SVCs. While many methodologies in the past determine static and dynamic VAR support sequentially, this method simultaneously determines the optimal allocation of static and dynamic VAR [31].

### *4.2.1. Original mixed integer programming*

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 short-term post-fault transient voltage characteristics.

Minimize

168 Induction Motors – Modelling and Control

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

information [29] is used to reduce the computational burden [30].

**4.2. Formulation of coordinated control planning algorithm** 

Continuation power flow (CPF) [17] based tools and time domain simulation are used to perform the contingency analysis. The post-contingency state of the system is checked for performance criteria violations, and a list of critical contingencies is formed. In the case of steady state analysis the process of contingency screening using margin sensitivity

This section presents the formulation of optimization developed to address the coordinated VAR planning problem for all the contingencies that have either one or both the planning problems shown in Table 1. The information required for the optimization algorithm are reactive device cost and maximum capacity limit at various voltage levels, performance measures and their sensitivities with respect to MSCs and SVCs under each critical contingencies, and an initial set of candidate locations for MSCs and SVCs. While many methodologies in the past determine static and dynamic VAR support sequentially, this method simultaneously determines the optimal allocation of static and dynamic VAR [31].

**4.1. Contingency analysis** 

$$\sum\_{i \in \Omega} \left[ \mathbf{C}\_{vi\\_MSC} \mathbf{B}\_{i\\_MSC} + \mathbf{C}\_{fi\\_MSC} \mathbf{q}\_{i\\_MSC} + \mathbf{C}\_{vi\\_svc} \mathbf{B}\_{i\\_svc} + \mathbf{C}\_{fi\\_svc} \mathbf{q}\_{i\\_svc} \right] \tag{12}$$

Subject to

$$\sum\_{i \in \Omega} \mathbf{S}\_{M,i}^{(k)} \mathbf{I} \mathbf{B}\_{i \\_M \mathbf{S} \mathbf{C}}^{(k)} + \mathbf{B}\_{i \\_ {\text{suc}}}^{(k)} \mathbf{I} + \mathbf{M}^{(k)} \ge \mathbf{M}\_{\mathbf{r} \prime} \,\forall k \tag{13}$$

$$\sum\_{\text{i}\neq\Omega\_{\text{src}}} S^{(k)}\_{\tau,n,i} B^{(k)}\_{i\\_\text{src}} + \tau^{(k)}\_{\text{dip},n} \le \tau\_{\text{dip},n,\tau'} \; \forall n,k \tag{14}$$

$$0 \le B\_i^{(k)} \\_{\text{MSC}} \le B\_{i \\_\text{MSC}} \; \forall k \tag{15}$$

$$0 \le B\_{i\\_svc}^{(k)} \le B\_{i\\_svc'} \,\forall k \tag{16}$$

$$B\_{\text{i-min\\_MSC}} q\_{i\\_MSC} \le B\_{\text{i\\_MSC}} \le B\_{\text{i\\_max\\_MSC}} q\_{i\\_MSC} \tag{17}$$

$$B\_{i\min\\_svc}q\_{i\\_svc} \le B\_{i\\_svc} \le B\_{i\max\\_svc}q\_{i\\_svc} \tag{18}$$

$$\neq q\_{i\\_MSC'}\neq q\_{i\\_svc} = 0\,\text{1}\tag{19}$$

The decision variables are Bi (k)\_MSC, Bi\_MSC, qi\_MSC, Bi (k)\_svc, Bi\_svc, and qi\_svc.

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,

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

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

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

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

the superscript k represents the contingency causing insufficient voltage stability margin and/or slow voltage recovery problems,

ΩMSC: set of candidate locations to install MSCs,

Ωsvc: set of candidate locations to install SVCs,

Ω: union of ΩMSC and Ωsvc,

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

( ) \_ svc *<sup>k</sup> Bi* : size of the SVC at location i under contingency k,

( ) , *k <sup>M</sup> <sup>i</sup> S* : sensitivity of the voltage stability margin with respect to the shunt susceptance of MSC at location i under contingency k,

Role of Induction Motors in Voltage Instability and Coordinated Reactive Power Planning 171

recovery. It will terminate once the post-contingency performance criteria are satisfied for all concerned contingencies, and there is no significant change in decision variables from the

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

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

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

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

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

characterized by generators outside of that MZ, but within the study area.

previous MIP solution.

**5. Numerical illustration** 

loading and 239 generators producing 37946.7 MW.

performance for all considered contingencies.

impedance for reactive power load.

( ) , , *k <sup>n</sup> <sup>i</sup> S* : sensitivity of the voltage recovery time duration at bus n with respect to the size of the SVC at location i under contingency k,

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

Mr: required voltage stability margin,

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

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

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

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

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

## *4.2.2. Updated successive mixed integer programming*

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 programming problem.

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 recovery. It will terminate once the post-contingency performance criteria are satisfied for all concerned contingencies, and there is no significant change in decision variables from the previous MIP solution.
