Examining Large-Scale Low Inertia Power System Stability

Examining Large-Scale Low Inertia Power System Stability

Duncan Callaway

UC Berkeley and Lawrence Berkeley National Lab

January 14, 2021

”Low inertia”?

I Power systems typically have “rotating machines” thatproduce voltage waveforms and deliver current to loads.

I These machines couple their rotors to grid voltagewaveforms by powerful magnetic fields.

I In this way, the inertia of the machine is available to thegrid to smooth out disturbances

I As power electronic converters (inverters) supply morepower, the total inertia on the system decreasesI That means a reliable source of energy for grid

stabilization decreases

I But fast time scale dynamics (voltage and frequency)change, too!

Fernandez-Guillamon et al (2020)

1 / 19

A crash course on inverter control

Conventional (“grid following”) inverters (GFL):

I Setpoints: real and reactive power

I Sense grid frequency and voltage with a phase-locked loop (PLL)

I Track grid frequency exactly, and adjust voltage to provide real and reactivepower setpoints exactly.

I Cannot work without a grid connection.

“Grid forming” inverters (GFM)

I Setpoints: real and reactive power; voltage magnitude and frequency.I Adjust voltage and frequency setpoints in response to how much the actual real

and reactive power differ from the predefined setpoints.I “Droop control” example: Reduce voltage and frequency in proportion to amount

that real and reactive power exceed setpoints

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GFL vs GFM head-to-head

Grid-following (GFL) Grid-following (GFL)

Controls real and reactive power andfault currents

Instantaneously balances loads with-out centralized coordination

Cannot operate standalone Can operate standalone

Cannot achieve 100% penetration Can achieve 100% penetration

Upshot:

I GFL easy to think of as “negative load,” more comfortable for system operators

I GFM needed for “low inertia” conditions, exciting from a systems perspective.

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Exploring stability: Different measures for need to be explored

Low inertia: Early grid-scale work hasfocused on swing dynamics

I RoCoF

I Nadir

I Steady state

Undrill et al 2010

0 1 2 3 4 5

Time (seconds)

1.5

2

2.5

Voltage

(V)

×104

StableUnstable

Fig. 10: Time domain response of the original nonlinear model.The blue solid line represents a stable case where inverterpenetration level of 44%; the red dashed line represents anunstable case where inverter penetration level of 55%.

5) Machine frequency loop: Given that the inertia of therotor system has a direct impact on system frequency dynam-ics, the impact of varying levels of system inertia on power-system dynamic performance has recently attracted significantattention [12]. With this in mind, in Fig. 8 we plot max(Re(λ))as a function of the inverter penetration level (19) when theinertia of the machine is scaled by factors of 1/10 and 100.While the stability margins are improved at higher inertialevels, surprisingly, these large variations in rotor inertia donot yield correspondingly large impacts on small-signal systemstability.

6) Power controller: Lastly, we disable the inverter powercontroller by fixing the current reference values, idq∗

l , that areprovided to the inverter inner current controller. As demon-strated in Fig. 9, we observe that the eigenvalues are similarto the base case and, accordingly, we conclude that the powercontroller has minimal impact on system stability.

D. Validation of Small-signal Model

In an attempt to validate the small-signal model, we con-sider two time-domain models where in each case max(Re(λ))is either positive or negative, and in both cases the systemis given a small perturbation in the form of a 5% load stepat t = 0.1 s. Given that we are particularly interested invalidating penetration levels near the instability boundary, wefocus our time-domain models at operating conditions near thistransition point. Starting at an equilibrium point and followingthe aforementioned load step, the responses in Fig. 10 illustratethe machine terminal voltage response at stable and unstablepenetration levels of 44% and 55%, respectively, from Fig. 4.As confirmed in the time-domain plots, the nonlinear dynam-ical model is indeed stable and unstable when max(Re(λ)) isnegative and positive, respectively, thus validating the small-signal model.

IV. CONCLUSION AND FUTURE WORK

In this paper, we evaluated the stability of a coupledmachine-inverter system as a function of the ratio of inverter-to-machine power ratings. To enable this, we obtained ascalable inverter model where the control gains and physical

parameters of a baseline inverter model are scaled as a functionof voltage and power ratings. The scalable inverter modelallowed us to formulate a system model with a tunable inverterrating level. After linearizing and analyzing the eigenvalues ofthe system, it was found that the AVR and excitation systemof the machine exhibits a destabilizing interaction with the in-verter current controllers. Furthermore, the gains of the invertercurrent controllers have a large impact on the stability of themachine-inverter system. We believe the analysis reported inthis paper serves as a first step to charactering the stabilityof a power system with significant inverter-based generation.Future work includes extending this analytical framework toinvestigate multi-inverter multi-machine systems.

ACKNOWLEDGMENT

All authors were supported by the U.S. Department of En-ergy (DOE) Solar Energy Technologies Office under ContractNo. DE-EE0000-1583. Y. Lin, B. Johnson, and V. Gevorgianwere also supported by the DOE under Contract No. DE-AC36-08-GO28308 with NREL; and S. Dhople was also sup-ported by the National Science Foundation under the CAREERaward, ECCS-1453921, and grant ECCS-1509277.

REFERENCES

[1] J. A. Taylor, S. V. Dhople, and D. S. Callaway, “Power systems withoutfuel,” Renewable and Sustainable Energy Reviews, vol. 57, pp. 1322–1336, 2016.

[2] P. Kundur, N. J. Balu, and M. G. Lauby, Power system stability andcontrol. McGraw-hill New York, 1994, vol. 7.

[3] N. Pogaku, M. Prodanovic, and T. C. Green, “Modeling, analysis andtesting of autonomous operation of an inverter-based microgrid,” IEEETransactions on Power Electronics, vol. 22, no. 2, pp. 613–625, March2007.

[4] M. Rasheduzzaman, J. A. Mueller, and J. W. Kimball, “Reduced-order small-signal model of microgrid systems,” IEEE Transactions onSustainable Energy, vol. 6, no. 4, pp. 1292–1305, 2015.

[5] J. Guerrero, L. de Vicuna, J. Matas, M. Castilla, and J. Miret, “Awireless controller to enhance dynamic performance of parallel invert-ers in distributed generation systems,” IEEE Transactions on PowerElectronics, vol. 19, no. 5, pp. 1205–1213, Sept. 2004.

[6] J. W. Simpson-Porco, F. Dorfler, and F. Bullo, “Synchronization andpower sharing for droop-controlled inverters in islanded microgrids,”Automatica, vol. 49, no. 9, pp. 2603–2611, 2013.

[7] D. Dong, B. Wen, D. Boroyevich, P. Mattavelli, and Y. Xue, “Anal-ysis of phase-locked loop low-frequency stability in three-phase grid-connected power converters considering impedance interactions,” IEEETransactions on Industrial Electronics, vol. 62, no. 1, pp. 310–321,2015.

[8] F. Katiraei, M. Iravani, and P. Lehn, “Small-signal dynamic modelof a micro-grid including conventional and electronically interfaceddistributed resources,” IET Generation, Transmission & Distribution,vol. 1, no. 3, pp. 369–378, 2007.

[9] J. Tan, X. Wang, Z. Chen, and M. Li, “Impact of a direct-drivepermanent magnet generator (DDPMG) wind turbine system on powersystem oscillations,” in Power and Energy Society General Meeting,2012 IEEE. IEEE, 2012, pp. 1–8.

[10] N. W. Miller, B. Leonardi, R. DAquila, and K. Clark, “Western wind andsolar integration study phase 3A: low levels of synchronous generation,”NREL/TP-5D00-64822. Golden, Colorado: National Renewable EnergyLaboratory, Tech. Rep., 2015.

[11] A. Yazdani and R. Iravani, Voltage-sourced converters in power systems:modeling, control, and applications. John Wiley & Sons, 2010.

[12] A. Ulbig, T. S. Borsche, and G. Andersson, “Impact of low rotationalinertia on power system stability and operation,” IFAC ProceedingsVolumes, vol. 47, no. 3, pp. 7290–7297, 2014.

Lin et al 2017

However higher order system dynamics areimportant, too:

I Small signal stability

I Transient stability

4 / 19

Exploring stability: Different measures for need to be explored

Low inertia: Early grid-scale work hasfocused on swing dynamics

I RoCoF

I Nadir

I Steady state

Undrill et al 2010

0 1 2 3 4 5

Time (seconds)

1.5

2

2.5

Voltage

(V)

×104

StableUnstable

Fig. 10: Time domain response of the original nonlinear model.The blue solid line represents a stable case where inverterpenetration level of 44%; the red dashed line represents anunstable case where inverter penetration level of 55%.

5) Machine frequency loop: Given that the inertia of therotor system has a direct impact on system frequency dynam-ics, the impact of varying levels of system inertia on power-system dynamic performance has recently attracted significantattention [12]. With this in mind, in Fig. 8 we plot max(Re(λ))as a function of the inverter penetration level (19) when theinertia of the machine is scaled by factors of 1/10 and 100.While the stability margins are improved at higher inertialevels, surprisingly, these large variations in rotor inertia donot yield correspondingly large impacts on small-signal systemstability.

6) Power controller: Lastly, we disable the inverter powercontroller by fixing the current reference values, idq∗

l , that areprovided to the inverter inner current controller. As demon-strated in Fig. 9, we observe that the eigenvalues are similarto the base case and, accordingly, we conclude that the powercontroller has minimal impact on system stability.

D. Validation of Small-signal Model

In an attempt to validate the small-signal model, we con-sider two time-domain models where in each case max(Re(λ))is either positive or negative, and in both cases the systemis given a small perturbation in the form of a 5% load stepat t = 0.1 s. Given that we are particularly interested invalidating penetration levels near the instability boundary, wefocus our time-domain models at operating conditions near thistransition point. Starting at an equilibrium point and followingthe aforementioned load step, the responses in Fig. 10 illustratethe machine terminal voltage response at stable and unstablepenetration levels of 44% and 55%, respectively, from Fig. 4.As confirmed in the time-domain plots, the nonlinear dynam-ical model is indeed stable and unstable when max(Re(λ)) isnegative and positive, respectively, thus validating the small-signal model.

IV. CONCLUSION AND FUTURE WORK

In this paper, we evaluated the stability of a coupledmachine-inverter system as a function of the ratio of inverter-to-machine power ratings. To enable this, we obtained ascalable inverter model where the control gains and physical

parameters of a baseline inverter model are scaled as a functionof voltage and power ratings. The scalable inverter modelallowed us to formulate a system model with a tunable inverterrating level. After linearizing and analyzing the eigenvalues ofthe system, it was found that the AVR and excitation systemof the machine exhibits a destabilizing interaction with the in-verter current controllers. Furthermore, the gains of the invertercurrent controllers have a large impact on the stability of themachine-inverter system. We believe the analysis reported inthis paper serves as a first step to charactering the stabilityof a power system with significant inverter-based generation.Future work includes extending this analytical framework toinvestigate multi-inverter multi-machine systems.

ACKNOWLEDGMENT

All authors were supported by the U.S. Department of En-ergy (DOE) Solar Energy Technologies Office under ContractNo. DE-EE0000-1583. Y. Lin, B. Johnson, and V. Gevorgianwere also supported by the DOE under Contract No. DE-AC36-08-GO28308 with NREL; and S. Dhople was also sup-ported by the National Science Foundation under the CAREERaward, ECCS-1453921, and grant ECCS-1509277.

REFERENCES

[1] J. A. Taylor, S. V. Dhople, and D. S. Callaway, “Power systems withoutfuel,” Renewable and Sustainable Energy Reviews, vol. 57, pp. 1322–1336, 2016.

[2] P. Kundur, N. J. Balu, and M. G. Lauby, Power system stability andcontrol. McGraw-hill New York, 1994, vol. 7.

[3] N. Pogaku, M. Prodanovic, and T. C. Green, “Modeling, analysis andtesting of autonomous operation of an inverter-based microgrid,” IEEETransactions on Power Electronics, vol. 22, no. 2, pp. 613–625, March2007.

[4] M. Rasheduzzaman, J. A. Mueller, and J. W. Kimball, “Reduced-order small-signal model of microgrid systems,” IEEE Transactions onSustainable Energy, vol. 6, no. 4, pp. 1292–1305, 2015.

[5] J. Guerrero, L. de Vicuna, J. Matas, M. Castilla, and J. Miret, “Awireless controller to enhance dynamic performance of parallel invert-ers in distributed generation systems,” IEEE Transactions on PowerElectronics, vol. 19, no. 5, pp. 1205–1213, Sept. 2004.

[6] J. W. Simpson-Porco, F. Dorfler, and F. Bullo, “Synchronization andpower sharing for droop-controlled inverters in islanded microgrids,”Automatica, vol. 49, no. 9, pp. 2603–2611, 2013.

[7] D. Dong, B. Wen, D. Boroyevich, P. Mattavelli, and Y. Xue, “Anal-ysis of phase-locked loop low-frequency stability in three-phase grid-connected power converters considering impedance interactions,” IEEETransactions on Industrial Electronics, vol. 62, no. 1, pp. 310–321,2015.

[8] F. Katiraei, M. Iravani, and P. Lehn, “Small-signal dynamic modelof a micro-grid including conventional and electronically interfaceddistributed resources,” IET Generation, Transmission & Distribution,vol. 1, no. 3, pp. 369–378, 2007.

[9] J. Tan, X. Wang, Z. Chen, and M. Li, “Impact of a direct-drivepermanent magnet generator (DDPMG) wind turbine system on powersystem oscillations,” in Power and Energy Society General Meeting,2012 IEEE. IEEE, 2012, pp. 1–8.

[10] N. W. Miller, B. Leonardi, R. DAquila, and K. Clark, “Western wind andsolar integration study phase 3A: low levels of synchronous generation,”NREL/TP-5D00-64822. Golden, Colorado: National Renewable EnergyLaboratory, Tech. Rep., 2015.

[11] A. Yazdani and R. Iravani, Voltage-sourced converters in power systems:modeling, control, and applications. John Wiley & Sons, 2010.

[12] A. Ulbig, T. S. Borsche, and G. Andersson, “Impact of low rotationalinertia on power system stability and operation,” IFAC ProceedingsVolumes, vol. 47, no. 3, pp. 7290–7297, 2014.

Lin et al 2017

However higher order system dynamics areimportant, too:

I Small signal stability

I Transient stability

4 / 19

I “Can heterogeneous systemscontaining GFL inverters, GFMinverters, and machines operatetogether to guarantee frequencyregulation and stability?”

I “What are the interactions betweenmachine excitation systems andinverters with either GFM or GFLcontrols?

I “assumptions that underlie currentgeneration design and controlapproaches must bereexamined,...modified or evenredefined... ”

5 / 19




5 / 19




5 / 19




5 / 19

Talk structure

Exploring stability in low (but not zero) inertia systems

I Exploring factors that drive small-signal instability at high VSC penetrations

I Key finding (1): Control loops on SM and VSC together destabilize hybridsystems at high (65%+) VSC penetrations

I Key finding (2): Network structure and VSC types greatly alter stability. Worststability margins happen with SM, GFM and GFL.

I Work with Uros Markovic, Ognjen Stanojev, Petros Aristidou, Evangelos Vrettosand Gabriela Hug

Some defintions:

I VSC = voltage-source converter; think PV and storage inverters.

I SM = synchronous machine

I AVR = automatic voltage regulator

I PSS = power system stabilizer

6 / 19

Low inertia small signal stability – prior research

I Small signal analysis w/ network dynamicsI Katiraei, Iravani and Lehn (2007). SM and

VSC; focus on model validation, exploringlinearizations.

I Exploring intersections between VSCdesign factors and stabilityI M. Erickson PhD (2012; Lasseter): Small

signal analysis and participation factors.I Du, Lasseter and Khalsa (IEEE TSG 2018):

Filter XL and droop gains can destabilize.I Did not study network design and VSC

composition

I Exploring maximum penetration of VSCsI Lin et al (NAPS 2017): Small signal analysis

with different VSC penetrationI Omitted network dynamics, only studied GFL.

1949-3053 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSG.2018.2829438, IEEETransactions on Smart Grid

4

less damped with the increase of kppmax. When kppmax is greater than 95 rad/s, the system becomes unstable. Fig. 9 (b) shows the trace of eigenvalues as XL increases from 0.02 pu to 0.14 pu, where kppmax= 3 rad/s.

Fig. 9 Traces of eigenvalues when (a) kppmax increases from 1 rad/s to 120 rad/s, where XL=0.12 pu and (b) XL increases from 0.02 pu to 0.14 pu, where kppmax=3 rad/s.

Furthermore, an approximately linear stability boundary is obtained through a kppmax vs. XL two dimensional eigenvalue scan, Fig. 10 [20]. It shows that a larger XL allows a larger kppmax to maintain the stability, which indicates a well-designed coupling reactance XL is very important for the robustness of droop control, as mentioned in Section II A. When designing the controller parameter kppmax, both the dynamic response and stability should be considered. A larger kppmax can mitigate the overload faster, but the stability might be affected if kppmax is too large. Therefore, there is a trade-off between the dynamic response and stability when selecting kppmax. The red ‘+’ icons in Fig. 9(a), (b), and Fig. 10 represent the parameters used in the field test of Fig. 7. It can be seen that the field test selects a very conservative gain of kppmax.

Fig. 10 Small signal stability boudary.

B. Under Frequency Load Shedding

In some more severe circumstances such as loss of generation in an islanded microgrid or when a grid-connected microgrid goes into islanded mode with more loads than sources, the whole microgrid can become overloaded. Since all the sources are overloaded, one natural way is to trip some non-critical loads. With the proposed overload mitigation controller, the frequency of each grid-forming source continues dropping during overload. Therefore, the traditional under frequency load shedding strategy can be used. The frequency relays installed at the non-critical loads’ buses can detect the Load Shedding Frequency (LSF) and trip those loads within about 2 or 3 cycles’ delay. The LSF should be set lower than the normal operational frequency of an islanded microgrid. It should be noted that although under frequency load shedding is a

conventional approach that has been used in power system for decades, it cannot be directly applied in an inverter-based microgrid with conventional droop control. This is due to an inverter’s frequency will not drop continuously with conventional droop control when an inverter becomes overloaded. The proposed overload mitigation controller enables the inverter’s frequency to drop continuously during overload. Therefore, the under frequency load shedding can be applied in a microgrid.

Assume a grid-forming source is outputting its maximum power, its frequency drop ∆ω after a load step is shown in equation (4). The initial frequency drop is decided by the term (mp + kppmax)∆P, and after that the frequency drops at a speed of kipmax∆P, where ∆P is the overload power. If ∆P or the proportional gain kppmax is very large, the term (mp + kppmax)∆P will be larger than ∆ωLSF, which is the frequency difference between the rated frequency and LSF, as shown in Fig. 11 (a) and (b). This will cause the frequency to directly drop below LSF after the load step, thus the load will be tripped immediately after the relay’s time delay, as shown in Fig. 11 (a). If ∆P or the proportional gain kppmax is relatively small, the integral gain can guarantee that the frequency finally drops below the LSF, as shown in Fig. 11 (b).

max max( )p pp ipm k P k Pdtω∆ = + ∆ + ∆∫ (4)

(mp+kppmax) ΔP

Freuquency drop speed: kipmax ΔP

Tdelay

Ttrip

Power

Frequency

ΔP

LSF

Pmax

(mp+kppmax) ΔPTdelay

Ttrip

Power

Frequency

ΔP

LSF

Pmax

ΔωLSF ΔωLSF

(a) (b)

Fig. 11 Frequency drop of a grid-forming source during overload. (a) The overload power ∆P is large. (b) The overload power ∆P is small.

IV. FIELD TEST & SIMULATION The proposed overload mitigation controller has been

verified at the CERTS/AEP test bed [2]. The CERTS/AEP microgrid test bed has conducted extensive microgrid tests over ten years. The current circuit is shown in Fig. 12. The system rated voltage is 480 V and the rated frequency is 60 Hz. A1 and A2 are commercialized inverter-based natural gas sources, and their rated capacities are 100 kW and 60 kW, respectively [21]. B1 is a 93 kW natural gas synchronous generator [22]. ESS is an inverter-based energy storage with rated 100 kW discharge and 50 kW charge capacity [23]. All the above sources are CERTS droop-controlled, grid-forming sources. There are four adjustable load banks in the microgrid, and there is a frequency relay installed at load bank 4. It should be noted that the frequency relay needs about 2 or 3 cycles’ time delay to trip the load. The microgrid is connected to the grid through a static switch [3].

-100 -50 0-80

-60

-40

-20

0

20

40

60

80

Real[rad/s]

Osc

illatio

n Fr

eque

ncy[

Hz]

-300 -200 -100 0-80

-60

-40

-20

0

20

40

60

80

Real[rad/s]

Osc

illatio

n Fr

eque

ncy[

Hz]

0.02 0.04 0.06 0.08 0.1 0.12 0.140

20

40

60

80

100

120

XL [pu]

k ppm

ax [r

ad/s

]

Unstable

Stable

Du et al (2018)

0 0.2 0.4 0.6 0.8 1

Pi / (Pi + Pm)

-2

0

2

4

6

max

(Re(λ))

Fig. 4: Nominal case: Small-signal stability is ensured forpenetration levels approaching 50%.

0 0.2 0.4 0.6 0.8 1

Pi / (Pi + Pm)

-0.8

-0.6

-0.4

-0.2

0

0.2

max

(Re(λ))

Fig. 5: Bypassing the machine AVR and exciter circuitsignificantly improves stability margins.

0 0.2 0.4 0.6 0.8 1

Pi / (Pi + Pm)

-2

0

2

4

6

max

(Re(λ))

[kpi , k

ii ]

1/5[kpi , k

ii ]

5[kpi , k

ii ]

Fig. 6: Reduced current-controller gains adversely impactsmall-signal stability.

0 0.2 0.4 0.6 0.8 1

Pi / (Pi + Pm)

-1

-0.5

0

0.5

max

(Re(λ))

Fig. 7: Bypassing the PLL (assuming δi = δm) guaranteesstability for penetration levels greater than 80%.

0 0.2 0.4 0.6 0.8 1

Pi / (Pi + Pm)

0

5

10

max

(Re(λ))

M0.1M100M

Fig. 8: System eigenvalues after modifying the machine rotorinertia.

0 0.2 0.4 0.6 0.8 1

Pi / (Pi + Pm)

-5

0

5

10

max

(Re(λ))

Fig. 9: Eigenvalues with the inverter power controller by-passed.

2) Machine Exciter: The machine terminal voltage is thesignal that the inverter synchronizes to as it injects a controlledcurrent. Accordingly, interactions between automatic voltageregulator (AVR) and associated exciter circuitry within themachine and the inverter controllers may be critical for systemstability. To determine whether this is the case, we bypassthe AVR and exciter circuit dynamics in the machine modeland fix the machine voltage amplitude to its the nominalvalue. Figure 5 plots max(Re(λ)) as a function of the inverterpenetration level (19) for this setting, and we see that small-signal stability of the machine-inverter system is preservedacross the full range of inverter penetration levels considered.Remarkably, stability is retained even with an inverter rating10 times larger than the machine rating. This finding indicatesthat the presence of the AC4A-type AVR considered here has amajor impact on the stability of machine-inverter systems andto the authors’ knowledge is not reported in prior literature.

3) Current Controller: Next, we investigate impacts of theinverter current control gains on system stability. Here, the full-order model is considered and we modify the current control

gains, kpi and ki

i , from the baseline values in Table II. Inparticular, Fig. 6 illustrates results for the case where kp

i andkii are scaled by a factor of 5 and 1/5 from the nominal value.

As shown, small-signal instability is encountered at a smallerpenetration level of approximately 30% when the gains aredecreased by a factor of 1/5, and an increased penetration levelof 65% when the gains are increased by a factor of 5. Hence,more aggressive current control gains increase the range ofstable penetration levels.

4) PLL: The PLL is responsible for synchronizing the in-verter with the terminal ac voltage and its performance impactsdownstream inverter controllers. Therefore, it is plausible thatit would play a major role in system stability. To explore this,we bypass the PLL by assuming that the inverter has perfectknowledge of the rotor angle (i.e., δi = δg). Although thisoperating condition is not physically possible, this assumptionis a simple way to investigate the impact of the PLL onsmall-signal stability. As shown in Fig. 7, the system with noPLL and ideal angle-tracking at the inverter has a significantlyenlarged region of stability.

Lin et al (2017)

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Gap and Core ApproachWe want to study...

I Models with GFM, GFL and SM at different buses

I Importance of network dynamics

I High order SM model including AVR and PSS

I What physics or control loops are implicated

Approach:I VSC, SM modeled as differential-algebraic equations

I Each source represented in per unitI Validated against Simulink model

I Small-signal analysisI Includes computing participation factors

I Parameteric analysis:I Fraction of generation from GFM and/or GFLI Network dynamics, network topology

South Australian Power System Example

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50.1f

[Hz]

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Ognjen Stanojev MPC-Based FFC of VSCs in Low-Inertia Systems April 17, 2019 8 / 17

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Averaged VSC model: 3φ balanced, control in dq

Cdc vdc

vt

Grid

dq

abc

dq

abc

idqg

PowerCalculation

Unitedq

g

PhaseLockedLoop

SRFVoltage

Controller

SRFCurrent

ControllerPWM

ReactivePower

Controller

ActivePower

Controller

VirtualImpedance

Block

q⇤cv⇤c

p⇤c

!0

hedq

g , idqg

i

✓c

Rt Lt

vm

idqs

Rf

Lf

Cf

iabcg

eabcg

is vm m

qcpc

!cvdc

v

vc

!⇤ = [!0, !pll]

Inner control loopOuter control loop

idqt

Fig. 1: General configuration of the implemented VSC control structure.

For droop control the measured active power signal ispassed through a first-order Low-Pass Filter (LPF) with a cut-off frequency !z = 2⇡fz . Subsequently, the active powerdroop gain Rp regulates the output frequency !c based onthe mismatch between the filtered power measurement signalpc and the external setpoint p⇤c , as follows:

!c = !⇤c + Rp(p

⇤c � pc) (12)

˙pc = !z(pc � pc) (13)

On the other hand, the virtual inertia is based on a linearizedform of the conventional swing equation, representing the re-lation between physical inertia and damping of a synchronousmachine [18], [19]. Hence, the frequency can be expressed viaa differential equation of the form:

!c =1

Mc(p⇤c � pc)| {z }

pm�pe

� 1

McDc(!c � !⇤

c )| {z }pd

(14)

where the mechanical (pm) and electrical (pe) power of a syn-chronous machine are replaced by the active power setpoint p⇤

and the converter output power pc, respectively. The dampingpower term pd is incorporated through a feedback loop, witha damping constant Dc being the feedback gain imposed onthe frequency mismatch, while a normalized inertia constantMc determines the initial RoCoF during transients. The twocontrol architectures can be proven mathematically equivalentunder certain steady-state conditions [4], as follows:

Mc =1

!zRp, Dc =

1

Rp(15)

Nonetheless, virtual inertia control offers overall better be-havior during frequency transients [20]. Furthermore, thecorresponding phase angle ✓c is used as a reference angle forthe (dq)-transformation of the entire converter control system,with exclusion of the PLL, i.e., ✓c = !c!b.

Analogous to APC, the strong coupling of reactive powerand voltage enables a droop-based implementation of Reactive

Power Control (RPC). The desired output voltage vc is com-puted as an adjustment of the voltage setpoint v⇤ accordingto a mismatch in the reactive power:

vc = v⇤c + Rq(q⇤c � qc) (16)

˙qc = !z(qc � qc) (17)

with qc, qc and q⇤c denoting the actual, LPF and setpoint valueof the reactive power, respectively, and Rq being the reactivepower droop gain.

D. Virtual ImpedanceVirtual impedance is embedded as an additional degree of

freedom for active stabilization and disturbance rejection, asit splits the voltage reference into (dq)-components beforepassing it to the inner control loop. Despite maximizing theactive power output when set to zero, a non-zero q-componentis necessary to allow for “acceleration” and “deceleration” ofthe virtual machine [21]. Therefore, a minor cross-couplingof d- and q-components is included via the resistive (rv) andinductive (lv) elements. While the former is set to rv = 0for simplicity, the latter should be kept as small as possible,yielding the respective d-axis and q-axis voltage components:

vdc = vc � rvidg + !clviqg (18)

vqc = �rviqg + !clvidg (19)

which are directly used as reference inputs for the decouplingSRF voltage controller.

E. Inner Control Loop and ModulationThe computed voltage and frequency references are passed

to the inner control loop in order to impose a controlledsaturation of the converter’s currents and voltages [22].

A structure of the SRF voltage controller follows the similarprinciples as the controllers in [4], [22]:

is = Kvp (v � eg) + Kv

i ⇠ + j!ccfeg + Kif ig (20)

⇠ = v � eg (21)

GFL converter model

I Includes PLL

I 15 states in total

GFM converter model

I p − ω and q − v droop.

I 13 states in total

9 / 19

Synchronous generator model

TABLE I: Converter Control Parameters

Parameter Symbol Value UnitActive power droop gain Rp 2 %

Reactive power droop gain Rq 0.1 %

LPF cut-off frequency fz 5 HzRLC filter resistance rf 0.03 p.u.RLC filter inductance lf 0.08 p.u.RLC filter capacitance cf 0.074 p.u.

P-gain of SRF current control Kip 0.74 -

I-gain of SRF current control Kii 1.19 -

FF-gain of SRF current control Kif 1 -

P-gain of SRF voltage control Kvp 0.52 -

I-gain of SRF voltage control Kvi 1.16 -

FF-gain of SRF voltage control Kvf 1 -

PLL proportional gain Kpllp 0.4 -

PLL integral gain Kplli 4.69 -

Virtual impedance resistance rv 0 p.u.Virtual impedance inductance lv 0.2 p.u.

where Kvp and Kv

i are the proportional and integral gains ofthe SRF voltage PI controller, and ⇠ is the integrator state.Furthermore, a feed-forward signal of the measured currentscan be enabled or disabled by changing the gain Ki

f 2 [0, 1].The output current reference is is then used as an inputsetpoint to the current controller.

Similar to its voltage counterpart, the configuration of theSRF current controller is based on a PI control with decouplingterms:

vm = Kip(is � is) + Ki

i� + j!clf is + Kvf eg (22)

� = is � is (23)

with Kip, Ki

i and Kvf being the respective controller gains, and

� the integrator state. The generated output voltage referencevm is used to determine the averaged modulation signal

mabc = (T pT c)�1

mdq = (T pT c)�1 vm

vdc(24)

which reduces the AC side sensitivity to DC voltage (vdc)oscillations [4]. Due to high complexity and very fast internaldynamics, the converter switching and DC-side buffers are notincluded in the model, since their impact on the small-signalstability is rather negligible.

With inclusion of the current and voltage dynamics associ-ated to the electrical system interface in (3)-(5), the completemathematical model consists of 13 states for the grid-formingand 15 states for the grid-following converter unit. All relevantconverter parameters used in this paper are presented inTable I, whereas more details on the overall converter controlstructure, potential operation modes and respective transientproperties can be found in [5], [6], [20].

III. SYNCHRONOUS GENERATOR MODEL

We consider a traditional two-pole synchronous generatorequipped with a prime mover and a governor, described inper unit. Additionally, a voltage excitation system consisted ofAutomatic Voltage Regulator (AVR) and PSS is incorporated.A detailed block diagram is showcased in Fig. 2, wherethe synchronous generator is connected to the grid trough atransformer. The main parameters are listed in Table II. Thisis a well-established configuration used both for academic andindustrial purposes [2], [23], [24].

A. Electrical InterfaceA synchronous generator is interfaced through a transformer

(rt, lt) to the grid, and modeled in the SRF defined by itssynchronous velocity !s:

is =!b

lt(es � vn)�

✓rt

lt!b + j!b!s

◆is (25)

with es and is denoting the stator voltage and current respec-tively, and vn representing the nodal voltage at the PCC.

B. Internal Machine DynamicsThe internal machine dynamics are characterized by the

transients in the rotor circuits, as transients in the statorwindings decay rapidly and can thus be neglected. Rotor

AVR

PSS

Frequency Control

Ks

stab. gain

sTw

1 + sTw

washout

1 + sT1

1 + sT2

1 + sT3

1 + sT4

phase compensation

1

1 + sTe

transducer

k·k2norm

v⇤

Ke

exciter saturation

daiqs � q

aids df , D1 , Q1 , Q2

internal rotor dynamicselectrical torque

1

Rg

droop

p⇤

1

1 + sTg

governor

Km(1 + sFhTr)

1 + sTr

turbine

1

sMg + Dg

swing dynamics

!0

1

s

internal(dq)-frame

Grid

v2 v3

+vs

edqs

vt

�v1

+ vdf

dqa , idq

s

Rt Lt

��pm pg

+

pm �pe

+

pe [ ⌧e

�

+

�!r

+

!s ✓g abc

dq

Fig. 2: Block diagram of a synchronous generator equipped with a prime mover, governor and voltage excitation system.I 6th order machine physics (includingsubtransient reactance)

I Plus 8 add’l (mostly control) states

I dq representation

I round rotor, steam turbine primemover

I TGOV1 governor, PSS1A PSS, SEXSexciter.

10 / 19

Network model and modeling approach

TABLE II: Synchronous Generator Parameters

Parameter Symbol Value UnitDroop control gain Rg 2 %

Governor time constant Tg 0.5 sReheat time constant Tr 10 s

Mechanical power gain factor Km 0.85 -Turbine power fraction factor Fh 0.1 -Normalized inertia constant Mg 13 sNormalized damping factor Dg 1 p.u.Transducer time constant Te 0.05 sAVR exciter control gain Ke 200 -

Saturation minimum output V minf 0 p.u.

Saturation maximum output V maxf 4 p.u.

PSS stabilization gain Ks 5 -Washout time constant Tw 2 s

1st lead-lag derivative time constant T1 0.25 s1st lead-lag delay time constant T2 0.03 s

2nd lead-lag derivative time constant T3 0.15 s2nd lead-lag delay time constant T4 0.015 s

dynamics originate in the armature reaction, i.e., in the effectof the stator field on the rotor currents, which can be describedthrough flux linkage dynamics:

df =

!0rf

xda,u

vdf �

!0rf

xf

� d

f � da

�(26a)

D1= �!0rD1

xD1

� D1� d

a

�(26b)

Q1 = �!0rQ1

xQ1

( Q1 � qa) (26c)

Q2= �!0rQ2

xQ2

( Q2� q

a) (26d)

Here, subscripts f , D1, Q1 and Q2 stand for the quantitiesof the field circuit, d-axis damping circuit and two q-axisdamping circuits respectively, whereas , r and x denote therespective flux linkage, resistance and reactance of a circuit;vf is the exciter output voltage, !0 designates the synchronousangular velocity, and xd

a,u stands for the unsaturated mutualreactance. Superscripts d and q are omitted from dampingcircuit quantities for simplicity. The armature flux linkagesare expressed as follows:

da = xd

a,s

�ids +

df

xf+ D1

xD1

!(26e)

qa = xq

a,s

✓�iqs +

Q1

xQ1

+ D1

xQ2

◆(26f)

with the subtransient, saturated, mutual reactances xda,s and

xqa,s defined in the form:

xda,s =

✓1

xda,s

+1

xf+

1

xD1

◆�1

(26g)

xqa,s =

✓1

xqa,s

+1

xQ1

+1

xQ2

◆�1

(26h)

Finally, the inclusion of stator circuit balance completes theset of differential-algebraic equations in (26) describing theinternal generator dynamics:

eds = �raids + xli

qs � q

a (26i)

eqs = �raiqs + xli

ds � d

a (26j)

Stator voltages and currents are denoted by edqs and idq

s ,while ra and xl represent the armature resistance and leakagereactance respectively. Combining (25)-(26) with 8 controllerstates depicted in Fig. 2 yields a 14th order model. Formore details regarding the generator modeling and internalparameter computation we refer the reader to [2].

IV. UNIFORM POWER SYSTEM FORMULATION

A. Network Modeling

Modeling of the transmission network is described for ageneric system depicted in Fig. 3, composed of generatorssupplying local RL loads and the interconnecting transmissionlines modeled as ⇡-sections. In order to establish a consistentmathematical formulation, all variables have to be definedwithin a single, uniform SRF. For this purpose, the terminalcurrents (itu

) and voltages (vtu) of each generator unit u 2 U

are mapped to the network nodes j 2 Ju ⇢ J with generatorconnection, and subsequently aligned to the network SRFrotating at nominal angular speed !n:

xnj= xtu

e�j(✓n�✓j), 8j, u (27)

where xnj 2 {inj , vnj } denotes the “nodal” metrics describedin the nominal reference frame, ✓n = !n!b is the uniformSRF angle, and ✓j corresponds to the internal SRF angle ofthe respective unit, i.e., ✓s for the synchronous and ✓c for theconverter-based generator. Several benefits of using a uniformreference frame for describing the power system dynamicshave already been indicated in [25]. The nodal voltage andcurrent dynamics can now be expressed as follows:

vnj=!b

clj

icj� j!n!bvnj

, 8j (28)

ilj =!b

lljvnj �

✓rlj

llj!b+ j!b!n

◆ilj , 8j (29)

vtj vtk

vtj

itj

vnj

inj

\✓j 7! \✓n

vnk

ink

vtk

itk

\✓n [ \✓k

itj

ilj

Zlj

ilk

Zlk

icj

Cj

Rjk Ljkijk

ick

Ck

itk

dq

dq

dq

dq

Fig. 3: Generic network model with line dynamics and therespective (dq)-frame alignment.

6

50 55 60 65 70 75 80 85 900

50

100

150

200

250

⌘ [%]

<(�)

�

�23

�20,21

�19

�18

�13,14

VSCf

control

VSCF

control

stable reg. frequency dynamics

PLL&

APC

APC

PLL

SRFinner

control

Fig. 6: Impact of different controllers on system stability underhigh grid-following inverter penetration in Scenario III.

through interactions with the LC segments of the transmissionlines, which slows down the frequency dynamics and enablesthe grid-following VSCs to more accurately detect the weakglobal frequency signal. Similar conclusions have been drawnfor autonomous microgrids [26] and networks comprised ofa specific class of grid-forming inverters [24], as well as in[2] using a mechanical analogue of swings in a multi-machinesystem.

On the other hand, the scenarios experiencing instabilityrelated to voltage dynamics are for the most part unaffectedby the transmission line dynamics. The voltage control in-teraction between the synchronous and PE-based generationis somewhat mitigated, due to the time constants of theline dynamics and the SRF inner control loops being of thesame order of magnitude. However, the line dynamics do nothave any impact on the slower modes associated with thesynchronous machines. Therefore, the stability in Scenario Icannot be preserved for the grid-following penetration above70 %, as it is associated solely with the AVR and PSS controldesign. The predominant impact of SGs on system stability iseven higher in Scenario II, with grid-forming inverters onlymarginally affecting the critical modes within a narrow range

10 20 30 40 50 60 70 80 90

0

5

10

⌘ [%]

<(�)

Scenario I

Scenario II

Scenario III

Fig. 7: Impact of inverter penetration on system stability fordifferent unit configuration, with inclusion of transmissionline dynamics. Transparent dashed lines indicate results forscenarios without line dynamics from Fig. 3.

of ⌘ 2 [78, 79] %. Hence, the maximum admissible VSCpenetration after the inclusion of transmission lines remainsthe same in this case.

As a mean of model validation, we investigate the same2-bus test case comprised only of synchronous generators orgrid-forming converters. Understandably, networks with suchhomogeneous configurations of generators with standalonecapabilities face no stability problems under any ratio ofinstalled powers between the two units. Therefore, these resultsare not included in Fig. 7 for simplicity, nor will the respectivescenarios be discussed in the remainder of the study.

C. Stability Margins of the IEEE 9-bus SystemIn order to investigate the simultaneous interactions between

all three unit types, as well as to increase the network com-plexity, the IEEE 9-bus system given in Fig. 8 is consideredin this case study. A grid-following VSC is placed at node2 and a grid-forming VSC at node 3. The transmission linedynamics are also included in the model.

The stability mapping for the IEEE 9-bus system is shown inFig. 9, considering also different levels of network connectiv-ity. Each triangular axis denotes a penetration of the respectiveunit type, more precisely ⌘SG refers to synchronous genera-tors, ⌘F to grid-forming and ⌘f to grid-following inverter-based units; colored areas indicate a predominant penetration(� 50 %) of a single generation type. It should be noted thatthe system is always comprised of three generation units, i.e.,axes points indicating a 0 % penetration of a single generatortype are not considered, and the individual penetration levelsare varied in discrete steps of 1 %. We modify the originalsystem by gradually adding transmission lines, first betweenthe generator buses and subsequently between the load buses,as indicated by the red and blue lines in Fig. 8 respectively.Such procedure allows us to increase the network connectivity,defined as "g = 2Nb/(N2

n � Nn), from 40 % in the originalmodel to 60 % and 80 % in the modified system.

Fig. 9 shows that, for the generator configuration in thistest case, system stability cannot be maintained with lessthan 47 % synchronous generation. Reducing connectivity (inparticular removing generator bus network connections) makesthe system even more reliant on synchronous generation. Itis clear that stability erodes at high VSC penetration levelsindependent of the grid-forming share, which is consistentwith our previous findings in Section IV-A. Interestingly, the

3 6 7 8 2

5 9

4

1

Fig. 8: Diagram of the IEEE 9-bus system. The blue and redlines indicate the additionally incorporated transmission lines.

Page 6 of 8IEEE PES Transactions on Power Systems

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

I Generators supplying local RLloads

I Lines modeled as π sections

I All variables defined within asingle, uniform referenceframe.

I First results: 2 bus system

I Later in talk: 9 bus system

11 / 19

Network model and modeling approach











df =

!0rf

xda,u

vdf �

!0rf

xf

� d

f � da

�(26a)

D1= �!0rD1

xD1

� D1� d

a

�(26b)

Q1 = �!0rQ1

xQ1

( Q1 � qa) (26c)

Q2= �!0rQ2

xQ2

( Q2� q

a) (26d)



da = xd

a,s

�ids +

df

xf+ D1

xD1

!(26e)

qa = xq

a,s

✓�iqs +

Q1

xQ1

+ D1

xQ2

◆(26f)



xda,s =

✓1

xda,s

+1

xf+

1

xD1

◆�1

(26g)

xqa,s =

✓1

xqa,s

+1

xQ1

+1

xQ2

◆�1

(26h)



qs � q

a (26i)


ds � d

a (26j)




A. Network Modeling




xnj= xtu

e�j(✓n�✓j), 8j, u (27)


vnj=!b

clj

icj� j!n!bvnj

, 8j (28)

ilj =!b

lljvnj �

✓rlj

llj!b+ j!b!n

◆ilj , 8j (29)

vtj vtk

vtj

itj

vnj

inj

\✓j 7! \✓n

vnk

ink

vtk

itk

\✓n [ \✓k

itj

ilj

Zlj

ilk

Zlk

icj

Cj

Rjk Ljkijk

ick

Ck

itk

dq

dq

dq

dq


6

50 55 60 65 70 75 80 85 900

50

100

150

200

250

⌘ [%]

<(�)

�

�23

�20,21

�19

�18

�13,14

VSCf

control

VSCF

control


PLL&

APC

APC

PLL

SRFinner

control




10 20 30 40 50 60 70 80 90

0

5

10

⌘ [%]

<(�)

Scenario I

Scenario II

Scenario III









3 6 7 8 2

5 9

4

1



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

I Generators supplying local RLloads

I Lines modeled as π sections

I All variables defined within asingle, uniform referenceframe.

I First results: 2 bus system

I Later in talk: 9 bus system

11 / 19

Losing stability: SM with increasing GFL penetration, algebraic network 5

50 55 60 65 70 75 80 85 900

10

20

30

40

⌘ [%]

<(�)

�

�24,25

�23

�19,22

�16,17

�14,15

�20,21

VSCf

control

SGcontrol

stable region voltage dynamics frequency dynam.

PSS

SRFinner

control

AVR&

PSS

PLL APC

swingdynam.

Fig. 4: Impact of different controllers on system stability underhigh grid-following inverter penetration in Scenario I.

respective control gains could restore system stability withinthis range. While the increase in PSS stabilization gain provesto be relatively beneficial for the system, the adjustment ofinner loop PI controllers has no impact on the eigenvaluespectrum. This is an expected outcome due to a large timescaleseparation between the two feedback loops, which in turnhinders the synchronous generation from providing a stiffvoltage at the terminal. Furthermore, inner converter controlloops are often predefined by the manufacturer and optimallydesigned for providing fast and accurate voltage and powerreference tracking, implying that any parameter changes woulddistort its original purpose and effectiveness. Beside the SGvoltage regulators, the dynamics of the swing equation alsoprove to be relevant to a certain extent for the overall stability.However, in contrast to the popular belief that low inertialevels lead to vulnerability, it is in fact the insufficient dampingthat propagates the problem. Similar to the PSS stabilizationgain, the higher damping constant facilitates the integrationof converters, whereas the inertia constant has no impact onthe overall penetration; an observation that was also made in[14]. Nonetheless, damping is related to physical properties ofsynchronous generators, while droop gains, essentially corre-sponding to damping, are prescribed within narrow ranges bythe grid codes. This suggests that the most viable and practicalsolution would be to improve the PSS design, i.e., increase itsresponsiveness, in order to accommodate a high penetrationof grid-following inverter-based generation.

2) Scenario II: The issues pertaining to timescale separa-tion between different voltage controllers already highlightedin the previous scenario remain also in this scenario. It canbe noticed that the “forming” inverter property bolsters thevoltage vector at the respective bus and drastically improvesthe stability margin of the system. Nonetheless, for ⌘ > 78 %the AVR and PSS controllers cannot achieve adequate voltagestabilization, as the eigenvalues depicted in Fig. 5 drive deeplyinto the right-hand side of the root loci spectrum. Hence,the maximum feasible penetration of VSCs can hardly beimproved. Another important observation is that the frequencydynamics are not contributing to instability anymore, since

70 75 80 85 900

2

4

6

8

10

⌘ [%]

<(�)

�

�21�23

�20

�17

�18

VSCF

control

SGcontrol

stable region voltage dynamics

SRFinner

control

PSS AVR

fluxdynamics

Fig. 5: Impact of different controllers on system stability underhigh grid-forming inverter penetration in Scenario II.

both units independently establish an adequate frequencysignal and subsequently synchronize.

3) Scenario III: We tackle the 100 % PE-based scenarioby increasing the share of grid-following units reflected inan increasing factor ⌘. The results are presented in Fig. 6.The main distinction of this scenario is the elimination ofvoltage control issues in the presence of SGs and inverters,as both VSCs regulate voltage on the same timescale. Inspite of improving the voltage dynamics in the system, thesynchronization problems are aggravated due to an exclusionof a synchronous generator. In other words, the dedicated“forming” capability of a grid-forming converter is inferior tothe one of a traditional generator. As a result, for penetrationlevels above ⇡ 59 % the PLL units of grid-following inverterscannot accurately estimate the frequency signal, leading tofailure in the active power control and preventing system-level synchronization. Similar properties are also reflectedwhen analyzing ⌘ as a function of the proportional PLL gain,with a more responsive PLL potentially facilitating a higherpenetration factor. However, this approach does not solve thefundamental problem at hand, and provides only a marginalimprovement of few percent. It should be noted that theintegral gain of the respective PI controller does not affectthe system stability margin, indicating that the original PLLtime constant could be maintained in the process.

B. Inclusion of Transmission Line DynamicsWe broaden the scope of our analysis by including the

transmission line dynamics described in Section III. The samethree scenarios are re-evaluated and compared against theprevious case study, with the respective results illustratedin Fig. 7. A noticeable difference in the stability marginof a 100 % inverter-based system can be observed, wherethe inclusion of line dynamics significantly increases themaximum penetration ratio ⌘. This can be explained by theinductive and capacitive components of the lines acting asenergy buffers and augmenting the synchronization betweenthe two units. More precisely, the frequency issues associatedwith a large proportion of PLL-based generation are alleviated

Page 5 of 8 IEEE PES Transactions on Power Systems

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

I Early instability due tointeraction between PSS andVSC inner control loops.I This is consistent with Lin

et al’s finding: removingAVR dynamics stabilizessystem.

I Increasing PSS gainimproves margins

I “Non-forming” aspect ofGFM voltage apparentlycounteracts the voltage at theSM terminal

I Inertia constant has nobearing

12 / 19

Losing stability: SM with increasing GFM penetration, algebraic network5

50 55 60 65 70 75 80 85 900

10

20

30

40

⌘ [%]

<(�)

�

�24,25

�23

�19,22

�16,17

�14,15

�20,21

VSCf

control

SGcontrol

stable region voltage dynamics frequency dynam.

PSS

SRFinner

control

AVR&

PSS

PLL APC

swingdynam.

Fig. 4: Impact of different controllers on system stability underhigh grid-following inverter penetration in Scenario I.

respective control gains could restore system stability withinthis range. While the increase in PSS stabilization gain provesto be relatively beneficial for the system, the adjustment ofinner loop PI controllers has no impact on the eigenvaluespectrum. This is an expected outcome due to a large timescaleseparation between the two feedback loops, which in turnhinders the synchronous generation from providing a stiffvoltage at the terminal. Furthermore, inner converter controlloops are often predefined by the manufacturer and optimallydesigned for providing fast and accurate voltage and powerreference tracking, implying that any parameter changes woulddistort its original purpose and effectiveness. Beside the SGvoltage regulators, the dynamics of the swing equation alsoprove to be relevant to a certain extent for the overall stability.However, in contrast to the popular belief that low inertialevels lead to vulnerability, it is in fact the insufficient dampingthat propagates the problem. Similar to the PSS stabilizationgain, the higher damping constant facilitates the integrationof converters, whereas the inertia constant has no impact onthe overall penetration; an observation that was also made in[14]. Nonetheless, damping is related to physical properties ofsynchronous generators, while droop gains, essentially corre-sponding to damping, are prescribed within narrow ranges bythe grid codes. This suggests that the most viable and practicalsolution would be to improve the PSS design, i.e., increase itsresponsiveness, in order to accommodate a high penetrationof grid-following inverter-based generation.

2) Scenario II: The issues pertaining to timescale separa-tion between different voltage controllers already highlightedin the previous scenario remain also in this scenario. It canbe noticed that the “forming” inverter property bolsters thevoltage vector at the respective bus and drastically improvesthe stability margin of the system. Nonetheless, for ⌘ > 78 %the AVR and PSS controllers cannot achieve adequate voltagestabilization, as the eigenvalues depicted in Fig. 5 drive deeplyinto the right-hand side of the root loci spectrum. Hence,the maximum feasible penetration of VSCs can hardly beimproved. Another important observation is that the frequencydynamics are not contributing to instability anymore, since

70 75 80 85 900

2

4

6

8

10

⌘ [%]

<(�)

�

�21�23

�20

�17

�18

VSCF

control

SGcontrol

stable region voltage dynamics

SRFinner

control

PSS AVR

fluxdynamics

Fig. 5: Impact of different controllers on system stability underhigh grid-forming inverter penetration in Scenario II.

both units independently establish an adequate frequencysignal and subsequently synchronize.

3) Scenario III: We tackle the 100 % PE-based scenarioby increasing the share of grid-following units reflected inan increasing factor ⌘. The results are presented in Fig. 6.The main distinction of this scenario is the elimination ofvoltage control issues in the presence of SGs and inverters,as both VSCs regulate voltage on the same timescale. Inspite of improving the voltage dynamics in the system, thesynchronization problems are aggravated due to an exclusionof a synchronous generator. In other words, the dedicated“forming” capability of a grid-forming converter is inferior tothe one of a traditional generator. As a result, for penetrationlevels above ⇡ 59 % the PLL units of grid-following inverterscannot accurately estimate the frequency signal, leading tofailure in the active power control and preventing system-level synchronization. Similar properties are also reflectedwhen analyzing ⌘ as a function of the proportional PLL gain,with a more responsive PLL potentially facilitating a higherpenetration factor. However, this approach does not solve thefundamental problem at hand, and provides only a marginalimprovement of few percent. It should be noted that theintegral gain of the respective PI controller does not affectthe system stability margin, indicating that the original PLLtime constant could be maintained in the process.

B. Inclusion of Transmission Line DynamicsWe broaden the scope of our analysis by including the

transmission line dynamics described in Section III. The samethree scenarios are re-evaluated and compared against theprevious case study, with the respective results illustratedin Fig. 7. A noticeable difference in the stability marginof a 100 % inverter-based system can be observed, wherethe inclusion of line dynamics significantly increases themaximum penetration ratio ⌘. This can be explained by theinductive and capacitive components of the lines acting asenergy buffers and augmenting the synchronization betweenthe two units. More precisely, the frequency issues associatedwith a large proportion of PLL-based generation are alleviated


123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

I GFM provides greaterstability margin than GFLI But instability still exists

I Instability ultimately fromsame modes as GFM

I Swing dynamics modeeliminated (vs GFMinstability)

13 / 19

Adding the network (still two bus)

10 20 30 40 50 60 70 80 90

0

5

10

⌘ [%]

<(�)

SG � VSCF

SG � VSCf

VSCF � VSCf

Fig. 12: Impact of inverter penetration on system stabilityfor different unit configuration, with inclusion of transmissionline dynamics. Transparent lines indicate results for scenarioswithout line dynamics.

3 6 7 8 2

5 9

4

1


solely to the AVR and PSS control design. The predominantimpact of SMs on the system stability is even higher in Sce-nario II, with grid-forming inverters only marginally affectingthe critical modes within a narrow range of ⌘ 2 [78, 79] %.Hence, the maximum admissible VSC penetration after theinclusion of transmission lines remains the same in this case.

D. Stability Margins of the IEEE 9-bus System

In order to investigate the simultaneous interactions betweenall three unit types, as well as to increase the networkcomplexity, the IEEE 9-bus system presented in Fig. 13 isconsidered in this case study. The converter-based generationis placed at nodes 2 and 3, with a grid-following VSC beingthe former and a grid-forming the latter one. The transmissionline dynamics are also included in the model.

The stability mapping for the IEEE 9-bus system is show-cased in Fig. 14, while simultaneously considering differentlevels of network connectivity. Each triangular axis denotesa penetration of the respective unit type, more precisely ⌘SG

refers to synchronous generators, ⌘F to grid-forming and ⌘f

to grid-following inverter-based units. We modify the originalsystem by gradually adding transmission lines, first betweenthe generator buses and subsequently between the load buses,as indicated by the red and blue lines in Fig. 13 respectively.Such procedure allows us to increase the network connectivity,

defined as "g = 2Nb/(N2n � Nn), from 40 % in the original

model to 60 % and 80 % in the modified system.It can be observed in Fig. 14 that there exists a minimum

required level of synchronous generation in order to preservesystem stability. For "g = 40% this amounts to approximately57 %, whereas for more meshed networks this limit can bereduced by 8�10 %. Nonetheless, it is clear that the productionportfolios involving high penetration of converter-interfacedgeneration can drastically distort the system, independent ofthe grid-forming share. Interestingly, the permissible VSCinstallation margin significantly reduces when ⌘F /⌘f ratiodiverges from 1, as maximum value of ⌘ = ⌘F + ⌘f dropsfrom 44 % to 27 % in case of ⌘F ⇡ 0 % or ⌘f ⇡ 0 %. Thisis justified by different instabilities occurring between variousunit types, specifically voltage issues for a system comprisedof synchronous and inverter-based generators (Scenarios I andII) and synchronization obstacles related to a 100 % PE-basedsystem (Scenario III), as described in Section V-B. For arather balanced portfolio, all of these problems are somewhatconfined within reasonable limits, while an imbalance in⌘F /⌘f ratio tends to emphasize the voltage instability andendanger the whole network. Another valid point can bemade regarding the beneficial impact of transmission linedynamics. As previously indicated in Section V-C, a directconnection between units experiencing frequency instabilitymitigates the synchronization issues, thus facilitating a highershare of PE-based devices. Nonetheless, this is not the casefor transmission lines between the load nodes, as increasingconnectivity from 60 % to 80 % has a marginal impact on thestability margin.

We extend this analysis by differentiating between threedifferent generation portfolios: (i) P0 - a mix of all three unittypes, as previously discussed; (ii) Pf - a system comprisedonly of synchronous generators and grid-following VSCs; and(iii) PF - a system comprised only of synchronous generatorsand grid-forming VSCs; in the latter two cases, the totalinverter penetration ratio is equal to either ⌘f or ⌘F . Theresults for the IEEE 9-bus system presented in Fig. 15 suggestthat more homogeneous portfolios, such as Pf and PF , can

0 20 40 60 80 1000

20

40

60

80

1000

20

40

60

80

100

⌘f [%]

⌘ F[%

]

⌘S

G[%

]

stableregions

"g

100 %

80 %

60 %

40 %

20 %

Fig. 14: Stability mapping for the IEEE 9-bus test caseunder different levels of network connectivity. Colored arrowsindicate the appropriate reading on the respective axes.

I Darker lines: with linedynamics; greyed out lines:no line dynamics

I Line dynamics → significant stability improvement for 100% systemI This system’s first instabilities due to frequency related issues → lines dynamics delayI Colombino talk yesterday found network dynamics erode stabilityI However we put all GFM at one bus, and mix with GFL → a very different setting.

I Adding line dynamics has less impact on margins for SM+X systemsI These instabilities more strongly tied to voltage dynamics.

14 / 19

IEEE 9 bus results: Adding network connections improves stability margins

6

50 55 60 65 70 75 80 85 900

50

100

150

200

250

⌘ [%]<(

�)

�

�23

�20,21

�19

�18

�13,14

VSCf

control

VSCF

control


PLL&

APC

APC

PLL

SRFinner

control




10 20 30 40 50 60 70 80 90

0

5

10

⌘ [%]

<(�)

Scenario I

Scenario II

Scenario III









3 6 7 8 2

5 9

4

1



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

I Black network only ↔ εg = 40%

I Black and red lines ↔ εg = 60%

I Black, red, blue lines ↔ εg = 80%

7

0 20 40 60 80 1000

20

40

60

80

1000

20

40

60

80

100

⌘f [%]

⌘ F[%

]

⌘S

G[%

]

stableregions

"g

100 %

80 %

60 %

40 %

20 %

Fig. 9: Stability mapping for the IEEE 9-bus test case underdifferent levels of network connectivity. Colored arrows indi-cate the appropriate reading direction on the respective axes.

permissible converter installation margin significantly reduceswhen the ⌘F /⌘f ratio diverges from 1, as the maximum valueof ⌘ = ⌘F +⌘f drops from 44 % to 27 % in case of ⌘F ⇡ 0 %or ⌘f ⇡ 0 %. This is due to different instabilities occurringbetween various unit types, specifically voltage issues for asystem comprised of synchronous and inverter-based genera-tors (Scenarios I and II) and synchronization obstacles relatedto a 100 % PE-based system (Scenario III), as described inSection IV-A. For a rather balanced portfolio, all of theseproblems are somewhat confined within reasonable limits,while an imbalance in the ⌘F /⌘f ratio tends to emphasizethe voltage dynamics issues and endanger the whole network.Another valid point can be made regarding the beneficial im-pact of transmission line dynamics. As previously indicated inSection IV-B, a direct connection between units experiencingfrequency instability mitigates the synchronization issues, thusfacilitating a higher share of PE-based devices. Nonetheless,this is not the case for transmission lines between the loadnodes, as increasing connectivity from 60 % to 80 % has anegligible impact on the stability margin.

We extend this analysis by differentiating between threedifferent generation portfolios: (i) P0 - a mix of all three unittypes, as previously discussed; (ii) Pf - a system comprisedonly of synchronous generators and grid-following VSCs; and(iii) PF - a system comprised only of synchronous generatorsand grid-forming VSCs; in the latter two cases the portfolioconsists of two synchronous and one inverter-based generator,i.e., one of the VSCs from the original portfolio P0 is replacedby an SG, whereas the total inverter penetration ratio isequal to either ⌘f or ⌘F . The results for the IEEE 9-bussystem presented in Fig. 10 suggest that more homogeneousportfolios, such as Pf and PF , can withstand drastically higherpenetrations of renewables, with ⌘max reaching up to 75 %and 82 % respectively. Understandably, portfolio PF has ahigher ⌘ margin due to less accentuated voltage instabilitycompared to the Pf scenario. However, increasing networkconnectivity balances these penetration ratios out, as directtransmission line connections resolve the timescale separationbetween the voltage controllers of synchronous and converter-based generators (see Fig. 2). As a result, the system canfacilitate up to ⌘max = 88% of PE-interfaced units; an increaseof approximately 70 % compared to the original portfolio P0.

Alternatively, a different configuration with a less heteroge-

P0 Pf PF

40

60

80

100

44

7582

53

86 86

52

88 88

⌘m

ax

[%]

"g = 40 %

"g = 60 %

"g = 80 %

Fig. 10: Maximum permissible inverter penetration under dif-ferent system portfolios (P) and levels of network connectivity("g).

neous portfolio comprised of two grid-forming and one grid-following inverter is investigated. For higher levels of networkconnectivity the system faces no stability issues under anyshare of installed powers, whereas in case of "g = 40 % themaximum permissible penetration of grid-following generationis 93 %. This highlights the fact that the small-signal stabilityof low-inertia systems is drastically more vulnerable comparedto the one of 100 % PE-based systems.

D. Key Takeaway MessagesIt should be noted that the obtained permissible penetration

levels of inverter-based generation might vary drastically be-tween systems of different sizes and complexity. The resultsin this paper highlight some important qualitative properties offuture low-inertia systems. Therefore, several key conclusionscan be drawn from this study:

1) Transition might be hard: The existing literature focusesmostly on 100% inverter-based power systems and the associ-ated frequency stability and synchronization issues, assumingan easy transition along the way. While this might be the casefrom the perspective of achieving a stiff frequency signal in thenetwork, the overall stability of the interim system is signifi-cantly more vulnerable and harder to ensure. These issues havealready begun to emerge in large-scale networks such as SouthAustralia [12] and fast frequency control solutions have beenproposed [27], however our results indicate that such solutionsmay increase the maximum allowable VSC penetration, butcannot stabilize all possible VSC penetration levels. In otherwords, they are rather a temporary mitigation of a much deeperand more convoluted problem than a permanent solution.

2) Need for uniform regulation: Throughout this studywe assume a perfectly balanced converter control design,i.e., compatible control schemes and parameter tuning whichensure stable synchronization and safe operation of the system.However, this might not be the case in the future, especiallyconsidering current grid codes and lack of a uniform regulationfor VSC manufacturers. As a matter of fact, the portfoliosPf and PF could have even underperformed compared toP0 under such circumstances. Furthermore, even the soledefinition and classification of converter operation modes,e.g., grid-forming, grid-following and grid-supporting, lacksfoundation and tends to drastically differ between differentoperators and industries.

3) New premise of timescale separation: One of the maincontrasts between traditional and low-inertia systems is thetimescale separation between the respective controllers ofsynchronous and inverter-based generators, which leads tofrequency and voltage instabilities under high penetration of


123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

I ηmax = max stable percentage VSCs

I P0: Hybrid system (SM, GFM, GFL)

I Pf : SM and GFL only

I PF : SM and GFM only

15 / 19

IEEE 9 bus results: Adding network connections improves stability margins

6

50 55 60 65 70 75 80 85 900

50

100

150

200

250

⌘ [%]<(

�)

�

�23

�20,21

�19

�18

�13,14

VSCf

control

VSCF

control


PLL&

APC

APC

PLL

SRFinner

control




10 20 30 40 50 60 70 80 90

0

5

10

⌘ [%]

<(�)

Scenario I

Scenario II

Scenario III









3 6 7 8 2

5 9

4

1



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

I Black network only ↔ εg = 40%

I Black and red lines ↔ εg = 60%

I Black, red, blue lines ↔ εg = 80%

7

0 20 40 60 80 1000

20

40

60

80

1000

20

40

60

80

100

⌘f [%]

⌘ F[%

]

⌘S

G[%

]

stableregions

"g

100 %

80 %

60 %

40 %

20 %

Fig. 9: Stability mapping for the IEEE 9-bus test case underdifferent levels of network connectivity. Colored arrows indi-cate the appropriate reading direction on the respective axes.

permissible converter installation margin significantly reduceswhen the ⌘F /⌘f ratio diverges from 1, as the maximum valueof ⌘ = ⌘F +⌘f drops from 44 % to 27 % in case of ⌘F ⇡ 0 %or ⌘f ⇡ 0 %. This is due to different instabilities occurringbetween various unit types, specifically voltage issues for asystem comprised of synchronous and inverter-based genera-tors (Scenarios I and II) and synchronization obstacles relatedto a 100 % PE-based system (Scenario III), as described inSection IV-A. For a rather balanced portfolio, all of theseproblems are somewhat confined within reasonable limits,while an imbalance in the ⌘F /⌘f ratio tends to emphasizethe voltage dynamics issues and endanger the whole network.Another valid point can be made regarding the beneficial im-pact of transmission line dynamics. As previously indicated inSection IV-B, a direct connection between units experiencingfrequency instability mitigates the synchronization issues, thusfacilitating a higher share of PE-based devices. Nonetheless,this is not the case for transmission lines between the loadnodes, as increasing connectivity from 60 % to 80 % has anegligible impact on the stability margin.

We extend this analysis by differentiating between threedifferent generation portfolios: (i) P0 - a mix of all three unittypes, as previously discussed; (ii) Pf - a system comprisedonly of synchronous generators and grid-following VSCs; and(iii) PF - a system comprised only of synchronous generatorsand grid-forming VSCs; in the latter two cases the portfolioconsists of two synchronous and one inverter-based generator,i.e., one of the VSCs from the original portfolio P0 is replacedby an SG, whereas the total inverter penetration ratio isequal to either ⌘f or ⌘F . The results for the IEEE 9-bussystem presented in Fig. 10 suggest that more homogeneousportfolios, such as Pf and PF , can withstand drastically higherpenetrations of renewables, with ⌘max reaching up to 75 %and 82 % respectively. Understandably, portfolio PF has ahigher ⌘ margin due to less accentuated voltage instabilitycompared to the Pf scenario. However, increasing networkconnectivity balances these penetration ratios out, as directtransmission line connections resolve the timescale separationbetween the voltage controllers of synchronous and converter-based generators (see Fig. 2). As a result, the system canfacilitate up to ⌘max = 88% of PE-interfaced units; an increaseof approximately 70 % compared to the original portfolio P0.

Alternatively, a different configuration with a less heteroge-

P0 Pf PF

40

60

80

100

44

7582

53

86 86

52

88 88

⌘m

ax

[%]

"g = 40 %

"g = 60 %

"g = 80 %

Fig. 10: Maximum permissible inverter penetration under dif-ferent system portfolios (P) and levels of network connectivity("g).

neous portfolio comprised of two grid-forming and one grid-following inverter is investigated. For higher levels of networkconnectivity the system faces no stability issues under anyshare of installed powers, whereas in case of "g = 40 % themaximum permissible penetration of grid-following generationis 93 %. This highlights the fact that the small-signal stabilityof low-inertia systems is drastically more vulnerable comparedto the one of 100 % PE-based systems.

D. Key Takeaway MessagesIt should be noted that the obtained permissible penetration

levels of inverter-based generation might vary drastically be-tween systems of different sizes and complexity. The resultsin this paper highlight some important qualitative properties offuture low-inertia systems. Therefore, several key conclusionscan be drawn from this study:

1) Transition might be hard: The existing literature focusesmostly on 100% inverter-based power systems and the associ-ated frequency stability and synchronization issues, assumingan easy transition along the way. While this might be the casefrom the perspective of achieving a stiff frequency signal in thenetwork, the overall stability of the interim system is signifi-cantly more vulnerable and harder to ensure. These issues havealready begun to emerge in large-scale networks such as SouthAustralia [12] and fast frequency control solutions have beenproposed [27], however our results indicate that such solutionsmay increase the maximum allowable VSC penetration, butcannot stabilize all possible VSC penetration levels. In otherwords, they are rather a temporary mitigation of a much deeperand more convoluted problem than a permanent solution.

2) Need for uniform regulation: Throughout this studywe assume a perfectly balanced converter control design,i.e., compatible control schemes and parameter tuning whichensure stable synchronization and safe operation of the system.However, this might not be the case in the future, especiallyconsidering current grid codes and lack of a uniform regulationfor VSC manufacturers. As a matter of fact, the portfoliosPf and PF could have even underperformed compared toP0 under such circumstances. Furthermore, even the soledefinition and classification of converter operation modes,e.g., grid-forming, grid-following and grid-supporting, lacksfoundation and tends to drastically differ between differentoperators and industries.

3) New premise of timescale separation: One of the maincontrasts between traditional and low-inertia systems is thetimescale separation between the respective controllers ofsynchronous and inverter-based generators, which leads tofrequency and voltage instabilities under high penetration of


123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

I ηmax = max stable percentage VSCs

I P0: Hybrid system (SM, GFM, GFL)

I Pf : SM and GFL only

I PF : SM and GFM only

15 / 19

Some future directions

I What if there are many devices at each bus, operating at different setpoints?

(Henriquez et al, IECON 2020)

16 / 19

Some future directions, ctd

I Is it possible to intentionally destabilize GFM systems?

(Roberts et al, in review)

17 / 19

Closing thoughts

Takeaways

I Perhaps 100% is easier than 80%?

I Line dynamics critical

I Heterogenous systems more challenging

I Stability assessments need to model inner voltage and current loops in converters

I The only generic result (across many recent papers) is that small signalinstabilities are realistically possible.

What are the solutions?

1. Perhaps we can just slow down the VSC inner control loops?

2. Need to understand if VSM, VOC, matching control would do similar things

3. Adjust gains for VSC PI controllers – but can this be done to guarantee stability?

4. Add damping to block diagrams – much like a PSS

18 / 19

Closing thoughts

Takeaways

I Perhaps 100% is easier than 80%?

I Line dynamics critical

I Heterogenous systems more challenging

I Stability assessments need to model inner voltage and current loops in converters

I The only generic result (across many recent papers) is that small signalinstabilities are realistically possible.

What are the solutions?

1. Perhaps we can just slow down the VSC inner control loops?

2. Need to understand if VSM, VOC, matching control would do similar things

3. Adjust gains for VSC PI controllers – but can this be done to guarantee stability?

4. Add damping to block diagrams – much like a PSS

18 / 19

Enormous amounts of work fromLBNLEvangelos Vrettos

ETH ZurichGabriela HugUros MarkovicOgnjen Stanojev

Leeds Petros Aristidou

Contact info:Duncan Callaway ([email protected])

Many thanks to the organizers for awonderful symposium!

19 / 19

Parameters

TABLE I: Converter Control Parameters

Parameter Symbol Value UnitActive power droop gain Rp 2 %

Reactive power droop gain Rq 0.1 %

LPF cut-off frequency fz 5 HzRLC filter resistance rf 0.03 p.u.RLC filter inductance lf 0.08 p.u.RLC filter capacitance cf 0.074 p.u.

P-gain of SRF current control Kip 0.74 -

I-gain of SRF current control Kii 1.19 -

FF-gain of SRF current control Kif 1 -

P-gain of SRF voltage control Kvp 0.52 -

I-gain of SRF voltage control Kvi 1.16 -

FF-gain of SRF voltage control Kvf 1 -

PLL proportional gain Kpllp 0.4 -

PLL integral gain Kplli 4.69 -

Virtual impedance resistance rv 0 p.u.Virtual impedance inductance lv 0.2 p.u.

where Kvp and Kv

i are the proportional and integral gains ofthe SRF voltage PI controller, and ⇠ is the integrator state.Furthermore, a feed-forward signal of the measured currentscan be enabled or disabled by changing the gain Ki

f 2 [0, 1].The output current reference is is then used as an inputsetpoint to the current controller.

Similar to its voltage counterpart, the configuration of theSRF current controller is based on a PI control with decouplingterms:

vm = Kip(is � is) + Ki

i� + j!clf is + Kvf eg (22)

� = is � is (23)

with Kip, Ki

i and Kvf being the respective controller gains, and

� the integrator state. The generated output voltage referencevm is used to determine the averaged modulation signal

mabc = (T pT c)�1

mdq = (T pT c)�1 vm

vdc(24)

which reduces the AC side sensitivity to DC voltage (vdc)oscillations [4]. Due to high complexity and very fast internaldynamics, the converter switching and DC-side buffers are notincluded in the model, since their impact on the small-signalstability is rather negligible.

With inclusion of the current and voltage dynamics associ-ated to the electrical system interface in (3)-(5), the completemathematical model consists of 13 states for the grid-formingand 15 states for the grid-following converter unit. All relevantconverter parameters used in this paper are presented inTable I, whereas more details on the overall converter controlstructure, potential operation modes and respective transientproperties can be found in [5], [6], [20].

III. SYNCHRONOUS GENERATOR MODEL

We consider a traditional two-pole synchronous generatorequipped with a prime mover and a governor, described inper unit. Additionally, a voltage excitation system consisted ofAutomatic Voltage Regulator (AVR) and PSS is incorporated.A detailed block diagram is showcased in Fig. 2, wherethe synchronous generator is connected to the grid trough atransformer. The main parameters are listed in Table II. Thisis a well-established configuration used both for academic andindustrial purposes [2], [23], [24].

A. Electrical InterfaceA synchronous generator is interfaced through a transformer

(rt, lt) to the grid, and modeled in the SRF defined by itssynchronous velocity !s:

is =!b

lt(es � vn)�

✓rt

lt!b + j!b!s

◆is (25)

with es and is denoting the stator voltage and current respec-tively, and vn representing the nodal voltage at the PCC.

B. Internal Machine DynamicsThe internal machine dynamics are characterized by the

transients in the rotor circuits, as transients in the statorwindings decay rapidly and can thus be neglected. Rotor

AVR

PSS

Frequency Control

Ks

stab. gain

sTw

1 + sTw

washout

1 + sT1

1 + sT2

1 + sT3

1 + sT4

phase compensation

1

1 + sTe

transducer

k·k2norm

v⇤

Ke

exciter saturation

daiqs � q

aids df , D1 , Q1 , Q2

internal rotor dynamicselectrical torque

1

Rg

droop

p⇤

1

1 + sTg

governor

Km(1 + sFhTr)

1 + sTr

turbine

1

sMg + Dg

swing dynamics

!0

1

s

internal(dq)-frame

Grid

v2 v3

+vs

edqs

vt

�v1

+ vdf

dqa , idq

s

Rt Lt

��pm pg

+

pm �pe

+

pe [ ⌧e

�

+

�!r

+

!s ✓g abc

dq

Fig. 2: Block diagram of a synchronous generator equipped with a prime mover, governor and voltage excitation system.











df =

!0rf

xda,u

vdf �

!0rf

xf

� d

f � da

�(26a)

D1= �!0rD1

xD1

� D1� d

a

�(26b)

Q1 = �!0rQ1

xQ1

( Q1 � qa) (26c)

Q2= �!0rQ2

xQ2

( Q2� q

a) (26d)



da = xd

a,s

�ids +

df

xf+ D1

xD1

!(26e)

qa = xq

a,s

✓�iqs +

Q1

xQ1

+ D1

xQ2

◆(26f)



xda,s =

✓1

xda,s

+1

xf+

1

xD1

◆�1

(26g)

xqa,s =

✓1

xqa,s

+1

xQ1

+1

xQ2

◆�1

(26h)



qs � q

a (26i)


ds � d

a (26j)




A. Network Modeling




xnj= xtu

e�j(✓n�✓j), 8j, u (27)


vnj=!b

clj

icj� j!n!bvnj

, 8j (28)

ilj =!b

lljvnj �

✓rlj

llj!b+ j!b!n

◆ilj , 8j (29)

vtj vtk

vtj

itj

vnj

inj

\✓j 7! \✓n

vnk

ink

vtk

itk

\✓n [ \✓k

itj

ilj

Zlj

ilk

Zlk

icj

Cj

Rjk Ljkijk

ick

Ck

itk

dq

dq

dq

dq


20 / 19

Documents

Examining Large-Scale Low Inertia Power System Stability