Impedance Modeling With Stability Boundaries for Constant Power Load During Line Failure

Harbor cranes particularly use induction motors (IM) as the main prime mover, which are powered by the grid through the machine side converter (MSC) followed by a grid side converter (GSC). To supply power to multiple cranes in parallel, double-circuit lines are utilized. Failure of a single feeder causes voltage instability in the load bus. To analyze the voltage stability on the load bus, this article proposes a comprehensive model of the GSC while simplifying the MSC as constant power loads (CPL). When used to describe the CPL behavior of the connected IM load, the proposed modeling shows how input admittance behaves as a negative incremental, growing voltage instability on the load bus. This study uses Nyquist-based stability analyzes to address the voltage stability issue caused by a double-circuit line failure and a negative incremental input admittance. The feasibility of creating a phase-locked loop (PLL) for such grid disturbances is investigated. The possibility of installing a static VAR compensator (SVC) with a battery energy storage system (BESS) on the load bus is explored if there is no equilibrium point in the <inline-formula><tex-math notation="LaTeX">$P_{e}$</tex-math></inline-formula> - <inline-formula><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula> curves during line failure.

grid can result in two types of instability: interharmonic and subsynchronous oscillations [2].However, because of their negative incremental input impedance and regulated output voltage, VSCs exhibit constant power load (CPL) behavior.An electrical system may collapse if there are too many critical power loads (CPL), such as induction motor (IM) loads [3].The most common causes of this problem are overvoltage and oscillations.Because it is time-dependent, the impedance mismatch between the source and load has an effect on the small signal stability of both the DC and AC systems.The results of the small signal stability analysis have a significant impact on the design of three-phase power conversion systems for such IM loads.These practical considerations require the advancement of VSC-based system control methods.The impedance-based method [4] gained its popularity and can be calculated theoretically or measured using a variety of state-of-the-art techniques.The voltage stability of the GSC, as well as that of any other connected system with source-load equivalency [5], can be examined using the Nyquist Criterion (NC) [5].The harmonic linearization method is used to generate the sequence impedance in the phase domain, whereas traditional linearization methods are used to obtain an approximate impedance in the dq domain.In this case, the impedance is represented as symmetric 2 × 2 matrices with nonzero off-diagonal elements.Examining the eigenvalues of such matrices is a common technique for assessing the dependability/stability of power systems [6].Both physical components and control parameters can be fully dynamically specified using this method.There is a substantial amount of research on load modeling and how voltage phenomena are affected by load dynamics [7].However, there are few resources available to model the dynamic loads of interconnected VSCs.
When there is a large number of CPLs, power grids experience harmonic stability and loss of synchronization (LOS) [8].Due to the control dynamics of the GSCs, the phase-locked loop (PLL) experiences LOS, posing a serious risk to the stability of the power grid.Second-order adaptive PLLs are introduced in [9], although they still pose a risk of LOS when operating in the presence of large disturbances in the grid.Several control techniques were used in [10] to avoid LOS with PLL-synchronized GSCs.According to a study by the North American Electric Reliability Corporation (NERC), freezing the PLL at a certain bandwidth is the fastest way to restore a grid fault [11].However, the use of zero current injection, adaptive current injection based on the X/R ratio of the grid impedance, and adaptive injection based on the frequency detected by the PLL can improve the transient stability of the GSC [12].In the event of a severe fault, the grid code requires the GSC to support the grid with 1 pu of reactive current.This is inconsistent with the zero current injection strategy.An equally improbable alternative method that requires prior knowledge of the impedance of the grid is the injection of X/R-based adaptive current [2].Based on the output frequency of the PLL, 1 pu of reactive current can be injected during grid faults, which is not enough to prevent the grid fault from occurring if the grid impedance is not entirely inductive.It is possible to improve the transient voltage stability of the GSC without changing the injected current profile by increasing the damping ratio of the SRF-PLL.The PLL design requirement is arbitrary because the damping ratio required to stabilize GSCs in these studies cannot be quantified.This study aims to assess the impact of PLL under GSC control on transient stability from a design point of view.Because LOS is unavoidable when there is no equilibrium point during significant disturbances, a GSC with an equilibrium point can be considered temporarily stable [1].
If there is no equilibrium point available on the power angle in a post-fault scenario, a Static VAR Compensator (SVC) combined with a battery energy storage system (BESS) at the load bus can provide consistent power to IM loads.It improves the synchronization performance in weak grids and the grid frequency change rate, while delivering both active and reactive power to the grid as a grid following converter [13].As a result, the SVC will use a control architecture similar to traditional GSC control, but the P − f and Q − V droop approaches will be much easier to implement [14], [15].Such a control with power synchronization is desirable to regulate the grid frequency and voltage during weak-grid scenarios.The addition of a low-pass filter (LPF) to the power control loop allows for inertia control.However, this method is not robust enough to withstand grid disturbances.This is because small signal disturbances can be linearly analyzed near an equilibrium operating point.This analysis remains valid until the equilibrium operating point is shifted by a large signal disturbance, such as a transmission line fault, a severe grid voltage dip, or a large CPL swing.Small-signal disturbances can be linearly analyzed near an equilibrium operating point because linear analysis is possible near an equilibrium operating point.Such concerns can be investigated through the transient stability of the BESS-SVC system [16], which is the main motivation for the SVC to maintain grid synchronization.According to the available literature, voltage instability in the load bus during line failure is a problem that the GSC control has yet to solve.r As the constant power load (crane load) influences the input admittance of the GSC, voltage instability may occur at the load bus.Therefore, a detailed impedance analysis must be investigated for such loads with connected converters.r During a line failure in a double-circuit transmission sys- tem, it also severely affects the voltage stability on the load bus.Therefore, a proper PLL must be designed for the GSC to encounter such disturbances through stability analysis.
r In addition, a BESS with SVC can be integrated into the load bus to support both reactive and active power, so that the voltage stability issue can be solved.r To validate the control strategy for the voltage stability of the defined problem through real-time implementation.The impedance modeling of MSC and GSC along with voltage stability analysis at the load bus is studied in [3].However, it lacks detailed transient voltage stability analysis, which is covered in this article with the following original contributions.
r On the basis of detailed modeling in Section II, instability scenarios during line failure conditions are investigated for CPL operation.
r A method to adjust the bandwidth of the PLL in the pres- ence of an equilibrium point is proposed in Section III-A, which is further supported by a Nyquist-based analysis of transient stability.
r In the presence of no equilibrium point during line failure, a method of integrating SVC with BESS on the load bus is proposed in Section III-B to enhance the transient stability of the overall system.Further, in Section IV, extensive hardware-in-loop validation scenarios are examined.Finally, conclusions are drawn based on prior discussions in Section V.

A. System Description and Modeling
Fig. 1 depicts the single-line diagram with IM as a connected load through the MSC followed by a GSC.The load bus B2 supplies power to the GSC through the filter impedance Z f .A double-circuit transmission line with a pair of step-down transformers (T 1 and T 2) connects the supply bus (B1) and the load bus (B2).A grid impedance Z g is considered between the infinite bus and the supply bus B1.In this article, the voltage instability issue on the bus B2 is investigated during a single feeder fault or any of the transformers malfunction.Fig. 1 illustrates the transformer failure in line 2, while Fig. 2 provides information on the timing of the fault scenario.To evaluate the fault scenario most effectively, impedance modeling is suggested in this section.In addition, the equilibrium point in the power angle curve during such a fault scenario is also investigated later in this section.
1) IM Impedance Modeling: The generalized model of IM in the dq-frame can be represented as (1) [17]. where where R s and L s are the stator resistance and inductance of IM, respectively.ω r (= pω m ) and ω m denote the electrical and mechanical angular speed of the IM with the number of pole pairs p. ψ d m denotes the d-axis stator flux of the motor.s is considered as the Laplace transform operator.
2) MSC Impedance Modeling: The motor side of the MSC can be modeled in the dq-frame from the control perspective as (3).
where G c and G sc are the transfer functions of the MSC current and speed controller.Considering the average model of the GSC, the modulation switching function of MSC in dq-frame can be represented by m dq MSC = [m d MSC m q MSC ] T .i dcm and v dc represent the DC side current and voltage of the MSC.The subscript '0' represents the steady-state value of the corresponding variable.Substituting i dq m from (1) into (3), the expression for v dq m can be obtained as (5).
Now, the expression for the DC side modeling of MSC can be written as (6).
The relation for A dq MSC and C MSC can be obtained as (7).
Using ( 5) and ( 7), the final relation for i dcm can be written as (8).
Now, the equivalent dc side admittance (Y dcm ) of MSC can be represented as (9), which can be further utilized to obtain the load admittance (Y dc ).
3) Grid Impedance Modeling: The equivalent grid impedance in the dq-frame can be modeled as where R e and L e are the equivalent grid resistance and inductance, respectively.ω s (= 2πf s ) and f s are the grid angular frequency and the grid frequency, respectively.Let Y dq e =([Z dq e ] −1 ) be the equivalent admittance of the grid.Now, the grid current (i dq c ) can be deduced as (11).
where v dq g and v dq B2 are the voltages on the infinite bus and on the load bus B2.
4) GSC Impedance Modelling: According to Fig. 1, the GSC is connected to the grid through a line filter, which can be modeled in a dq-frame as in (12).
where R f and L f are the resistance and inductance of the line filter, respectively.Taking into account the GSC control strategy mentioned in Section III-A, the GSC can be modeled in the dq frame as (13). where are the current and voltage on the ac side of the GSC in the dq frame.The GSC admittance (Y dq GSC ) and input matrix (B dq GSC ) in the dq frame can be modeled as ( 14) [7].
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.where Υ (= )) is a variable and I is the identity matrix of order 2. G dc , and G pll represent the transfer function of the dc-link voltage controller and the PLL controller.Taking into account the average model of the GSC, the modulation switching function of the GSC in the dq-frame can be represented by 11) in ( 13), the voltage at B2 can be obtained as (15).
According to the power balance relation on the ac and DC side of the GSC, the expression for the dc side current (i dcg ) can be obtained as (16).
where variables A dq GSC and Y dc,GSC can be deduced from the GSC parameters.Using ( 15) in ( 16), the final expression for i dcg can be obtained as (17).(17) Considering the perturbation due to v dc only, the relation in (17) can be reduced to (18).This assumption plays a crucial role in estimating the DC-side equivalent admittance (Y dcg ) by utilizing the principles of a two-port network.
Now, dc-side equivalent admittance (Y dcg ) of the GSC can be represented as (19), which can be used to obtain the impedance of the source (Z dc ).

B. Constant Power Loads and Instability
Fig. 3 illustrates the ac/dc side impedance-admittance network in the dq-frame for the system presented in Fig. 1.It represents the impedance/admittance of the above-mentioned impedance modeling.On the input side or on the ac side of the GSC, the loop gain L dq ac can be written as The load bus voltage v dq B2 phenomena are fully dependent on this ac-side loop gain L dq ac and can be represented as IM as CPL behaves as a negative incremental resistance in the low-frequency zone that also causes severe instability issue at the load bus.A high source impedance (Z dq e ) can induce an unfavorable behavior of the negative incremental resistance of the input admittance (Y dq c ) of the GSC, leading to overall system instability.As discussed in Section III-A2, the ac loop gain can be monitored by the phase locked loop (PLL) bandwidth to discover this instability.This is because PLL in the GSC controller greatly affects the input admittance (Y dq c ).It is evident that research is necessary on a well-designed PLL for GSC operation during grid disruption.
Before and after a fault in a double-circuit transmission line occurs, the equivalent grid impedance Z e (= R e + jX e ) can be evaluated as where X e (i.e.ω s L e ) represents the equivalent grid reactance; Z g is the grid impedance; Z T 1 and Z T 2 denote the impedance of the transformer T 1 and T 2, respectively, which takes into account the resistance and leakage inductance of the windings.Specifically, if a fault arises on one of the transmission lines, Z e may increase as per ( 22).This has resulted in a voltage instability at the load bus.To deal with such situations, multiple perspectives are adopted in Section III to evaluate the operational reliability of a double-circuit transmission line.The stability analysis for the double circuit transmission line can be evaluated from the power flow equation mentioned in (23).
where V B2 and V g denote the RMS voltage at the load bus-B2 and the infinite grid, respectively.P e and P m represent the available electrical power and the required mechanical power without considering any loss components.The phase difference between grid voltage and load bus voltage measures the power angle (δ).The investigation of transient stability during line failure is illustrated through power angle curves and phase portraits mentioned in Fig. 4. The stable equilibrium point (SEP) and the unstable equilibrium point (UEP) on the power angle curve can be evaluated based on whether or not the relation P e ≈ P m satisfies the relationship, respectively.Depending on whether or not the system possesses equilibrium points at the moment of line failure, two distinct groups of transient stability challenges can be recognized, 1) presence of an equilibrium point and 2) no equilibrium point exists.1) Presence of an Equilibrium Point During Line Failure: Referring to Fig. 4(a), there exists a SEP at the point a during the prefault condition.The blue line represents P e − δ performance under such a typical condition.Due to the power flow through the transmission line-2 failing due to a transformer malfunction, X e doubles (approximately) unexpectedly.Such situation can be reflected in the P e − δ curve as a solid green line.In this case, the SEP and the UEP are represented as points c and d, respectively.Because the required mechanical power is greater than the available electrical power, i.e., P m > P e , the dynamic equation occurs.The IM rotor accelerates from point b to point c before decelerating as shown in (24).
where J is the moment of inertia of IM and B is the damping constant.
If the IM rotor speed exceeds the rated speed, the power angle increases.The system will remain stable until the normal rotor speed is restored before the point d.If, on the other hand, P m < P e , the rotor returns to its original speed after the diffraction point, and the IM gradually loses phase lock with the power grid.The stability of the system can be observed more clearly from the phase portraits shown in Fig. 4(c) for various values of X e .The solid green line ascends from point a to point b, where the open circle represents the unstable point.The trajectory stabilizes as the system approaches the point c, as indicated by the solid green dot.For clear observation, a number of potentially unstable fixed points on the trajectory are not labeled on the plot.Both δ 0 and δ u power angles are unstable in this situation, but only δ 1 is stable, as shown in Fig. 4(a) and (c).The transient response of the power angle reflects an oscillation with substantial overshoot before stabilizing at δ 1 .The value of delta at the point b increases as X e increases.GSC synchronization fails when delta exceeds a critical clearing power angle, denoted by δ c , and no stable fixed points can be identified in this situation.
2) No Equilibrium Point During Line Failure: The red solid line in Fig. 4  a failure occurs, a protective relay must be activated to open circuit breakers at both ends of the transmission line-2.With this approach, the fault can be cleared to prevent the overall system from collapsing.Even postfault clearance, there exist no equilibrium points on the red dashed line because P m > P e always persists.Therefore, the IM rotor speed cannot be restored unless and until an additional power source is integrated into the load bus B2.As a solution, a BESS-SVC unit is more flexible to meet the additional active and reactive power requirements of the IM.With BESS-SVC integration, a continuous green line can be observed in Fig. 4(b).Therefore, the rotor speed can be restored with the power angle δ u corresponding to the point d; however, the point d is an UEP.Furthermore, the IM rotor will move to UEP e and oscillate until the SEP appears at c for the restoration of the equilibrium point of the system, where the critical clearing angle (δ c ) plays a critical role in maintaining the system's transient stability.
Such illustration can also be made from the phase portraits of Fig. 4(c).It can be observed that after the line fault occurs, the phase portrait trajectory begins at the point a and continues to rise with the possibility of losing synchronism at higher of X e .The phase portrait initially reaches an UEP e if the fault has been cleared earlier at b.The trajectory undergoes multiple oscillations before settling down at SEP c.However, if the faultclearing angle δ c is as large as at point e, the phase portrait has no fixed point and approaches infinity with loss of synchronism.

A. Control of GSC With PLL Bandwidth Adjustment
Impedance modeling is explored in the previous section for PLL-enabled GSC and MSC.The control architecture used for GSC and MSC has considered the decouple control strategy mentioned in [7].Such a control strategy utilizes conventional PLL which has an issue with the bandwidth adjustment during line failure.Therefore, an adaptive PLL is utilized in the control architecture of [7], as illustrated in Fig. 5.This adaptive PLL does not directly indicate the PLL transfer function mentioned in Section II-A4; however, it is the PLL transfer function with the variable k i pll , as discussed in Fig. 6.A detailed study of the adaptive PLL is presented in this section, while an equilibrium point exists in the post-fault scenario.Nevertheless, if there is no equilibrium point in the post-fault scenario, a BESS-SVC integration at the load bus is discussed later in this section.

1) Adaptive PLL During Equilibrium Point Exists:
The line voltage on bus B2 is utilized for the detection of PLL phase.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Therefore, consider that the bus-B2 voltage phase (θ B2 ) is the estimated phase by the PLL, i.e., θ B2 = θpll = ω s t + δ B2 , which can be denoted as where ω s is the angular frequency of the grid and Δω pll denotes the change in the estimated angular frequency of the PLL ωpll .k p pll and k i pll are the proportional and integral constants of the PI controller used in PLL.δ B2 and δ g denote the phase angle of the bus B2 voltage and the grid voltage, respectively.Now, the q-axis bus-B2 voltage (v q B2 ) can be represented as where v q Z e is the imaginary part of the voltage drop across Z e .v q B2 in (26) can be further simplified as v q B2 = X e i d e + R e i q e − V g sin δ (27) by denoting the power angle, δ = δ B2 − δ g = θ B2 − θ g = θpll − θ g .Here, θ g = ω s t + δ g is the phase of the grid voltage v g .Now, the dynamics of the power angle ( δ) can be derived from (25) as Now, the stable equilibrium point (SEP) exists if v q B2 = 0, i.e., X e i e + R e i q e = V g sin δ ≤ V g , as 0 < δ ≤ 1. (29) The magnitude of the grid voltage, grid impedance, and the amount of active and reactive current injected into the grid affect the equilibrium points during a fault, according to the relation (29).Due to the requirement for a significant amount of active and reactive current injected into the weak-grid state (large X e and R e ), severe faults (small V g ) are more likely to result in no equilibrium point, i.e., loss of synchronization (LOS).Further simplification of (28) using δ ≈ sin δ (as higher-order terms become insignificant in a Taylor series expansion) results in a second-order dynamic system, which can be represented as with the damping ratio (ζ pll ) and the setting time (t s pll ) of where V g is considered to be nominal grid voltage.The transient stability of the GSC can be achieved during a fault with v q Z e = −R e i q e,max = −V g,f .Here, V g,f is the grid voltage under fault scenario.To keep the PLL in synchronization with the grid, the unstable equilibrium point (UEP) b should be merged with SEP c, as shown in Fig. 4(a).In both dynamic and stable grid fault operations, the relationship −R e i q e,max ≤ V g,f sin δ holds for δ ≤ 0. For k i pll > 0, UEP at b always occurs during fault scenarios, while k i pll = 0 brings SEP at c. Again, V B2 changes abruptly during large disturbances, leading to a large variation in Δω pll .To stabilize the grid frequency and the PLL phase tracking error, it is necessary to use an adaptive PLL.In the adaptive PLL of Fig. 6, the value of k i pll can be changed according to where x 1 and x 2 represent the threshold value of the rate of change of angular frequency.| ωpll | is the rate of change of angular frequency estimated by PLL, which can be obtained by processing d|Δω pll |/dt through an LPF with a time constant of T f to attenuate the unwanted high-frequency noise.The time constant T f must be chosen in the range of 180-200 ms [19].
| ωpll | signal can provide the appropriate information on abrupt change in V B2 .The threshold value of x 1 can be deduced as follows.
From the relation in ( 31), (33) can be further modified as By choosing R e as 0.02 pu for the transmission line, the transient duration Δt as 10 ms, the PLL settling time t s pll as 100 ms [19], i q e,max and V g as 1 pu, (34) can be estimated to x 1 ≤ 8.8 Hz/s.Once SEP is achieved by setting k i pll = 0, ωpll always converges to zero, which means that SEP can be achieved at every instant unless another large disturbance occurs.However, considering the practical noise scenario in ωpll , the value of x 2 is set at 0.5 Hz/s.

2) Robustness and Stability Analysis
With PLL Bandwidth Adjustment: By analyzing Fig. 3, the DC side loop gain can be represented as (35) using the effect of Z dc and Y dc on the DC link side of both GSC and MSC.
Now, the marginal stability of the DC side loop gain can be written as (36) using the Nyquist stability criteria mentioned in [20].
Furthermore, the marginal stability of the DC side can be derived as (37) by fulfilling the condition mentioned in (36).
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The stability of the DC-link side of GSC can be explored through the eigenvalue s dc illustrated in Fig. 7, which is obtained by solving (37).It can be observed that the admittance of the ac side of the GSC can also be derived from (37) as Here, the ac-side loop gain of the GSC with its marginal stability criteria can be represented as where I denotes identity matrix of order 2.This type of modeling is only possible with a symmetric grid.Because ac-side Nyquist plots have only two eigenvalues (s ac1 , s ac2 ), it is difficult to estimate and quantify the stability margin.The suggested modeling and stability studies may not be applicable to an asymmetric grid due to potential discrepancies between the ac and DC components of the analysis.One way to assess the system robustness is to examine the ac and DC side Nyquist plots of the GSC in Fig. 7. Depending on the severity of the line fault, these plots converge simultaneously to the critical point.In Fig. 7, the absence of a boundary surrounding the critical point (−1, j0) demonstrates the stability of the system.The phase margin of the system is minimal when the s ac1 locus tends to the critical point.In the same instant, the s ac2 locus corresponds to the large phase margin.However, the entire phase margin cannot be predicted logically.Given that there is only one eigenvalue locus, s dc , the DC side analysis can provide more precise phase margin information.The accurate phase margin can be estimated at the point where the locus s dc intersects the unit circle.Fig. 7 shows a minor increase in stability with increasing PLL bandwidth.When X e (given by II-B) rises during a line fault scenario, the loci s dc and s ac1 move closer to the critical point (−1, 0).When the loci of s ac1 , s dc converge near the critical point, the system approaches marginal stability.If X e is allowed to increase further, the system will enter an unstable region where synchronization fails and the critical point (−1, 0) will be encircled by the s ac1 loci.The PLL's bandwidth must be set in order to acquire the loci s ac1 , s dc close to the critical point and avoid encirclement.It is theoretically possible to meet such bandwidth requirements by adjusting the PLL  conditions considered here; as a result, expanding the PLL's bandwidth is suggested to improve the stability of the GSC.
3) Influence of PLL Bandwidth Adjustment on Grid Strength: SCR is a useful indicator to assess grid reliability, which can be calculated by dividing the short circuit capacity of the PCC by the rated capacity of the GSC.When the SCR is equal or greater than 20, the grid is robust; when it is less than 6, the grid becomes weak; and when it is less than 2, the grid becomes severely weak.The endurance of the grid is critical to the overall reliability of the system.The Nyquist curves for s ac1 , s ac2 at 39 Hz, 80 Hz and 120 Hz, respectively, are shown in Fig. 8(a) and (b).Because neither the s ac1 nor s ac2 Nyquist curves pass through the point (−1, j0), the system is stable at SCR={20, 10}, as shown in Fig. 8.However, the power oscillation occurs at SCR={3, 2} during lower PLL bandwidth, as shown in Fig. 8(a), while Fig. 8(b) depicts the system resistance to variations in grid strength for SCR={3, 2}.For each of the cases, the Nyquist curves of s ac1 and s ac2 do not cluster at the same location (−1, j0) and the system is stable.As a result, it can be concluded that the adaptation of the PLL bandwidth adjustment to a weak grid is poor when the PLL bandwidth is small but excellent when it is large.
4) Influence of PLL Bandwidth Adjustment on Grid Power: Changing the available electrical power on the bus B2 alters the steady-state operating point of the system, affecting the admittance characteristics of the GSC and the stability of the system.Fig. 8 shows the Nyquist curves of s ac1 and s ac2 for various SCR and PLL bandwidth values as P e varies.Because the Nyquist curves of s ac1 and s ac2 do not intersect at (−1, j0) for the values of P e , it can be concluded that the system is resistant to variations in P e , i.e., 0.4pu, 0.8pu, and 1.0 pu.The power oscillation occurs at output powers of 0.8 pu and 1.0 pu, where the Nyquist curves of s ac1 and s ac2 are very close to the point (−1, j0).It can also be observed that the power oscillation occurs when P e is greater than 0.4pu, but the system is stable above 0.4pu.In light of this, it is clear that the GSC is more susceptible to power oscillation when the PLL bandwidth is narrow, whereas it is less susceptible when the PLL bandwidth is broad.

B. SVC With BESS Integration
Without the BESS unit, the SVC can only provide reactive power in bus B2, but with BESS, it can provide control in the four quadrants of the P-Q plane.BESS are made up of a battery string, a thermal cooling system, and DC filter circuits.The frequency of the load can be regulated using quick active power compensation, and the bus voltage can be restored using reactive power compensation.The detailed BESS-SVC unit control mechanism and the transient stability investigation of various line failure situations are discussed below.
1) Control Methodology: At PCC of Fig. 9, the injected active power (P i ) and the reactive power (Q i ) can be expressed as where V s and V B2 represent the magnitude of the SVC voltage and the Bus-B2 voltage phasor with phase angle of δ s and δ B2 , respectively.δ i represents power angle of the injected power.The variable X i denotes the SVC interface reactance.In power synchronization control, the control law for the power angle δ i can be written as where k p i is the proportional constant.P * i denotes the reference active power to be injected.To evaluate the transient stability, the dynamic corresponding to the power angle can be simplified as where Δω g represents the change in the angular frequency of the grid due to SVC power injection.Similarly, the control law for the bus-B2 voltage V B2 can be represented as where k q i is the proportional constant and ΔV B2 represents the change in the bus voltage V B2 .V * B2 represents the reference bus-B2 voltage and Q * i denotes the reference reactive power to be injected.Further simplifying (44), the bus B2 voltage can be written as Considering the low-pass filter (LPF) in the active power loop, the relation in (42) can be modified as where ω p i is the cutoff frequency of the active power loop LPF.For a lower value of δ i , sinδ i ≈ δ i ; hence, the relation in (40) can be simplified to where G p i (=V s V B2 /X i ) is the gain corresponding to active power.Using (47), (46) can be further simplified as Equation ( 48) represents a second-order system with a damping ratio (ζ p i ) and an angular frequency (ω n i ) of Taking into account the LPF in the reactive power loop of Fig. 10, the relation in ( 44) can be modified as where ω q i is the cutoff frequency of the reactive power loop LPF.Expanding (50), the bus-voltage dynamics can be represented as where Q i can be used from (41).

2) Transient Stability Analysis:
The transformer-based circuit model for the integration of SVC in bus B2 is illustrated in Fig. 11(a).Here, the input admittance of the BESS-SVC Y dq i can be computed on the basis of the discussion made in Section II-A4.DC side impedance analysis for BESS-SVC is not essential here to overcome the complexity in modeling BESS.Furthermore, to simplify the impedance model of the complete network, a Thevenin's theorem is applied to Fig. 11(b), which yields the Thevenin's equivalent voltage (v dq th ) and Thevenin's equivalent impedance (Z dq th ) as where k dq th is a reduction factor due to installation of the SVC in bus B2 and can be represented as where is the input impedance of the SVC.As seen from ( 53), the factor k dq th is always less than 1, which leads to decreased equivalent grid impedance.Hence, the locus of s ac1 and s ac2 shift away from the critical point (−1, j0) and avoid encircling it, as shown in Fig. 7.It can be concluded that the overall system regains more stability and existence of the equilibrium point.Such an equilibrium point during post fault situation can be observed from Fig. 4(b) and (c).With installation of SVC, i.e., through supply of both active and reactive power, P e ∼ δ the curve shifts from red line (faulty condition) to green line (post-fault condition).Initially, the system approaches UEP e and oscillates until it reaches the SEP c.At the SEP c, the demand for IM power (P m ) is met through SVC integration with restoration of the rotor speed.To meet such transient stability criteria, the appropriate estimation of active power reference (P * i ) and reactive power reference (Q * i ) are most important.Again, tuning of k p i must strike a balance between the setting time in (49) and other stability requirements.To achieve the best value of k q i , the commonly used damping ratio of ζ p i =0.707 is a better approach.The low-pass filter's bandwidth must be greater than that of the PLL bandwidth of the SVC with a filter time constant of 10 ms while remaining below the switching frequency.Following a grid fault, the suggested transient stability augmentation methodology activates and automatically shuts down on clearance.It is reassuring to see the BESS-SVC unit resynchronized with the grid after a fault because the grid voltage amplitude provides sufficient buffer zones and a significant positive damping coefficient.If the PLL angle exceeds the maximum buffer region and enters the negative damping zone, it will be difficult to accurately measure transient stability.It will become unstable if you push into the adjacent reverse regulation zone.Because of its complexity, stability evaluations must be performed using numerical computations, which is not covered with this article.

A. Experimental Prototype and System Descriptions
A simulation study is conducted through the MATLAB / Simulink environment to validate the proposed impedance modeling and the corresponding stability analysis.With simulation verification, the system is tested through hardware-in loop (HIL) using an OPAL-RT 5700 real-time emulator, as shown in Fig. 12. OPAL-RT 5700 system has two different processing units (the digital signal processing unit (DSP) and the field programming gate array (FPGA) unit) organized as a primary-secondary configuration, respectively.The control unit is processed through the DSP unit, which is configured as a primary processor.The overall system is designed in the FPGA unit through the eHS Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.solver of the OPAR-RT 5700 system, which is regarded as a secondary processor.The overall system is configured using the FPGA eHS solver with a sampling time of 1 μs, while the DSP control is programmed at 10 μs.IM with specifications of 11 kV , 1.716 MW , J = 21.5 kgm 2 , 1500 rpm, p = 2 is considered for simulation study.Transformers T 1 and T 2 with rating 2 MV A, 132 kV /11 kV are considered for the double-circuit transmission line.The control parameters are tabulated in Table I.

B. HIL Experimental Results
As per Fig. 2, a 30 s time window is considered for this study.A line-to-ground fault is simulated at t = 10 s and the fault is cleared at t = 20 s.Fig. 13 shows the simulation results of the RMS voltage of phase-a on bus B2 (V a B2 ) and the RMS value of phase-a compensation current (I a c ).Furthermore, to verify the effect of the dynamics of the system, the instantaneous phase-a voltage (v a B2 ) on the bus B2 and the phase a compensation current (i a c ) are provided.Two different scenarios without and with compensation for voltage instability are considered to maintain voltage stability at the load bus.Fig. 13(a) represents the performance analysis during line-2 failure.At the time of fault occurrence, V a B2 swings between 1.4 pu to 0.6 pu approximately.As the available electrical power in the load bus reaches its new maximum, I a c behaves as expected as V a B2 .The magnitude of the voltage is observed with an increase in swinging frequency that confirms the instability of the line voltage in the load bus.The zoomed view of the waveforms is shown in Fig. 13(b) and (c) during fault occurrence and fault clearance, respectively.1) Scenario 1: Post-Fault Operation With Change in PLL Bandwidth: This scenario analyses the impact of voltage stability in the load bus due to change in PLL bandwidth, considering SCR=3.The HIL validation result in Fig. 14(a) shows such performances with PLL bandwidth ranging from 30 Hz∼120 Hz.During normal operation, the PLL bandwidth for the GSC controller is set to 30 Hz.When a fault occurs at t = 10 s, which can be seen in Fig. 13, the grid voltage and current start oscillating with such bandwidth of 30 Hz.However, by changing the bandwidth from 39 Hz∼120 Hz, the magnitude of the swing voltage continues to decrease.It is seen from Fig. 14(b) that at PLL bandwidth of 177 Hz, the system becomes unstable, as mentioned in Section III-A2.Therefore, it is recommended that the PLL bandwidth be changed to the range of 80 Hz∼150 Hz.It is also observed from Fig. 14(b) that the system becomes more prone to instability at a lower SCR value (SCR<2) during the PLL bandwidth of 150 Hz∼176 Hz.Therefore, an appropriate trade-off for PLL bandwidth at SCR > 2 should be chosen for stable operation of the overall system.With the integration of the BESS-SVC unit into the load bus at t = 18 s, the voltage in the load bus B2 is completely compensated for considering the PLL bandwidth of the GSC controller at 100 Hz.During such a period, no such swing in grid voltage and current is observed.Using an adaptive PLL with a re-tuned bandwidth at different line reactance values X e is a practical solution to real-time grid issues.At various SCR and grid voltage magnitudes, this pretuned bandwidth value must be classified for stable, marginally stable, and unstable system operation.
2) Scenario 2: Post-Fault Operation With Installation of the BESS-SVC Unit: As observed in Fig. 4(b), there may be the possibility of no equilibrium point on the power angle curve after the post-fault situation.However, by interconnection of Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
the BESS with the SVC in the load bus, the equilibrium point can be reached and the load bus voltage can regain its normal operating value of ∼1 pu.Hence, this scenario investigates the BESS-SVC integration through Fig. 15(a).Fig. 15(b) and (c) are the zoomed view of Fig. 15(a) at fault occurrence and fault clearance situation.It can be seen that the load bus voltage stabilizes to ∼ 1 pu during the post-fault and post-compensation period.It can be seen from Fig. 15(b) and (c) that there is significant overshoot and undershoot in the load bus voltage during post-fault compensation and fault clearance.This dynamic performance is controlled by the controller gains mentioned in Table I.

V. CONCLUSION
This article presents a comprehensive impedance modeling of the IM, MSC, GSC, and the grid to analyze system stability during CPL operation and transmission line failure mode operation.CPLs might have a destabilizing impact, resulting in large voltage swings or perhaps the complete collapse of the power system.These oscillations can be minimized or stabilized by adjusting either the PLL bandwidth or by integrating the BESS-SVC unit into the load bus.The appropriate regulation of PLL bandwidth and/or BESS-SVC controller parameters can be achieved through Nyquist stability criteria of both the AC and DC side eigenvalues.Because PLL performance affects the stability of grid disturbances, an adaptive PLL is designed for the test system.Setting the PLL bandwidth of 39 Hz∼ 176 Hz on the AC side of the GSC's equivalent impedance yields adequate performance.The benefits of boundary control for the adjustment of the PLL bandwidth are also studied.Sometimes, setting the PLL bandwidth has different degrees of impracticality, considering the various SCR into account.Therefore, in this article, we investigate the practicality of integrating the BESS-SVC unit on the load bus with improved performance in comparison to PLL bandwidth variation during LOS.Both methods are validated through simulation and real-time HIL implementation by demonstrating the expected improvements in voltage stability at the load bus while the failed transmission line is restored.As a future perspective, the impact of PLL bandwidth on the SVC controller along with BESS modeling can be studied to better understand the utilization of BESS resources.Despite the CPL's voltage stability issues, it experiences frequency stability, transient stability, and small-signal stability, which are areas to be addressed in future studies using the investigated method.

Fig. 1 .
Fig. 1.Single IM connected to infinite bus through MSC and GSC.
(b) represents the P e − δ curve during transmission line-2 failure, without an equilibrium point.When such

Fig. 7 .
Fig. 7. Stable case of AC and DC side Nyquist plot.

Fig. 11 .
Fig. 11.(a) Equivalent impedance-admittance model in dq-frame with BESS integration and (b) its Thevenin equivalent model at grid side.

Fig. 13 .
Fig.13.results of load bus voltage and ac side current of GSC without compensation.

Fig. 14 .
Fig. 14.HIL results of load bus voltage and ac side current of GSC with compensation through PLL bandwidth adjustment (a) with 39 Hz, 80 Hz, and 120 Hz, (b) with 120 Hz, 176 Hz and 177 Hz at SCR = 2.

Fig. 15 .
Fig. 15.HIL results of load bus voltage and ac side current of GSC with compensation.

TABLE I CONTROLLER
PARAMETERS FOR HIL STUDY