Optimal Placement of Virtual Inertia in Power GridsBala Kameshwar Poolla Saverio Bolognani Florian Dörfler∗

April 14, 2021

AbstractA major transition in the operation of electric power gridsis the replacement of bulk generation based on synchronousmachines by distributed generation based on low-inertiapower electronic sources. The accompanying “loss of ro-tational inertia” and the fluctuations by renewable sourcesjeopardize the system stability, as testified by the ever-growing number of frequency incidents. As a remedy, nu-merous studies demonstrate how virtual inertia can be em-ulated through various devices, but few of them addressthe question of “where” to place this inertia. It is howeverstrongly believed that the placement of virtual inertia hugelyimpacts system efficiency, as demonstrated by recent casestudies. In this article, we carry out a comprehensive anal-ysis in an attempt to address the optimal inertia placementproblem. We consider a linear network-reduced power sys-tem model along with an H2 performance metric accountingfor the network coherency. The optimal inertia placementproblem turns out to be non-convex, yet we provide a set ofclosed-form global optimality results for particular probleminstances as well as a computational approach resulting inlocally optimal solutions. We illustrate our results with athree-region power grid case study and compare our locallyoptimal solution with different placement heuristics in termsof different performance metrics.

1 IntroductionAs we retire more and more synchronous machines and re-place them by renewable sources interfaced with power elec-tronic devices, the stability of the power grid is jeopardized,which has been recognized as one of the prime concerns bytransmission system operators [1, 2]. Both in transmissiongrids as well as in microgrids, low inertia levels togetherwith variable renewable generation lead to large frequencyswings.

Not only are low levels of inertia troublesome, but par-ticularly spatially heterogeneous and time-varying inertiaprofiles can lead to destabilizing effects, as shown in an in-teresting two-area case study [3]. It is not surprising thatrotational inertia has been recognized as a key ancillary ser-vice for power system stability, and a plethora of mecha-nisms have been proposed for the emulation of virtual (or

∗This material is supported by ETH start-up funds and the SNFAssistant Professor Energy Grant #160573. B.K. Poolla, S. Bolognani,and F. Dörfler are with the Automatic Control Laboratory at the SwissFederal Institute of Technology (ETH) Zürich, Switzerland. Emails:{bpoolla,bsaverio,dorfler}@ethz.ch.

synthetic) inertia [4–6] through a variety of devices (rangingfrom wind turbine control [7] over flywheels to batteries [8]),as well as inertia monitoring schemes [9] and even inertiamarkets [10]. In this article, we pursue the questions raisedin [3] regarding the detrimental effects of spatially hetero-geneous inertia profiles, and how they can be alleviated byvirtual inertia emulation throughout the grid. In particu-lar, we are interested in the allocation problem “where tooptimally place the inertia” ?

The problem of inertia allocation has been hinted at be-fore [3], but we are aware only of the study [11] explicitlyaddressing the problem. In [11], the grid is modeled by thelinearized swing equations, and eigenvalue damping ratiosas well as transient overshoots (estimated from the systemmodes) are chosen as optimization criteria for placing vir-tual inertia and damping. The resulting problem is highlynon-convex, but a sequence of approximations led to someinsightful results.

In comparison to [11], we focus on network coherency asan alternative performance metric, that is, the amplificationof stochastic or impulsive disturbances via a quadratic per-formance index measured by the H2 norm [12]. As perfor-mance index, we choose a classic coherency criterion penal-izing angular differences and absolute frequencies, which hasrecently been popularized for consensus and synchronizationstudies [13–18] as well as in power system analysis and con-trol [19–21]. We feel that this H2 performance metric is notonly more tractable than spectral metrics, but it is also verymeaningful for the problem at hand: it measures the effectof stochastic fluctuations (caused by loads and/or variablerenewable generation) as well as impulsive events (such asfaults or deterministic frequency errors caused by markets)and quantifies their amplification by a coherency index di-rectly related to frequency volatility. Finally, in comparisonto [11], the damping or droop coefficients are not decisionvariables in our problem setup, since these are determinedby the system physics (in case of damping), the outcomeof primary reserve markets (in case of primary control), orscheduled according to cost coefficients, ratings, or grid-coderequirements [22].

The contributions of this paper are as follows. We providea comprehensive modeling and analysis framework for theinertia placement problem in power grids to optimize an H2

coherency index subject to capacity and budget constraints.The optimal inertia placement problem is characteristicallynon-convex, yet we are able to provide explicit upper andlower bounds on the performance index. Additionally, weshow that the problem admits an elegant and strictly convex

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reformulation for a performance index reflecting the cost ofprimary control which is often advocated as a remedy tolow-inertia stability issues. In this case, the optimal inertiaplacement problem reduces to a standard resource allocationproblem, where the cost of each resource is proportional tothe ratio of expected disturbance over inertia.

A similar simplification of the problem is obtained undersome reasonable assumptions on the ratio between the dis-turbance and the damping coefficient at every node. For thecase of a two-area network, a closed-form global allocationis derived, and a series of observations are discussed.

Furthermore, we develop a computational approach basedon a gradient formula that allows us to find a locally op-timal solution for large networks and arbitrary parameters.We show how the combinatorial problem of allocating a lim-ited number of inertia-emulating units can be also incorpo-rated into this numerical method via a sparsity-promotingapproach.

A detailed three-region network has been adopted as casestudy for the presentation of the proposed method. Thenumerical results are also illustrated via time-domain sim-ulations, that demonstrate how an optimization-based al-location exhibits superior performance (in different perfor-mance metrics) compared to heuristic placements, and, per-haps surprisingly, the optimal allocation also uses less effortto emulate inertia.

From the methodological point of view, this paper extendsthe H2 performance analysis of second-order consensus sys-tems to non-uniform damping, inertia, and input matrices(disturbance location). This technical contribution is es-sential for the application that we are considering, as theseparameters dictate the optimal inertia allocation in an in-tertwined way.

The remainder of this section introduces some notation.Section 2 motivates our system model and the coherencyperformance index. Section 3 presents numerical inertia al-location algorithms for general networks and provides ex-plicit results for certain instances of cost functions and prob-lem scenarios. Section 4 presents a case study on a three-region network accompanied with time-domain simulationsand a spectral analysis. Finally, Section 5 concludes thepaper.

Notation We denote the n-dimensional vectors of all onesand zeros by 1n and 0n. Given an index set I with cardi-nality |I| and a real-valued array {x1 . . . x|I|}, we denote byx ∈ R|I| the vector obtained by stacking the scalars xi andby diag{xi} the associated diagonal matrix. The vector eiis the i-th vector of the canonical basis for Rn.

2 Problem Formulation

2.1 System model

Consider a power network modeled by a graph with nodes(buses) V = {1, . . . , n} and edges (transmission lines) E ⊆V × V. We consider a small-signal version of a network-reduced power system model [23, 24], where passive loads

are eliminated via Kron reduction [25], and the network isreduced to the sources i ∈ {1,. . . , n} with dynamics

miθ̈i + diθ̇i = pin,i − pe,i , i ∈ {1, . . . , n} , (1)

where pin,i and pe,i refer to the power input and electricalpower output, respectively. If bus i is a synchronous ma-chine, then (1) describes the electromechanical swing dy-namics for the generator rotor angle θi [23, 24], mi > 0is the generator’s rotational inertia, and di > 0 accountsfor frequency damping or speed droop control (neglectingramping limits). If bus i connects to a renewable or bat-tery source interfaced with a power electronics inverter op-erated in grid-forming mode [26, 27], then θi is the volt-age phase angle, di > 0 is the droop control coefficient,and mi > 0 accounts for power measurement time con-stant [28] or arises from virtual inertia emulation througha dedicated controlled device [4–6]. Finally, the dynamics(1) may also arise from frequency-dependent or actively con-trolled frequency-responsive loads [24].

Under the assumptions of constant voltage magnitudes,purely inductive lines, and a small signal approximation,the electrical power output at the terminals is given by [24]

pe,i =∑n

j=1bij(θi − θj), i ∈ {1, . . . , n} , (2)

where bij ≥ 0 is the susceptance between nodes {i, j} ∈ E .The state space representation of the system (1)-(2) is

then[θ̇ω̇

]=

[0 I

−M−1L −M−1D

] [θω

]+

[0

M−1

]pin , (3)

where M = diag{mi} and D = diag{di} are the diagonalmatrices of inertial and damping/droop coefficients, and L =LT ∈ Rn×n is the network Laplacian (or susceptance) matrixwith off-diagonal elements lij = −bij and diagonals lii =∑nj=1,j 6=i bij . The states (θ, ω) ∈ R2n are the stacked vectors

of angles and frequencies and pin ∈ Rn is the net power input– all of which are deviation variables from nominal values.

2.2 Coherency performance metricWe consider the linear power system model (3) driven bythe inputs pin,i accounting either for faults or non-zero ini-tial values (modeled as impulses) or for random fluctuationsin renewables and loads. We are interested in the energyexpended in returning to the steady-state configuration, ex-pressed as a quadratic cost of the angle differences and fre-quency displacements:∫ ∞

0

∑n

i,j=1aij(θi(t)− θj(t))2 +

∑n

i=1si ω

2i (t) dt . (4)

Here, si are positive scalars and we assume that the nonneg-ative scalars aij = aji ≥ 0 induce a connected graph – notnecessarily identical with the power grid itself. We denoteby S the matrix diag{si}, and by N the Laplacian matrixof the graph induced by the aij . In this compact notation,N = L would be an example of short-range error penaliza-tion [13, 14], while N = In − 1n1

Tn/n penalizes long-range

errors.

Aside from consensus and synchronization studies [13–18]the coherency metric (4) has recently also been also used inpower system analysis and control [19–21]. Following the in-terpretation proposed in [19], the above metric (4) can repre-sent a generalized energy in synchronous machines. Indeed,for aij = gij (where gij are the power line conductances) andsi = mi, the metric (4) accounts for the heat losses in thegrid lines and the mechanical energy losses in the generators.

Adopting the state representation introduced in (3), theperformance metric (4) can be rewritten as the time-integral∫∞

0y(t)Ty(t)dt of the performance output

y =

[N

12 0

0 S12

]︸ ︷︷ ︸

=C

[θω

]. (5)

In order to model the localization of the disturbances inthe grid, we parametrize the input pin as

pin = T12 η, T = diag{ti}, ti ≥ 0

We therefore obtain the state space model[θ̇ω̇

]=

[0 I

−M−1L −M−1D

]︸ ︷︷ ︸

=A

[θω

]+

[0

M−1T 1/2

]︸ ︷︷ ︸

=B

η. (6)

In the following, we refer to the input/output map (5), (6)as G = (A,B,C). If the inputs ηi are Dirac impulses, then(4) measures the squared H2 norm ‖G‖2 of the system [12].

There is a number of interpretations of the H2 norm ‖G‖2of a power system [19]. The relevant ones in our context are:

1. The squared H2 norm of G measures the energy ampli-fication, i.e., the sum of L2 norms of the outputs yi(t),for unit impulses at all inputs ηi(t)=δ(t):

‖G‖22 =∑n

i=1

∫ ∞0

yTi (t) yi(t)dt.

These impulses can model faults or initial conditions.

2. The squared H2 norm of G quantifies the steady-statetotal variance of the output for a system subjected tounit variance stochastic white noise inputs ηi(t):

‖G‖22 = limt→∞

E{yT(t) y(t)

}.

In our case, the white noise inputs can model stochasticfluctuations of renewable generation or loads.

In general, the H2 norm of a linear system can be cal-culated efficiently by solving a linear Lyapunov equation.In our case an additional linear constraint is needed toaccount for the marginally stable and undetectable modev0 = [1T

n 0Tn]T corresponding to an absolute angle reference

for the grid.

Lemma 1. (H2 norm via observability Gramian) Forthe state-space system (A,B,C) defined above, we have that

‖G‖22 = Trace(BTPB) , (7)

where the observability Gramian P ∈ R2n×2n is uniquely de-fined by the following Lyapunov equation and an additionalconstraint defined by v0 = [1T

n 0Tn]T:

PA+ATP + CTC = 0 , (8)Pv0 = 02n . (9)

Proof. Following the typical derivation of theH2 norm for state-space systems [12], we have‖G‖22 = Trace(BTP̂B), where P̂ is the observabilityGramian P̂ =

∫∞0eA

TtCTCeAt dt. Note from (5) thatthe mode v0 = [1T

n 0Tn]T associated with the marginally

stable eigenvalue of A is not detectable, i.e., it holds thatCeAtv0 = Cv0 = 02n for all t ≥ 0. Because the remainingeigenvalues of A are stable, the indefinite integral exists.

Next, we show that P̂ is a solution for both (8) and (9).By taking the derivative of eA

TtCTCeAt with respect to t,and then by integrating from t = 0 to t = +∞, we obtain

ATP̂ + P̂A =[eA

TtCTCeAt]∞

0.

Using the fact that Cv0 = Av0 = 02n, we conclude that[eA

TtCTCeAt]∞

0= −CTC and therefore (8) holds for P̂ .

The fact that P̂ satisfies (9) can be verified by inspection,as

P̂ v0 =

∫ ∞0

eATtCTCeAtv0 dt =

∫ ∞0

eATtCTCv0 dt = 02n.

It remains to show that the P̂ is the unique solution of (8)and (9). To this end, note that rank

(AT)

= 2n− 1 and therank–nullity theorem imply that the kernel of AT is given bya vector ξ ∈ R2n. It can be verified that ATξ = 02n holdsfor ξ = [(D1n)T (M1n)T]T, and it directly follows that allsolutions of (8) are parametrized by

P (τ) = P̂ + τξξT,

for τ ∈ R. Finally, (9) holds if (P̂ + τξξT)v0 = 02n. Incombination with P̂ v0 = 02n this implies τ = 0. With thischoice of τ , P equals the positive semidefinite matrix P̂ .

3 Optimal inertia allocationWe assume that each node i ∈ {1, . . . , n} has a nonzero1

inertial coefficient mi > 0 and we are interested in optimallyallocating additional virtual inertia in order to minimize theH2 norm (4), subject to upper bounds mi at each bus, anda total budget constraint mbdg, accounting for the availableinstallation space and the total cost of the storage devices.

This problem statement is summarized as

minimizeP ,mi

‖G‖22 = Trace(BTPB

)(10a)

subject to∑n

i=1mi ≤ mbdg (10b)

mi ≤ mi ≤ mi , i ∈ {1, . . . , n} (10c)

PA+ATP + CTC = 0, Pv0 = 02n , (10d)1Observe that the case mi = 0 leads to an ill-posed model (1)

whose number of algebraic and dynamic states depend on the systemparameters.

where (A,B,C) are the matrices of the input-output sys-tem (5)-(6). Observe the bilinear nature of the Lyapunovconstraint (10d) featuring products of A and P , and recallfrom (6) that the decision variables mi appear as m−1

i in A.Hence, the problem (10) is highly non-convex and typicallyalso large-scale.

In the following, we will provide general lower and upperbounds, a simplified formulation under certain parametricassumptions, a detailed analysis of a two-area power system,and a numerical method determine locally optimal solutionsin the fully general case.

3.1 Performance boundsTheorem 2. (Performance bounds) Consider the powersystem model (5)-(6), the squared H2 norm (7), and theoptimal inertia allocation problem (10). Then the objective(10a) satisfies

t

2d

(Trace(NL†) +

n∑i=1

simi

)

≤ ‖G‖22 ≤t

2d

(Trace(NL†) +

n∑i=1

simi

), (11)

where t = mini{ti}, t = maxi{ti}, d = mini{di}, and d =maxi{di}.

Proof. Let us express the observability Gramian P as theblock matrix

P =

[X1 X0

XT0 X2

].

With this notation, the squared H2 norm (7), ‖G‖22 reads as

Trace(BTPB) = Trace(TM−2X2) =

n∑i=1

tiX2,ii

m2i

, (12)

where we used the ring commutativity of the trace and thefact that T 1/2 andM−1 are diagonal and therefore commute.

The constraint (10d) can be expanded as[X1 X0

X0T X2

]A+AT

[X1 X0

X0T X2

]+

[N 00 S

]= 0. (13)

By right-multiplying the (1,1) equation of (13) by theMoore-Penrose pseudo-inverse L† of the Laplacian L, weobtain

−X0M−1LL† − LM−1XT

0 L† = −NL†.

By the constraint (9) we have that[1Tn 0T

n

]P =

[0Tn 0T

n

]which implies 1T

nX0 = 0Tn. This fact together with the iden-

tity LL† = (In−1n1Tn/n), implies that LL†X0 = X0. Then,

by using the ring commutativity of the trace, and its in-variance with respect to transposition of the argument, weobtain

2 Trace(M−1X0) = Trace(NL†). (14)

On the other hand, equation (2,2) of (13) implies that

XT0 +X0 = X2M

−1D +DM−1X2 − S.

Similarly as before we left-multiply byM−1, use trace prop-erties and the commutativity of M−1 and D, and obtain

2 Trace(M−1X0 −DM−2X2) = −Trace(M−1S). (15)

Thus, (14) and (15) together deliver

Trace(DM−2X2) =1

2Trace(M−1S +NL†). (16)

From (12) we obtain the relations

t

n∑i=1

X2,ii

m2i

≤ ‖G‖22 ≤ tn∑i=1

X2,ii

m2i

,

which can be further bounded as

t

d

n∑i=1

diX2,ii

m2i

≤ ‖G‖22 ≤t

d

n∑i=1

diX2,ii

m2i

. (17)

The structural similarity of (16) and (17) allows us to stateupper and lower bounds by rewriting (17) as in (11).

Notice that in the bounds proposed in Theorem 2, the net-work topology described by the Laplacian L enters only as aconstant factor, and is decoupled from the decision variablesmi. Moreover, in the case N = L (short-range error penaltyon angles differences), this offset term becomes just a func-tion of the grid size: Trace(NL†) = Trace(LL†) = n− 1.

Theorem 2 (and its proof) sheds some light on the na-ture of the optimization problem that we are considering,and in particular on the role played by the mutual relationbetween disturbance strengths ti, damping coefficients di,their ratios ti/di, frequency penalty weights si, and the de-cision variables mi. These insights are further developed inthe next section.

3.2 Noteworthy casesIn this section, we consider some special choices of the per-formance metric and some assumptions on the system pa-rameters, which are practically relevant and yield simplifiedversions of the general optimization problem (10), enablingin most cases the derivation of closed-form solutions.

We first consider the performance index (4) correspondingto the cost of primary control. As a remedy to mitigate low-inertia frequency stability issues, additional fast-rampingprimary control is often put forward [3]. The primary con-trol effort can be accounted for by the integral quadraticcost ∫ ∞

0

θ̇(t)TDθ̇(t)dt . (18)

Hence, the cost of primary control (18) mimics the H2 per-formance where the performance matrices in (5) are chosenas N = 0 and S = D. This intuitive cost functions allows aninsightful simplification of the optimization problem (10).

Theorem 3. (Primary control cost minimization)Consider the power system model (5)-(6), the squared H2

norm (7), and the optimal inertia allocation problem (10).For a performance output characterizing the cost of primary

control (18): S = D and N = 0, the optimization problem(10) can be equivalently restated as the convex problem

minimizemi

n∑i=1

timi

(19a)

subject ton∑i=1

mi ≤ mbdg (19b)

mi ≤ mi ≤ mi, i ∈ {1, . . . , n} , (19c)

where, we recall, ti describes the strength of the disturbanceat node i.

Proof. With N = 0 and S = D, the Lyapunov equation (13)together with the constraint (9) is solved explicitly by

P =

[X1 X0

XT0 X2

]=

1

2

[L 00 M

].

The performance metric as derived in (12) therefore becomes

‖G‖22 =

n∑i=1

tiX2,ii

m2i

=1

2

n∑i=1

timi

.

This concludes the proof.

The equivalent convex formulation (19) yields the follow-ing important insights. First and foremost, the optimal so-lution to (19) is unique (as long as at least one ti is greaterthan zero) and also independent of the network topologyand the line susceptances. It depends solely on the locationand strength of the disturbance as encoded in the coeffi-cients ti. For example, if the disturbance is concentrated ata particular node i with ti 6= 0 and tj = 0 for j 6= i, then theoptimal solution is to allocate the maximal inertia at nodei: mi = min{mbdg,mi}. In absence of capacity constraints(19c) and presence of the budget constraint (19b), the opti-mal inertia allocation is proportional to the square root ofthe disturbance

√ti.

We now consider a different assumption that also allowsto derive a similar simplified analysis in other notable cases.

Assumption 1. (Uniform disturbance-damping ra-tio) The ratio λ = ti/di is constant for all i ∈ {1, . . . , n}.�

Notice that the droop coefficients di are often scheduledproportionally to the rating of a power source, to guaranteefair power sharing [22]. Meanwhile, it is reasonable to ex-pect that the disturbances due to variable renewable fluctua-tions scale proportionally to the size of the renewable powersource. Hence, Assumption 1 can be justified in many prac-tical cases, including of course the case where both dampingcoefficient and disturbances are uniform across the grid. Un-der this assumption, we have the following result.

Theorem 4. (Optimal allocation with uniformdisturbance-damping ratio) Consider the power systemmodel (5)-(6), the squared H2 norm (7), and the optimal in-ertia allocation problem (10). Let Assumption 1 hold. Then

the optimization problem (10) can be equivalently restated asthe convex problem

minimizemi

n∑i=1

simi

(20a)

subject ton∑i=1

mi ≤ mbdg (20b)

mi ≤ mi ≤ mi, i ∈ {1, . . . , n}, (20c)

where we recall that si is the penalty coefficient for the fre-quency deviation at node i.

Proof. From Assumption 1, let λ = ti/di > 0 be constantfor all i ∈ {1, . . . , n}. Then we can rewrite (12) as

‖G‖22 =

n∑i=1

tiX2,ii

m2i

= λ

n∑i=1

diX2,ii

m2i

.

This is equal, up to the scaling factor λ, to the left handside of (16). We therefore have

‖G‖22 =λ

2Trace(M−1S +NL†), (21)

which is equivalent, up to multiplicative factors and constantoffsets, to the cost of the optimization problem (20a).

Again, as in Theorem 3, Theorem 4 reduces the originaloptimization problem to a simple convex problem for whichthe optimal inertia allocation is independent of the networktopology, and in most cases can be derived as a closed formexpression of the problem parameters.

Under Assumption 1, we can also identify an interestingspecial case. Assume that the frequency penalty S is chosenproportional to inertia coefficients, S = cM for some c ≥ 0:∫ ∞

0

∑n

i,j=1aij(θi(t)− θj(t))2 + c ·

∑n

i=1mi ω

2i (t) dt .

This choice corresponds to penalizing the change in kineticenergy – a reasonable and standard penalty in power sys-tems. We have the subsequent result, that follows directlyby evaluating (21) for this specific choice of S = c·M (whichalso includes the case where no frequency penalty is consid-ered, i.e. c = 0, and therefore only angle differences arepenalized).

Corollary 5. (Kinetic energy penalization with uni-form disturbance-damping ratio) Let Assumption 1hold, and let the penalty on the frequency deviations be pro-portional to the allocated inertia, that is, S = c ·M . Thenthe performance metric ‖G‖22 is independent on the inertiaallocation, and assumes the form

‖G‖22 =λ

2

(c · n+ Trace(NL†)

),

where λ = ti/di > 0, for all i ∈ {1, . . . , n}, is the uniformdisturbance-damping ratio.

m1

0 1 2 3 4 5 6 7 8 9 10

f(m

1)

0

1

2

3

4

5

6

dissimilar t/d

identical t/d

Figure 1: Cost function profiles for identical and weaklydissimilar ti/di ratios for the two-area case.

3.3 Explicit results for a two-area networkIn this subsection, we focus on a two-area power grid as in [3]to obtain some insight on the nature of this optimizationproblem. We also highlight the role of the ratios ti/di, whichplay a prominent role in Assumption 1 and the bounds (11).

In the case of a two-area system, it is possible to derive ananalytical solution P (m) of the Lyapunov equation (10d), asa closed form function of the vector of inertia allocationsmi.We thus obtain an explicit expression for the objective (10a)as

‖G‖22 = f(m) := Trace(B(m)TP (m)B(m)

)(22)

where f(m) is a rational function of polynomials of orders4 in the numerator (respectively, 6 in the denominator) interms of inertial coefficients mi.

As the explicit expression is more convoluted than insight-ful, we will not show it here, but only report the followingstatements which can be verified by a simple but cumber-some analysis of the rational function f(m):

1. The problem (10) admits a unique minimizer.

2. For sufficiently large bounds mi, the budget constraint(10b) becomes active, that is, the optimizers satisfym∗1+m∗2 = mbdg. In this case, m2 can be eliminated as m2 =mbdg−m1, and (10) can be reduced to a scalar problem.

3. In case of no capacity constraints and for identical ti/diratios and frequency penalties S, the optimal inertial co-efficients are identical m∗1 = m∗2 (as predicted by The-orem 4). In case that ti/di > tj/dj , then m∗i > m∗j(see the example in Figure 1, where we eliminated m∗2 =mbdg −m∗1).

4. For sufficiently uniform ti/di ratios, the problem (10) isstrongly convex. We observe that the cost function f(m)is fairly flat over the feasible set (see Figure 1).

5. For strongly dissimilar ti/di ratios, we observe a less flatcost function. If the disturbance affects only one node,for example, t1 = 1 and t2 = 0, strong convexity is lost.

From the above facts, we conclude that the input scalingfactors ti play a fundamental role in the determination ofthe optimal inertia allocation. To obtain a more complete

d1 = 6 > d2 = 1, mbdg = 25, a12 = 1

t1=1-t

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Op

tim

al in

ert

ia a

lloca

tio

n

0

5

10

15

20

25

d1 = d2 = 1, mbdg = 11, a12 = 1

t1=1-t

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Op

tim

al in

ert

ia a

lloca

tio

n

0

2

4

6

8

10

12

m∗

1

m∗

2

mbdgm1

∗ +m2∗

Figure 2: Optimal inertia allocation for a two-area systemwith identical frequency penalties S = I2, with non-identicaland identical damping coefficients di, with disturbances in-puts varying from [t1, t2] = [0, 1] to [t1, t2] = [1, 0], and fortwo choices of budget mbdg.

picture, we linearly vary the disturbance input matrices from[t1, t2] = [0, 1] to [t1, t2] = [1, 0], that is, from a disturbancelocalized at node 2 to a disturbance localized at node 1. Theresulting optimizers are displayed in Figure 2 showing thatinertia is allocated dominantly at the site of the disturbance,which is in line with previous case studies [3,11]. Notice alsothat depending on the value of the budgetmbdg, the capacityconstraints mi, and the ti/di ratios, the budget constraintmay be active or not. Thus, perhaps surprisingly, sometimesnot all inertia resources are allocated. Overall, the two-areacase paints a surprisingly complex picture.

3.4 A numerical method for the generalcase

In Subsections 3.2 and 3.3, we considered a subset of sce-narios and cost functions that allowed the derivation oftractable reformulations and solutions of the inertia allo-cation problem (10). In this section, we consider the opti-mization problem in its full generality. Similarly as in Sec-tion 3.3, we denote by P (m) the solution to the Lyapunovequation (10d), and we express the cost function ‖G‖22 as afunction f(m) of the vector of inertia allocations mi, i.e.

‖G‖22 = f(m) := Trace(B(m)TP (m)B(m)

). (23)

The proposed computational approach consists of deriv-ing an efficient algorithm for the computation of the gradi-ent ∇f(m) of f(m). Information on the gradient ∇f(m) isessential for many numerical optimization methods, for ex-ample, the problem (10) can be approached via the partialLagrangian

L(m, ρ, ρ, ς) = f(m) + ρT(m−m)

+ ρT(m−m) + ς

(n∑i=1

mi −mbdg

),

where ρ, ρ, ς > 0 are multipliers, and we do not dualizethe Lyapunov constraint (10d). If the gradient of f(m) isexplicitly available, then a locally optimal solution can becomputed, for example, via the iterative primal-dual algo-rithm [29]

m(k + 1) = m(k)

− α(k)(∇f(m(k)) + ρ(k)− ρ(k) + ς(k)1n

),

ρ(k + 1) = [ρ(k) + α(k) (m(k)−m)]+ ,

ρ(k + 1) =[ρ(k) + α(k) (m−m(k))

]+,

ς(k + 1) =

[ς(k) + α(k)

(n∑i=1

mi(k)−mbdg

)]+

,

where k ∈ Z≥0 is the iteration index, α(k) > 0 is an ap-propriate step size, and [·]+ = max{0, ·}. In general, mostcomputational approaches can be sped up tremendously ifthe large-scale set of nonlinear (in the decision variables)Lyapunov equations (10d) can be eliminated and includedinto the gradient information, as we are proposing. In thefollowing, we provide an algorithm that achieves so usingthe routine Lyap(A,Q), which returns the matrix P thatsolves PA+ATP +Q = 0 together with Pv0 = 02n.

Algorithm 1: Gradient computationInput current value m of the decision variablesOutput numerical evaluation g of the gradient ∇f(m)

A(0) ←[

0 I−M−1L −M−1D

];

B(0) ←[

0M−1T 1/2

];

P (0) ← Lyap(A(0), CTC

);

for i = 1, . . . , n doΦ← eieTi ;

A(1) ←[

0 0ΦM−2L ΦM−2D

];

B(1) ←[

0−ΦM−2T 1/2

];

P (1) ← Lyap(A(0), P (0)A(1) +A(1)TP (0)

);

gi ← Trace(

2B(1)TP (0)B(0) +B(0)TP (1)B(0));

Theorem 6. (Gradient computation) Consider the ob-jective function (23), where P (m) is a function of m viathe Lyapunov equation (10d). The objective function is dif-ferentiable for m ∈ Rn>0, and its gradient at m is given byAlgorithm 1.

The proof of Theorem 6 is partially inspired by [16] andrelies on a perturbation analysis of the Lyapunov equation(10d) combined with Taylor and power series expansions.

Proof. In order to compute the gradient of (23) atm ∈ Rn>0,we make use of the relation

∇µf(m) = ∇f(m)Tµ , (24)

where ∇µf(m) is the directional derivative of f in the di-rection µ ∈ Rn, defined as

∇µf(m) = limδ→0

f(m+ δµ)− f(m)

δ, (25)

whenever this limit exists. From (23) we have that

f(m+ δµ) = Trace(B(m+ δµ)TPB(m+ δµ)

), (26)

where P is a solution of the Lyapunov equation

PA(m+ δµ) +A(m+ δµ)TP + CTC = 0 (27)

and where by A(m + δµ) we denote the system matrix de-fined in (6), evaluated at m+ δµ. The matrices A(m+ δµ)and B(m + δµ) viewed as functions of scalar δ can thus beexpanded in a Taylor series around δ = 0 as

A(m+ δµ) = A(0)(m,µ) +A

(1)(m,µ)δ +O(δ2) ,

B(m+ δµ) = B(0)(m,µ) +B

(1)(m,µ)δ +O(δ2)

(28)

with coefficients A(i)(m,µ) and B(i)

(m,µ), i ∈ {0, 1}. To compute

the coefficients of the Taylor expansion in (28), we recall thescalar series expansion of 1/(mi + δµi) around δ = 0:

1

(mi + δµi)=

1

mi− δµim2i

+O(δ2).

Using the shorthand Φ = diag(µi), we therefore have

A(0)(m,µ) =

[0 I

−M−1L −M−1D

]A

(1)(m,µ) =

[0 0

ΦM−2L ΦM−2D

]B

(0)(m,µ) =

[0

M−1T 1/2

]B

(1)(m,µ) =

[0

−ΦM−2T 1/2

].

Accordingly, the solution to the Lyapunov equation (27)can be expanded in a power series as

P = P (m+ δµ) = P(0)(m,µ) + P

(1)(m,µ)δ +O(δ2), (29)

and therefore the Lyapunov equation (27) becomes

(P (0) + δP (1) +O(δ2))(A(0) + δA(1) +O(δ2))+

(A(0) +δA(1) +O(δ2))T(P (0) +δP (1) +O(δ2))+CTC = 0,

where we dropped the subscript (m,µ) for readability. Bycollecting terms associated with powers of δ, we obtain twoLyapunov equations determining P (0) and P (1):

P (0)A(0) +A(0)TP (0) + CTC = 0 , (30a)

P (1)A(0) +A(0)TP (1) + (P (0)A(1) +A(1)TP (0)) = 0 . (30b)

By the same reasoning as used for equation (8), the firstLyapunov equation (30a) is feasible with a positive semidef-inite P (0) satisfying P (0)v0 = 02n. The second Lyapunovequation (30b) is feasible by analogous arguments. Finally,by using (26) together with (28) and (29), we obtain

f(m+ δµ) = f(0)(m,µ) + f

(1)(m,µ)δ +O(δ2) ,

where f (0)(m,µ) = f(m) and

f(1)(m,µ) = Trace

(2B

(1)(m,µ)

TP

(0)(m,µ)B

(0)(m,µ)

+B(0)(m,µ)

TP

(1)(m,µ)B

(0)(m,µ)

).

(31)

From (25), it follows that ∇µf(m) = f(1)(m,µ) as defined

in (31), thereby implicitly establishing differentiability off(m).

This concludes the proof, as the algorithm computes eachcomponent of the gradient ∇f(m) by using the relation (24)with the special choice of µ = ei for i ∈ {1, . . . , n}.

3.5 The planning problem: economic allo-cation of resources

In this subsection, we focus on the planning problem ofoptimally allocating virtual inertia when economic reasonssuggest that only a limited number of virtual inertia de-vices should be deployed (rather than at every grid bus).Since this problem is generally combinatorial, we solve amodified optimal allocation problem, where an additional`1-regularization penalty is imposed, in order to promote asparse solution [30].

The regularized optimal inertia allocation problem is then

minimizeP ,mi

Jγ(m,P ) = ‖G‖22 + γ‖m−m‖1 (32a)

subject to∑n

i=1mi ≤ mbdg (32b)

mi ≤ mi ≤ mi , i ∈ {1, . . . , n} (32c)

PA+ATP + CTC = 0, Pv0 = 02n , (32d)

where γ ≥ 0 trades off the sparsity penalty and the originalobjective function.

As in (32c) the allocations mi are lower bounded by apositive mi, the objective (32a) can be rewritten as:

Jγ(m,P ) = Trace(BTPB

)+∑n

i=1γ (mi −mi) . (33)

Observe that the regularization term in the cost (33) is lin-ear and differentiable. Thus, problem (33) fits well into our

gradient computation algorithm, and a solution can be de-termined within the fold of Algorithm 1 by incorporatingthe penalty term. Likewise, our analytic results in Section3.2 can be re-derived for the cost function (33). We high-light the utility of the performance-sparsity trade-off (33) inSection 4.

4 Case study: 12-Bus-Three-RegionSystem

25 km 10 km 25 km10 km

25 km

110 km

110 k

m

110 km

1

2

3 4

5

6

78

910 11

12

1570 MW

1000 MW100 Mvar

567 MW100 Mvar

400 MW 490 MW

611 MW164 Mvar

1050 MW284 Mvar

719 MW133 Mvar

350 MW69 Mvar

700 MW208 Mvar

700 MW293 Mvar

200M

var

350M

var

Figure 3: A 12 bus three-region test case adopted from[11, 24]. Transformer reactance 0.15 p.u., line impedance(0.0001+0.001i) p.u./km.

In this section, we investigate a 12-bus case study illus-trated in Figure 3. The system parameters are based on amodified two-area system from [24, Example 12.6] with anadditional third area, as introduced in [11]. After Kron re-duction of the passive load buses, we obtain a systems of 9buses, corresponding to the nodes where inertia can be al-located.We investigate this example computationally using Algo-rithm 1 to drive standard gradient-based optimization rou-tines, while highlighting parallels to our analytic results. Weanalyze different parametric scenarios and compare the in-ertia allocation and the performance of the proposed nu-merical optimization (which is a locally optimal solution)with two plausible heuristics that one may deduce from theconclusions in [3, 11] and the special cases discussed follow-ing Theorem 3: namely the uniform allocation of the avail-able budget, in the absence of capacity constraints, that is,mi = muni = mbdg/n; or the allocation of the maximum in-ertia allowed by the bus capacity constraints in the absenceof a budget constraint, that is, m = m (which we set asmi = 4mi).

Uniform disturbance We first assume that the distur-bance affects all nodes identically, T = In/9. In Figure 4 weconsider the case where there are only capacity constraintsat each bus, and we compare the different allocations vis-à-vis: the initial inertia m, a locally optimal solution m∗, andthe maximum inertia allocation m. Figure 5 compares theresults in the case where there is only a budget constraint onthe total allocation. We compare the initial inertia m, thelocally optimal allocation m∗, and the uniform placementmi = muni.

Localized disturbance We then consider the scenariowhere a localized disturbance affects a particular node, inthis example, node 4 with T = diag{0, 0, 1, 0, 0, 0, 0, 0, 0}.As in Figures 4 and 5, a comparison of the different inertialallocations and the performance values is presented in Fig-ures 6 and 7 for the cases of capacity and budget constraints.

We draw the following conclusions from the above testcases – some of which are perhaps surprising and counterin-tuitive.

1. First, our locally optimal solution achieves the best per-formance among the different heuristics in all scenarios.

2. In the case of uniform disturbances with only capacityconstraints on the individual buses (Figure 4), the opti-mal solution does not correspond to allocating the max-imum possible inertia at every bus.

3. In the case of uniform disturbances with only the totalbudget constraint (Figure 5), the optimal solution is re-markably different from the uniform allocation of inertiaat the different nodes.

4. In case of uniform disturbances, the performance im-provement with respect to the initial allocation and thedifferent heuristics is modest. This confirms the intuitiondeveloped for the two-area case (Section 3.3) regardingthe flatness of the cost function.

5. In stark contrast is the case of a localized disturbance,where adding inertia dominantly to disturbed node isan optimal choice in comparison to heuristic placements.The latter is also in line with the results presented for thetwo-area case and the closed-form results in Theorem 3.

6. In the case of a localized disturbance, adding inertia toall undisturbed nodes may be detrimental for the perfor-mance, even for the same (maximal) allocation of inertiaat the disturbed node, as shown in Figure 6.

7. In Figure 8, we show the optimal allocations for a convexcombination of the angle and frequency penalties, that is,when the coherency performance metric (4) is modifiedas ∫ ∞

0

(1− ρ) · θ(t)TNθ(t) + ρ · ω(t)TSω(t) dt ,

where ρ ∈ [0, 1]. The optimal allocation appears moreuniform across the network when penalizing only fre-quency violations, as suggested by Theorem 3 for thespecial case (N,S) = (0, D).

8. The sparsity-promoting approach proposed in Section 3.5is examined in Figure 9. For a uniform disturbance with-out a sparsity penalty, inertia is allocated at all ninebuses of the network. For γ = 6 × 10−5 an alloca-tion at only seven buses is optimal with hardly a 1.3%degradation in performance. For sparser allocations, thetrade-off with performance becomes more relevant. Thesparsity effect is significantly pronounced for localizeddisturbances. The optimal solution for γ = 0 for a lo-calized disturbance at node 4, requires allocating inertia

node

1 2 4 5 6 8 9 10 12

ine

rtia

0

50

100

150

m

m∗

m

tra

ce

0

0.05

0.1

0.15

Figure 4: Optimal inertia allocation for a uniform distur-bance subject to capacity constraints (10c).

node

1 2 4 5 6 8 9 10 12

ine

rtia

0

30

60

90

tra

ce

0

0.05

0.1

0.15m

m∗

muni

Figure 5: Optimal inertia allocation for a uniform distur-bance subject to budget constraint (10b).

node

1 2 4 5 6 8 9 10 12

ine

rtia

0

40

80

120

160

m

m∗

m

tra

ce

0

0.05

0.1

0.15

0.2

0.25

Figure 6: Optimal inertia allocation for a localized distur-bance at node 4 subject to capacity constraints (10c).

node

1 2 4 5 6 8 9 10 12

ine

rtia

0

40

80

120

160m

m∗

muni

tra

ce

0

0.05

0.1

0.15

0.2

0.25

Figure 7: Optimal inertia allocation for a localized distur-bance at node 4 subject to budget constraint (10b).

node

1 2 3 4 5 6 7 8 9

ine

rtia

0

20

40

60

80

100

120

140

ξ=1 ξ=.90 ξ=.83 ξ=.66 ξ=.50 ξ=.33 ξ=.16 ξ=.09 ξ=.01

Figure 8: Optimal inertia allocation for a uniform distur-bance, subject to budget constraints, and with a convexcombination of penalties on angle differences N and fre-quency excursions S.

γ×10

-40 0.5 1 1.5 2 2.5 3

Card

inalit

y

0

2

4

6

8

10

Cardinality-Loc Cardinality-Uni Performance-Loc Performance-Uni

Rela

tive P

erf

orm

ance L

oss (

%)

0

20

40

60

80

100

0

20

40

60

80

100

Figure 9: Relative performance loss (%) as a function ofsparsity promoting penalty γ for (a) uniform disturbance,(b) localized disturbance at node 4 with capacity con-straints. 0% performance loss corresponds to the optimalallocation, 100% performance loss corresponds to no addi-tional inertia allocation.

Time(s)0 50 100 150

∆θ

1-∆

θ4

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05(a)

Time(s)0 50 100 150

∆ω

4

-0.1

-0.05

0

0.05

0.1

0.15(b)

Time(s)0 50 100 150

∆ω

5

×10-3

-2

-1

0

1

2

3

4

5(c)

Time(s)0 50 100 150

Co

ntr

ol e

ffo

rt-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2(d)

m

muni

m∗

Figure 10: Time-domain simulations of angle differences, select frequencies, and control effort for a localized disturbanceat node 4.

at buses (4, 6). However, for γ = 2 × 10−4, we observean allocation of inertia only at bus 4 does not affect theperformance significantly, while being preferable from aneconomic perspective.

9. Figure 10 shows the time domain responses to a localizedimpulse at node 4, modeling a post-fault condition. Sub-figure (a) (respectively, (b)) detail the superior perfor-mance of an optimal inertia allocation with regards to thepeak (overshoot) for angle differences (respectively, fre-quencies). Subfigure (c) displays the frequency responseat node 5 of the system. Note that the scale of this plotsis of order 10−3 rendering the deviations as potentiallyinsignificant. Similar comments apply to other signalswhich are not displayed here. Finally subfigure (d) showsthe control effort m · θ̈i expended by the virtual inertiaemulation at the disturbed bus i. Perhaps surprisingly,observe that the optimal allocation m = m∗ requires theleast control effort.

10. Figure 11 plots the eigenvalue spectrum for different in-ertia profiles. The case of no additional inertia alloca-tion, m = m, marginally outperforms with respect toboth the best damping asymptote (most damped nonzeroeigenvalues) as well as the best damping ratio (narrowestcone). This is the optimization criterion in [11]. As ap-parent from the time-domain plots in Figure 10, this casealso leads to the worst time-domain performance (withrespect to overshoots) compared to the optimal alloca-tion m = m∗, which has slightly poorer damping asymp-tote and ratio. These observations reveal that the spec-trum holds only partial information, and they stronglyadvocate the use of the H2-norm as opposed to spectralperformance metrics.

Real Axis-0.18 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02

Imag

inar

y Ax

is

-3

-2

-1

0

1

2

3

conemmunim∗

Figure 11: The eigenvalue spectrum of the state matrix A for different inertia profiles, where m∗ has been optimized fora localized disturbance at node 4.

5 Conclusions

We considered the problem of placing virtual inertia inpower grids based on an H2 norm performance metric re-flecting network coherency. This formulation gives rise toa large-scale and non-convex optimization program. Forcertain cost functions, problem instances, and in the low-dimensional two-area case, we could derive closed-form so-lutions yielding some, possibly surprising insights. Next, wedeveloped a computational approach based on an explicitgradient formulation and validated our results on a three-area network. Suitable time-domain simulations demon-strate the efficacy of our locally optimal inertia allocationsover intuitive heuristics. We also examined the problemof allocating a finite number of virtual inertia units via asparsity-promoting regularization. All of our results show-case that the optimal inertia allocation is strongly dependenton the location of disturbance.

Our computational and analytic results are well alignedand suggest insightful strategies for the optimal allocationof virtual inertia. We envision that these results find appli-cation in stabilizing low-inertia grids through strategicallyplaced virtual inertia units. As part of our ongoing and fu-ture work, we also consider the extension to more detailedsystem models.

Acknowledgements

The authors wish to thank Mihailo Jovanovic, Andreas Ul-big, Theodor Borsche, Dominic Gross, and Ulrich Münz fortheir comments on the problem setup and analysis methods.

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