Integrated T&D Voltage Stability Assessment Considering Impact of DERs and Distribution Network Topology

Electric distribution systems are going through a major upgrade with increasing penetration of distributed energy resources (DERs) and impacting transmission system operational analysis. Traditionally, the transmission system’s voltage stability margin is computed by performing a continuation power flow (CPF) analysis. In transmission-CPF, loads in distribution systems are aggregated at the transmission-distribution (T&D) interface buses, and the distribution system’s electrical characteristics, like feeder length, increasing penetration of DERs, and associated control, are not considered. The traditional CPF thus considers the distribution network as a purely passive network without active network management and distributed energy resources. In this paper, two integrated T&D CPF methods are implemented using a co-simulation framework that uses individual transmission and distribution solvers. The first method is iteratively-coupled T&D CPF (IC-TDCPF), where the iterative coupling concept forces convergence of boundary variables at T&D PCC (Point of common coupling) at every CPF step. The second method, Loosely Coupled T&D CPF (LC-TDCPF), relaxes the boundary convergence constraints at PCC to achieve faster calculation speed. Moreover, a reduced distribution network equivalent is also developed to enable faster computation of VSM by reducing the number of integrated T&D buses. The developed T&D CPF co-simulation framework uses an open-source platform MATPOWER for solving transmission system and open-source software OPENDSS for solving distribution system. Three large integrated T&D systems with 4196, 10587, and 12295 buses are developed to demonstrate the impact of DERs, voltage-dependent load models, and distribution network topology on integrated voltage stability margin using the proposed T&D CPF and co-simulation.


I. INTRODUCTION
Load in the power system can be supplied until the voltage collapse point is reached with a certain margin. Estimating the changing collapse point and the margin between the operating point and collapse point is critical for the secure operation of the power system. The voltage stability margin (VSM) is normally computed by performing continuation power The associate editor coordinating the review of this manuscript and approving it for publication was Sarasij Das .
flow [1], [2], [3], [4] or running successive power-flow simulations. This analysis is normally performed by modeling distribution networks in the form of aggregated loads at T&D boundary buses.
Over the years, transmission and distribution analysis tools have evolved separately, leading to decoupled or very weakly coupled analysis using individual solvers. Most transmission system operators (TSOs) and distribution system operators (DSOs) rely on legacy individual solvers without active communication or sharing of data. Individual T&D solvers make several assumptions to simplify and decouple power-flow analysis. Transmission solvers model distribution either using a simplified circuit or in the form of aggregated load. Distribution solvers, on the other hand, model transmission system interface using an ideal voltage source connected in series with Thevenin impedance. This type of analysis is incapable of considering characteristics of the transmission system e-g the impact of generator Q-limits. The increasingly active nature of the distribution system due to greater penetration of DERs, more prevalence of power electronic devices, and more meshed topology demands the development of integrated simulation tools for both static and dynamic studies.
Many researchers have focused on the development of T&D co-simulation tools for integrated power-flow analysis. Co-simulation-based approaches actively exchange boundary data between T&D systems to properly consider the mutual impact of systems. T&D co-simulation approaches can be broadly subdivided into two categories: a) Iteratively coupled co-simulation approach [5], [6], [7], [8], [9] and b) Loosely coupled co-simulation approach [10], [11]. Iteratively coupled co-simulation approach ensures convergence of T&D boundary variables to arrive at an accurate integrated T&D solution. Loosely coupled approaches, on the other hand, relax T&D boundary convergence and compromise accuracy to enhance computational efficiency. Standalone tools can model T&D systems together but are computationally expensive [12], [13], [14]. Researchers have shown that the accuracy achieved by the iteratively coupled approach is comparable to that of standalone tools at much less computational expense [7].
This paper provides the integrated T&D voltage stability assessment using the developed co-simulation. Most of the researchers have focused on integrated power flow, but not much work has been reported regarding co-simulationbased integrated T&D voltage stability assessment. In [15], researchers developed a MATLAB-based tool using the primary/secondary concept to perform T&D CPF. The limitation of this tool is that it doesn't consider a co-simulation framework to use individual legacy solvers, which is essential because user-developed tools can lack the ability to model all the system components accurately. In [16], which is pioneering work on co-simulation based T&D voltage stability assessment, the authors do not consider the mathematical details of co-simulation interface and the impact of different co-simulation approaches on integrated voltage stability assessment. Moreover, in [16], authors do not consider predictor/corrector-based implementation to compute the maximum loading point, which can lead to conservative assessment. There is a need to develop mathematically robust T&D voltage stability assessment tools based on co-simulation for accurate integrated voltage stability assessment. The limitations of the existing research on integrated T&D voltage stability assessment can be summarized as follows: • Existing standalone T&D simulation tools does not allow integrated voltage stability assessment in a computationally efficient with detailed modeling of DERs.
• Most of the existing co-simulation research focuses on integrated T&D power flow analysis with limited focus on integrated voltage stability assessment.
• Existing research on integrated T&D voltage stability assessment lacks mathematically robust implementation of T&D continuation power flow leading to conservative margin assessment and possible numerical issues near the maximum loading limit.
• There needs to be more focus on the development of suitable reduced models for the distribution networks to enhance the computational efficiency of integrated T&D voltage stability assessment while conserving the accuracy achieved through co-simulation.
In this paper T&D integrated voltage stability analysis algorithms are developed using iteratively coupled and loosely coupled co-simulation methods. Balanced representation is used for the transmission system, which is solved using open-source transmission analysis package MATPOWER [17]. Three-phase representation is used for the distribution system, which is solved using open-source distribution analysis software OPENDSS [18]. The proposed analysis assumes that unbalanced distribution feeders are distributed so that the transmission system is nearly balanced. This assumption helps make use of legacy transmission analysis software and three-phase distribution solvers for integrated analysis. The main contributions in this work are as follows: • Developing a mathematically robust co-simulation framework for integrated transmission and distribution (T&D) voltage stability assessment.
• Developed an iteratively coupled transmission distribution continuation power flow (IC-TDCPF) algorithm for integrated T&D voltage stability assessment considering active nature of distribution networks.
• Developed a loosely coupled transmission distribution continuation power flow (LC-TDCPF) algorithm for integrated T&D voltage stability assessment considering distribution system characteristics.
• Developing a distribution network reduction approach to reduce the computational burden of IC-TCPF.
• Developed large integrated T&D system models using standard IEEE transmission and distribution systems to demonstrate the performance of developed tool in various operating conditions.
• Voltage stability assessment case studies considering impact of distribution system losses, distribution load voltage dependency, DER penetration, DER voltage regulation capacity, DER operating conditions and control modes. The rest of the paper is organized as follows. Section II presents preliminary background on transmission and distribution standalone and integrated modeling. Section III discusses the existing co-simulation approaches used for integrated T&D analysis. Section IV presents the developed algorithms for integrated T&D voltage stability assessment using predictor/corrector based T&D CPF. Section V presents a network reduction methodology to reduce the computational cost of T&D integrated voltage stability assessment. Section VI presents the case-studies considering large synthetic integrated T&D networks based IEEE test systems. Section VII provides a detailed discussion on the main takeaways from the case studies, and Section VIII provides the concluding remarks.

II. T&D SYSTEM MODELING
This section reviews the background literature on transmission system modeling, distribution system modeling, and integrated T&D modeling.

A. TRANSMISSION SYSTEM MODELING
In this work, MATPOWER, an open-source transmission system analysis software, is used to model the transmission system. The transmission system is assumed to be balanced, and per-phase transmission power-flow equations are represented as follows: Here, V T represents the vector of transmission system voltage magnitudes, and δ T vector represents the transmission system voltage angles. Together these vectors define the power system state, and S inj T defines the vector of complex power injections at transmission buses. The equations g T can be efficiently solved by traditional power-flow solution methods like Newton and Fast decoupled.

B. DISTRIBUTION SYSTEM MODELING
Distribution systems in power systems are heavily unbalanced, and in order to perform analysis, they have to be modeled in three-phase. In this work OPENDSS is used to model the distribution system in three-phase. As a result, the distribution system power-flow problem can be mathematically represented as follows: Here V B represents the voltage at the transmission and distribution boundary bus, which is assumed as a slack bus in distribution power-flow analysis. S inj,abc D defines the vector of complex power injections at distribution buses, and S BD is the total complex power flowing into the distribution system. The equations g D can be solved by distribution power flow techniques like forward-backward or current injection methods.

C. INTEGRATED T&D MODELING
In this work, transmission and distribution systems are modeled together with boundary coupling. This representation given by the set of equations in (4) divides the system into a transmission system, boundary system, and distribution networks connected to transmission through boundary buses. The first couple of equations in (4) mathematically describe transmission power flow equations which are identical to (1) and (2), but these equations are coupled with the distribution when integrated T&D simulation is considered. The third and fourth Equations in (4), which are identical to (2) and (3), represent the distribution system power flow equations. In equation (4), g v B and g s B represent boundary equations that ensure same boundary values of T&D systems when the set of equations given below are solved together. In equation (4), the T&D systems are coupled by g v B and g s B .
The set of equations in (4) can be solved by a stand-alone simulator in which distribution and transmission systems can be modeled together or using co-simulation based techniques.

III. APPROACHES FOR INTEGRATED T&D CO-SIMULATION
Due to time and mathematical complexity associated with stand-alone simulators, T&D systems in the integrated form are normally solved by adopting co-simulation approaches in literature which consider either a loosely coupled co-simulation approach or an iterative approach for coupling. In this section, the mathematical basics of iteratively coupled co-simulation and loosely coupled co-simulation are briefly discussed as a basis for the formulation of integrated T&D CPF approaches presented in Section IV.

3) Solve for boundary injections (S (k)
BD ) from transmission boundary buses to distribution systems for k th iteration by solving distribution network using distribution solvers. Distribution solvers use boundary voltages from transmission power flow results: 4) Compute residue R by computing the maximum difference between boundary injections of iteration k and k − 1.
5) If R < Th, save solution otherwise increment iteration counter and go to step 1.

B. LOOSELY COUPLED CO-SIMULATION
In the loosely coupled co-simulation method [10], the constraints regarding the convergence of boundary variables are relaxed, and T&D integrated systems are solved in a non-iterative manner. The algorithm for loosely coupled co-simulation for specified loading conditions is given as follows: 1) Solve distribution system using flat start or initially specified boundary voltages.
2) Solve transmission system using obtained boundary injections using transmission solver to obtain final solution.

IV. INTEGRATED T&D VOLTAGE STABILITY ASSESSMENT
In this Section three different CPF methods for integrated T&D voltage stability assessment are presented. The first method is the traditional CPF method, in which loads in the aggregated form are assumed to be directly connected at boundary buses. The second method considers iteratively coupled co-simulation for CPF implementation, and the third method considers a loosely coupled method of co-simulation for CPF implementation.

A. TRANSMISSION CONTINUATION POWER FLOW (T-CPF)
Boundary injections can be computed at transmission boundary buses (S (0) BD ) by solving distribution system equations (2) at base case loading (λ = 1). These equations can be solved by any distribution power flow method assuming the boundary bus as a slack bus. Once the aggregate boundary injection values at the base case are available, modified transmission system parameterized equations are formed. The predictor-corrector steps are performed until the nose point is reached to compute the CPF results.
Eq. (10) shows that base case boundary complex power flowing from transmission to the distribution system is a sum of injections at distribution buses (loads and DER) and the losses in the distribution network. T-CPF uses this value at base-case and linearly extrapolates the load at T&D boundary bus as shown in eq. (11). This linear approximation gives an approximately accurate value of actual complex power flowing from transmission to distribution system at loading point VOLUME 11, 2023 λ i because actual distribution system losses do not change in a linear manner when the load is incremented. This results in optimistic VSM computation because of non-consideration of distribution system losses at CPF steps with loading greater than base-case.
B. IC T&D CONTINUATION POWER FLOW (IC-TDCPF) 1) PREDICTOR Distribution system equations (2) are solved at base case loading (λ = 1) to obtain the approximate initial boundary injections (S (0) BD ). These equations can be solved by any distribution power flow method assuming the boundary bus as a slack bus. The purpose of using the predictor is to provide the corrector step a better set of initial values to solve for an integrated T&D system state at a specified loading condition. Once we have base case boundary injections, the predictor can be set up as follows to predict the next solution on the PV curve: Equation (14) is used to make tangential prediction of the next CPF solution. Since, the co-simulation is an iterative process, the availability of accurate initial conditions obtained from the predictor enhances the mathematical robustness.

2) CORRECTOR
The corrector in the iteratively coupled CPF approach solves for the current T&D state by ensuring iterative convergence of boundary variables. Once λ pred along with predicted boundary and transmission voltages are available, the corrector solves the following equations for specified loading (λ pred ) until boundary variables converge.

3) STEP LENGTH CONTROL
In (14) σ stands for the step used in the CPF formulation. The step length is automatically adjusted if the algorithm faces convergence issues in either distribution or transmission network solution. As the system approaches, the collapse point T&D corrector might not be able to converge for a certain predicted value. In this case, step length is reduced so that the corrector step can be run with a smaller difference between the next state and the previous state. In this work, whenever convergence issues arise, step length (σ ) is reduced by half for every non-convergence. Algorithm-1 explains the IC-TDCPF algorithm in detail, where predictor and corrector steps are repeated until the step length becomes smaller than a specified threshold. Predictor: Using eq. (13)-(14) make tangential prediction of next CPF solution (h + 1 th ) from previous solution (h th ).

9
If T&D co-simulation converges set Con = 0, otherwise set Con = 1. The predictor step in LC-TDCPF is identical to IC-TDCPF, which uses approximate initial-boundary injections to make the predictions. The difference lies in the corrector implementation where instead of iteratively solving until the convergence of boundary variables, the boundary variables are 1 Specify CPF step length parameter σ . 2 Initialize loading parameter λ = 1.0 and initialize boundary voltages (V T (B) (0) ). Set CPF step counter h = 1 3 Solve eq. (2) using initialized values to compute the initial boundary injections (S (0) BD (h)). 4 while σ > Th σ do 5 Predictor: Using eq. (13)-(14) make tangential prediction of next CPF solution (h + 1 th ) from previous solution (h th ).

9
If transmission power flow converges for computed boundary injections, set Con = 0 otherwise set Con = 1. only exchanged once, relaxing the boundary convergence condition. Algorithm 2 describes the LC-TDCPF for voltage stability assessment.

V. T&D INTEGRATED CONTINUATION POWER FLOW WITH REDUCED DISTRIBUTION MODEL
The integrated T&D CPF analysis, despite the accuracy achieved, can be very time-consuming. The complexity encourages considering a reduced distribution network model for integrated analysis. In this work, we consider a simple four-bus reduced model to perform integrated T&D CPF. The reduced distribution network is solved using OPENDSS due to the high R/X ratio of the equivalent, and the transmission side is solved using MATPOWER. Either LC-TDCPF or IC-TDCPF can be used to compute VSM using a reduced distribution model and the transmission network. To simplify the computation of the reduced network model, we consider the equivalent distribution feeder to be balanced. where, The values of A TD ,A DG ,A LD ,P loss , and Q loss can be computed by solving the distribution network in OPENDSS. If multiple instances of these values are available, the unknown parameters of the reduced distribution model can be obtained by solving the over-determined optimization problem.

VI. T&D CO-SIMULATION FRAMEWORK
T&D Co-simulation framework developed in this work for implementing CPF methods discussed in Section IV uses two open-source software for transmission and distribution analysis. For the transmission system, MATPOWER is used as the power flow solver, and for the distribution system OPENDSS, an open-source distribution system solver, is used. The co-simulation interface is set up in MATLAB where the two individual solvers MATPOWER and OPENDSS exchange the boundary variable information.

VII. CASE STUDIES
This Section presents the voltage stability assessment case studies using the developed T&D CPF methods. This Section VOLUME 11, 2023 first describes the integrated test systems used for case studies, followed by the demonstration of T&D CPF for different use cases. The considered use cases include consideration of distribution topology impact, the impact of distribution load composition, the impact of DER penetration and placement, and the impact of DER operating conditions on the voltage stability assessment. Voltage stability assessment is also demonstrated using the proposed reduced distribution network model. A computational complexity comparison is also performed among the proposed methods. VSM (Voltage stability margin) is used as the voltage stability assessment index in this work. VSM is computed considering the base load and maximum load at the integrated T&D buses. Mathematically VSM is computed as follows:

B. IMPACT OF DISTRIBUTION NETWORK TOPOLOGY ON VSM
In this section voltage stability margin for the developed test systems is computed using IC-TDCPF, LC-TDCPF, and T-CPF. To perform a voltage stability assessment using IC-TDCPF, Algorithm-1 is used, and Algorithm-2 is used to perform an assessment using LC-TDCPF. Results for T-CPF are obtained using MATPOWER based CPF implementation. The purpose of analysis in this section is to motivate the use of developed T&D CPF algorithms for accurate voltage stability assessment of integrated T&D power systems.

1) MODIFIED IEEE 14 BUS SYSTEM
Voltage stability margins are computed for TS-1 using T-CPF and the proposed T&D CPF algorithms. Fig. 2(a) shows PV curves plotted using different CPF methods for transmission bus 12, 13, and 14 in TS-1.

2) MODIFIED IEEE 9 BUS SYSTEM
Similarly, for TS-2 IC-TDCPF, LC-TDCPF, and T-CPF methods are used to compute voltage stability margins. Fig. 2(b) shows PV curves for transmission buses 5,7, and 9 in the modified IEEE 9 bus system.

3) MODIFIED IEEE 118 BUS SYSTEM
For the modified IEEE 118 bus system (TS-3), voltage stability margins computed using different CPF algorithms for considering distribution system impact are presented. PV curves in Fig. 2(c) show the PV curves obtained using different CPF methods for transmission buses 118, 95, and 38 in TS-3. It can be observed that T-CPF based voltage stability margin is optimistic because of the linear modeling of distribution system losses. Thus it is important to consider co-simulationbased integrated T&D CPF algorithms for accurate voltage stability margin computation.

C. IMPACT OF LOAD MODELS
In this section impact of load models on T&D-VSM is discussed. Loads in default IEEE 123 node feeders are replaced by ZIP loads having different compositions. T&D-VSM for the developed test systems is computed using the proposed methods and compared with T-CPF-based VSM to emphasize the need for consideration of distribution networks in detail. The motivation behind case studies in this section is to emphasize the usage of TD-CPF algorithms for accurately capturing the impact of voltage-dependent load in distribution feeders on integrated VSM because T-CPF by design, can not consider the impacts of voltage drops in distribution.
For IEEE 14 bus system, VSM is computed using IC-TDCPF, LC-TDCPF, and T-CPF methods. It is observed that the voltage dependency of the load greatly impacts the voltage stability margin. Default load models represent IEEE 123 node feeder with actual load models in OPENDSS provided standard file. Table 2 compares the voltage stability margins computed using the proposed T&D CPF methods to T-CPFbased VSM. Fig. 3(a) shows the PV curves obtained for bus-4 using the IC-TDCPF algorithm for considered load models.

2) MODIFIED IEEE 9 BUS SYSTEM
For the modified IEEE 9 bus system, VSM is computed using IC-TDCPF, LC-TDCPF, and T-CPF methods. Table 3 shows VSM computed using the considered CPF methods. Fig. 3(b) TABLE 4. Impact of load models-modified IEEE 9 bus system. shows the PV curves obtained for bus-5 using the IC-TDCPF algorithm for considered load models.

3) MODIFIED IEEE 118 BUS SYSTEM
For the modified IEEE 118 bus system, VSM is computed using IC-TDCPF, LC-TDCPF, and T-CPF methods. Table 4 compares the voltage stability margin using the considered CPF methods. Fig. 3(c) shows the PV curves obtained for bus-118 using the IC-TDCPF algorithm for considered load models.
It is observed that T-CPF provides an optimistic estimate of VSM for all the developed test systems as compared to IC-TDCPF and LC-TDCPF methods. This happens because T-CPF uses aggregated load at base-case, leading to the assumption that distribution losses vary linearly.

D. CONSIDERATION OF GENERATOR Q-LIMITS
In this section impact of generator reactive power limits on VSM using T-CPF, IC-TDCPF and LC-TDCPF are compared for the developed test systems. Whenever a generator hits Q-limits, the slope of the PV curve changes and leads to a different behavior of voltage. Co-simulation framework with MATPOWER has an inbuilt feature to consider Q-limits when solving corrector power flow equations. These studies can allow operators to efficiently control reactive power reserves for enhancing voltage security.

1) MODIFIED IEEE 14 BUS SYSTEM
There are 4 generators in IEEE 14 bus transmission system other than the slack bus. The Q-limits of these generators are provided in MATPOWER transmission case file. Q-limits, when enforced while performing T-CPF, will affect the system loading limit. It can be seen from Table 5 that IC-TDCPF and LC-TDCPF can consider the impact of Q-limits on VSM. Fig. 4(a) shows the impact of Q-limits on the PV curve for Bus 4 computed using T&D CPF algorithms.

2) MODIFIED IEEE 9 BUS SYSTEM
There are 2 generators in IEEE 9 bus system other than the slack bus. The Q-limits of generators are modified to show the impact of generator Q-limits. Max reactive power capacity of the generator at bus-2 is modified to be equal to 70 MVARs, and that of the generator at bus 3 is modified to be equal to 25 MVARs. It can be observed from Table 5 that IC-TDCPF and LC-TDCPF are able to capture the impact generator Q-limits. Fig. 4(b) shows the impact of Q-limits on the PV curve for Bus 5 in the modified IEEE 9 bus system.

3) MODIFIED IEEE 118 BUS SYSTEM
The modified IEEE 118 bus system has 54 generators other than the slack bus. These generators exhaust their Q-limits one by one as system loading is increased. These Q-limits lead to a change in the behavior of the PV curve computed using T&D CPF algorithms, as demonstrated in Fig. 4(c), which shows the PV curves for Bus 118. Table 5 shows that IC-TDCPF and LC-TDCPF are able to capture the impact generator Q-limits on integrated T&D VSM.

E. IMPACT OF DER PENETRATION AND PLACEMENT
In this section impact of DER penetration level and placement on voltage stability is considered using IC-TDCPF and  LC-TDCPF methods. Four different cases shown in Table 6 are created by modifying the IEEE 123 node distribution systems used to form the integrated T&D sysems. Table 6 shows the buses at which DERs are installed along with power injection at by each DER. In Case-1 total distributed generation of 500 KW is installed with 100 KW DER installation each at four (04) different buses as specified in Table 6. Similarly, for Case-2 and Case 3, total DER installation is 500 kW, but different buses are selected for installation. In Case-4, the total DER installation is increased to 1000 KW. For all the cases, OPENDSS provided generator model-3 is used. The maximum KVAR of all the generators in Table 6 is specified as 100 KVAR. T&D integrated voltage stability analysis using proposed approaches is performed for modified IEEE 14 and modified IEEE 118 bus system with default load models. Voltage stability margins reported in Table 6 consider the impact of generator Q-limits in both T&D systems. The impact of DER installation on integrated T&D VSM is investigated for the modified IEEE 14 bus system. It can be observed from Table 7 that the installation of DER improves the VSM for all cases in Table 6 as compared to cases without DER installation.

2) MODIFIED IEEE 118 BUS SYSTEM
In this section impact of DER installation on modified IEEE 118 voltage stability margin (VSM) is investigated. The impact of Q-limits is also taken into account when computing VSM. It can be observed from Table 8 that VSM with the installation of DER improves for all the cases in Table 6.

F. IMPACT OF DER VOLTAGE REGULATION CAPACITY
In this section impact of the voltage regulation capacity of DERs in the distribution system on overall T&D VSM is considered. OPENDSS using model 3 generator allows modeling distributed generation in PV mode similar to transmission PV buses. In order to demonstrate the impact of DER voltage regulation ability, generators in Case-I are modeled with various allowed maximum KVAR. This puts a limit on the voltage regulation ability of DERs. It can be observed from Table 8 that allowed maximum KVAR of DERs impacts the voltage maintaining ability, which in turn affects the overall T&D voltage stability margin.

G. IMPACT OF DER OPERATING CONDITIONS ON VSM
Although DERs provide substantial benefits, the power output of DERs is hugely dependent on environmental variables like temperature, wind speed, solar irradiance, etc. It is important to consider the impact of these variables on VSM computed at the integrated level. The proposed co-simulation methodology for VSM can be used to compute VSM, considering the varying operating conditions. In particular, in this study, the DER is modeled as an array of photo-voltaic panels (PV-panel). The amount of irradiance varies with the time of the day leading to different PV generation values. The VSM is computed as a function of solar irradiance. Fig 6 shows the VSM as a function of solar irradiance, where VSM is computed using IC-TDCPF. The VSM is computed in Fig 6 considering the UPF (unity power factor) mode of operation. VSM can be similarly computed considering reactive power contribution from DER according to reactive power control modes in IEEE-1547 standard. Table 10 shows a comparison of VSM with UPF and VVC (Volt-Var Control) for the IEEE-14 bus and IEEE-118 bus network using IC-TDCPF.

H. INTEGRATED T&D CPF USING REDUCED DISTRIBUTION NETWORK MODEL
In this section, first, the reduced model of the IEEE 123 node feeder with 500 kW of DG (Corresponding to Case-1 in Table 6) is estimated. Table 11 shows the estimated reduced network parameters. The estimated model is then used to perform IC-TDCPF for VSM computation. It is observed that the considered 4 bus reduced model in this work brings down the number of integrated T&D buses immensely, which leads to better timing performance. Figure 7 shows that the reduced model provides optimistic VSM compared to IC-TDCPF with the full distribution model. This study also motivates the development of more accurate reduced models for VSM computation that can bring down the computational complexity while maintaining the accuracy of fully integrated assessment.

I. COMPUTATIONAL COMPLEXITY AND ROBUSTNESS
This sub-section demonstrates the mathematical robustness of the IC-TDCPF in terms of the number of co-iterations at each step of CPF. The number of co-iterations is reduced by using a predictor-corrector approach. Figures 8 and 9 show the comparison of co-iterations between IC-TDCPF and the traditional co-simulation approach without the predictorcorrector implementation.  Despite being mathematically robust, IC-TDCPF is computationally demanding, and this work proposes two relaxed approaches to deal with the computational complexity. This sub-section also compares the simulation times required to compute the VSM using the considered CPF methods. Fig. 10 shows that the traditional CPF provides the fastest computation, followed by LC-TDCPF, and IC-TDCPF. IC-TDCPF is time-consuming but provides accurate solution, whereas LC-TDCPF provides a trade-off between speed and accuracy. A reduced distribution network can be used to alleviate the computational complexity of IC-TDCPF by reducing the number of integrated T&D buses with some compromise on the accuracy of computed VSM.

J. DISCUSSIONS
• Traditional transmission-level CPF (T-CPF), which models the distribution system as a lumped load or as a simplified equivalent network is unable to provide an accurate VSM because of the over-simplification of the distribution network electrical characteristics. He is currently a Lane Professor and the Chairperson at the Computer Science and Electrical Engineering Department, West Virginia University, and an Adjunct Professor of electric power engineering at Washington State University. He also has a joint appointment as a Senior Scientist, with the Pacific Northwest National Laboratory (PNNL). He is the author of more than 300 technical publications, including a book on power system security and four patents. His research interest includes data-driven algorithms for power system operation and control, including resiliency analysis. He is serving or served as an Editor for the IEEE TRANSACTIONS ON SMART GRID, IEEE TRANSACTIONS ON POWER SYSTEMS, and IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS.