Grid-Forming Inverters vs. Synchronous Generators: Disparate Power Conversion Systems and the Impacts on Frequency Dynamics
11 Frequency Dynamics with Grid Forming Inverters:A New Stability Paradigm
R. W. Kenyon,
Student Member, IEEE , A. Sajadi,
Senior Member, IEEE , B. M. Hodge,
Senior Member, IEEE
Abstract —Traditional power system frequency dynamics aredriven by Newtonian physics, where a synchronous generator(SG), the historical primary source of power, follows a decel-eration frequency trajectory upon power imbalances accordingto the swing equation. Subsequent to a disturbance, an SG willmodify pre-converter, mechanical power as a function of fre-quency; these are reactive, second order devices. The integrationof renewable energies is primarily accomplished with invertersthat convert DC power into AC power, and which hitherto haveemployed grid-following control strategies that require otherdevices, typically SGs, to establish a voltage waveform and elicitpower imbalance frequency dynamics. A 100% integration of thisparticular control strategy is untenable and attention has recentlyshifted to grid-forming (GFM) control, where the inverter directlyregulates frequency; direct frequency control implies that a GFMcan serve power proactively by simply not changing frequency.With analysis and electromagnetic transient domain simulations,it is shown that GFM pre-converter power has a first orderrelation to electrical power as compared to SGs. It is shown thatthe traditional frequency dynamics are dramatically altered withGFM control, and traditional second-order frequency trajectoriestransition to first-order, with an accompanying decoupling of thenadir and rate of change of frequency.
Index Terms —synchronous generators, grid forming inverters,frequency response, nadir, rate of change of frequency
I. I
NTRODUCTION
Frequency dynamics in AC power systems have foreverbeen govnerned by Newtonian physics, where a synchronousgenerator (SG), the conventional primary source of power,follows a deceleration frequency trajectory upon mechanical–electrical power imbalances. The integration of variable re-newable energy generation into power systems is primarilyaccomplished with inverters (i.e., inverter based resources(IBRs)), which hitherto have employed grid-following (GFL)control strategies that rely on other devices to establish thevoltage profile [1]. As renewable shares continue to grow at anaccelerating pace, small to medium sized power systems withsignificant instantaneous penetrations of IBRs have becomecommonplace in many systems across the globe [2]–[4]. Asthe penetration of GFL devices increases, overall systeminertia decreases due to the supplanting of SGs [5]; it is
R. W. Kenyon and B. M. Hodge are with the Electrical Com-puter and Energy Engineering (ECEE) and the Renewable and Sus-tainable Energy Institute (RASEI) at the University of Colorado Boul-der, Boulder, CO 80309, USA, and the Power Systems EngineeringCenter, National Renewable Energy Laboratory (NREL), Golden, CO80401, USA, email: { richard.kenyonjr,BriMathias.Hodge } @colorado.edu, { richard.kenyon,Bri-Mathias.Hodge } @nrel.govA. Sajadi is with the Renewable and Sustainable Energy Institute (RASEI)at the University of Colorado Boulder, Boulder, CO 80309, USA, email:[email protected] well documented that instability can occur for high/completepenetrations of GFL inverters (i.e., low–zero inertia systems)[5]–[8]. As a result, attention has shifted to grid-forming(GFM) control, where instead of regulating to active powerset points as a current source, the IBR establishes a voltageand frequency at the point of interconnection to the grid.A primary challenge concerning the operation of these low-inertia power systems is the maintenance of system stability, inparticular the frequency response when SGs are displaced byGFL IBRs; recent work has pointed towards the potential ofGFM inverters to mitigate these stability challenges [9]–[16].The authors of [9]–[12] have extensively studied the small-signal stability of power systems with integrated GFMs bydeveloping high-fidelity differential-algebraic models; often,non-zero, minimum SG quantities are declared in order topreserve system stability. Conversely, the feasibility of op-erating bulk power systems with 100% GFM-based genera-tion has been computationally demonstrated in the electro-magnetic transient domain [13], and positive sequence [14].[13] investigated the dynamic interactions between GFMs andSGs and have identified that the integration of GFMs improvesthe system frequency response; however, these studies fail tolink the fundamental shift in power conversion order to theseimproved dynamics. Other studies have started to identify thedamping-like contribution of droop controlled GFM invertersto frequency dynamics [13]–[15], but cast these conclusionson the basis of similar frequency trajectories across an entiresystem.This paper investigates the power conversion dynamics ofSGs and GFMs (i.e., converters) and the associated frequencydynamics in bulk electric power systems along a shiftinggeneration trajectory; namely, from a 100% SG, standardinertia system, to a 100% GFM, inertia-free system. The mainconclusions of this paper are summarized: • It is analytically derived that, with respect to pre-converter power, the primary frequency dynamics ofa GFM are first order, and second order for an SG.Frequency-power portrait analyses of test system loadstep responses with varied GFM and SG quantities con-firm these lower order dynamics. • The standard assertion of the fast frequency response ofIBRs, as applicable to pre-converter power of a GFM, isrefuted. A GFM serves power simply by not changingthe local frequency. • Common average frequency metrics, typically used toapproximate an overall system frequency response, aredemonstrated as inadequate with GFM devices, and thedecoupling of traditional nadir and rate of change of a r X i v : . [ ee ss . S Y ] F e b frequency relations with GFMs is demonstrated.II. M ATHEMATICAL M ODELS OF C ONVERTERS
This section focuses on the mathematical models that de-scribe the power converter devices of interest; namely, themechanics and associated time scales in which a convertercan modulate the power throughput. To facilitate the com-parison of frequency dynamics between GFMs and SGs, thenotation relating the flow of power through either device isfirst summarized in Fig. 1. As shown, both devices have apre-converter power ( p m ) flowing into the device, which isanalogous to the mechanical torque applied to the shaft ofan SG, or the power supplied by an energy storage systemto the GFM . Each type of converter stores a quantity ofenergy, either as kinetic energy for the SG E int,G = Iω mech with I and ω being the moment of inertia and shaft rotationrotational speed, respectively, or within the GFM primarilyas electrical capacitive storage, E int,I = C DC V DC , with C DC and V DC being the DC link capacitance and capacitorvoltage, respectively. Generally, E int,G >> E int,I [17], [18].Electrical power is delivered to the grid by the converter,represented as p e . The distinction between GFM pre-converterpower, p m,I , and electrical power, p e,I , is made to create thecomparison analogy; these values are only distinguished by alow-pass filter in this particular GFM control design ( p m,I isoften referred to as filtered power, p avg , in the literature [10],[19]). Here, it is assumed that p m,I is readily available withinthe low-pass filter rise time on account of standard energystorage/inverter response times [20], [21]. Neglecting losses,conservation of energy requires that ˙ E (cid:54) = 0 if p m (cid:54) = p e .Fig. 1: Converter topology showing the relation between thedevice internal energy ( E int ) , pre-converter power ( p m ), andelectrical power ( p e ). A. Grid Forming Inverter
Although a variety of GFM control strategies have beenpresented in the literature, such as the droop [22], multi-loop droop [15], virtual synchronous machine [23], and virtualoscillator control [24], they all approximate a similar objective;namely, the construction of a voltage and frequency at thepoint of interconnection with dynamics associated via a power other sources of power exist for the GFM, such as curtailed photovoltaicoutput, although here an energy storage system with available energy isassumed for simplicity dot notation indicates time derivative; ˙ x = ddt x export relationship. For this work, we focus on the frequencydynamics of the multi-loop droop control GFM, with thesimplified control block diagram shown in Fig. 2. Henceforth,GFM implies this particular control strategy. In this diagram,from the left, p m is injected into the converter via the DCvoltage source, where it is modified with a pulse widthmodulation (PWM) controller that outputs a series of modifiedamplitude square waves that are filtered and interfaced withthe power system. The LC output filter meets with a couplinginductor to create the LCL topology shown in Fig. 2. Thecurrent ( i f ) and capacitor voltage ( E I ) are regulated byproportional-integral (PI) controllers operated in the direct-quadrature reference frame. Additional detail on the controllerdesign and implementation can be found in [25]. + - [ v o , ω I ][ v set ,ω set ] e [ v , ω ] Voltage
Controller Current
Controller
PWM - + v o E I v i L f L I C f R I R f i o i f Fig. 2: Control scheme of multi-loop droop grid-forminginverter.In this paper, where the interest is in frequency dynamics,the voltage can be assumed to be stiffly regulated and thereforeconstant within the subsequent mathematical formulations.This complements the SG model approximation to be dis-cussed in Section II-B. The low pass filter dynamical relationbetween p m,I and p e,I is provided in (1). ˙ p m,I = 2 π ( p e,I − p m,I ) τ I (1)where τ I,fil is the filter time constant; as cutoff frequency, ω I,fil = 2 π/τ I . The work in [10] found a limit of ω I,fil >ω n , which indicates that ω I,fil should be greater than 75 rad/sin a 60 Hz system, and therefore τ I ≤ . s; i.e., on the orderof rapid energy system response times. By (1), p m,I has a firstorder relation to p e,I . The active power–frequency governingequations, which are the control basis of the multi-loop drooprelation, are shown in (2) and (3): ˙ δ I = M P ( p m,I,o − p m,I ) (2) ˙ ω I = 2 πM P ( p m,I − p e,I ) τ I (3)where δ I is the GFM phase angle, M P is the droop gain, p m,I,O is the pre-converter power set point, p m,I is the pre-converter power, ω I is the GFM frequency, and p e,I is thepower transferred to the grid; (3) is formulated by combining(2) and (1). Note that (2) is expressed in relative form (i.e.,the steady state value is zero).Based on (1), the p m,I evolves based only on p e,I . Follow-ing, δ I evolves according to (2). It can therefore be stated thatthe GFM frequency is a function of the pre-converter power;the device meets the increase in power demand ( p e,I ), and then adjusts the frequency according to the droop relation. Thisdistinction indicates that a GFM does not require frequencydeviations in order for p m,I to evolve. In this control scheme,frequency is changed to accomplish power sharing, but theunderlying throughput power mechanics indicate an entirelydifferent, and inverted, relationship with power differentialsas compared to the SG. B. Synchronous Generator
The SG is a well understood and documented device;see [17], [26] for in-depth discussions. Here, ideal voltageregulation is assumed and the machine/voltage dynamics areneglected. While governor models are generally non-linearwith higher order filtering, deadbands, and saturation, a validapproximation for frequency dynamics can be made with a firstorder system [17]. The result is a 3rd order dynamical systemconsisting of the swing equation (shown in (4) and (5)) andthe first order governor dynamics (6). The swing equation,i.e., Newton’s second law in rotational form, relates the SGrotation speed with p m,G – p e,G power imbalance. ˙ δ G = ω G − ω s (4) ˙ ω G = 1 M (cid:16) p m,G − p e,G − D ˙ δ G (cid:17) (5)where δ G is the SG phase angle, ω G is the SG rotor speed, ω s is the synchronous speed (equivalent to ω for a twopole machine), M = Hω s where H is the inertia of themachine, p m,G is the pre-converter power, p e,G is the electricalpower, and D is the damper winding component. The governordynamics of the SG can be approximated by the slowest actionas: ˙ p m,G = R D − ( ω o − ω G ) − ( p m,G − p m,G,o ) τ G (6)where R D is the droop setting of the device (it includes afactor of π ), p m,G,o is the pre-converter power set point, and τ G is the governor response time. The value of τ G can varysubstantially depending on type and model, but is generallynot less than 0.5 s [17]. Taking (5) within the derivative of(6), it can be concluded that p m,G has a second order relationto p e,G : ¨ p m,G = − R D − ˙ ω G − ˙ p m,G τ G (7) = − ( R D M ) − ( p m,G − p e,G ) − ˙ p m,G τ G (8)Therefore, p m,G has a second order relation to p e,G , ascompared to the first order relation seen with the GFM device.Furthermore, it can be stated that p m,G is a function offrequency, the inverse of the GFM pre-converter–frequencyrelationship. More simply, it can be said that whereas the SG isa reactive device, the GFM is a proactive device, with respectto frequency. III. S INGULAR P ERTURBATION A NALYSIS
A common practice in the mathematical formulation andanalysis of power systems is the utility of singular perturbationtheory, wherein dynamical equations are reduced to algebraicexpressions because the associated dynamics are too fast to beof interest [27]. Within the context of power system dynamicsanalysis, this is most transparent in the algebraic treatmentof the transmission system. Here, we apply this tool to thefrequency dynamic analysis of the GFM and SG. Note thatboth (1) and (6) are first order differential equations, withtime constants τ I and τ G . There is, in general, an order ofmagnitude of separation between these values; i.e. τ I << τ G .Therefore, where the time scale of interest is the settlingof frequency dynamics associated with the SG, using thefoundations of singular perturbation analysis, ˙ p m,I can beassumed ; thus, according to (1), p m,I ≈ p e,I .Due to the relatively slower response of the SG governor,immediately following a disturbance the difference between p m,G,o and p m,G is negligible; i.e., p m,G,o ≈ p m,G . Applyingthese approximations, the frequency dynamics relevant beforesubstantial SG governor action of the GFM ((2) and (3)) andSG ((4) and (5)) can be reformulated. The approximated GFMfrequency dynamics are given in (9): ˙ δ I = M P ( p m,I,o − p e,I ) ∝ − M P p e,I (9)The SG dynamics are given in (10) and (11): ˙ δ G = ω G − ω o (10) ˙ ω G = 1 M ( p m,G,o − p e,G ) ∝ − p e,G M (11)From (9), it can be concluded that GFM frequency is al-gebraically related to p e,I ; therefore, considering the firstorder relation of p e,I and p m,I in (1), GFM frequency hasa first order relation with pre-converter power. From (10) and(11), with respect to p e,G the frequency dynamics of the SGfollow a first order response; therefore, with (6), the frequencydynamics of the SG have a second order relation with pre-converter power. A. Device Step Response
The conclusion that the SG manifests a second-order fre-quency dynamic is extensively studied and well understood[17], [26]. However, our analytical model suggests that theGFM exhibits a first order response. For verification, wesimulate the frequency response of the SG and GFM followinga load. All simulations are performed in the power systemcomputer aided design (PSCAD) [28] simulation platform.Frequency, rate of change of frequency (ROCOF), and inertiametrics are defined as in the Appendix.The GFM model is a full order, averaged model thatwas created based on the state of the art [10], [29] and ismade available open-source at [30]. A full text on the modelimplementation is available at [25], with all default parameterslisted. Parameters of note are M P = 0 . Hz, pu/S, pu ) and τ I = 0 . s . The SG model uses the standard PSCADmachine model with default prime–double prime reactanceand time constants, two damper windings sans saturation andresistive windage losses. The exciter is an AC7B model withdefault parameters. All instances of SGs share a commonset of parameters; for all 9 and 39 bus system simulations, R D = 0 . Hz/S base ) , H = 4 s (inertia seconds), and τ G = 0 . s . The single device tests examine varied H valuesto exhibit the traditional nadir (lowest frequency excursion)and rate of change of frequency relationship.The frequency and power step response of each device fora 10% load step are presented in Figs. 3 and 4. In this model,each device is operated in standalone fashion, dispatched at50% with a constant power load connected directly to theterminals. The SG frequency trace in Fig. 3 shows the secondorder negative step response following the load step withovershoot and subsequent damped oscillations that settle tothe droop determined steady state. The GFM frequency tracesfollows a standard first order response; steady state is achievedwith no overshoot and a far smaller response time as comparedto the SG. A summary of the nadir, and ROCOF are presentedin Table I. Note the inverse proportionality between inertia andROCOF and the correlation between a lower nadir and reducedinertia. Conversely, note the larger ROCOF of the GFM devicebut a resultant higher nadir.TABLE I: Single Device Step Response Results ROCOF Nadir Settling FrequencyDevice (Hz/s) (Hz) (Hz)SG ( H = 4 s ) 0.48 59.77 59.85SG ( H = 3 s ) 0.63 59.73 59.85SG ( H = 2 s ) 0.95 59.67 59.85SG ( H = 1 s ) 1.90 59.52 59.85GFM 1.50 59.85 59.85 Fig. 3: Frequency response of isolated GFM and SG devicesto a 10% load step at a 50% initial loading.The p e and p m responses of each device are presented inFig. 4. Note that the p e response is identical for each device;because the voltage dynamics are near ideal, the electricalpower is primarily determined by network changes and notthe device dynamics. The slight p e overshoot is a relic ofthe constant power load modeling in PSCAD load impedancevalues are modulated every half cycle. The first order relationof p m,I and p e,G is obvious in the GFM traces, and at amuch faster rate than the SG response, which corroborates the singular perturbation application. The p m,G follows a secondorder response complete with overshoot and oscillations, aswell as the initial acceleration and inflection change.Fig. 4: Power response of isolated GFM and SG devices toa 10% load step at a 50% initial loading. p e is identical foreach device. The acceleration period of p m,G , a second ordercharacteristic, is evident in the magnified window.The results of two standard benchmarks, the IEEE 9 and 39bus test systems, that were used to validate our analysis arepresented in the next two sections.IV. T EST C ASE
I: IEEE 9 B US S YSTEM
Simulations on the IEEE 9 bus test system were performedin the PSCAD simulation environment. The network and allassociated dynamic elements are available open source at [30],[31]. Buses 4-9 are 230 kV, buses 1-3 are 16.5 kV, 18.0 kV,and 13.8 kV, respectively. All SG or GFM devices are ratedat 200 MVA, with other pertinent parameters as establishedin Section III. The load is modelled as constant power withno frequency or voltage dependence. A 10% load step (31.5MW, 11.5 MVar) occurs at bus 6. Prior to the perturbation, thesystem is brought to steady state by initiating all devices asideal sources, and then systematically releasing the associateddynamics in a manner conducive to maintaining steady statestability. Greater detail on this startup process can be foundin [32]. Different interconnection scenarios of SG and GFMsinto the system are created by systematically supplanting anSG with a GFM, as shown in Table II.TABLE II: 9 Bus Configuration and Results
Device at Bus Inertia ROCOF NadirScenario 1 2 3 (s) (Hz/s) (Hz)A SG SG SG 4.0 0.50 59.72B GFM SG SG 2.6 0.73 59.76C GFM GFM SG 1.3 1.12 59.79D GFM GFM GFM 0.0 1.61 59.83
The results in Fig. 5 show the results for four simulations(scenarios A, B, C, and D - explained in Table II) where thethe SGs are systematically replaced by the GFMs, resultingin a incremental decrease in H. These reduced values arecaptured in Table II, along with the the resultant nadir andROCOF statistics for each scenario. It is evident that as themechanical inertia is decreased, the nadir increases, which isindicative of the dominant first order response of the additionalGFMs. Additionally, although the ROCOF increases as inertia is reduced, it does not correlate with a lower nadir, as wouldbe expected in a second order system.Fig. 5: Average system frequency response for varied quan-tities of GFM/SGs. Although the peak ROCOF grows withfewer online SGs, the nadir is simultaneously reduced.The system frequency oscillation period with all SGs (Sce-nario A) matches the single machine step response; i.e.,0.4 Hz oscillations. Individual device frequencies for eachScenario are presented in Fig. 6, where it is evident that forScenario A, all three SGs have similar frequency trajectories;the three devices maintain broad synchronization following theperturbation. Herein lies the motivation for center of inertiaand average frequency metrics.Fig. 6: Initial frequency response of each device for the foursimulated scenarios on the 9 bus system. Note the contrarymotion present with mixed systems (scenario B and C),indicating an initial lack of broad synchronization.The Scenario B system frequency follows a standard secondorder step response with a damping value around 50%, witha brief inversion between 0.5–0.75s; this corroborates theassertion that droop controlled GFMs add to the dampingof the system [15]. From Fig. 6, it is obvious that the threedevices are not broadly synchronized immediately followingthe disturbance. The large GFM frequency changes are thecause of the average frequency inversion (Fig. 5) just afterthe perturbation. The bedrock assumption of average systemfrequency metrics is that all devices have similar frequency trajectories [15], [17]; these results show this assumption isno longer valid with GFM devices.The Scenario C frequency exhibits overshoot, but asmoother recovery; i.e. the concavity in the green trace from t= 1.5–2.5s is the inverse of the expected second order recovery.This system frequency trace is not representative of an over-damped second order system, nor a critically damped systemwhich would follow a far more gradual trajectory to the settlingfrequency. The GFM devices are acting too quickly for allof the devices to remain synchronized during the first 0.5s following the perturbation. In Scenario D, with all GFMdevices, the system frequency follows a typical first orderresponse where overshoot and similar frequency oscillationsare absent. The device frequencies show that with all GFM,the devices maintain a general synchronization throughout therecovery.Fig. 7: Comparison of pre-converter and electrical powers ofeach device for the four 9 bus simulation scenarios. Note that P e and GFM P m are overlapped at these resolutions.The p e and p m response for each scenario are presentedin Fig. 7. The electrical powers show inter-area oscillations( f = 1 . Hz) between SG 1 and SG 2 in Scenario A. Notethe time separation between these p e,G oscillations and p m,G .Scenario B shows the very rapid changes in p m,I – p e,I (theseare indistinguishable at this resolution) of GFM 1, whichexacerbates a larger peak p e,G of SG 2 and 3, while the sub-sequent oscillations are more damped. The peak p e,G outputof SG 3 in Scenario C is further increased, although thereare no oscillations following this overshoot. The conclusionis that the rapid frequency changes of the GFMs, evidentin Fig. 6, cause the network conditions to change rapidly,while the SG frequency follows the slower second orderresponse and resulting in a larger p e,G extraction. Therefore,the fast frequency change of the GFM forces the SG intolarger oscillations because of its slow response and exacerbatesthe dynamic excursions. Scenario D power outputs show a large reduction in power oscillations (including interarea), withminimal overshoot and a relatively rapid arrival to settlingoutputs. Frequency–power portraits are omitted for the 9 bus,due to strong similarities with the 39 bus results subsequentlypresented (Fig. 9).These 9 bus system results depict that the presence ofGFM inverters reduces the average system frequency nadir,while increasing ROCOF. In a second order system, these twodirectional changes would not be correlated; the cause is dueto the first order response of the GFM devices. The traditionalaverage frequency determination methods therefore do notproduce an appropriate metric with GFMs because the rapidchanges of the GFM frequency result in device frequencies thatare at times contrary and divergent to adjacent SGs, negatingthe fundamental assumption of these metrics.V. T EST C ASE
II: IEEE 39 B US S YSTEM
The IEEE 39 bus test system [33] is also presented asa larger case study, with the entire system as simulated inPSCAD and supporting Python code available open-sourceat [30]. The network has been partitioned into 6 subsystemsusing the Bergeron parallelization components; a valuablecontribution to future research. The network elements areunchanged. All buses operate at 230 kV, with a 230/18 kVgenerator step–up unit installed to connect all generationelements at 18 kV. All generation elements are rated at 1000MVA. Dispatch and voltage set points are unchanged fromthe test system configuration. All ten initial SG devices aresystematically replaced by GFMs, with a scenario definingeach iteration; i.e. scenarios 0–10 in Table III. The 10% loadstep (600 MW/141 Mvar) occurs at bus 15.TABLE III: 39 Bus Configuration and Results
Inertia ROCOF NadirScenario GFMs at Buses (s) (Hz/s) (Hz)0 n/a 4.0 0.567 59.6901 30 3.6 0.587 59.7122 30–31 3.2 0.669 59.7173 30–32 2.8 0.808 59.7244 30–33 2.4 0.930 59.7305 30–34 2.0 1.071 59.7386 30–35 1.6 1.225 59.7487 30–36 1.2 1.396 59.7488 30–37 0.8 1.525 59.7569 30–38 0.4 1.648 59.77210 All GFM 0.0 1.852 59.808
Figure 8 shows the average frequency for each of the 11scenarios simulated on the 39 bus system. Along with thefrequency statistics presented in Table III, it is concluded thatwhile ROCOF increases with larger quantities of GFMs anda resultant decrease in system mechanical inertia, the nadir israised. Additionally, the the nadir occurs sooner after the loadstep. The damping of the frequency oscillations increases witha larger quantity of GFMs. The average frequency shows morevariance immediately following the disturbance with largerquantities of GFM. These unusual frequency traces are theresult of the at times contrary GFM frequency as compared tothe SG units, further diminishing the bedrock assumptions ofaverage frequency metrics. Here, it is noted that the simulation of the 39 bus test system in PSCAD with only inverters is asignificant testament to the viability of zero inertia systems.Fig. 8: Average frequency response of 39 bus system for variedquantities of GFMs and SGs and a 10% load step at bus 15.The f – p m,I portraits for a selection of 39 bus system simu-lations are presented in Fig. 9, where the subtitles correspondwith the Table III entry. With no GFMs, the SGs followthe trajectory of an initial frequency deviation prior to pre-converter changes. Prior to convergence on the steady statevalues, the trajectories exhibit oscillations in the form of con-verging spirals. With half GFMs in scenario 5, the first orderrelation between frequency and pre-converter power is evident,while the SG trajectories are shortened with less overshoot.With only a single SG online in scenario 9, the SG exhibitsno pre-converter power overshoot. This is due to the fasterfrequency oscillations, where the large governor response timedoes not permit a reaction. Scenario 10, with only GFMsonline, exhibits the first order/quasi linear relationship betweenfrequency and p m,I .Fig. 9: Frequency–power portraits of each device for scenario0, 5, 9, and 10 for the 10% load step on the 39 bus test system.VI. D ISCUSSION
With only reactive devices such as SGs (see Section II-B)matching system aggregate p m to p e and the associated slow governor response (6), a larger ROCOF generally yields adeeper nadir for the same magnitude power imbalance [34].Consider the following approximating equation: ∆ f prior = α ROCOF × t response (12)where ∆ f prior is the frequency deviation prior to substantive p m,G changes, α ROCOF is the ROCOF for a particular system,and t response is the pre-converter power response time (as usedin (1) and (6)). Evidently, a relatively larger ROCOF for thesame t response (such as τ G , in (6)), yields a larger ∆ f prior before substantial p m changes take place. This is the well-known inertial response period when rotational kinetic energy( E int,G ) is extracted from the SGs; for SG frequency responsedominated systems, less inertia yields larger α ROCOF values,which are susceptible to lower nadirs potentially triggeringfrequency load shedding [35]. This is corroborated by thesimulation results presented in Table I for a single device.A defining feature in this relationship is the reactive natureof SGs to frequency; the frequency must change prior to achange in p m,G . This relation is inverted with a GFM device;when p e,I changes due to varying network conditions, p m,G isdirectly impacted prior to any change in frequency. Howevera GFM changing frequency is a control response, ostensiblydesigned to achieve some type of load sharing, and not theresult of a necessary chain of events to match p e,I and p m,I as for an SG. Therefore and as derived in Section III, the pre-converter–frequency relationship is of lower order with GFMs.Resulting from this lower order relationship is a reduction infrequency dynamics in the presence of GFMs as presented inthe simulation results of Sections IV and V, where it is evidentthat the standard inertial frequency response of SG dominatedsystems is no longer to be expected with GFM devices.Fig. 10: Nadir and ROCOF as a function of mechanical inertiafor the single device, 9 bus, and 39 bus systems following a10% load step.Figure 10 summarizes the relation between ROCOF andNadir of the single SG device, 9 bus, and 39 bus test systemsas a function of mechanical inertia. The data presented is fromTables I II and III. Evident is the standard anti-correlationbetween larger ROCOF and lower nadirs in the traditional,SG dominated power systems; these are approximated with the single device system, which is de facto the center of inertiaresponse. In the presence of GFM devices, this relation isno longer present, as shown by the now correlated nadir andROCOF data for the 9 and 39 bus systems. Here, it is notedthat there are potential broader issues due to larger ROCOFssuch as relay tripping, device disconnection, and machine shaftstrain, but the analysis of these considerations is beyond thescope of this work. However, the ability to greatly reducedthe severity in nadirs with GFM may out weigh these otherpotential issues. VII. C ONCLUSION
This work investigated the power transfer dynamics ofsynchronous generators and grid forming inverters and thedriving factors associated with these dynamics. It was shownanalytically that grid forming devices have a lower order rela-tion between electrical and pre-converter power as comparedto the synchronous generator, which results in a reductionin device frequency dynamics. From these relations, it wasdemonstrated that the multi-loop droop grid forming inverter isa proactive device with respect to frequency and pre-converterpower, contrasting with the reactive nature of the synchronousgenerator. As implemented in the 9 and 39 bus test systems,it was shown that the traditional frequency metric relations oflarger rate of change of frequency and nadir are decoupled inthe presence of these grid forming devices.R
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Renewable Energy
Standard PRC-006-2Automatic Underfrequency Load Shedding . Atlanta, GA: NERC, Aug.2017. A PPENDIX
A. Average System Frequency
The frequency at a generation bus is defined as the shaftrotation speed for an SG, or the control system derivedfrequency for a GFM. The center of inertia system frequencyformulation, where an aggregate system frequency responsecan be derived based on the summation of SG parameterssystem wide [15], [17], is not used because the method willnot capture the frequency regulation of the zero inertia GFMs;i.e. this metric lacks applicability to these zero inertia devices.To arrive at a system average frequency, f ( t ) , each devicefrequency is weighted according to the device rating, as shownin (13): f ( t ) = (cid:80) ni =1 ( M V A i ∗ f i ( t )) (cid:80) ni =1 M V A i (13)where f i ( t ) is the frequency of device i at time t , M V A i isthe device i rating, and n is the number of devices. B. Rate of Change of Frequency
ROCOF, with respect to the rotation speed of a SG, isa continuous function; however, for practical purposes suchas device action (i.e., protection, inverter response, etc.) it iscalculated with a sliding window averaging method, as shownin (14): ˙ f ( t ) := f ( t + T R ) − f ( t ) T R (14)where f is the frequency, and T R is the size, in seconds,of the sliding window. A T R = 100 ms window is used,in accordance with [34]. The largest absolute ROCOF valueduring a particular event is the result presented. C. Mechanical Inertia
An aggregate mechanical inertia value is calculated as: (15).This serves the purpose of quantifying the changeover frominertial devices, SGs, to non-inertial devices, GFMs. Lowerinertia here implies a greater quantity of generation supplantedby GFMs. H = (cid:80) ni =1 H i S B,i (cid:80) ni =1 S B,i (15)where H i is the inertia rating (in s ) of device i , S B,i is theMVA rating of device i , and nn