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Placing Grid-Forming Converters to Enhance SmallSignal Stability of PLL-Integrated Power Systems

Chaoran Yang, Linbin Huang, Huanhai Xin, and Ping Ju

Abstract—The modern power grid features the high penetra-tion of power converters, which widely employ a phase-lockedloop (PLL) for grid synchronization. However, it has been pointedout that PLL can give rise to small-signal instabilities underweak grid conditions. This problem can be potentially resolved byoperating the converters in grid-forming mode, namely, withoutusing a PLL. Nonetheless, it has not been theoretically revealedhow the placement of grid-forming converters enhances thesmall-signal stability of power systems integrated with large-scale PLL-based converters. This paper aims at filling this gap.Based on matrix perturbation theory, we explicitly demonstratethat the placement of grid-forming converters is equivalent toincreasing the power grid strength and thus improving thesmall-signal stability of PLL-based converters. Furthermore, weinvestigate the optimal locations to place grid-forming convertersby increasing the smallest eigenvalue of the weighted and Kron-reduced Laplacian matrix of the power network. The analysis inthis paper is validated through high-fidelity simulation studies ona modified two-area test system and a modified 39-bus test system.This paper potentially lays the foundation for understanding theinteraction between PLL-based (i.e., grid-following) convertersand grid-forming converters, and coordinating their placementsin future converter-dominated power systems.

Index Terms—Generalized short-circuit ratio (gSCR), grid-forming converters, grid strength, phase-locked loop (PLL),small-signal stability.

I. INTRODUCTION

Power converters are extensively integrated into modernpower systems as the grid interfaces of renewables, HVDCsystems, energy storage systems, and so on [1]–[3]. Currently,most of the converters adopt a phase-locked loop (PLL)for grid synchronization. The mechanism of PLL is easy tounderstand, as it can be considered as tracking the angle andfrequency of the power grid with second-order dynamics [4].For this reason, PLL-based converters are also widely knownas “grid-following” converters [5], [6]. The large-scale inte-gration of PLL-based converters gives rise to unprecedentedchanges to power systems as the synchronization mechanismof PLL-based converters is totally different from conventionalpower sources, i.e., synchronous generators (SGs).

Compared with conventional power systems, the synchro-nization dynamics of such PLL-integrated power systems aremore complex and new types of instability issues may arise.For example, it has been reported that PLL-based converterscould become unstable under weak grid conditions, which

Manuscript received July 8, 2020. The authors are with the Col-lege of Electrical Engineering at Zhejiang University, Hangzhou, China.(Emails: yang chaoran@zju.edu.cn, huanglb@zju.edu.cn, xinhh@zju.edu.cn,pju@zju.eu.cn)

This work was supported by the National Natural Science Foundation ofChina (No. 51922094, No. 52007163).

belongs to small-signal instability issues [7]–[10]. This typeof instability is dominated by the dynamics of PLL, while theoscillation frequency and stability margin are also pertinentto the design of other loops [7]. Such instability/oscillationshould be prevented in practice because it endangers the secureoperation of power systems and may cause economic loss oncethe converters get tripped.

Recent works have shown that grid-forming converters arenaturally immune from the PLL-induced instabilities since aPLL is not needed to realize grid synchronization [11], [12].Grid-forming converters are supposed to have the capabil-ity of forming a local grid without connected to an extravoltage source. According to this definition, currently thereexist many control strategies that can be classified as grid-forming, e.g., droop control, virtual synchronous machines(VSM), synchronverters, and virtual oscillator control [13]–[15]. Without loss of generality, in this paper we considerVSM as a prototypical type of grid-forming control, as it alsocovers another popular type, i.e., droop control (which can beconsidered as VSM with zero virtual inertia).

The initial motivations of using grid-forming control were torealize islanded operation, inertia emulation, voltage support,etc., while it turns out that another significant advantage is therobustness against various power grid strength, namely, it fitswell with weak grid conditions [2], [16]–[18]. Hence, grid-forming control can be considered as a promising techniqueto accommodate large-scale power converters. Moreover, grid-forming converters also achieve better performance than grid-following converters in terms of virtual inertia provision [5].

However, currently almost all the installed power convertersin practice have been equipped with PLL-based controllers,and thus it could be unrealistic to change all of them into grid-forming converters. As an alternative, one can change someof the installed converters into grid-forming type, or requirethat the converters to be installed should employ grid-formingcontrol. That is to say, future power systems will compriseboth PLL-based converters and grid-forming converters.

There have been research works using a single-converter-infinite-bus system to demonstrate that grid-forming converterscan maintain desired stability margin even under very weakgrid conditions [16], [19]. By comparison, PLL-based convert-ers may become unstable which feature sustained oscillationsof the frequency output (i.e., PLL-induced instabilities) [7],[8]. It has also attracted an increasing research interest tostudy the interaction between grid-forming converters andPLL-based converters [1], [5]. For example, Ref. [5] inves-tigated how the virtual inertia implementations in PLL-basedconverters and grid-forming converters affect the performance

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of the system’s frequency responses, and an optimal placementalgorithm was proposed to improve the frequency responses.

However, in terms of the (PLL-induced) small-signal sta-bility problem, it has not been studied yet how grid-formingconverters and PLL-based converters interact with each otherand affect such stability of multi-converter systems. In otherwords, it remains unclear whether the placements of grid-forming converters can help improve the small-signal stabilityof power systems that integrated with large-scale PLL-basedconverters and reduce the chance of PLL-induced oscillations.Moreover, it is also unclear how to optimally place the grid-forming converters with regards to the stability margin.

This paper aims at filling these gaps by theoretically explor-ing the interactions between grid-forming converters and PLL-based converters and analyzing the resulting (PLL-induced)small-signal stability.

In a first step, we model the system which describes howthe grid-forming converters interact with PLL-based convertersvia the power network. Then, by using matrix perturbationtheory, we explicitly analyze how the integration of grid-forming converters affects the small-signal stability of PLL-based converters. Our analysis is based on our previous findingthat the small-signal stability of PLL-based converters is deter-mined by the power grid strength which can be characterizedby the generalized short-circuit ratio (gSCR) [20], [21]. Thisenables us to study the impacts of grid-forming converterson the stability of PLL-based converters by simply focusingon how the grid-forming converters equivalently change thegrid strength. We will explicitly show that the integrationof grid-forming converters is equivalent to placing voltagesources in the power network and thus enhance the gridstrength, which is attributed to the voltage regulation insidegrid-forming controls. Moreover, based on the analysis, weinvestigate how to optimally place the grid-forming convertersto enhance the overall system stability, which is based onincreasing the smallest eigenvalue of the weighted and Kron-reduced Laplacian matrix of the power network (i.e., gSCR ofthe system).

The three substantial contributions of this paper are sum-marized as follows:

1) By using matrix perturbation theory, it is theoreticallyshown that the placement of grid-forming converters isequivalent to changing the grid strength (characterized bygSCR) from the perspective of the PLL-based converters.

2) It is rigorously proven that the placement of grid-formingconverters has positive effects on the (PLL-induced)small-signal stability of multi-converter systems, and thelocations of the grid-forming converters determine towhat extent the stability can be improved.

3) Based on the theoretical analysis of how grid-formingconverters affect PLL-based converters, we formulate atractable optimization problem in order to find the optimallocations to place grid-forming converters and enhancethe (PLL-induced) small-signal stability of the system.

The rest of this paper is organized as follows: Section IIpresents the system modeling considering grid-forming andPLL-based converters. Section III analyzes the impacts ofgrid-forming control on the stability of PLL-based converters

AC Grid

Three-phase Converter LCL Filter

Power Part

Grid-Forming Control

Voltage Control(& Virtual Impedance)

CurrentControl

Reactive Power/ Voltage Regulation

abcabc

dqdq

PWM

Swing Equation

Power Calculation

PLL-based Control

Active & Reactive Power Control

CurrentControl

PI

abcabc

dqdq

PWM

Power Calculation

PLL

Control Scheme Switch

Fig. 1. One-line diagram of a grid-connected converter with grid-formingcontrol or PLL-based control.

by focusing on the grid strength. Section IV investigatesthe optimal placement of grid-forming converters. Simulationresults are given in Section V. Section VI concludes the paper.

II. MULTI-CONVERTER SYSTEM MODELING

In this section, we will briefly introduce the admittancemodels of grid-forming converters and PLL-based converters,and then develop the closed-loop model of a multi-convertersystem in order to illustrate how grid-forming convertersinteract with PLL-based converters via the network. As men-tioned before, we consider VSM as a prototypical grid-formingcontrol without loss of generality.

Fig. 1 shows a three-phase converter which is connected tothe ac grid via an LCL filter. The converter can be operatedin PLL-based mode or grid-forming mode, with the controldiagrams given in Fig. 1.

A. Admittance Modeling of PLL-based Converters

As labelled in Fig. 1, Vabc is the three-phase capacitorvoltage, Iabc is the converter-side three-phase current, Igabc

is the three-phase current injected into the ac grid, U?abc is the

converter’s voltage determined by the modulation, and Uabc

is the ac grid voltage. Let ~V = Vd + jVq , ~I = Id + jIq ,~Ig = Igd + jIgq , ~U? = U?d + jU?q , and ~U = Ud + jUq berespectively the space vectors of Vabc, Iabc, Igabc, U?

abc,and Uabc in the controller’s dq-frame.

3

By using complex transfer functions, the dynamic equationsof the LCL can be formulated as [22]

~U? − ~V = (sLF + jωLF ) ~I , (1)

~I − ~Ig = (sCF + jωCF ) ~V , (2)

~V − ~U = (sLg + jωLg) ~Ig , (3)

where LF is the converter-side inductance, Lg is the grid-sideinductance, and CF is the capacitance of the LCL filter.

The dynamic equation of the current loop is

~U? = PICC(s)(~Iref − ~I

)+ jωLF ~I + fVF(s)~V , (4)

where PICC(s) is the transfer function of the PI regulator,~Iref = ~Irefd +j~Irefq is the current reference vector which comesfrom the power control loops, fVF(s) = KVF/(TVFs + 1)is a first-order filter which eliminates the high-frequencycomponents of the voltage feed-forward signals.

The power control loops can be formulated as

Irefd = PIPC(s)(P ref − PE

),

Irefq = PIQC(s)(QE −Qref

),

(5)

where PIPC(s) and PIQC(s) are the transfer functions of thePI regulators, P ref and Qref are the power reference values,PE and QE are the active and reactive powers calculated by

PE = VdIgd + VqIgq ,

QE = VqIgd − VdIgq.(6)

The converter is synchronized with the grid via the PLL,which determines the angle of the controller’s dq-frame as

θ =ω

s=

1

sPIPLL(s)Vq , (7)

where PIPLL(s) is the transfer function of the PI regulator, θand ω are respectively the angle and angular frequency of thecontroller’s dq-frame.

The voltage and current vectors can be transformed intothe global dq-frame (whose angular frequency is ωg = ω0 =100πrad/s and the angle is θg) as

~I ′g = I ′gd + jI ′gq = ~Igejδ ,

~U ′ = U ′d + jU ′q = ~Uejδ ,(8)

where δ = θ − θg , ~I ′g and ~U ′ are respectively the grid-sidecurrent and the grid voltage in the global dq-frame.

We note that the above equations based on space vectorsand complex transfer functions can be transformed into theirmatrix forms by considering [22]

yd + jyq = [Gd(s) + jGq(s)] (xd + jxq)

⇔[ydyq

]=

[Gd(s) −Gq(s)Gq(s) Gd(s)

] [xdxq

].

(9)

Then, by linearizing (1)-(8) and combining them, we obtainthe admittance model of PLL-based converters denoted by

−[

∆I ′gd∆I ′gq

]= YPLL(s)

[∆U ′d∆U ′q

], (10)

where ∆ denotes the perturbed value of a variable, YPLL(s)

PLL-based

PLL-based

PLL-based

Grid-Forming

Grid-Forming

AC Grid

1

2

n

n+1

n+m

n+m+1

n+m+2

n+m+3

n+m+k‐2

n+m+k‐1

n+m+k

n+m+k+1Power Network

Fig. 2. Illustration of a multi-converter system.

is the 2×2 admittance matrix. For the detailed derivation andexpression of YPLL(s) we refer to [8], [21], etc.

B. Admittance Modeling of Grid-Forming Converters

Different from PLL-based converters, the current referencevector ~Iref of in the grid-forming controller in Fig. 1 comesfrom the voltage loop as

~Iref = PIVC(s)(~V ref − ~V ) + jωCF ~V + kF ~Ig , (11)

where PIVC(s) is the transfer function of the PI regulator,~V ref = 1 + j0 is the voltage reference vector, and kF is thecurrent feed-forward coefficient. Note that ~V ref can also beprovided by a reactive power control loop if needed.

Moreover, the converter in Fig. 1 achieves grid synchroniza-tion by emulating the swing equation as{

sθ = ω ,Jsω = P0 − PE −D(ω − ω0) ,

(12)

which determines the angle and frequency of the controller’sdq-frame.

By combining the linearized form of (1)-(4), (6), (8), (11),and (12), we derive the admittance model of grid-formingconverters (VSMs in this paper) as

−[

∆I ′gd∆I ′gq

]= YGF(s)

[∆U ′d∆U ′q

],

YGF(s) = −

[Y (s) 0

Y 2(s)V 2d0−(I

′gd0)

2

Js2+Ds Y (s)

],

(13)

where YGF(s) is the 2 × 2 admittance matrix, the subscript0 denotes the steady-state value of a variable, and

Y (s) = GVF(s)+GCC(s)PIVC(s)+sCF+jωCF [1−GCC(s)]kFGCC(s)−1 ,

GCC(s) = PICC(s)sLF+PICC(s) ,

GVF(s) = 1−fVF(s)sLF+PICC(s) .

(14)We note that generally Y (s) has large magnitudes due tothe voltage control, or in other words, the voltage control issupposed to have limited output impedance with appropriatecontrol designs [23].

C. Closed-Loop Dynamics of Multi-Converter Systems

Now we are ready to formulate the closed-loop dynamicsof multi-converter systems. Consider a multi-converter systemas depicted in Fig. 2 (the network topology is for illstration),

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which contains n PLL-based converters (connected Nodes 1 ∼n), m grid-forming converters (connected to Nodes n + 1 ∼n+m), k interior nodes (Nodes n+m+1 ∼ n+m+k), andan infinite bus (Node n+m+ k + 1). The interior nodes arenot directly connected to the converters and will be eliminatedthrough Kron reduction by assuming that the currents injectedinto these nodes remain constant [24]. The infinite bus can beconsidered an “grounded” node in small-signal modeling [21].

For a transmission line that connects Node i and Node j,its dynamic equation can be expressed as[

∆I ′d,ij∆I ′q,ij

]= BijF (s)

[∆U ′d,i −∆U ′d,j∆U ′q,i −∆U ′q,j

],

F (s) =1

(s+ τ)2/ω0 + ω0

[s+ τ ω0

−ω0 s+ τ

],

(15)

where[

∆I ′d,ij∆I ′q,ij

]is the current from i to j and

[∆U ′d,i∆U ′q,i

]is

the voltage at i (in the global dq-frame), Bij = 1/(Lijω0) isthe susceptance between i and j, and τ is the identical R/Lratio of all the lines.

Let Q ∈ R(n+m+k)×(n+m+k) be the grounded Laplacianmatrix of the electrical network which can be calculated by

Qij = −Bij(i 6= j) and Qii =n+m+k∑j=1,i6=j

Bij +Bi,n+m+k+1. By

performing Kron reduction, we eliminate the interior nodesand obtained the Kron-reduced Laplacian matrix as

Qred = Q1 −Q2Q−14 Q3 , (16)

where Q1 ∈ R(m+n)×(m+n), Q2 ∈ R(m+n)×k, Q3 ∈Rk×(m+n), Q4 ∈ Rk×k, Q =

[Q1 Q2

Q3 Q4

].

Then, similar to [20], [21], [25], it can be deduced from(15) and (16) that the network dynamics can be formulated as

∆I′g = Qred ⊗ F (s)∆U′g , (17)

where ∆I′g ∈ R2m+2n is the stacked current vector of then + m converters (in the global dq-frame) injected into thenetwork, ∆U′g ∈ R2m+2n is the stacked voltage vector of then+m converters, and ⊗ denotes the Kronecker product.

On the other hand, considering (10) and (13), the dynam-ics of the n PLL-based converters and the m grid-formingconverters can be formulated as

−∆I′g = S⊗ I2[In ⊗YPLL(s) 0

0 Im ⊗YGF(s)

]∆U′g ,

(18)where In ∈ Rn×n denotes the identity matrix, 0 denotes thezero matrix with a proper dimension, S ∈ R(n+m)×(n+m)

is a diagonal matrix whose ith diagonal element Si is thecapacity ratio of the ith node’s rated capacity to the basecapacity of per-unit calculation (we use the same base valueswhen performing per-unit calculations for all the converters).In the above formulation, for simplicity we ignore the powerangle differences of the converters (which are generally smallenough [20]) such that the admittance matrices are the samewhen applying the same control scheme.

We remark that the above modeling is based on dividingthe system into two parts, namely, the power network and

PLL‐based ConverterEq. (10)

PLL‐based ConverterEq. (10)

Network

Grid‐forming ConverterEq. (13)

Grid‐forming ConverterEq. (13)

1

n

n+1

n+m

Network DynamicsConverters’Dynamics

Eq. (18) Eq. (17)

Fig. 3. Modeling of the converters’ dynamics and the network dynamics.

PLL-based

PLL-based

PLL-based

Grid-Forming

Grid-Forming

AC Grid

1

2

n

n+1

n+m

n+m+1

n+m+2

n+m+3

n+m+k‐2

n+m+k‐1

n+m+k

n+m+k+1Power Network

Fig. 4. Closed-loop dynamics of multi-converter systems.

the combination of all the converters, and then deriving thetransfer function matrices of these two parts, as illustrated inFig. 3. The converters’ dynamics are reflected in (18), whichhas a block diagonal structure and each block is either (10)or (13) as it represents the converter’s dynamics. The networkdynamics are reflected in (17), which is derived based on theKron-reduced Laplacian matrix of the power network, similarto the derivations in [21] and [25].

By combining (17) and (18), we obtain the closed-loopdynamics of the multi-converter system as shown in Fig. 4.We note that the block diagonal structure of the converters’dynamics facilitates the analysis of the impacts of grid-formingconverters, which will be elaborated upon in the next section.

III. IMPACTS OF GRID-FORMING CONVERTERS

Based on the above system modeling, in this section wefocus on analyzing the impacts of grid-forming converters onthe small-signal stability of the system integrated with PLL-based converters. For simplicity in the following analysis, weassume m = 1, i.e., only one grid-forming converter is placed.We will show that the case of multiple grid-forming converterscan be analyzed by repeating our analysis on m = 1.

It can be deduced from Fig. 4 that the characteristic equation(with m = 1) of the system is

0 = det

([SB ⊗YPLL(s) 0

0 YGF(s)

]+Qred ⊗ F (s)

)= det [CGF(s)] det{SB ⊗ [YPLL(s)F−1(s)] +Qn ⊗ I2−Qn,1 ⊗ F (s) C−1GF(s) Q1,n ⊗ I2} det [In ⊗ F (s)] ,

(19)

where det(·) denotes the determinant, SB is the capacityratio matrix for PLL converters (without loss of generality,

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we assume the capacity of the grid-forming converter equalsthe base capacity such that its capacity ratio is 1, i.e., S =diag{SB, 1}), CGF(s) = YGF(s)+Qn+1F (s), Qn ∈ Rn×n,Q1,n ∈ R1×n, Qn,1 ∈ Rn×1, and Qn+1 ∈ R satisfy

Qred =

[Qn Qn,1Q1,n Qn+1

]. (20)

Notice that det[CGF(s)] is in fact the closed-loop charac-teristic equation of a grid-forming converter system connectedto the infinite bus via a susceptance Qn+1, which should bedesigned as stable. Moreover, as mentioned before, generallyYGF(s) has large magnitudes due to the grid-forming design,so CGF(s) ≈ YGF(s). Hence, it can be deduced from(19) that we can instead focus on the following characteristicequation for evaluating the system stability

0 = det{In ⊗ [YPLL(s)F−1(s)] + (S−1B Qn)⊗ I2−(S−1B Qn,1Q1,n)⊗ [F (s)Y−1GF(s)]} .

(21)

According to the matrix perturbation theory [26], (21) canbe reformulated as

0 = [x⊗ a(s)]>{In ⊗ [YPLL(s)F−1(s)] + (S−1B Qn)⊗ I2− (S−1B Qn,1Q1,n)⊗ [F (s)Y−1GF(s)]}[y ⊗ b(s)]+ o(‖F (s)Y−1GF(s)‖2) ,

(22)

where x and y are the left and right eigenvectors correspondingto the smallest eigenvalue of S−1B Qn (denoted by λ′1) withnormalization x>y = 1, a(s) and b(s) are the normalizedleft and right eigenvectors of YPLL(s)F−1(s) which satisfya>(s)[YPLL(s)F−1(s)]b(s) = γ(s) pertinent to the dominantpoles of the system, o(‖F (s)Y−1GF(s)‖2) is the second-orderapproximation error [26, Theorem 2.3]. By ignoring thisapproximation error it can be further derived that

0 ≈ [x⊗ a(s)]>{In ⊗ [YPLL(s)F−1(s)] + (S−1B Qn)⊗ I2− (S−1B Qn,1Q1,n)⊗ [F (s)Y−1GF(s)]}[y ⊗ b(s)]

= [a>(s)YPLL(s)F−1(s)b(s)] + (x>S−1B Qny)

− (x>S−1B Qn,1Q1,ny)[a>(s)F (s)Y−1GF(s)b(s)]

= γ(s) + λ′1 + ∆λ(s) ,(23)

where

∆λ(s) = −(x>S−1B Qn,1Q1,ny)[a>(s)F (s)Y−1GF(s)b(s)] .(24)

Eq. (23) describes how the grid-forming converter interactswith PLL-based converters via the power network and thusaffects the dominant poles of the whole system. In the fol-lowing, we introduce two propositions in order to provide aintelligible interpretation on (23).

Proposition III.1 (Single Converter System). The dominantpoles of a PLL-based converter that is connected to an infinitebus (with line susceptance being λ1) are determined by

0 = γ(s) + λ1 , (25)

where the definition of γ(s) has been given above.

Proof. It can be deduced from (17) and (18) that the charac-teristic equation of such a system (n = 1,m = 0) is

0 = det[YPLL(s) + λ1F (s)]

= det[YPLL(s)F−1(s) + λ1I2]det[F (s)] .(26)

According to the definitions of a(s) and b(s) as given above,the dominant poles can be obtained by solving

0 = a>(s)[YPLL(s)F−1(s) + λ1I2]b(s) = γ(s) + λ1 , (27)

which concludes the proof.

The following proposition will show that (25) also de-termines the dominant poles of multi-PLL-based-convertersystems under certain circumstances.

Proposition III.2 (Multi-Converter system [20]). Considera power network (with the Kron-reduced Laplacian matrixQred ∈ Rp×p) which interconnects p (p ∈ Z+) PLL-basedconverters (with the capacity ratio matrix being S). Thedominant poles of this system can be obtained by solving(25) (wherein λ1 is the smallest eigenvalue of S−1Qred, alsodefined as the generalized short-circuit ratio (gSCR) in [20]to evaluate the power grid strength).

Proof. It can be deduced from (17) and (18) that the charac-teristic equation of such a system (m = 0) is

0 = det[Ip ⊗YPLL(s) + (S−1Qred)⊗ F (s)] , (28)

which is equivalent to

0 = det{(T−1 ⊗ I2)[Ip ⊗YPLL(s)

+ S−1Qred ⊗ F (s)](T ⊗ I2)}= det[Ip ⊗YPLL(s) + Λp ⊗ F (s)]

=

p∏i=1

det[YPLL(s) + λiF (s)] ,

(29)

where T ∈ Rp×p diagonalizes S−1Qred as T−1S−1QredT =Λp = diag{λ1, λ2, ..., λp} (λ1 < λ2 < ... < λp).

Hence, the system that has p PLL-based converters can bedecoupled into p subsystems as indicated by (29). Moreover,it can be deduced that the dominant poles is determined by theweakest system, which is in fact a single PLL-based converterconnected to an infinite bus with susceptance λ1 [20], [21].Then, by Proposition III.1, the dominant poles of this weakestsystem are determined by (25), which concludes the proof.

Remark 1 (Power Grid Strength). The value of λ1 in (25)reflects the network connectivity and thus the power gridstrength, also defined in [20] as the generalized short-circuitratio (gSCR) of the system. Moreover, λ1 determines the sta-bility margin of PLL-based multi-converter systems (a largerλ1 indicates a larger stability margin), and a low λ1 may giverise to PLL-induced instabilities [21].

Based on the above results, it can be deduced by comparing(23) and (25) that placing grid-forming converters is equivalentto the change of power grid strength (i.e., gSCR) from λ1to λ′1 + ∆λ(s). We note that when the capacity of the grid-forming converter is sufficiently large (i.e., λ′1 is relatively

6

large in such cases), ∆λ(s) in (24) has magnitudes smallenough to be ignored in the frequency range of control effectsbecause YGF(s) has large magnitudes due to the voltagecontrol. We will verify this issue when conducting case studies.Hence, we have the following statement.

Remark 2 (Grid-Forming Control and Power Grid Strength).The main impact of placing grid-forming converters (i.e.,changing the control scheme of a converter from PLL-basedcontrol to grid-forming control) can be interpreted as changingthe power grid strength (i.e., gSCR) from λ1 to λ′1.

For clarity, we recall that we consider a multi-convertersystem which contains n PLL-based converters and one grid-forming converter as in (19), and λ1 is the smallest eigenvalueof S−1Qred ∈ R(n+1)×(n+1) while λ′1 is the smallest eigen-value of S−1B Qn (Qn is defined in (20)). Note that according to[20, Lemma 1], there holds λ′1 > 0 and λ1 > 0. The followinglemma is given to compare λ′1 with λ1.

Lemma III.3. Consider two weighted Kron-reduced Lapla-cian matrices S−1Qred ∈ R(n+1)×(n+1) and S−1B Qn ∈ Rn×n,where S−1B Qn is a submatrix of S−1Qred considering (20) andS = diag{SB, 1}. It holds that λ1 < λ′1, where λ1 and λ′1 arerespectively the smallest eigenvalues of S−1Qred and S−1B Qn.

Proof. The claimed result can be obtained by considering theinterlacing theorem in [27, Theorem 4.3.8].

With Remark 2 and Lemma III.3, it can be further deducedthat the placement of a grid-forming converter is equivalentto increasing the power network strength (characterized bygSCR) from λ1 to λ′1. Moreover, since λ′1 is the smallesteigenvalue obtained by deleting the (n+1)th row and (n+1)thcolumn of S−1Qred, the case of multiple grid-forming con-verters can be analyzed by repeating the above analysis andcalculating the smallest eigenvalue of S−1Qred after deletingthe rows and columns corresponding to the nodes of grid-forming converters. Note that if the grid-forming converter islabelled as the ith converter in (19), then λ′1 should be thesmallest eigenvalue obtained by deleting the ith row and ithcolumn of S−1Qred. We summarize the above finding in thefollowing proposition.

Proposition III.4 (Placement of Grid-Forming Converters).Consider a multi-converter system whose closed-loop dynam-ics are described by Fig. 4 (the number of grid-formingconverters could be 0), the PLL-induced small-signal stabilityof the system is improved by changing the control scheme ofany PLL-based converter to grid-forming control.

Proof. The above statement rigorously follows from Re-mark 1, Remark 2 and Lemma III.3.

So far, we have theoretically explained how the placement ofgrid-forming converters enhances the power grid strength andthus the (PLL-induced) small-signal stability. One remainingquestion is how to optimally place the grid-forming convertersto improve the stability, which will be explored in the follow-ing section.

IV. OPTIMAL PLACEMENT OF GRID-FORMINGCONVERTERS

This section investigates the problem of how to optimallyplace grid-forming converters to improve the system stability.To be specific, we consider a power system that is integratedwith p PLL-based converters, and q (q < p) of them willbe changed to use grid-forming control instead of PLL-based control. Define a symmetric weighted Laplacian matrixL = S−

12QredS

− 12 , which shares the same eigenvalues with

S−1Qred because they are similar matrices. According to theanalysis and results in the previous section, the optimal loca-tions to place these grid-forming converters can be obtainedby solving the following optimization problem

maxI⊂V

λmin[RV\I(L)] , (30)

where V = {1, 2, ..., p} is the set that denotes the converternodes, I ⊂ V is the set of the q nodes to place the grid-formingconverters which will be obtained by solving the optimizationproblem, λmin(·) denotes the smallest eigenvalue of a sym-metric matrix, L ∈ Rp×p is the symmetric weighted Laplacianmatrix to represent the power network and the capacities of theconverters, and RV\I(L) denotes the remaining matrix afterdeleting the rows and columns included in the set I. That isto say, (30) aims at selecting the locations of grid-formingconverters to equivalently enhance the power grid strength(i.e., gSCR) and thus the system stability.

Since (30) is a combinatorial optimization problem, one cansolve it by simply enumerating all possible results, which isdoable if the system is small-scale. However, if the systemis integrated with large-scale converters, the computationalburden of the enumeration will be unacceptable. As a remedy,we propose a greedy method to obtain a suboptimal solutionfor the placement of grid-forming converters, and we will showby case studies that this suboptimal solution is identical tothe optimal solution in some cases and has very satisfactoryperformance to be used in practice. To be specific, we considernow the following iterative optimization problem that will besolved for q times

maxαi∈V

λmin[RV\αi(L[i])] , (31)

where i ∈ {1, 2, ..., q} denotes the iteration number, αi ∈ Vis the node to place a grid-forming converter that will bedetermined by the ith iteration, RV\αi(·) is a function todelete the row and the column that are corresponding tothe Node αi defined in L, L[i+1] = RV\αi(L[i]), withL[1] = L ∈ Rp×p. Hence, by iteratively solving (31) for qtimes using a greedy heuristic, the locations for placing the qgrid-forming converters can be obtained. We summarize thesolving process in the following.

In Step 2, a trivial way to obtain the solution of (31) (withoutloss of generality we assume i = 1 here) is to enumeratethe smallest eigenvalues of RV\α1

(L) with α1 = 1, 2, ..., pand then pick the largest one. This method requires to doeigenvalue calculation for p times, which may also have highcomputational burden in large-scale systems. In fact, deletingthe α1th row and the α1th column can be interpreted as

7

Algorithm 1 Greedy Solution to (30)Input: Kron-reduced Laplacian matrix: Qred ∈ Rp×p, ca-pacity ratio matrix: S ∈ Rp×p, the number of grid-formingconverters to be placed: q1) Initialize i = 1, L[i]|i=1 = L = S−

12QredS

− 12 .

2) Solve (31) for αi.3) Calculate L[i+1] = RV\αi(L[i]).4) Iterate through Steps 2-3 for i = 2, 3, ..., q.

Output: The (sub)optimal locations for the q grid-formingconverters, i.e., αi (i ∈ {1, 2, ..., q}).

connecting Node α1 directly to the grounded node (i.e., theinfinite bus). Hence, the solution of (31) can be consideredas the “farthest” node from the grounded node, consideringthat connecting the farthest node to the grounded node willincrease the network connectivity (reflected by the smallesteigenvalue [24], [28]) to the most extent.

This farthest node can be approximately located by checkingthe participation factors of the nodes on the smallest eigen-value of L (i.e., λ1). According to [21], the participation factorof Node i on λ1 equals the sensitivity of λ1 to the self-susceptance of Node i (included in Lii), formulated as

p1,i =∂λ1∂Lii

= v1,iu1,i (32)

where p1,i denotes the participation factor of Node i on λ1, Liidenotes the ith diagonal entry of L, v1 ∈ Rp and u1 ∈ Rp arerespectively the left and right eigenvectors corresponding to λ1(i.e., v>1 Lu1 = λ1), v1,i and u1,i are the ith entries of v1 andu1. Then, we pick the node that has the largest participationfactor among all the p nodes, which can be considered asan suboptimal solution to (31) (we will showcase that thissuboptimal solution leads to very satisfactory results and insome case it would be identical to the optimal solution). Notethat this method only requires to do the eigendecompositiononce, which is much more efficient than the enumerationmethod.

V. SIMULATION RESULTS

In this section, we provide detailed simulation results toillustrate the validity of our analysis and the effectiveness ofthe proposed algorithm for the optimal placement of grid-forming converters.

A. Case studies on a four-converter test system

Without loss of generality, we consider a two-area four-converter test system as shown in Fig. 5. Also note that ouranalysis and conclusions in this paper are general to othermulti-converter systems with arbitrary network topology. InFig. 5, where Nodes 1 ∼ 4 are converter nodes, Nodes 5 ∼ 9are interior nodes (which will be eliminated via Kron reductionas in (16)), and Node 10 is the infinite bus (considered as thegrounded node in the small-signal modeling). For simplicity,the capacities of the converters are assumed to be the same,i.e., S is an identity matrix and thus L = Qred. The main

Load 1 Load 2

1

2

56 7 8

93

4

Converter 1

Converter 2

Converter 3

Converter 4

Area 1 Area 2

10

Infinite Bus

Fig. 5. A two-area test system.

Fig. 6. The Bode diagram of ∆λ(s) with the parameters in the Appendix.

parameters of the systems are given in the Appendix, and theKron-reduced Laplacian matrix is calculated as

Qred = L =

[8.07 −3.86 −0.20 −0.27−3.86 12.27 −0.41 −0.54−0.20 −0.41 4.04 −1.27−0.27 −0.54 −1.27 4.9675

],

with the smallest eigenvalue being λ1 = 3.00.To begin with, we give the Bode diagram of ∆λ(s) (as

defined in (23) and (24)) to verify our claim in Section IIIthat the magnitudes of ∆λ(s) are small enough to be ignored.It can be seen from Fig. 6 that in the frequency rangeof interest (within 1Hz ∼ 200Hz regarding PLL-inducedinstabilities), the magnitudes of ∆λ(s) are around 0.01, whichare small enough to be ignored. Hence, ∆λ(s) can be ignoredwhen analyzing PLL-induced instabilities, which leads to theclaimed Remark 2 and the subsequent results in Section IV.

Based on the proposed method in Section IV, we explorenow the optimal placement of grid-forming converters in thetwo-area test system. Assume that two (out of four) converterswill be changed from PLL-based control to grid-formingcontrol, i.e., q = 2. According to the analysis in Section IV,the optimal locations of these two grid-forming converters canbe obtained by solving combinatorial optimization problem in(30). In Table I, we enumerate all the possible combinations.Obviously, the optimal locations are {3, 4}, that is, by chang-ing the control schemes of Converter 3 and Converter 4 fromPLL-based control to grid-forming control, the (PLL-induced)stability margin will be increased to the most extent.

TABLE ITHE VALUE OF λmin[RV\I(L)] WITH DIFFERENT I

I {1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4}

λmin[RV\I(L)] 3.1504 4.9270 4.0239 4.9437 4.0338 5.7711

8

On the other hand, solving (30) through enumeration willresult in unacceptable computational burden in large-scale sys-tems. To alleviate this problem, the greedy solution obtainedby Algorithm 1 can be used.

In the following, we still consider the two-area system inFig. 5 and q = 2, and use Algorithm 1 to obtain the twooptimal locations. For a first step, we solve (31) for the firstlocation α1 (i.e., i = 1 and L[1] = L). Table II enumeratesall the possible results, and obviously, the solution is α1 =3. As we discussed in Section IV, another convenient (andcomputationally efficient) way to obtain α1 is to check theparticipation factors, as shown in (32). Table III shows theparticipation factors of the converter nodes on the smallesteigenvalue of L[1]. It can be seen that Node 3 has the largestparticipation factor and thus α1 = 3, which is consistent withthe enumeration results in Table II.

TABLE IITHE VALUE OF λmin[RV\α1

(L[1])] WITH DIFFERENT α1

α1 1 2 3 4

λmin[RV\α1(L[1])] 3.1046 3.1292 4.7091 3.9570

TABLE IIIPARTICIPATION FACTORS OF L[1]

α1 1 2 3 4

Participation factor 0.0284 0.0193 0.6231 0.3292

After selecting the first optimal location α1, we have

L[2] = RV\α1(L[1]) =

8.07 −3.86 −0.27−3.86 12.27 −0.54−0.27 −0.54 4.9675

.Note that the row 3 and column 3 of the above matrix arecorresponding to the Converter 4 in Fig. 5. Then, we solve(31) for the second location α2. Table IV enumerates all theremaining converter nodes, and it can be seen that the optimalsolution is α2 = 4. To illustrate that this optimal locationcan also simply obtained by checking the participation factorsof the remaining converters nodes on the smallest eigenvalue,Table V gives the participation factors of the Nodes 1, 2 and 4,which indicates that Converter 4 has the largest participationfactor and thus the second optimal location is α2 = 4,consistent with the result obtained in Table IV.

TABLE IVTHE VALUE OF λmin[RV\α2

(L[2])] WITH DIFFERENT α2

α2 1 2 4

λmin[RV\α2(L[2])] 4.9270 4.9437 5.7711

TABLE VPARTICIPATION FACTORS OF L[2]

α2 1 2 4

Participation factor 0.1283 0.0614 0.8103

It can be seen from the above results that the greedysolution obtained by employing Algorithm 1 (i.e., α1 = 3and α2 = 4) is fully consistent with the optimal solution

0.0 0.2 0.4 0.6 0.8 1.0

0.95

1.00

1.05

0.0 0.2 0.4 0.6 0.8 1.0

0.95

1.00

1.05

0.0 0.2 0.4 0.6 0.8 1.0

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0

0.8

1.0

1.2

Time (s) Time (s)

Time (s) Time (s)

Fig. 7. Time-domain responses of the four converters when placing grid-forming controls at different spots (an overload event occurs at 0.2s and iscleared after 0.02s). — : Case 1 (All the converters use PLL-based control);— : Case 2 (Grid-forming control is applied in Converters 1 and 2); — :Case 3 (Grid-forming control is applied in Converters 3 and 4).

by directly solving (30) (i.e., I = {3, 4}), which verifies theeffectiveness of the greedy algorithm. Although the greedyalgorihm is not guaranteed to reach the optimal solution of(30), i.e., sometimes it would lead to a suboptimal solution, itcan generally satisfy the practical expectation and the obtainedlocations to place the grid-forming converters can significantlyimprove the system stability. We consider the development ofrigorously optimal and computationally efficient solutions to(30) as future works.

In the following, we provide time-domain simulation resultsand eigenvalue analysis on the two-area test system in Fig. 5to further verify the effectiveness of the optimal placementof grid-forming converters. Fig. 7 shows the converters’ re-sponses under different control settings (an overload eventoccurs at 0.2s and is cleared after 0.02s). When all the fourconverters apply PLL-based control (Case 1), the system isoscillating (critically stable) which is caused by the interactionbetween PLLs and weak grid condictions. When applyinggrid-forming control in Converters 1 and 2 (Case 2), thedamping ratio is improved and the oscillation is restrained,that is, the placements of grid-forming converters improve thesystem stability. Moreover, by applying grid-forming controlin Converters 3 and 4 (Case 3), the system has very highdamping ratio, indicating a satisfactory stability margin, whichis consistent with the above results on the optimal placementsof grid-forming converters.

Fig. 8 plots the dominant poles of the four-converter test sys-tem corresponding to the above control settings (Cases 1 ∼ 3).Compared to Case 1 (which has very low damping ratio), thedamping ratio is increased to about 0.05 when applying grid-forming control in Converters 1 and 2, and is increased toabout 0.4 when applying grid-forming control in Converters 3and 4. The above eigenvalue results are fully aligned withthe time-domain responses in Fig. 7, which again verifies thevalidity of the analysis in this paper and the effectivenessof the proposed algorithm for optimally placing grid-formingconverters.

9

-60 -50 -40 -30 -20 -10 0-200

-100

0

100

200

-60 -50 -40 -30 -20 -10 0-200

-100

0

100

2000.05

0.05

0.1

0.1

0.2

0.2

0.4

0.4

0.8

0.8

Case 1 Case 2 Case 3

Real Axis

Imaginary Axis

Fig. 8. Dominant eigenvalues of the of the four-converter test system whenplacing grid-forming controls at different spots.

Infinite Bus

1

2

3

4

5

67 8

9 3932

31

30

33

25

28

2919

20

21

22

23

24

15

16

17

18 14

10

1112

13

34

2726

36

35

37

38

Converter 9

Converter 2

Converter 3

Converter 5

Converter 4

Converter 1

Converter 7

Converter 8

Converter 6

Fig. 9. A nine-converter test system.

B. Case studies on a nine-converter test system

In the following, we consider a nine-converter system in or-der to further test the effectiveness of the proposed algorithm.As shown in Fig. 9, The nine converters are interconnectedvia a 39-bus network, where Bus 39 is connected to aninfinite bus. Note that the topology of the this network isactually the same as the standard England IEEE 39-bus system,and the network parameters are the same as those in [21].The converter parameters are basically the same as thoselisted in Appendix A (with the PLL parameters changed to{147, 10773} in order to provide a critically stable base case).

The smallest eigenvalue of the Kron-reduced Laplacianmatrix of the network (with the network parameters givenin [21] and the capacities of the converters are identical)is λ1 = 3.31. Then, we use the algorithms presented inSection IV to choose two converters to be operated as grid-forming converters. To find the globally optimal solution, wesolve the optimization problem in (30) by enumerating all thepossible results and pick the best solution, and the obtainedoptimal set is I = {1, 4}. The smallest eigenvalue becomesλmin[RV\I(L)] = 14.43 with the optimal set I = {1, 4},which indicates that grid strength is dramatically increased byapplying grid-forming controls in Converter 1 and Converter 4.

For comparison, we also use Algorithm 1 (wherein thesolution to (31) is obtained by checking the participationfactors) to calculate the suboptimal solution to (30), and theobtained locations are α1 = 9 and α2 = 8. The smallesteigenvalue becomes λmin[RV\{α1,α2}(L)] = 12.03 with thissuboptimal set {α1, α2} = {8, 9}. It can be seen that althoughthis solution is a suboptimal one, it also effectively increasesthe grid strength, which validates the effectiveness of the

0.0 0.2 0.4 0.6 0.8 1.0

0.96

0.98

1.00

1.02

1.04

0.0 0.2 0.4 0.6 0.8 1.0

0.96

0.98

1.00

1.02

1.04

0.0 0.2 0.4 0.6 0.8 1.0

0.96

0.98

1.00

1.02

1.04

0.0 0.2 0.4 0.6 0.8 1.0

0.96

0.98

1.00

1.02

1.04

0.0 0.2 0.4 0.6 0.8 1.0

0.96

0.98

1.00

1.02

1.04

0.0 0.2 0.4 0.6 0.8 1.0

0.96

0.98

1.00

1.02

1.04

Converter 1

Converter 2

Converter 3

Converter 4

Converter 5

Converter 6

Converter 7

Converter 8

Converter 9

Time (s) Time (s)

Time (s) Time (s)

Time (s) Time (s)

(a)

(b)

(c)

Converters 1~5 Converters 6~9

Converters 1~5 Converters 6~9

Converters 1~5 Converters 6~9

Fig. 10. Time-domain responses of the nine converters when placing grid-forming controls at different spots (an overload event occurs at 0.2s and iscleared after 0.02s). (a) All the converters use PLL-based control. (b) Grid-forming control is applied in Converters 1 and 4. (c) Grid-forming control isapplied in Converters 8 and 9.

proposed algorithm.Fig. 10 shows the active power responses of the nine

converters under different control settings (an overload eventoccurs at 0.2s and is cleared after 0.02s). In Fig. 10 (a),all the converters apply PLL-based control and the systemhas very low damping ratio (low PLL-induced small-signalstability margin), which is caused by the interaction betweenPLL-base converters and weak grid conditions [20], [21].By comparison, it can be seen from Fig. 10 (b) that whenapplying grid-forming controls in Converters 1 and 4 (whichis the optimal solution as discussed above), the multi-convertersystem has high damping ratio and superior performance. InFig. 10 (c), grid-forming controls are applied in Converters 8and 9 (which is the suboptimal solution to (30) obtained byAlgorithm 1 as discussed above), and it can be seen that thesystem also has superior damping performance, which againverifies the effectiveness of Algorithm 1. The above simulationresults are again consistent with the analysis in this paperwhich shows the placements of grid-forming converters caneffectively improve the (PLL-induced) small-signal stabilityof multi-converter systems.

VI. CONCLUSIONS AND DISCUSSIONS

This paper investigated the impacts of grid-forming con-verters on the small-signal stability of power systems that are

10

integrated with PLL-based converters. By deriving the small-signal model of multi-converter systems, we explicitly revealedthat the placement of grid-forming converters is equivalentto enhancing the power grid strength characterized by thegeneralized short-circuit ratio (i.e., the smallest eigenvalue ofthe weighted and Kron-reduced Laplacian matrix), therebyimproving the (PLL-induced) small-signal stability. Our anal-yses in this paper focus on the characterization of power gridstrength and thus provide a convenient way to study howthe locations of grid-forming converters influence the systemstability. On this basis, we investigated the optimal placementof grid-forming converters for improving the system stability.After explicitly formulating the optimization problem, weelaborated on how to solve it in a computationally efficientfashion such that it can be used in large-scale systems. Simula-tions based on a high-fidelity two-area test system and a high-fidelity 39-bus test system verify that placing grid-formingconverters in proper locations can significantly improve thesmall-signal stability of PLL-integrated power systems.

Our work in this paper is based on characterizing the (PLL-induced) small-signal stability of multi-converter systems fromthe perspective of power grid strength, which provides aconvenient way to rigorously show that the placements ofgrid-forming converters enhance the overall system stability(as discussed in Proposition III.4). Moreover, by focusing onthe power grid strength, the optimal locations to place grid-forming converters can be conveniently determined by solvingan optimization problem which is related only to the networkparameters and the capacities of the converters. In this way, weavoid directly optimizing the damping ratio of the dominantpoles of the multi-converter system, which is intractable as itwould require to derive the state matrix of the whole system.

Based on our theoretical analysis in this paper, possiblefuture works can include the optimal placements of grid-forming converters considering both small-signal stabilityand frequency stability, and also stability-constrained networkplanning of multi-converter systems.

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APPENDIX ASYSTEM PARAMETERS

See Table VI.

11

TABLE VIPARAMETERS OF THE TWO-AREA TEST SYSTEM

Base Values for Per-unit Calculationfbase = 50Hz ωbase = 2πfbase Ubase = 690V Sbase = 1.5MVA

Power Network Parameters (per-unit values)B15 = 0.10 B25 = 0.05 B39 = 0.2 B49 = 0.15

B56 = 0.015 B67 = 0.015 B78 = 0.035 B7,10 = 0.018

B89 = 0.02 τ = 0.1 C1 = 0.05 C2 = 0.05

Load 1: 0.5 Load 2: 0.5LCL Parameters of the Converter (per-unit values)

LF = 0.05 CF = 0.05 Lg = 0.06

Parameters of the PLL-Based Control (per-unit values)PI parameters of the current control Loop: 0.3, 10PI parameters of the active power control Loop: 0.5, 40PI parameters of the reactive power control Loop: 0.5, 40PI parameters of the PLL: 104, 5390P ref = 1.0 Qref = 0 TVF = 0.02 KVF = 1.0

Parameters of the Grid-Forming Control (per-unit values)PI parameters of the current control Loop: 0.3, 10PI parameters of the voltage control Loop: 4, 30TVF = 0.02 KVF = 1.0 J = 2 D = 50

kF = 0.1 P0 = 1.0