Dynamic Performance Evaluation of the Secondary Control in Islanded Microgrids Considering Frequency-Dependent Load Models

Abstract

Performance analysis is challenging in modern power distribution networks due to the increasing penetration of distributed energy resources (DERs) interfaced by voltage source converters (VSCs). Parameter uncertainty, black-box models of the components, and low inertia are some of the issues that must be addressed. The latter can represent high vulnerabilities to sudden load changes in a microgrid (MG). Additionally, the islanded operation represents a challenge for inverter-based (DERs), where secondary control must regulate the microgrid frequency and voltage to its nominal values. When a secondary control strategy is implemented, it is necessary to validate its performance under several conditions. Most existing research papers focus on the microgrid’s small-signal and transient stability. At the same time, little has been done on the influence of the load model on voltage stability. This type of analysis is required to ensure an adequate transition between the grid-connected and stand-alone modes. This paper provides a voltage stability analysis of a microgrid, considering secondary control based on receding horizon and frequency-dependent load models. Simulation results demonstrate the robust performance of the secondary control and validate the importance of considering or adapting voltage stability indices to ensure adequate microgrid performance.

1. Introduction

Variable renewable energy (VRE), inverter-connected resources (ICR), and distributed energy resources (DER) represent a challenge for analyzing and operating microgrids and active distribution networks (ADN) [1,2,3]. Stability studies in these grids require the extension of classic concepts to this new context, including components such as power converters, small wind turbines, photovoltaic modules, batteries, and micro-gas turbines. Moreover, the grid-connected and islanded mode must be considered, as well as new protection schemes, energy management systems, and several control strategies used for voltage and frequency regulation in grids with low inertia, weak damping, and small output impedance [3,4,5,6].

The current scientific literature about microgrid stability mainly focuses on small-signal and transient stability, and few consider voltage stability. The latter is defined as the system’s capability to maintain and restore voltage on all buses after a small or large disturbance [7]. Numerous factors affect the voltage stability, such as the impedance, the $r/x$ factor, the amount of load, and the position of the transformers’ taps. The model of the load may also affect the voltage stability. However, most microgrid studies are performed with constant power models (CPLs). Considering that a microgrid may be exposed to frequency and voltage changes in isolated mode, it is necessary to analyze the impact on the load model and the performance of secondary control.

The revised literature shows numerous references related to small-signal [8,9,10,11,12] and transient stability analysis on microgrids [13,14,15,16]. However, there are few works related to voltage stability. Some studies proposed strategies to prevent the occurrence of failures in smart grids [17,18]; other studies apply tools to characterize the voltage stability as an eigenvalue analysis [19], energy function [20], PQ curve [21], or voltage stability indices (VSI) [22,23,24,28]. Although VSI plays an important role in exploring the stability condition of the electric system, few proposals include both islanded and connected operation modes [24,28]. As shown in

Table 1. Comparison of voltage stability analysis on ADN.

Ref.	Voltage Stability Analysis				Load Model	Operating Mode
	PQ Curve	Stability Index	Eigenvalue Analysis	Energy Function
[22]		SI			Constant	Connected
[23]		FVSI, MVSI			ZIP and constant	Connected
[19]					PQ type	Islanded
[20]					PQ type	Islanded
[21]					PQ type	Islanded
[24]		k${}_{QU}$, k${}_{PU}$			Load profile	Hybrid
[25]		M${}_{n/d}$			Constant power	Connected
[26]		PTSI, VCPI, VSI			Constant power	Connected
[27]		SI			ZIP	Connected
[28]		A-VSI, E-VSI			ZIP, composite	Hybrid
Proposal		VCPI_1			Frequency-dependent	Hybrid

, most papers considered CPLs in the analysis of the connected mode without considering the influence of the frequency. In this mode, the main grid imposes this variable and the voltage at the point of common coupling (PCC). However, maintaining voltage stability in islanded mode is more challenging than in grid-connected mode because the MG is not supported by the utility and requires the control actions of inverters to regulate this variable. Due to the susceptibility of islanded MGs to voltage and frequency variations being higher in grid-connected mode, this requires the proper modeling of voltage and frequency-dependent loads. To the best of the authors’ knowledge, this aspect has been little studied in the literature.

In [22] a new real-time stability index was proposed based on estimation of the Thevenin impedance; however, this work only considered CPLs. In [23], a modified voltage stability index was used, assuming a two-tier load model (ZIP or constant impedance model) according to voltage ranges. The authors in [19] explored the voltage instability/collapse problem in ac–dc hybrid microgrids (HMGs) during islanding and under contingencies considering CPLs. Constant power loads were also used in [20], where a small-signal voltage stability analysis of the microgrid was achieved. An alternative was presented in [21], where an energy function was used to monitor the microgrid security margin. The authors in [24] analyzed the influence of the microgrid operating condition on the voltage stability margin indexes. A new voltage stability index based on the network–load admittance ratio was exposed and tested in [25]; however, it only considers CPL. This load type assumption also is considered in [26], where the performance of three traditional indices on a microgrid test system is analyzed. Other references, such as [28], propose two indices to define the voltage stability in radial distribution networks when constant and ZIP load models are included. These load types are also considered in [27], which analyzes the effect of DG capacity and location on the voltage stability analysis of a radial distribution system. Although other load types than constant power are considered in the above references, the influence of frequency is not considered. To fill this research gap, we extend the application of a centralized secondary control based on the receding horizon previously proposed in [29], considering the influence of frequency-dependent load models to analyze the impact on microgrid voltage stability.

The rest of the paper is organized as follows. Section 2 describes the theoretical aspects: the hierarchical control strategy, with particular consideration of secondary control, and the proposed technique; the voltage stability in microgrids, load modeling influence, and evaluation of the stability of the islanded microgrid through the application of voltage stability indices are presented in Section 4. Simulation results are presented and discussed in Section 4. Finally, conclusions are presented in Section 5.

2. Review of Microgrid Control

2.1. Hierarchical Control

Unlike traditional power systems, microgrids can operate with high voltage and frequency deviations. The lack of inertia for systems with DERs in islanded mode produces significant changes in how the frequency behaves [30]. For this reason, all control layers of a microgrid must respond adequately to guarantee stable operation. An AC microgrid uses a hierarchical control structure split into three main layers: primary, secondary, and tertiary control loops [31,32].

Although voltage and frequency stability is mainly dominated by primary control in electric systems and also microgrids, as control with faster response to disturbances, secondary control must be designed to produce feasible set points to primary control, guaranteeing the stability of all variables as they approach the nominal values while considering available resources from tertiary control. The objectives of the three-level hierarchical control scheme are described in brief in subsequent sections.

Primary control is the fastest control layer that executes local control actions at each power converter to carry the system to a stable frequency and voltage operation point. This control emulates the inertial behavior of synchronous machines by the droop control method. Figure 1 depicts the diagram of a droop-controlled converter, where active and reactive power references are, respectively, $p_{\mathrm{ref}}$ and $q_{\mathrm{ref}}$. The secondary control provides these references every time the frequency $\omega$ and/or the voltages $|v_k|$ are deviated from their nominal values, ${\omega }_o$ and $v_{\mathrm{ref}}$, respectively. $\gamma$ and $\lambda$ represent the droop constants given by primary control at each power converter.

The main objective of secondary control is to restore the frequency and voltages to the nominal values. This control provides new set-points for the primary control while considering the frequency and voltage variations.

Figure 2 shows the frequency behavior in a microgrid when there is a switch from connected to islanded mode, considering the effect of primary and secondary control.

A frequency variation leads primary control to stabilize it in a stable point, which can be different from the nominal frequency value $f^{*}$. Secondary control, which can be centralized or decentralized [31], considers the operation point found by primary control and the availability of the distributed energy sources, as well as the grid capabilities, and generates new set points in order to balance the system and return the frequency to $f^{*}$. Despite including power electronic devices and new renewable energy generation, a microgrid has low inertia properties, and its generation capacity is limited. Then, a small change in the microgrid load or small perturbations may create a powerful impact on the system’s frequency and voltage. However, when a decentralized strategy is considered, the power balance cannot be guaranteed since each distributed generator, connected to the grid by an electronic converter, can only stabilize its power injection, voltage, and frequency [33,34].

On the other hand, a centralized strategy for the secondary control must coordinate the generation units interfaced by a power electronic converter to keep their frequency at the same value. In the case of decentralized approaches, every decentralized secondary control has to communicate with each other to avoid each one stabilizing differently and to different values, which can be detrimental for the grid operation [35]. Additionally, in decentralized approaches, all generation units can reach different stable operation points. The stable operation point of frequency must be the same for all generation units. Instead, centralized approaches are efficient for electric systems with nearby generation units, such as microgrids. This paper focuses on the centralized secondary control of an islanded microgrid when frequency-dependent load models are considered.

Finally, the tertiary control, which is in charge of the steady-state optimal operation of the grid, takes the data of economic dispatching, loads, and primary resource availability, among others, into consideration to optimize the MG operation [36]. Once the optimization problem is solved, the tertiary control communicates the set-points of the power outputs of each distributed generator unit, through the secondary control.

2.2. Optimization-Based Secondary Control

Receding horizon approaches for control systems are classically studied problems [37]. These strategies are based on an optimization algorithm that solves a certain number of problems in a determined time interval to produce a control signal that optimally solves the problem in a time horizon. Optimization algorithms used to solve the problem must be reliable and guarantee a global optimum and uniqueness on the solution. For secondary control problems, considering that electronic converters have a limited capacity to adjust the injected power, an approach based on the concept of receding horizon is analyzed, considering the primary control effect, optimal steady-state values given by tertiary control, and grid limitations [29]. Moreover, this secondary control problem may guarantee optimal modification on the set-points to preserve the power balance and set-points that preserve the grid stability. Although primary control is the control layer in charge of preserving the grid’s stability, secondary control modifies it. Hence, secondary control may guarantee stability in each step-by-step solution of the problem.

The main objective of the proposed centralized control is to carry grid frequency to the nominal value. Suitable values of voltage will also be considered in the optimization problem. A frequency-dependent power flow is executed to determine the operative state of the grid in every time step until the frequency is restored. Afterward, the optimization finds the optimal control actions for secondary control to restore the frequency considering grid restrictions. For this reason, it must be a convex problem to guarantee a global optimum in each step. It is worth mentioning that the frequency will be returned to its rated value only if the converters’ capabilities and the primary source’s availability are enough; here, the tertiary layer of control is in charge of guaranteeing the availability of resources if a disturbance appears. Figure 3 shows an example of how a receding horizon strategy can be implemented to solve frequency deviation problems in islanded microgrids where any disturbance is considered in $t_o$. The secondary and primary control effects are shown, while the nominal frequency is reached in $t_1$. A steady state for primary control is reached in every step. However, the effect of the frequency from the primary is clearly considered in the secondary control.

2.3. Hierarchical-Optimal Control Model

In order to build a model for hierarchical control on islanded microgrids, a power flow algorithm should be considered. Then, a power flow as formulated in (9) includes typical elements suggested for typical power flow methods but also considering the variation in the frequency followed by reference active and reactive powers, the effect of primary control, and the variation given by a secondary control.

Here, J corresponds to the Jacobian matrix associated with the non-linear equations representing the active and reactive power injection at each node. Notice that new variables $\mathrm{\Delta }{\theta }_c$ and $\mathrm{\Delta }\omega$ are added. The former represents the centroid of the system, and the latter considers the frequency variations. The centroid concept is closely related to the center of inertia, frequently used in power system applications. Moreover, the centroid allows the definition of a reference point in a system with a lack of inertia using the droop constants of power converters. Thus, an optimization model as shown by Equations (1)–(9) can be built.

$$\begin{array}{cc}\hfill \mathrm{minimize}& \hfill ∥\mathrm{\Delta }\omega ∥\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \mathrm{subject}\mathrm{to}& \hfill V^{\min }\le V+\mathrm{\Delta }V\le V^{\max }\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & \hfill P^{\min }\le P+\mathrm{\Delta }{P}_{\sec }\le P^{\max }\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & \hfill ∥\mathrm{\Delta }{P}_{\sec }∥\le \mathrm{\Delta }{P}_{\sec }^{\max }\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & \hfill ∥\mathrm{\Delta }{Q}_{\sec }∥\le \mathrm{\Delta }{Q}_{\sec }^{\max }\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & \hfill E_{\mathrm{battery}}^{\min }\le E_{\mathrm{battery}}^k\le E_{\mathrm{battery}}^{\max }\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & \hfill P_{\mathrm{battery}}^{\min }\le P_{\mathrm{battery}}\le P_{\mathrm{battery}}^{\max }\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & \hfill E_{\mathrm{battery}}^k=E_{\mathrm{battery}}^{k-1}-P_{\mathrm{battery}}^k\mathrm{\Delta }t\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & \hfill \left(\begin{array}{c}\overline{P}+\mathrm{\Delta }{P}_{\sec }+{\mathrm{\Gamma }}^T\mathrm{\Delta }\omega \\ \overline{Q}+\mathrm{\Delta }{Q}_{\sec }+{\Lambda }^{\mathrm{T}}\mathrm{\Delta }V\\ \mathrm{\Delta }{\theta }_C\end{array}\right)=J\left(\begin{array}{c}\mathrm{\Delta }\theta \\ \mathrm{\Delta }V\\ \mathrm{\Delta }\omega \end{array}\right)\hfill \end{array}$$

(9)

Equations (2) and (3) represent the limits of voltage and power generated at each power converter. $\mathrm{\Delta }V$, $\mathrm{\Delta }{P}_{\sec }$, and $\mathrm{\Delta }{Q}_{\sec }$ represent the variation in the voltage and the variation in set-points for active and reactive power due to the secondary control response at every converter, respectively. Additionally, $\mathrm{\Delta }{P}_{\sec }^{\max }$ and $\mathrm{\Delta }{Q}_{\sec }^{\max }$ are the limitations for the power converters to modify their set-points in every k iteration due to the secondary control response. $E_{\mathrm{battery}}$ and $P_{\mathrm{battery}}$ represent the stored energy and power injected from all storage devices for every iteration considering a time window $\mathrm{\Delta }t$. Then, Equations (6)–(8) represent the limitations of stored energy, injected active power, and the state of internal energy considering power injected, respectively. A more detailed model description can be seen in [29]. The workflow of the optimization model can be seen in the diagram of Figure 4, where the considerations of the primary, secondary, and tertiary layers of the control model makes up a complete hierarchical control structure.

3. Stability in Microgrids

Although microgrids are similarly represented as electric power systems, the stability analysis differs since many differences exist between microgrids and conventional power systems. The main contrast lies in the low reactance to resistance ratio given by the radically smaller microgrids compared to power systems. Microgrids include renewable energy sources that depend on the stochastic behavior of the primary resource. The fitful behavior of renewable energy sources, the reduced loads, and the power injections made by electronic power converters with small inertia make the stability analysis even more challenging. As microgrids must also operate in islanded mode, a minor modification of the grid configuration may result in significant voltage and frequency deviations. Then, control and operation strategies must take voltage and frequency variables near their nominal value and guarantee frequency and voltage stability in microgrids, especially in islanded operations. According to [38], microgrid stability can be classified regarding the part of the observed system—the Stability of the Control System or the Power Supply and Balance Stability. The last one is related to the system’s ability to maintain an active power balance through the demand power sharing among DERs. This class of stability can be subcategorized into Frequency and Voltage Stability. Voltage stability refers to general system stability and DC Link Capacitor Voltage Stability. Regarding the former, specific characteristics of microgrids, such as inverter dynamics, Q-V droop control, and load dynamics, among others, can affect the microgrid voltage stability.

Therefore, it is worth analyzing the impact of load modeling, specifically frequency-dependent load models, on the stability when a hierarchical control is applied, particularly during islanded operation.

3.1. Load Modeling

As the accuracy of control and stability analysis largely depends on the quality of the models used in describing electric system behavior, the load modeling has to be carefully considered in studies of microgrids. It has been widely demonstrated that different load models have different impacts on simulation results for voltage stability [39]. Therefore, in this paper, we consider the exponential load model that depends on the bus frequency, according to (10) and (11):

$$\begin{array}{c}p=p_0{\left({\displaystyle \frac{v}{v_0}}\right)}^{{\nu }_{pu}}[1+{\kappa }_{pf}(f-f_0)]\end{array}$$

(10)

$$\begin{array}{c}q=q_0{\left({\displaystyle \frac{v}{v_0}}\right)}^{{\nu }_{qu}}[1+{\kappa }_{qf}(f-f_0)]\end{array}$$

(11)

where f is the frequency of the bus voltage, $f_0$ is the nominal frequency, ${\kappa }_{pf}$, ${\kappa }_{qf}$ are the frequency sensitivity parameters, $p_0$ and $q_0$ are the power consumed at the rated voltage $v_0$, and ${\nu }_{pu}$, ${\nu }_{qu}$ are voltage exponents for active and reactive powers, respectively. Generally, in electric power systems, the frequency factor f is often neglected since the voltage changes are much more frequent and more pronounced than the changes in system frequency; however, this assumption is not valid for isolated microgrids [38].

3.2. Voltage Stability Indices

Voltage stability indices (VSI) have been widely used to estimate proximity to stability problems in bulk systems, as these can determine how close a system is to a voltage collapse. The objective of these indices is to define a scalar magnitude to determine the status condition either for buses or line systems [7].

However, when these VSI are applied in MGs, there are many questions related to their applicability and validity. For instance, the X/R relation in MG is different compared to power systems and, therefore, the coupling between active power and frequency (P-f) and reactive power and voltage (Q-V) is less noticeable. Otherwise, some authors consider that voltage collapse has not been observed in MGs [38]; nevertheless, [19] argue that voltage collapse is possible in microgrids during serious contingency events. In this paper, a review of some VSI is realized and then they are applied, with the aim to determine the impact of the centralized secondary control strategy on the voltage and frequency stability of an isolated microgrid.

The voltage stability can be evaluated by line voltage stability indices (LVS) and bus stability indices (BSI). Many indices are considered in the literature, such as VCPI [40], L-index [41], NVSI [42], L${}_{ij}$ [43], and PTSI [44], among others; however, relatively little research is reported on the application of these VSI in microgrids, as shown in Table 1.

According to [45], many of the existing VSIs cannot be directly employed for microgrid stability analysis due to different assumptions, such as no shunt admittance, zero resistance, or no DERs included, among others. However, few papers have demonstrated the applicability of the proposed indices under simulation as shown in

Table 2. VSI applied in microgrids.

VSI	Reference	Assumption
$PTSI$	[26]	Shunt admittance is not considered
$SI$	[27]	Shunt admittance is not considered
$A-VSI$	[28]	Only voltage measurements are required
$FVSI$	[23]	Depends only on reactive power load
$MVSI$	[23]	P and Q of DG are considered

. This paper selects and tests an index to verify the robustness of the proposed secondary control approach and its influence on microgrid stability.

However, some indices can be adjusted by considering DERs in either the sending or receiving end of a two-bus circuit. The modification can be done considering that the active power is composed by DER power minus load power. For the reactive power, we can include the compensator’s reactive power.

3.3. Islanded Mode of Microgrids

A microgrid, such as the one shown in Figure 5, is considered stable following the previously mentioned concepts if, after a disturbance, all of the steady-state variables are preserved. Moreover, new steady-state variables can be achieved as long as they are kept in the operational constraints. A disturbance can be considered as any external input, such as load modifications, component breakdown, or even operational set-point adjustments for the power converters [38]. However, due to the particular characteristics of microgrids, stability can be analyzed under different approaches. In traditional power systems, voltage stability issues happen more frequently, as shown previously, than frequency stability issues. However, in islanded microgrids, it is harder to maintain the frequency due to the absence of inertia in the system and the high penetration of renewable generation. Some requirements of the islanded operating condition are related to the restoration process of the voltage and frequency as quickly as possible to their nominal values.

Although the primary control mainly dominates stability in electrical systems, the secondary control must be designed to generate feasible set-points for the primary control, which also guarantees the stability of all the variables, particularly for microgrids. This paper focuses on the voltage stability analysis of a microgrid with frequency-dependent load models when a secondary control strategy is applied.

4. Results

In order to analyze the influence of the hierarchical control over the voltage stability, a test system provided by CIGRE [46] is adapted as shown in Figure 5. As the microgrid topology has a considerable impact on load ability and voltage stability, it has been considered a radial network, which is more challenging than a meshed-networked microgrid concerning load ability limits. The test system considers energy storage systems, wind and solar energy generation, and different load types. Each node in the dotted boxes in Figure 5 considers the inclusion of a power converter. The test system parameters are shown in Appendix A and are given per unit with 400 V, 100 kW, and 60 Hz as base quantities.

The exponential load models (ELM) presented in Section 3 are considered in the simulation to analyze the influence of load modeling during the islanding transition of the microgrid. Load characterization is assumed as residential, small industry, and agricultural classes. Each category has a different composition, as shown in

Table 3. Load characteristics.

		Air Conditioner	Lighting	Refrigeration	Resistive	Large Motor	Small Motor	Pump
	1. Residential	50%	20%	10%	20%	0	0	0
Load types	2. Industry	0	20%	0	10%	60%	10%	0
	3. Agricultural	0	0	0	0	0	0	100%
	${\nu }_{pu}$	0.2	0.96	0.77	2	0.07	0.07	1.4
	${\nu }_{qu}$	2.3	7.38	2.5	0	0.5	0.5	1.4
ELM parameters	${\kappa }_{pf}$	0.9	1	0.53	0	2.5	2.5	5
	${\kappa }_{qf}$	−2.67	−26.6	−1.46	0	1.2	1.2	4
	PF	0.75	0.9	0.84	1	0.89	0.83	0.85

. The ELM parameters described in Section 3.1, i.e., ${\nu }_{pu}$, ${\nu }_{qu}$, ${\kappa }_{pf}$, ${\kappa }_{qf}$, for each component are also presented in Table 3, where $PF$ refers to the power factor. The values of the above constants for every load type were provided by [13]. Two different load scenarios are analyzed. Both cases consider constant frequency-dependent loads, with the same load composition but different load demand, as shown in

Table 4. Apparent power in VA for each case.

Node	Case 1	Case 2	Load Type
11	7000	0	1
13	40,000	0	1
14	18,000	18,000	1
18	12,000	12,000	2
19	1000	1000	3

.

4.1. Hierarchical Control

This section focuses on the analysis of the influence of the frequency-dependent load model as the composition load type on the stability of an islanded microgrid. To fulfill this objective, two cases are analyzed, as presented below.

4.1.1. Case 1

The microgrid is disconnected from the main grid in $t=0$ s. As can be observed from Figure 6a, the converter’s frequency values differ from the nominal values during 0.8 s, when the constant load model is considered. During this time, the secondary control strategy takes these values back to the rated values. Otherwise, when we consider the exponential load model, the results show that the secondary control needs more time to achieve the isolated microgrid frequency stabilization, as shown in Figure 6b. In Figure 7a,b, it can be analyzed how the larger voltage variations are obtained with a more detailed load model compared to the constant power model.

4.1.2. Case 2

In this case, the energy demand is less than in case 1, but with the same load composition in the three remaining buses. Figure 8a,b show the secondary control with the same frequency magnitude deviations (between 0.98 and 1.04 p.u.) for the constant load model as a frequency-dependent load; however, the stabilization time required for the latter model is 0.05 s.

Although this time is minimal for the system analyzed, in a more complex microgrid with many loads and several DERs interfaced by VSCs, this stabilization time could increase and represent a problem in the switching from the grid-connected mode to the islanded mode. The simulation results of Figure 9a,b prove that case 2 resulted in a better voltage profile throughout the system compared to case 1.

4.2. Stability Index

In order to prove the secondary control’s effectiveness for the microgrid voltage stability, we apply the line voltage stability VCPI_1 in both case 1 and case 2. The VCPI_1 is defined in [47] as:

$$\begin{array}{c}\mathrm{VCPI}\_1=v_{r}cos\left(\delta \right)-0.5v_s\end{array}$$

(12)

when VCPI_1 $\ge 0$, the line voltage is stable, and otherwise, it is unstable.

Although, in this index, the Thevenin network impedance at the sending bus is neglected, in this paper, it was selected as it does not make simplifications in its formulation concerning the active or reactive powers, as in other indices such as FVSI [23] or Lmn [48]. Due to variations in the active power in buses with DGs, these indices cannot be used to analyze the microgrid stability, as demonstrated in Section 4.

4.2.1. VSI for Case 1

For the constant load model, when the microgrid is isolated, the secondary control is able to keep the voltage $\pm 0.025$ p.u., as shown in Figure 7a. This validates the adequate control carried out by the secondary control (maximum limit $\pm 0.05$ defined in the algorithm); however, an adequate voltage profile does not guarantee voltage stability. For this reason, a stability index can be used during the operating mode transition. The microgrid transition to islanded mode results in a minor reduction in the voltage stability margin according to Figure 10a due to the secondary control effect. At each microgrid line (from 1 to 18, according to

Table A2. Node connection for the CIGRE low-voltage test system.

Line	From	To	Distance (m)	Line Type
1	1	2	35	1
2	2	3	35	1
3	3	4	35	1
4	4	5	35	1
5	5	6	35	1
6	6	7	35	1
7	7	8	35	1
8	8	9	35	1
9	9	10	35	1
10	3	11	30	2
11	4	12	30	3
12	6	13	30	4
13	10	14	30	3
14	4	15	35	1
15	15	16	35	1
16	16	17	35	1
17	17	18	30	1
18	9	19	30	2

), the index has been calculated in each iteration (it) that uses the secondary control (from it2 to it17, for this case). For visualization purposes, only the most relevant behaviors are shown. The first iteration (it1) corresponds to connected-mode operation of the microgrid. This allows a follow-up of microgrid stability behavior during the transition that can be used as a monitoring tool.

The simulation results show that line 18 that connects the buses 9 and 19 of Figure 5 has the lower index value at the last iteration (0.483) compared to the 0.505 value in the connected mode. Although this line is one of the furthest from the PCC, the converter control of the distributed generation resource (solar energy), and the ESS located in bus 19, it is enough to ensure a minimal variation that does not affect the stability.

It is noticed how the secondary control through each iteration improves the microgrid stability of 14 of 19 lines as the index values are greater than in connected mode, specifically after the fifth iteration. However, after the thirteenth iteration, one drop occurs in the index values. This represents a weaker system compared to connected mode, but for the operation condition analyzed, the stability of the microgrid is maintained. It is highlighted that line 12 (bus 6–13) presents an index value improvement in the last iteration compared to connected mode due to the contribution of the secondary control of the converter located on bus 13.

According to Figure 10b, when the frequency-dependent load model is considered in case 1, there is a greater effort of secondary control after iteration 4 to achieve the rated voltage and frequency values, which can be reflected in the greater VCPI final values compared to the values of Figure 10a. For instance, line 11 (bus 4–12) presents the highest index value. This is because there is no load in bus 12, unlike other buses interconnected with converters, which means less demand for the controller. As the aim is to assess the influence of frequency-dependent load in the operation of the microgrid, a comparison of the index values of buses before (it1) and after microgrid isolation is considered (it17). It can be observed that incorporating a frequency-dependent load achieves a better stability condition (highest index value) than when the constant load model is considered.

4.2.2. VSI for Case 2

The value index of lines for case 2 (where less demand is presented) is shown in Figure 11a,b. Under this light load condition, there are no significant differences between the stability conditions for both load types.

4.3. Comparison of Voltage Stability Indexes

Table 5. Comparison of VSIs with frequency-dependent load.

Operation Mode	VCPI_1 [47]		VSMI [49]		FVSI [23]
	Case 1	Case 2	Case 1	Case 2	Case 1	Case 2
	L18	L18	L9	L18	L18	L10
Connected	0.505	0.505	1.001	0.987	−0.03	0.055
Low value	0.484	0.476	0.446	0.839	0.03	0.054
Isolated	0.484	0.476	0.494	0.876	0.03	0.054

presents the results of other voltage stability indexes such as VSMI [49] and FVSI [23] applied in the test microgrid, intending to compare VCPI_1’s behavior when the frequency-dependent load model is considered. The two additional indexes are calculated when the microgrid is connected to the grid as an isolated mode, specifically for line 18, which is the weakest line (except case 2 for VSMI and FVSI). For the safe operation of the microgrid, the VSMI index must be greater than 0. It is an essential highlight for the formulation, where it is assumed that both the line resistance as well as the line shunt admittance are neglected. The fast voltage stability index (FVSI) depends only on the reactive power load, representing a significant limitation, especially in DGs with a unit power factor. This index must be below 1 for a stable line. The above indexes’ assumptions do not consider the real characteristics of microgrids; therefore, the results obtained with both indexes must be carefully examined. For instance, in Table 5, the variations in FVSI are minimal, and this does not allow the proper definition of the voltage stability condition of the system. This is a consequence of not considering the active power influence.

According to the results shown in Table 5, the most vulnerable line according to VSMI in case 1 corresponds to line 9 (L9) (bus 9 to bus 10) as the value index is 1.00 in connected mode but decreases to 0.494 in isolated mode. This behavior also is demonstrated in Figure 12a. However, line 9 is located between two buses with an inverter interfaced DG that allows recovery of the frequency and voltage levels by secondary control, and this is validated with the results given by VCPI_1. It is important to highlight that if this information is used in an online monitoring system, it can be interpreted as a warning condition that will avoid the transition operation. However, as previously mentioned, due to the assumptions of VSMI, this is not a recommended index to determine the microgrid voltage stability.

A remarkable result is shown in Figure 12b, where we present the VSMI values for case 2 with constant load. In this case, for line 18, the lowest index value corresponds to 0.83 in the fifth iteration; however, in the last iteration (it6), the index value increases to 0.94. This information can be misinterpreted, as described previously, if this index is incorporated into a monitoring system. Compared to the results of Figure 11a, there is a minimal variation in VCPI that does not affect the stability of the microgrid.

5. Conclusions

This paper presented an analysis of voltage stability for an islanded microgrid operation with different load models considering an optimization approach of a secondary control strategy. The analysis shows that, for the hierarchical control strategy considered, the microgrid remains stable in terms of voltage and frequency for a constant load model and a frequency-dependent load model; however, under some cases, the latter load model can improve the microgrid response. Moreover, the variation in all voltage indexes is shown for every iteration of the control strategy. Despite a deterioration in the indexes, the control strategy keeps the network within the limits of stable operation.

Several voltage stability indexes (VSI) have been applied in power systems; however, due to specific microgrid characteristics, it is possible that the same assumptions considered in their formulation do not make them suitable for analyzing microgrid stability. Simulation results have proven that indexes such as VSMI or FVSI, which do not consider line resistance, line shunt admittances, or the dependence of active power, can offer results that do not represent the actual microgrid condition. Therefore, the selection of voltage stability indexes must be analyzed carefully as their formulation can lead to incorrect or hasty conclusions on the stability of the microgrid.