Sensitivity-Based Impedance Modeling of AC Voltage-Controlled Converters

Loads affect the stability of grid-forming (GFM) converters, both in stand-alone and grid-connected modes. This has motivated the development of methods that study the stability of GFM converters using an impedance-based analysis. AC impedance-based stability analyses usually study the stability of GFM converters by applying the Nyquist criterion to an impedance ratio denoted as minor loop gain. A limitation of the previous approach when applied to GFM converters, which is not always clearly stated, is that the output impedance of a GFM converter depends on the impedance seen by the converter. This loading effect is not accurately modeled in the literature. This article proposes an impedance model that accounts for this loading effect on the GFM converter output impedance, without introducing approximations. As a direct consequence of the presented impedance model, the stability of the GFM converter when connected to different loads can be determined by checking the stability of a single transfer function, the sensitivity function. Using the sensitivity function, the impedance model presented also reveals a clear link between the voltage controller characteristics and the converter output impedance, which has been reported as a limitation of prior impedance-based analysis methods.


Sensitivity-Based Impedance Modeling of AC Voltage-Controlled Converters
Diego Pérez-Estévez , Member, IEEE, Diego Ríos-Castro, Member, IEEE, and Jesús Doval-Gandoy , Member, IEEE Abstract-Loads affect the stability of grid-forming (GFM) converters, both in stand-alone and grid-connected modes.This has motivated the development of methods that study the stability of GFM converters using an impedance-based analysis.AC impedance-based stability analyses usually study the stability of GFM converters by applying the Nyquist criterion to an impedance ratio denoted as minor loop gain.A limitation of the previous approach when applied to GFM converters, which is not always clearly stated, is that the output impedance of a GFM converter depends on the impedance seen by the converter.This loading effect is not accurately modeled in the literature.This article proposes an impedance model that accounts for this loading effect on the GFM converter output impedance, without introducing approximations.As a direct consequence of the presented impedance model, the stability of the GFM converter when connected to different loads can be determined by checking the stability of a single transfer function, the sensitivity function.Using the sensitivity function, the impedance model presented also reveals a clear link between the voltage controller characteristics and the converter output impedance, which has been reported as a limitation of prior impedance-based analysis methods.

I. INTRODUCTION
T HE steady increase in electric energy demand over the past half-century in combination with a growing load diversity has raised new challenges concerning the stability of microgrids and ac buses that are supported by power converters [1].Loads that were traditionally connected to a constant voltage and constant frequency grid supported by synchronous generators, for example, traditional ship grids, are now connected to ac buses supported by GFM converters, such as in ship grids with modern dc-based distribution [2], [3].Each ac bus is supported by a GFM converter operating in stand-alone or multiple parallel GFM converters operating isochronously (without a droop control) [4].These ac buses often feed different types of loads and experience variable loading configurations, such as hotel loads, HVAC systems, pumps, etc.In particular, two situations are specially troublesome for converter-based grids: the parallel aggregation of converter-based resources [5], [6] and the connection of impedance-varying loads such as nonlinear loads and constant power loads [7].The load that a GFM converter sees at its output affects its stability [8], [9] and its output impedance [10].AC impedance-based stability analyses study the stability of GFM converters by applying the Nyquist criterion to the impedance ratio between these two impedances, so called minor loop gain [11], [12], [13].This method was originally developed for designing passive filters for dc systems [14].In addition to study the stability of dc power converters [15], the method was also extended to study the stability of three-phase GFM converters, both in stand-alone and grid-connected modes [11], [13], including frequency coupling effects due to nonlinear effects [16].
Dynamic loads, such as motors, can significantly affect the voltage waveform quality and challenge the stability of GFM converters within a microgrid [17], [18], [19], [20], [21].In particular, this type of load can challenge the stability and normal operation of a grid [19], [20], [22], [23], [24], [25] due to its low equivalent impedance combined with the potential to generate harmonic and interharmonic current disturbances that depend on the point of operation.When the grid is supported by a GFM converter, these harmonic and interharmonic current disturbances can be amplified by the feedback controller of the GFM converter, depending on its disturbance rejection capability and cause instability.In order to understand these problems, impedance-based stability studies have been published in the literature [26], [27].Although different reference frames can be used to define an impedance model, it has been shown that equivalent relations between models can be defined [28].Impedance models used by prior art use fairly complex models, even when they are converted to single-input single-output (SISO) models, as pointed out by the authors in [26] and they are difficult to relate to physical parameters as reported in [29].
A common limitation of the previous proposals that provide impedance models, which is not always clearly stated, is that the output impedance of a GFM converter depends on the impedance seen by the converter from its output terminals.This loading effect is usually not accounted for in the impedance models found in the literature [12], [30].But, contrarily to the case of dc systems where a filter impedance is selected to stabilize the impedance seen by the dc converter and ensure a stable operation, the impedance seen by a GFM converter can experience very large changes, from infinity (no load) to much smaller than 1 p.u. when a load, such as a motor is connected.In order to assess the stability of a GFM converter for different load values, without incurring into large approximations, it becomes necessary to calculate multiple times the output impedance (once for every possible loading condition) and then evaluate multiple times the minor loop gain to check for stability.This loading effect was identified in [10] and associated to the instability problems found experimentally.
In contrast to prior art, this article proposes an impedance model that calculates the converter output impedance including the effect of loads or any other impedance connected to the point of common coupling (PCC), without introducing approximations.The impedance model expresses the impedance of the PCC as the product of two factors.These two factors are the open-loop impedance of the PCC bus and the sensitivity function of the controller.It should be noted that the impedance of a bus (or node) is the parallel equivalent impedance of all the elements connected to such bus.The impedance model given is not restricted to a specific controller design.As demonstrated, the model can be applied to both transfer-function based controllers or state-space controllers, including single-loop and double-loop architectures.
As a direct consequence of the presented impedance model, which decomposes the impedance of the PCC into two factors, and the fact that the impedance of the PCC is related to the stability of the GFM converter, the stability of the GFM converter when connected to different loads can be determined by checking the stability of a single transfer function, the sensitivity function.This avoids the need to calculate the closed-loop converter output impedance for multiple loading conditions, and then apply the Nyquist criterion to each minor loop gain.Last but not least, it is shown that the magnitude of the PCC impedance has a direct relation with the magnitude of the sensitivity function; therefore, the robustness of the system when feeding nonlinear loads that generate harmonic or interharmonic current disturbances, such as an induction motor with a soft starter, is directly related to the magnitude of the sensitivity function.This method leverages the convenience of linear control theory and the ease of calculation of the sensitivity function; hence it is a valuable solution for practicing power-electronic engineers and researchers due to its simplicity compared to prior art.
The results derived from the theoretical analysis are experimentally tested, and the theoretical impedance model is compared to experimental measurements of the GFM converter output impedance.

II. RELATION BETWEEN OUTPUT IMPEDANCE AND THE SENSITIVITY FUNCTION
Traditionally, ac impedance-based stability analyses study the stability of GFM converters by applying the Nyquist criterion to the impedance ratio Z cl out /Z th [11], [12].For the most simple Fig. 2. Flow chart that details the five steps of the proposed method followed to obtain the output impedance of a GFM converter.
scenario, the problem studies the stability of interconnecting two voltage-source systems represented by their Thevenin equivalents.If only the voltage of one of the voltage-source systems is considered, as shown in Fig. 1, the following relation emerges: When the ratio Z cl out /Z th is equal to negative one, the system becomes unstable.This stability problem is also revealed by analyzing the impedance of the PCC bus, and it occurs when such impedance is infinite: As a result, the impedance of the PCC is a transfer function that can also be used to study the stability.
The proposal follows a five-step method to obtain the output impedance of a GFM converter.The five-step method is summarized in Fig. 2. The method requires three inputs, namely, the parameters of the LC filter installed, a mathematical model of the controller that commands the GFM converter, and the impedance that the converter sees at its output.The first two steps of the method calculate two open-loop impedances seen at the output of the converter.The third step obtains the sensitivity function of the GFM converter.The last two steps (cf.Fig. 2) calculate the closed-loop impedances seen at the output of the converter, using the previously calculated open-loop impedances and the sensitivity function.The sensitivity function is used to establish a link between the closed-loop impedances and the open-loop impedances.This proposal offers a tool to close the reported gap between understandings of academia and industry regarding the relation between the closed-loop impedances and the controller characteristics of GFM converters [29].The sensitivity function is often used in the literature to study the stability and robustness of a controller to parameter variations.Unlike the classical use, in this article the sensitivity function is used to link the output impedance of a grid-forming converter to the open-loop impedances.

A. Grid-Forming Converter With Unit Sensitivity
Although GFM converters rarely operate using only feedforward control, it is convenient to describe the output impedance of feedback-controlled GFM converters in terms of the output impedance of an identical grid-forming converter but without any feedback loops, i.e., with a controller that has unit sensitivity at all frequencies.A linear model of an open-loop-controlled GFM converter is illustrated in Fig. 3.The current source I o models a load disturbance and it is used to calculate the impedance seen at the PCC Z ol g,PCC .The impedance Z ol g,PCC is equal to the transfer function v C /I o .This impedance depends both on the output impedance of the converter Z ol out and the impedance that the converter sees at its output due to loads or other converters operating in parallel.All these impedances external to the GFM converter are lumped together in a single transfer function Z th in this model.If the converter is not connected to any loads or to a grid, then the impedance that the converter sees at its output Z th is assumed infinite.
In the model presented in Fig. 3, three inputs affect the output voltage v C .These inputs are the commanded voltage u that determines the VSC output voltage u VSC , the output current disturbance I o , and the load or grid back-emf voltage v th .The voltage v th is an independent voltage source; therefore it does not affect the impedance models.Associated to the other two inputs, two discrete-time transfer functions are defined, as illustrated in Fig. 4(a).
The first transfer function G 1 (z) relates u with v C , assuming the rest of the model inputs are zero (I o and v th are zero).This model is represented in Fig. 4(b) and it includes a one-and-a-half sample delay that describes the computational and modulation delays between the commanded controller voltage u and the actual VSC output voltage u VSC .
The second transfer function G 2 (z) relates an output current disturbance I o with a voltage disturbance in the capacitor voltage d o , assuming the rest of the model inputs are zero (u and v th are zero).Since an open-loop controller is considered, This model is represented in Fig. 4(c) and it is equal to the impedance seen at the PCC: where Z ol out is the open-loop GFM converter output impedance.Such impedance is equal to: These two transfer functions G 1 (z) and G 2 (z) can be combined to obtain a block diagram representation of the electrical system composed by the GFM converter plus load, as shown in Fig. 4(a).
A notable fact about a GFM converter with unit sensitivity is that its output impedance only depends on the LC filter parameters, as indicated by ( 4), and it is not affected by the controller that is commanding the GFM converter or the impedance that the converter sees at the PCC Z th .This greatly simplifies the calculation of the converter output impedance in this case and guarantees a stable output impedance transfer function for any loading condition; however, these two characteristics are not true for a feedback-controlled GFM converter.The next section presents the three remaining steps of the proposed five-step method to obtain a model of a GFM converter output impedance.

B. Grid-Forming Converter With a Feedback Controller
Similarly to Fig. 3, Fig. 5(a) shows a simplified electrical diagram of a power converter, but now the GFM converter is commanded by a controller that feeds back the output voltage v C .The presentation that follows can be applied to any feedback control structure, including high-level controllers such as a droop control, provided they are linearized.Only the sensitivity function of the controller is relevant to the presented method.
Contrarily to the previous section, now the VSC is modeled by a dependent voltage source.The voltage generated by the VSC depends on the output voltage v C and this changes the output impedance of the converter, compared to the previous section where no voltage was generated by the VSC in response to output disturbances.Specifically, the voltage generated by the VSC is a function of the model G 1 (z), and this model ultimately depends on the loads connected at the output (the impedance seen by the converter at its output), as illustrated in Fig. 5(a).
From an analytical point of view, the proposed method to calculate the output impedance in a feedback-controlled GFM converter follows the five steps indicated in Fig. 2. First, the open-loop output impedance of the GFM converter Z ol out and the open-loop impedance at the PCC bus Z ol g,PCC are determined, as explained in the previous section.Next, the closed-loop impedance at the PCC Z cl g,PCC is calculated.A superscript "cl" is used to remark that now the converter is commanded by a feedback controller, compared to the previous section.In order to calculate the effect of the feedback controller on the output impedance, it is convenient to refer to the block diagram model shown in Fig. 5(b).Since the converter is modeled by a dependent voltage source u vsc , now the VSC output voltage u vsc is not zero, even when v * C = 0, as illustrated in Fig. 5(c).From this model, the impedance at the PCC seen by a disturbance current I o is: LC Filter and loads

• S(z)
Controller response , (7) where S(z) is the output sensitivity transfer function of the feedback controller, and G 2 (z) is the open-loop impedance of the PCC (3).Both the load Z th and the open-loop converter output impedance (4) are shaped by a common factor S(z): At frequencies where |S(z)| > 1, both impedances are increased.And at frequencies where |S(z)| < 1, the impedances are reduced.This interpretation establishes a clear and simple relation between open-loop impedances and closed-loop impedances, and it is used in the next section to guide the assesment of voltage controllers for GFM converters.Finally, the last step obtains the closed-loop converter output impedance combining (8) and Z cl g,PCC = Z cl out //Z th : Equation ( 9) quantitatively shows how the output impedance of a feedback-controlled converter depends on the load that it sees at its output and on the sensitivity function of the controller, but it lacks a clear and simple connection between these three parameters.On the other hand, (8) provides a more clear connection between the closed-loop impedances and the open-loop impedances of the setup.Equation ( 8) also provides a novel way to assess the stability of a GFM converter that is connected to a load.Specifically, the stability of the GFM converter when connected to different loads Z th can be determined by checking the stability of a single transfer function, the sensitivity function because, according to the presented impedance model, which decomposes the impedance of the PCC into two factors (8), one of the factors, the open-loop impedance of the PCC, is always a stable transfer function.

III. ANALYSIS OF VOLTAGE CONTROLLERS FOR GFM CONVERTERS FEEDING AC BUSES WITH VARIABLE LOADING CONDITIONS
This section analyzes the impedance of an ac bus that is supported by a GFM converter in standalone configuration or multiple parallel converters operating in isochronous mode with constant voltage and constant frequency references.The ac bus feeds a variable load configuration that is modeled by an equivalent Thevenin impedance that can change its value depending on the nature and number of loads connected.Examples of this load diversity are directly connected induction motors, motors fed from a variable frequency drive, and motors that use a soft starter.The output impedance of a GFM converter that feeds such an ac bus should meet two requirements: the impedance should be low at frequencies where load currents can flow, and Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.this low impedance value should be always maintained, even when the plant configuration changes due to the connection or disconnection of loads.These two requirements are defined in detail in the following.

A. A Model of Induction Motor Loads
AC buses can feed a wide range of motor loads, including HVAC systems, water chillers, cranes, pumps, fans, etc.These motors often operate at different loading regimes ranging from light load to full load depending on the vessel loading scenario.Induction motor loads can demand starting currents several times larger than the rating current of the motor.In order to limit the starting current of induction motors, a soft starter is a popular solution that can be used when starting a motor from a weak grid, such as a microgrid supported by a GFM converter.However, it has been observed that, under certain conditions, converters continue to manifest stability problems when feeding motor loads [19], [20], [21], [25], even when a soft starter is used to limit the starting current.Furthermore, the stability problems persist after the starting period.
A motor that experiences vibrations contains additional spectral components in the motor current, which are related to load torque variability, and can be modeled by a complex back electromotive force (BEMF) waveform.These spectral components appear as side bands of the fundamental grid frequency f g [31]: where p is the number of pole pairs, s is the slip in p.u., and m is a positive integer.
There are various models that describe the dynamics of an induction motor [32].Fig. 6(a) shows a commonly used electrical model of an induction motor.The model depends on several electrical parameters: the stator resistance R s , the rotor resistance R r , the magnetizing inductance L m , the leakage inductance of the stator L ls , of the rotor L lr , and the BEMF, which is responsible for the nonlinear effects described before.This model can be simplified by means of the Thevenin equivalent model displayed in Fig. 6(b), where It should be noted that, the Thevenin impedance Z th that models a motor mainly depends on the leakage inductance of the stator and rotor of the motor, which are small in value.
Therefore, a GFM converter that is connected to a motor has to maintain stable operation when |Z th | < 1 p.u. and also when Z th is an infinite impedance (no load).This extreme plant model variation that occurs when a motor load is connected, compared to other loads, challenges the system stability and also causes the output impedance of the GFM with a feedback controller to change because it causes a large parameter variation in the plant model G 1 [cf.Fig. 4(b)].

B. Sensitivity-Based Impedance Analysis for Voltage-Controlled Converters
This Section shows the sensitivity function of two voltage controllers which have different design goals, and as a result, have a different output impedance.One design [33] uses a single-loop architecture with a direct pole-placement strategy.The other design [34] uses a cascaded double-loop structure with a proportional controller in the inner current loop and a proportional-resonant (PR) controller in the outer voltage loop.In order to generate a low distortion voltage waveform and maintain the grid voltage quality when feeding motor loads, a low impedance in the supply voltage is required at the f load sidebands, in addition to the fundamental frequency and the main low-order harmonics.Since the frequency f load depends on the load slip and type of motor, the output impedance should be low in a wide range of frequencies around the supply frequency to account for different operating conditions.
Using (8), the prior two requirements for the output impedance can be directly translated into two requirements for the sensitivity function of the controller S(z), i.e, S(z) should not change when the plant model G 1 (z) changes and the magnitude of S(z) should be low at a wide range of frequencies around the supply frequency and at the main low-order harmonics.
The theory presented up to this point is applicable to any linear controller, and it does not impose any restrictions on the order or complexity of the controller transfer function.However, in order to shape the sensitivity function according to the two previous requirements and to accommodate further requirements, such as an integral or resonant action to eliminate steady-state errors, it becomes necessary to use a high-order controller.The selected high-order controller is calculated using the direct pole-placement technique proposed in [33], which establishes the desired location for the closed-loop poles, and, as a result shapes the sensitivity function.This is a state-space controller with a single-loop architecture.The controller includes a resonant action at the fundamental grid frequency.To use this controller, the designer should specify a nominal bandwidth.The detailed controller structure is shown in [33,Fig. 3].
A high-bandwidth controller manifests stability problems due to high sensitivity at high frequencies [33].A low controller bandwidth also degrades the voltage quality of the GFM converter during steady-state regime when feeding induction motor loads that produce interharmonic disturbances.This can be explained by calculating the sensitivity function at low frequencies for different loading conditions Z th and controller bandwidths f bw .Fig. 7 shows the magnitude of S(z) at low frequencies for three values of Z th and three values of f bw (30 Hz, 300 Hz, and

TABLE I PROTOTYPE SETUP PARAMETERS
3000 Hz).The three loading conditions displayed correspond to Z th = 0.01 + j0.01 p.u., Z th = 0.01 + j1 p.u., and Z th equal to the equivalent impedance of the motor used in the experimental results (Z th = 0.5 + j0.64 p.u.).The thin lines show the shape of the sensitivity function for intermediate values of L load between 1 p.u. and 0.01 p.u.The setup parameters used for this study are the same as the experimental setup parameters and they are summarized in Table I.The three bandwidth values presented are obtained by setting the frequency of the dominant pole to the corresponding bandwidth value.On the one hand, at low frequencies, Fig. 7 shows how the magnitude of S(z) decreases when f bw increases.The markers placed at the interharmonics h = 0.5 and h = 1.5 show that current disturbances located at such low frequencies [cf.(10)] perceive a higher sensitivity and, as a result, a higher output impedance when the controller bandwidth decreases.For f bw = 30 Hz, the sensitivity at low frequencies is larger than with a high bandwidth.The reason why the sensitivity function experiences larger variations at low frequencies in the case of the low-bandwidth controller design compared to the higher bandwidth designs, which seems counter-intuitive, is because the low-bandwidth design needs a larger controller effort in order to change the open-loop bandwidth of the plant, which can be approximated by the resonant frequency of the LC filter (1125 Hz), to the lower bandwidth determined by the controller.Specifically, the low bandwidth design moves a greater distance the open-loop plant poles in order to place a dominant pole at 30 Hz, compared to the higher bandwidth designs, which place the dominant pole at 300 Hz and 3000 Hz, respectively.This characteristic depends on the parameters of the LC filter.On the other hand, the magnitude of S(z) tends to one as the value of Z th decreases.The physical explanation for the latter result is that reducing the impedance of the PCC bus (a stiffer PCC bus) limits the influence of the feedback controller; hence, |S(z)| approaches one as in the openloop or feedforward control case, cf.Section II-A.A very high f bw value results in a very low sensitivity at low frequencies, as shown in Fig. 7(c), but it also results in a high sensitivity at high frequencies, which reduces the robustness to plant parameter variations, as explained in [33].Fig. 8 shows the magnitude of S(z) at high frequencies.Both designs with 30 Hz and 300 Hz bandwidth do not exhibit sensitivity peaks at high frequencies when Z th changes.But the design with 3000 Hz bandwidth does manifest sensitivity peaks, which result in a high impedance in the PCC bus at high frequencies, for some values of Z th .Fig. 9 shows the stability region of the controller as a function of the selected bandwidth f bw and the impedance seen by the GFM converter Z th = R th + jωL th .For the setup parameters used in this article, the converter becomes unstable when f bw > 400 Hz and the impedance Z th is small.The stability is calculated by determining the magnitude of the system closed-loop poles as L th diminishes and R th = 1%, which are the same poles as the poles of S(z).This stability problem is manifested in the sensitivity function in the form of a sensitivity peak that appears at high frequencies as the load seen by the GFM converter decreases, cf.Fig. 8(c).Fig. 10 shows the magnitude of S(z) at low frequencies for the same load variation when the GFM converter is commanded by a dual-loop voltage controller with a P inner current loop and a PR outer voltage loop (inner loop gain K p,i and outer loop gains K p,v and K r,v ).Compared to the state-space controller, this is Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
The transfer function for the voltage loop is Three different gain values are evaluated.In a low gain parametrization, the controller action is small and the sensitivity is close to one at all frequencies, similarly to an open-loop controller [see Fig. 10(a)].In the typical gain parametrization, the sensitivity is low only at a very narrow frequency region, which does not comprise low-order interharmonic and subharmonic frequencies [see Fig. 10(b)].Outside this narrow frequency region, a sensitivity higher than one is obtained.In order to increase the width of the low-sensitivity frequency region, there are two options: increase the controller gains or detune the resonant controller.By increasing the controller gains, the sensitivity significantly increases at all frequencies except at the resonant frequency of the PR controller [see Fig. 10(c)].And, if the PR controller is detuned by adding a small damping ζ to the resonant part of the controller, then a steady-state error appears in the output voltage because the sensitivity function is no longer zero.For all gain configurations, similarly to the state-space controller, the sensitivity function approaches one when the load seen by the GFM converter decreases, i.e., an scenario with a stiffer PCC bus.However, when the load impedance increases, the sensitivity function increases in a wide frequency range at low frequencies.The width and amplitude in this low frequency range cannot be independently controlled by tuning the controller gains.In summary, this type of controller does not have enough degrees of freedom to reduce the shape of the sensitivity at interharmonic frequencies close to the fundamental frequency, and increasing the control gains quickly degrades the robustness while not reducing the sensitivity at the frequencies of interest.This characteristic shows that this controller structure is not particularly suited for GFM converters that feed motor loads that generate interharmonic or subharmonic current components, and a higher order controller that permits to shape the sensitivity function becomes useful.

IV. EXPERIMENTAL RESULTS
This section assesses the agreement between the theory of the proposed sensitivity-based impedance model for GFM converters and the experimental results.The experiments are conducted in a voltage-controlled GFM converter feeding a chilled water system, as shown in Fig. 11.The high-order controller with the structure shown in [33,Fig. 3] is used for the experimental results.Two designs with different bandwidth (30 Hz and 300 Hz) are tested.The controller with a low bandwidth (30 Hz) has the sensitivity function shown in Figs.7(a) and 8(a).The controller with 300-Hz bandwidth has the sensitivity function shown in Figs.7(b) and 8(b).The dual-loop transfer-function-based controller is not selected for the experimental results because it has a high sensitivity at low frequencies, similarly to the low-bandwidth state-space design.The experimental results are carried out in a 5-kW VSC with a 700-V dc bus v dc that forms a 400-V line-to-line 50-Hz three-phase grid.The chilled water system contains several electric motors (pumps and compressors).The experimental results show the start-up and steady-state operation of a pump in the chilled water system using a soft starter.The soft starter changes the equivalent impedance Z th that the GFM converter sees at its output.During the soft-start period, the soft starter increases the equivalent impedance of the motor load seen by the GFM converter, which mitigates the loading effect  of the induction motor on the GFM converter.And although the soft starter introduces multiple harmonics, the stability is not affected because the converter sees a high load impedance.During steady-state operation the soft starter is bypassed.Therefore the equivalent impedance seen by the GFM converter decreases.The equivalent motor impedance is Z th = 0.5 + j0.64 p.u.This low impedance can cause stability problems, depending on the controller design and the resultant sensitivity function.The VSC switching frequency is 5 kHz.A double-update sampling strategy is adopted, which results in a sampling frequency of 10 kHz.The experimental waveforms shown in the following were captured using the same embedded hardware platform that was used to implement the digital controller.The GFM converter operates without a power-based synchronization loop.Parallel operation with other voltage sources is not considered in this work.Fig. 11 shows a simplified block diagram and a photograph of the setup.Table I details the experimental setup parameters.
First, the low-bandwidth controller is tested.Fig. 12 shows the load current in the time and the frequency domains during the load start.The oscilloscope capture also includes two zoomed regions that show in detail the current waveforms when the soft starter is operating and when the start-up process is completed.Fig. 12(b) is a vector spectrogram that shows the evolution over time of the load current during the start up and during steady-state operation in the frequency domain.During the softstart region, the load generates significant low-order harmonics mainly due to the soft starter nonlinear operation.This nonlinear distortion can also be observed in the current waveforms shown in Fig. 12(a).Fig. 12(c) shows two snapshots of the vector spectrogram that capture the harmonic distortions at two time instants (t = 2.5 s and t = 5 s), which are representative of the two regimes of operation: soft-start operation and steady-state operation after start up, respectively.
The region where the load is directly connected to the GFM converter corresponds to steady-state operation.In this region, the load current contains two interharmonics (h = 0.5 and h = 1.5) centered around the fundamental component, as shown in Fig. 12(c), which indicates a stability problem that is causing a large amplification of these two interharmonic frequencies and also the fundamental frequency.The amplification of this interharmonic frequencies is predicted by the proposed impedance model, which shows the magnitude of the sensitivity function is greater than one at such frequencies, as shown in Fig. 7(a).In the time domain, these interharmonics manifest as amplitude fluctuations (a modulation) of the fundamental component of the load current, cf.(10).As shown, the voltage waveform also contains the two interharmonics seen in the current, which indicates that the GFM converter has a high output impedance at such frequencies.From a control theory point of view, the controller exhibits a high sensitivity at these frequencies, which causes significant output voltage fluctuations.As shown Fig. 7(a), the magnitude of the sensitivity function is greater than one at such frequencies, which indicates the PCC bus has an impedance higher than the open-loop impedance, hence it is more susceptible to current disturbances.
The previous load start-up event is repeated using the same GFM converter and setup parameters, but increasing the controller bandwidth to the recommended value (300 Hz).Fig. 14(b) shows the evolution over time of the harmonic currents delivered to the load during the start-up.During the soft-start region, the load generates significant low order harmonics mainly due to the soft starter nonlinear operation.Such distortion is observed in the frequency domain, cf.Fig. 14(c), and in the time domain, cf.Fig. 14(a).During steady-state operation, the load current does not contain significant harmonics, as shown in Fig. 14(c).This response indicates a low GFM converter output impedance, as expected from a low sensitivity in the voltage controller.Similarly, Fig. 15 shows the evolution over time of the voltage applied to the load during the same start-up event.The voltage at the PCC has a low distortion when the soft-start operation ends, which is in accordance with the expected voltage at the load terminals when directly connected to a strong grid or a GFM converter with a low closed-loop output impedance.This result is in agreement with Fig. 7(b).This figure shows the magnitude of the sensitivity function is smaller than one in a wide frequency range at low frequencies, hence the PCC bus has an impedance lower than the open-loop impedance, and it is less susceptible to current disturbances.
The next test directly compares the measured output impedance to the proposed theoretical expression.The theoretical expression is modeled in the discrete-time domain using a sampling frequency equal to the sampling frequency of the GFM converter digital controller.The impedances are measured using a digital measurement system with a sampling frequency equal to the sampling frequency of the digital controller, which is more than an order of magnitude larger than the bandwidth of the digital controller or the measurement bandwidth.The experimental measurement of the output impedance is performed using a transient response measurement [35] with a maximum  length binary sequence (MLBS) excitation signal that enables impedance measurement in a wide frequency range with a 10-Hz resolution.A 200 − Ω resistor is connected at the output of the GFM converter, which in combination with the MLBS excitation signal, produces a current disturbance.The resistor value is large, compared to the impedance measured, hence it does not significantly affects the measurement.Due to the limited bandwidth of the voltage controller, not all frequency components of the MLBS excitation signal appear in the output voltage.For frequencies beyond the bandwidth of the voltage controller (f bw = 300 Hz) the amplitude of the MLBS excitation signal is greatly attenuated, and the impedance measurement is more dependent on the current and voltage sensors noise.Fig. 16(a) shows the measured converter output impedance Z meas and the calculated output impedance Z cl out using the proposed method, when the GFM converter operates commanded by a state-space closed-loop controller that includes a resonant action at both the positive and the negative sequences of the fundamental output frequency.Fig. 16(b) repeats the previous comparison between the proposed theory and obtained measurements, but now a third resonant action is added to the closed-loop controller at the negative-sequence fifth harmonic, in addition to the two resonant actions placed at both sequences of the fundamental output frequency.As illustrated in Fig. 16(b), the output impedance of the converter is zero at the negative-sequence fifth harmonic and has a value of approximately 8 Ω at the positive sequence fifth harmonic.The impedance at the frequencies where there is a resonant action becomes zero and the shape of both the measured magnitude and phase of the output impedance closely match the theoretical values.As the frequency increases, the measurement becomes noisier due to signal-to-noise ratio limitations of the impedance measurement method.The measurements show a good agreement with the theory.The "water bed effect" that applies to the sensitivity function can also be observed in the obtained output impedances.When the bandwidth of the controller is increased so as to reduce the output impedance at frequencies centered around −250 Hz, the output impedance of the converter increases at other frequencies, such as at 500 Hz and higher frequencies, as predicted by the theory.Due to signal-to-noise ratio limitations of the measurement technique the output impedance is displayed from −600 Hz to 600 Hz.

V. CONCLUSION
This article has proposed an impedance model that calculates the converter output impedance including the effect of loads or Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.any other impedance connected to the PCC, without introducing approximations.The impedance model expresses the impedance of the PCC as the product of two factors.These two factors are the open-loop impedance of the PCC bus, which includes all the loads connected to the PCC and the sensitivity function of the controller.The proposal also provides a clear link between the controller characteristics and the output impedance, which has been reported as a limitation of prior art.The impedance model given is not restricted to a specific controller design.As demonstrated, the model can be applied to both transfer-function based controllers or state-space controllers, including singleloop and double-loop architectures.A case study of a voltage controlled GFM converter feeding an ac load with a soft starter is experimentally tested, using two controllers with different sensitivities, to validate the theory.

APPENDIX SENSITIVITY FUNCTION OF STATE-SPACE-AND TRANSFER-FUNCTION-BASED CONTROLLERS
The sensitivity function of a SISO system with a closedloop controller is a transfer function that relates the controlled variable of the system with a disturbance in such variable.The sensitivity function is a tool widely used by researchers and practitioner engineers to study the disturbance rejection capability of a controller.In the presented application, the controlled variable is the output voltage of the GFM converter.
To calculate the sensitivity function, it is convenient to draw a block diagram of the system, which comprises the plant model and the controller.For the controller used to obtain the results included in Fig. 7, a block diagram is illustrated in Fig. 17.This figure is adapted from [33,Fig. 3] by performing the following changes.The reference input v * c is cleared.A new disturbance input d o is added to the controlled variable v c .The saturator, which is a nonlinear block, is modeled by the identity function, which assumes that the converter does not enter into overmodulation.
Since this controller is designed using state-space theory, a state-space model that relates d o and v c can be directly obtained from Fig. 17 and it is shown in ( 14) top of the next page.

⎡ ⎢ ⎣
x 2 (k + 1) This state-space model is converted into transfer-function form.The resultant sensitivity function is To calculate the sensitivity function of a GFM converter with a transfer-function based double-loop controller, a block diagram of the system is also defined, as illustrated in Fig. 18(a).This block diagram is modified, as in the previous case, to include a new disturbance input d o at the output and the reference input v * c is cleared.For the sake of clarity, the inner current loop, which is depicted in red, is replaced by an equivalent transfer function G i,cl (z).The simplified block diagram is shown in Fig. 18(b).The sensitivity function can be directly obtained from Fig. 18(b): (16)

Fig. 1 .
Fig. 1.Impedance-based equivalent model of a GFM converter for stability analysis using the impedance ratio Z cl out /Z th .

Fig. 3 .
Fig. 3. Measurement of the output impedance Z ol out of a GFM converter composed of a voltage source converter (VSC) with an open-loop voltage controller and an LC filter by injecting an output current disturbance I o .A load of value Z th with a back electromotive force v th is connected to the grid-forming converter at the PCC.

Fig. 4 .Fig. 5 .
Fig. 4. Detailed model of an open-loop controlled GFM converter that sees an impedance Z th at its output.(a) Block diagram representation using two transfer functions G 1 (z) and G 2 (z).(b) Detailed model of the transfer function G 1 (z), which describes the relation between the commanded controller voltage u and the output voltage v C .(c) Detailed model of the transfer function G 2 (z), which describes the effect of an output current disturbance I o on the output voltage v C .

Fig. 6 .
Fig. 6.Induction motor model.(a) Detailed model as a function of the motor electrical parameters: stator resistance R s , rotor resistance R r , magnetizing inductance L m , stator leakage inductance L ls , rotor leakage inductance L lr , and BEMF.(b) Thevenin equivalent of the detailed motor model.

Fig. 7 .
Fig. 7. Change in the sensitivity function S(z) at low frequencies when the converter sees a resistive-inductive load.(a) State-space controller with a low bandwidth design (30 Hz).(b) State-space controller with the recommended bandwidth design (300 Hz).(c) State-space controller with high bandwidth design (3000 Hz).

Fig. 8 .
Fig. 8. Change in the sensitivity function S(z) at high frequencies when the converter sees a resistive-inductive load.(a) State-space controller with a low bandwidth design (30 Hz).(b) State-space controller with the recommended bandwidth design (300 Hz).(c) State-space controller with high bandwidth design (3000 Hz).

Fig. 9 .
Fig. 9. Stability region of the controller as a function of the selected bandwidth f bw and the impedance seen by the GFM converter Z th = R th + jωL th (R th = 1%).

Fig. 10 .
Fig. 10.Change in the sensitivity function S(z) at low frequencies when the converter sees a resistive inductive load at its output.The GFM converter is commanded by a dual-loop voltage controller with a proportional (P) inner current loop and a PR outer voltage loop (inner loop gain K p,i and outer loop gains K p,v and K r,v ).(a) Low gains.(b) Typical gains.(c) High gains.

Fig. 12 .
Fig. 12.Current waveforms during the soft start of a pump connected to a GFM converter commanded by a low bandwidth (30 Hz) voltage controller.(a) Motor phase current waveforms i m,abc .(b) Spectrogram of the motor current in the αβ frame i m .(c) Spectrum of the motor current in the αβ frame i m when the soft starter is operating and after the motor has started.

Fig. 13 .
Fig. 13.Voltage waveforms during the soft start of a pump connected to a GFM converter commanded by a low bandwidth (30 Hz) voltage controller.(a) Phase voltage waveforms at the PCC v g,PCC,abc .(b) Spectrogram of the voltage at the PCC in the αβ frame v g,PCC .(c) Spectrum of the voltage at the PCC in the αβ frame v g,PCC when the soft starter is operating and after the motor has started.

Fig. 13
Fig.13shows the evolution over time of the voltage at the PCC during the same start-up event.Both time and frequency domain representations of the GFM converter output voltage are shown in Fig.13(a) and (b), respectively.Fig.13(c) shows two snapshots of the vector spectrogram that capture the voltage distortions at two time instants (t = 2.5 s and t = 5 s).As shown, the voltage waveform also contains the two interharmonics seen in the current, which indicates that the GFM converter has a high output impedance at such frequencies.From a control theory point of view, the controller exhibits a high sensitivity at these frequencies, which causes significant output voltage fluctuations.As shown Fig.7(a), the magnitude of the sensitivity function is greater than one at such frequencies, which indicates the PCC bus has an impedance higher than the open-loop impedance, hence it is more susceptible to current disturbances.The previous load start-up event is repeated using the same GFM converter and setup parameters, but increasing the controller bandwidth to the recommended value (300 Hz).Fig.14(b)shows the evolution over time of the harmonic currents delivered to the load during the start-up.During the soft-start region, the load generates significant low order harmonics mainly due to the soft starter nonlinear operation.Such distortion is observed in the frequency domain, cf.Fig.14(c), and in the time domain, cf.Fig.14(a).During steady-state operation, the load current does

Fig. 14 .
Fig. 14.Current waveforms during the soft start of a pump connected to a GFM converter commanded by a voltage controller with the recommended bandwidth (300 Hz).(a) Motor phase current waveforms i m,abc .(b) Spectrogram of the motor current in the αβ frame i m .(c) Spectrum of the motor current in the αβ frame i m when the soft starter is operating and after the motor has started.

Fig. 15 .
Fig. 15.Voltage waveforms during the soft start of a pump connected to a GFM converter commanded by a voltage controller with the recommended bandwidth (300 Hz).(a) Phase voltage waveforms at the PCC v g,PCC,abc .(b) Spectrogram of the voltage at the PCC in the αβ frame v g,PCC .(c) Spectrum of the voltage at the PCC in the αβ frame v g,PCC when the soft starter is operating and after the motor has started.

Fig. 16 .
Fig. 16.Measured GFM converter output impedance Z meas and theoretical impedance Z cl out provided by the proposed method.A 200-Ω resistor is connected at the output of the GFM converter to perform the measurement.(a) Controller with the recommended bandwidth (300 Hz) and resonant action at the positive and negative sequences of the fundamental output frequency.(b) Controller with the recommended bandwidth (300 Hz) and resonant action at the positive and negative sequences of the fundamental output frequency, and at the negative-sequence fifth harmonic.

Fig. 17 .
Fig. 17.Detailed scheme of the selected state-space voltage controller.