Decentralised Robust Inverter-based

Control in Power Systems

?

Richard Pates

⇤Enrique Mallada

⇤⇤

⇤ Lund University, Lund, Sweden (e-mail:richard.pates@control.lth.se).

⇤⇤ Johns Hopkins University, Baltimore, MD 21218 USA (e-mail:mallada@jhu.edu)

Abstract: This paper develops a novel framework for power system stability analysis, thatallows for the decentralised design of inverter based controllers. The method requires that eachindividual inverter satisfies a standard H1 design requirement. Critically each requirementdepends only on the dynamics of the components and inverters at each individual bus, andthe aggregate susceptance of the transmission lines connected to it. The method is both robustto network and delay uncertainties, as well as heterogeneous network components, and whenno network information is available it reduces to the standard decentralised passivity su�cientcondition for stability. We illustrate the novelty and strength of our approach by studying thedesign of inverter-based control laws in the presence of delays.

Keywords: Inverter-based control, Virtual-inertia, Robust Stability, Power Systems

1. INTRODUCTION

The composition of the electric gird is in state of flux (Mil-ligan et al., 2015). Motivated by the need of reducingcarbon emissions, conventional synchronous combustiongenerators, with relatively large inertia, are being replacedwith renewable energy sources with little (wind) or noinertia (solar) at all (Winter et al., 2015). Alongside, thesteady increase of power electronics on the demand sideis gradually diminishing the load sensitivity to frequencyvariations (Wood et al., 1996). As a result, rapid fre-quency fluctuations are becoming a major source of con-cern for several grid operators (Boemer et al., 2010; Kirby,2005). Besides increasing the risk of frequency instabilities,this dynamic degradation also places limits on the totalamount of renewable generation that can be sustained bythe grid. Ireland, for instance, is already resourcing to windcurtailment whenever wind becomes larger than 50% ofexisting demand in order to preserve the grid stability.

One solution that has been proposed to mitigate thisdegradation is to use inverter-based generation to mimicsynchronous generator behavior, i.e. implement virtualinertia (Driesen and Visscher, 2008a,b). However, whilevirtual inertia can indeed mitigate this degradation, itis unclear whether this particular choice of control isthe most suitable for this task. On the one hand, unlikegenerator dynamics that set the grid frequency, virtualinertia controllers estimate the grid frequency and itsderivative using noisy and delayed measurements. On theother hand, inverter-based control can be significantly

? This work was supported by the Swedish Research Council throughthe LCCC Linnaeus Center, NSF CPS grant CNS 1544771, JohnsHopkins E2SHI Seed Grant, and Johns Hopkins WSE startup funds.* R. Pates is a member of the LCCC Linnaeus Center and theELLIIT Excellence Center at Lund University.

faster than conventional generators. Thus using invertersto mimic generators behavior does not take advantage oftheir full potential.

Recently, a novel dynamic droop control (iDroop) (Mal-lada, 2016) has been proposed as an alternative to virtualinertia that seeks to exploit the added flexibility presentin inverters. Unlike virtual inertia that is sensitive tonoisy measurements (it has unboundedH2 norm (Mallada,2016)), iDroop can improve the dynamic performancewithout unbounded noise amplification. However, as moresophisticated controllers such as iDroop are deployed, thedynamics of the power grid become more complex anduncertain, which makes the application of direct stabil-ity methods harder. The challenge is therefore to designan inverter control architecture that takes advantage ofthe added flexibility, while providing stability guarantees.Such an architecture must take into account the e↵ect ofdelays and measurement noise in the design. It must be ro-bust to unexpected changes in the network topology. Andmust provide a “plug-and-play” functionality by yieldingdecentralised, yet not overly conservative, stability certifi-cates.

In this paper we leverage classical stability tools for theLur’e problem (Brockett and Willems, 1965) to develop anovel analysis framework for power systems that allows adecentralised design of inverter controllers that are robustto network changes, delay uncertainties, and heteroge-neous network components. More precisely, by modelingthe power system dynamics as the feedback interconnec-tion of input-out bus dynamics and network dynamics(Section 2), we derive a decentralised stability conditionthat depends only on the individual bus dynamics and theaggregate susceptance of the transmission lines connectedto it (Section 3). When no network information is avail-able, our condition reduces to the standard decentralised

Preprints of the 20th World CongressThe International Federation of Automatic ControlToulouse, France, July 9-14, 2017

Copyright by theInternational Federation of Automatic Control (IFAC)

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-

N

uP

dP

uN

yN

P = diag(pi

)

yP

Fig. 1. Input-output Power Network Model

passivity su�cient condition for stability. We illustratethe novelty and strength of our analysis framework bystudying the design of inverter-based control laws in thepresence of delays (Section 4).

2. INPUT-OUTPUT POWER NETWORK MODEL

In this section we describe the input-output representationof the power grid used to derive our decentralised stabilityresults. We use i 2 V := {1 . . . , n} to denote the ith bus inthe network and the unordered pair {i, j} 2 E to denoteeach transmission line.

As mentioned in the previous section, we model the powernetwork as the feedback interconnection of two systems,P := diag (p

i

, i 2 V ) and N shown in Figure 1. Eachsubsystem p

i

denotes the ith bus dynamics, and mapsuP,i

to yP,i

, where uP,i

denotes the ith bus exogenousreal power injection, i.e., the real power incoming to bus ifrom other parts of the network or due to unmodeled buselements. The output y

P,i

:= !i

denotes the bus frequencydeviation from steady state. Similarly, the system Ndenotes the network dynamics, which maps the vector ofsystem frequencies u

N

:= ! = (!i

, i 2 V ) to the vectorof electric power network demand y

N

= (yN,i

, i 2 V ), i.e.yN,i

is the total electric power at bus i that is being drainedby the network. Thus, if we let d

P

= (dP,i

, , i 2 V ) denotethe unmodeled bus power injection, then by definition weget u

P

= �yN

+ dP

.

This input-output decomposition provides a general mod-eling framework for power system dynamics that encom-passes several existing models as special cases. For exam-ple, it can include the standard swing equations (Shen andPacker, 1954), as well as several di↵erent levels of detailsin generator dynamics, including turbine dynamics, andgovernor dynamics, see e.g., (Zhao et al., 2016). The mainimplicit assumption in Figure 1, which is standard in theliterature and well justified in transmission networks (Kun-dur, 1994), is that voltage magnitudes and reactive powerflows are constant. However, a similar decomposition couldbe envisioned in the presence of voltage dynamics andreactive power flows. Such extension will be subject offuture research.

In this paper we focus on power system models that satisfythe following assumptions:Assumption 2.1. P =diag (p1, . . . , pn), with p

i

2 H 1⇥11 .

Assumption 2.2. N = 1s

LB

, where LB

2 Rn⇥n is aweighted Laplacian matrix.

Here H m⇥n

1 denotes the Hardy space of m by n matrixtransfer functions analytic on the open right-half plane

(Re {s} > 0) and bounded on the imaginary axis (jR). Weonly consider such a general class of transfer functions toallow for the consideration of delays, and use real rationaltransfer functions to model most components.

In the rest of this section we illustrate how di↵erentnetwork components can be modeled using this frameworkas well as the implications of assumptions 2.1 and 2.2.For concreteness, we make specific choices on the modelsfor generators and loads. We highlight however that ouranalysis framework can be extended to more complexmodels provided they satisfy assumptions 2.1 and 2.2.

2.1 Bus Dynamics

We model the bus dynamics using the standard swingequations (Shen and Packer, 1954). Thus the frequencyof each bus i evolves according to

pi

:M

i

!i

= �Di

!i

+ xi

+ uP,i

yP,i

= !i

, (1)

where the state !i

represents the frequency deviation fromnominal and x

i

denotes the power injected by inverter-based generation at bus i. The parameter M

i

� 0 denotesthe aggregate bus inertia. For a generator bus M

i

> 0,and D

i

> 0 represents the damping coe�cient. For a loadbus M

i

= 0 and Di

> 0 represents the load sensitivity tofrequency variations (Bergen and Hill, 1981).

We consider linear control laws for the inverter dynamicsthat depend solely on the local frequency. Thus, we modelthe inverter dynamics as a negative feedback law of theform

xi

(s) = �ci

(s) !i

(s) , (2)

where xi

(s) and !i

(s) denote the Laplace transform ofxi

(t) and !i

(t), respectively.

Combining (1) and (2) we get the following input-outputrepresentation of the bus dynamics

pi

(s) =1

Mi

s+Di

✓

1 +ci

(s)

Mi

s+Di

◆�1

. (3)

Whenever ci

(s) = 0, (3) represents a bus without invertercontrol, and by choosing either M

i

> 0 or Mi

= 0,(3) can model generator or load buses respectively. Thus,(3) provides a compact and flexible representation of thedi↵erent bus elements. It is important to notice thatAssumption 2.1 is rather mild and only requires that thetransfer function (3) of each bus is stable.

Finally, the control law ci

(s) defines a general modelingclass for inverter-based decentralised controllers. A num-ber of conventional architectures can be written in thisform. For example, virtual inertia inverters correspond to

ci

(s) = Ki

+K⌫

i

s (4)

whereKi

� 0 andK⌫

i

� 0 are the droop and virtual inertiaconstants, and by setting K⌫

i

= 0, we recover the standarddroop control. Similarly, the iDroop dynamic controller isgiven by

ci

(s) =K⌫

i

s+K�

i

Ki

s+K�

i

(5)

where K⌫

i

� 0, K�

i

� 0 are tunable parameters (Mallada,2016).

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2.2 Network Dynamics

The network dynamics, under Assumption 2.2, are givenby

N : ✓ = uN

yN

= LB

✓, (6)

where, as mentioned before, the matrix LB

2 Rn⇥n isthe B

ij

-weighted Laplacian matrix that describes how thetransmission network couples the dynamics of di↵erentbuses.

Thus Assumption 2.2 is equivalent to the DC power flowapproximation, where the parameter B

ij

usually denotesthe susceptance of line {i, j}, although more generallyrepresents the sensitivity of the power flowing throughline {i, j} due to changes on the phase di↵erence, i.e.,B

ij

= vivj

xijcos(✓⇤

i

� ✓⇤j

) where vi

and vj

are the (constant)

voltage magnitudes, xij

is the line inductance, and ✓⇤i

�✓⇤j

is the steady state phase di↵erence between buses i and j,see e.g., (Zhao et al., 2013).

To simplify the exposition, we refer here to Bij

as thetransmission line susceptance. Therefore,

[LB

]ii

=X

j:{i,j}2E

Bij

denotes the aggregate susceptance of the lines connectedto bus i. As we will soon see in the next section, anupper bound of this value, i.e. [L

B

]ii

1�i, can be used

to guarantee stability in a decentralised manner.

2.3 Connection with the Swing Equations

For sake of clarity we derive here a state space descriptionof the described models, and show how the standard swingequation model fits into our framework. Since u

N

= yP

=! and u

P

= dP

�yN

, then using (3) and (6), the state spacerepresentation of the feedback interconnection in Figure 1amounts to:

✓i

= !i

(7a)

Mi

!i

= �Di

!i

�X

j:{i,j}2E

Bij

(✓i

� ✓j

) + xi

+ dP,i

. (7b)

where the inverter-based power injection xi

evolves accord-ing to either

xi

= �Ki

!i

�K⌫

i

!i

(8)

for virtual inertia inverter-based control, or

xi

= �K�

i

(Ki

!i

+ xi

)�K⌫

i

!i

(9)

of the case of iDroop.

Equation (7) amounts to the standard swing equationsin which d

P,i

represents the net constant power injectionat bus i. One interesting observation, and perhaps alsoa peculiarity of our framework, is that while usually thephase of bus i (✓

i

) is considered to be part of the busdynamics, in our framework this state is part of thenetwork. This sidesteps the need to define, for example, areference bus to overcome the lack of uniqueness in theseangles.

3. A SCALABLE STABILITY CRITERION

This section consists of two main parts. First we give ageneralisation of a classical stability result for single-input-

single-output systems that can be applied to the struc-tured feedback interconnection in Figure 1. Second, wediscuss how it can be given a plug and play interpretation,and used to guide controller design in the context of theelectrical power system models from Section 2.

3.1 A Generalisation of a Classical Stability Criterion

Brockett and Willems (1965) gives a method for testingstability of a feedback interconnection in the presence of anuncertain constant gain. More specifically it is shown thatfor a real rational, stable, strictly proper transfer functiong (s), the feedback interconnection of g (s) and k 2 R isstable for all

0 k 1

k⇤

if and only if there exists a Strictly Positive Real (SPR)h (s) such that

h (s) (k⇤ + g (s)) (10)is SPR (see Dasgupta and Anderson (1996) for this precisestatement). The concept of an SPR transfer function isgiven by the following standard definition:Definition 3.1. A (not necessarily proper or rational)transfer functions g (s) is Positive Real (PR) if:

(i) g (s) is analytic in Re {s} > 0;(ii) g (s) is real for all positive real s;(iii) Re {g (s)} � 0 for all Re {s} > 0.

If in addition there exists an ✏ > 0 such that g (s� ✏) isPR, then g (s) is SPR.

The following theorem extends this result to the feedbackinterconnection in Figure 1, where the Laplacian matrixLB

plays the role of the uncertain gain. Our motivationfor trying to extend this result is driven by a desire fordecentralised stability conditions. Since ‘the gain’ of aLaplacian matrix is dependent on the network topology,in order to obtain a result that does not require exactknowledge of the network structure it is necessary to beable to handle a degree of uncertainty in this gain. Theresult in Brockett and Willems (1965) is then the naturalcandidate for extension, because it is the strongest possibleresult of this type in the scalar case.Theorem 3.2. Let �1, . . . , �n, be positive constants, andP 2 H n⇥n

1 satisfy Assumption 2.1. If there exists anonzero h such that sh is PR, and for each i 2 {1, . . . , n}:

h (s)⇣�

i

2s+ p

i

(s)⌘

(11)

is SPR, then for any

N 2⇢

1

sLB

: LB

meets Assumption 2.2 and [LB

]ii

1

�i

�

,

the feedback interconnection of P and N is stable 1 .

Before proving this result, observe that if we define h (s) =sh (s), then Theorem 3.2 asks if there exists an h 2 PRsuch that

h (s)⇣�

i

2+

pi

s

⌘

2 SPR.

1 Here stability is in the internal stability sense, that ish

PI

i

(I +NP )�1⇥

N I⇤

2 H1.

This definition does not require that ✓ (t) remains bounded inresponse to bounded disturbances, only that LB✓ (t) does. This ishow we avoid the need to define a reference bus.

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This representation makes clearer the connection with theoriginal result of Brockett and Willems (1965) (c.f. (10)).However, in order to satisfy the above, it is necessaryfor h (s) to have a zero at the origin, which motivatesour choice of statement. Also note that since N is PR,a standard passivity result for the interconnection inFigure 1 would require that p

i

is SPR. It can be shown(though we omit the proof due to space limitations) thatthis is a special case of Theorem 3.2. This is because thereexists an h such that if p

i

is SPR, then (11) is satisfied.

Proof. Since P 2 H n⇥n

1 , the interconnection of P and Nis stable if and only if

N (I + PN)�1 2 H n⇥n

1 .

We will now show that the conditions of the theorem aresu�cient for the above. Let D = diag

�

�1

2 , . . . , �n

2

�

, anddefine

A =⇣

D12L

B

D12

⌘

.

Hence N = D� 12

�

1s

A�

D� 12 , and A is a normalised

weighted Laplacian matrix. Factorise A as

A = QXQ⇤,

where X is positive definite with eigenvalues �(X) 1(i.e., I � X � ✏I), Q 2 Cn⇥(n�m), m > 0, Q⇤Q = I,✏ > 0. Hence N (I + PN)�1 equals

D� 12QXQ⇤D� 1

2

⇣

sI + PD� 12QXQ⇤D� 1

2

⌘�1,

=D� 12QX

�

sI +Q⇤D�1PQX��1

Q⇤D� 12 .

Clearly then it is su�cient to show that�

sI +Q⇤D�1PQX��1 2 H (n�m)⇥(n�m)

1 . (12)

The above can be immediately recognised as an eigenvaluecondition:

�s /2 ��

Q⇤D�1P (s)QX�

, 8s 2 C+. (13)

By Theorem 1.7.6 of Horn and Johnson (1991), for anys 2 C:��

Q⇤D�1P (s)QX�

⇢ Co�⇥

D�1⇤

ii

pi

(s) , i 2 N�

⇥ [✏, 1] .

In the above ⇥ denotes the product S1 ⇥ S2 = {ab : a 2S1, b 2 S2}. Therefore it is su�cient to show that

0 /2 Co�

s+⇥

D�1⇤

ii

pi

(s) , i 2 V�

⇥ [✏, 1], (14)

for all s = C+. Observe that since each pi

is bounded,this condition is trivially satisfied for large s. It is thereforeenough to check that this holds for s 2 C+, |s| < R, forsu�ciently large R. This can be done using the separatinghyperplane theorem, applied pointwise in s. In particular,(14) holds for any given s if and only if there exists anonzero h 2 C and � > 0 such that 8i 2 {i, . . . , n}:

Re�

h�

s+ k⇥

D�1⇤

ii

pi

(s)�

> �, 8 ✏ k 1. (15)

By a very minor adaptation of the argument in Theorem2 of Brockett and Willems (1965), we can use a functionto define this h pointwise in s. From the conditions ofthe theorem and the maximum modulus principle, for anyR � 0, there exists a � > 0 such that 8s 2 C+, |s| R:

Re�

h (s)�

s+⇥

D�1⇤

ii

pi

(s)�

> �. (16)

Since sh (s) is PR, for all k⇤ � 0,

Re�

h (s)�

s+⇥

D�1⇤

ii

pi

(s)�

+ k⇤sh (s)

> �.

Dividing through by (1 + k⇤) shows that under theseconditions

Re

⇢

h (s)

✓

s+1

1 + k⇤⇥

D�1⇤

ii

pi

(s)

◆�

> �.

Therefore (15) is satisfied on the entire right half planefor the required range of k values. Consequently (12) issatisfied, and the result follows.

3.2 A Plug and Play Interpretation

In order to give Theorem 3.2 a plug and play interpreta-tion, we have to introduce a little conservatism. In directanalogy with the approach pioneered in Lestas and Vin-nicombe (2006), we propose to make an a-priori choice ofthe function h (s). Observe that once we have done this,the conditions of Theorem 3.2 can be checked as follows:

(1) For each component, compute the smallest �i

suchthat (11) is satisfied.

(2) Check that 1�i

� [LB

]ii

.

Observe that the above process works completely indepen-dently of the size of the network, and is therefore highlyscalable. It is also entirely decentralised, giving a plug andplay requirement for individual components. In additionwe have the following appealing properties:

Robustness: Suppose we have an uncertain description ofour model p

i

, for example:

pi

(s) 2 {p (s) : p (s) = pi0 (s) +� (s) , k� (s)k1 < 1} .

Then provided we compute the smallest �i

such that (11)is satisfied for all p

i

in this set, we can guarantee stabilityin a way that is robust to this uncertainty. This hasnot compromised scalability, since this process remainsentirely decentralised.

Controller Design: Suppose pi

has some controller param-eters that can be specified, for example (as in (2))

pi

(s) =1

Mi

s+Di

✓

1 +ci

(s)

Mi

s+Di

◆�1

,

where ci

can be chosen. Then provided we can designci

so that pi

meets the plug and play requirement, thecomponent can be connected to the network withoutcompromising stability. Observe that this is an entirelydecentralised synthesis condition, and there is no needto redesign the controller in response to buses beingadded and removed elsewhere in the network. This is anapproach to synthesis that specifically addresses the needfor scalability.

The above are just observations about the structure ofthe stability tests. The fact that they are based on test-ing SPRness also brings a number of advantages. Morespecifically, provided the transfer functions in (11) arerational and proper, this condition can be e�ciently testedusing the Kalman-Yakobovich-Popov (KYP) lemma. Fur-thermore, the synthesis problem: design c

i

such that

h (s)

�i

2s+

1

Mi

s+Di

✓

1 +ci

(s)

Mi

s+Di

◆�1!

2 SPR;

is equivalent to an H1 optimisation problem, with statespace solution given in Sun et al. (1994). Finally if h (s)has relative degree 1, then (11) can be checked on the

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imaginary axis (for a rather complete list of the di↵erentcharacterisations of SPRness, see Wen (1988)). That is(11) is equivalent to

Ren

h (j!)⇣�

i

2j! + p

i

(j!)⌘o

> 0, 8! 2 R. (17)

This allows classical intuition from frequency responsemethods to be brought to bear on the above problems.

While obviously having many appealing features, all ofthe above rests on one crucial decision: the a-priori choiceof h (s). If an arbitrary choice is made, especially whenconducting stability analysis 2 , it is possible that thisapproach can be very conservative. We recommend thath (s) be chosen by defining a set of ‘expected models’

P = {p1 (s) , . . . , pm (s)} ,and then designing h (s) so that (11) is satisfied for all themodels in this set. Here p

k

are transfer functions describingthe dynamics of ‘typical’ bus models, perhaps obtainedby putting in average values of the model parameters.Provided m is not too large, this problem should beresolvable. The idea is that an h that is designed to workfor these transfer functions is a sensible ‘a-priori’ choice,since it guarantees that if the actual bus dynamics are ofthe ‘expected’ type, or can be designed to be close to theexpected type, they will satisfy the network protocol.

4. SCALABLE STABILITY ANALYSIS OF IDROOP

To motivate both the need for Theorem 3.2, and toillustrate its application, consider a two bus network.Assume that the buses have the same dynamics, given by

pi

=1

s+ 0.1

✓

1 +1

s+ 0.1ci

(s)

◆�1

, (18)

and the Laplacian matrix describing the transmissionnetwork is given by

LB

=

1 �1�1 1

�

.

We now consider the problem of how to design an iDroopcontroller subject to delay. That is, how to design

ci

(s) = e�s⌧iK⌫

i

s+K�

i

K

s+K�

i

,

where ⌧i

> 0 is the delay, such that the interconnection isstable. If the delay equals zero, then passivity theory canbe easily applied to answer the stability question. This isbecause (18) can be rearranged as

pi

=

✓

1

s+ 0.1

◆�1

+⇣

ci

(s)�1⌘�1

!�1

,

from which we see that pi

is the parallel interconnectionof c�1

i

and an SPR transfer function. This means thatif c

i

is SPR, so is pi

, which in turn implies that theinterconnection with the transmission network is stable(the transmission network is a PR transfer function, andit is well known that the feedback interconnection of a PRand SPR transfer function is stable). Hence we arrive atthe rather surprising conclusion that any possible choiceof iDroop controller will lead to a stable interconnection.2 This is of less concern for synthesis, because the extra degrees offreedom opened up by designing the controller can be exploited tomeet a wide range of protocols.

10�2 100 10210�3

10�2

10�1

100

! rad/s

Mag

nitude

10�2 100 102

�100

�50

0

! rad/s

Phase

Fig. 2. Bode diagram of pi

(solid curve), and first orderapproximation of the form in (19), with a = 1.37, b =1 (dashed curve).

This becomes even stranger when realise we can use thesame argument in networks of any size. Is it really truethat any possible set of heterogeneous buses, equippedwith arbitrarily chosen iDroop controllers, interconnectedthrough any possible network, is stable?

The answer is, at least in any practical sense, no. Tomake this absolutely concrete, let us return to the two busexample, and suppose we choose K⌫

i

= 1, K�

i

= 5, Ki

=30, as our iDroop parameters for both buses. A simpleNyquist argument shows that even a small value of ⌧

i

, forexample 0.05, will destabilise this two bus network. It is ofcourse not a limitation of iDroop that this can happen, noreven that surprising; the introduction of delays (and badlydesigned controllers) can similarly destabilise networkswhere the buses employ virtual inertia or droop controllers.It does however serve to illustrate the importance of robustdesign in the network setting. We will now demonstratethat, as claimed in Section 3.2, Theorem 3.2 can be usedto do this.

As discussed in Section 3.2, to obtain a plug and playcriterion, we need to make an a-priori choice of h. Let usjust pick

h (s) =1

s

!0+ 1

,

with !0 = 30 (!0 could be optimised based on expectedmodel parameters), and proceed with the robust synthesisof c

i

(s). Conducting a classical lead-lag design (this isanother advantage of the iDroop controller structure)allows us to achieve a p

i

which, frequency by frequency,is similar to a transfer function of the form

ai

s+ bi

. (19)

For a delay of ⌧i

= 0.5, this is illustrated in Figure 2,where the iDroop controller specified by K⌫

i

= 1.3, K�

i

=8, K

i

= 0.65, has been chosen. To conclude stability usingTheorem 3.2, we are required to verify that

1s

!0+ 1

✓

�i

2s+

ai

s+ bi

+�i

(s)

◆

2 SPR, (20)

for some �i

1. In the above �i

(s) is the di↵erencebetween (19) and the actual p

i

. A simple way to do this isto instead verify the condition

1s

!0+ 1

✓

�i

2s+

ai

s+ bi

◆

� ✏i

2 PR. (21)

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0 0.2 0.4 0.6 0.8 1 1.2 1.4

−0.8

−0.6

−0.4

−0.2

0

0.2

Fig. 3. Nyquist diagram of pi

(black curve), and theuncertainty set �

i

with ✏i

= 0.08 (shaded region).Since the Nyquist diagram lies inside the shadedregion, satisfying (21) with this ✏

i

is su�cient for (20).

Then, if�

�

�

�

�

1s

!0+ 1

�i

(s)

�

�

�

�

�

1

< ✏i

, () |�i

(j!)| < ✏i

s

1 +!2

!20

,

then (20) is satisfied. This relaxation is shown in Figure 3.

In addition to guaranteeing further levels of robustnessto unmodelled dynamics, (21) gives a simple way tocharacterises any allowable iDroop design in terms of onlythree parameters: (a

i

, bi

, ✏i

). This is an advantage becausethe PR condition in (21) can be rewritten as a set ofinequalities in terms of these parameters. This means thatour network protocol can be understood on the basis ofcoarse features of our iDroop design. Using the result ofe.g. Foster and Ladenheim (1963), it can be shown that(21) is equivalent to the following inequalities:

ai

� ✏i

bi

� 0bi

(�i

!0 � 2✏i

)� 2✏i

!0 � . . .

. . .!0

(bi

+ !0)

✓

p

ai

� ✏i

bi

�r

bi

⇣�i

!0

2� ✏

i

⌘

◆2

Provided the design of each individual iDroop controllersatisfies the above, the scalable stability tests are (ro-bustly) satisfied. We can easily check them for the twobus example. Substituting in (a

i

, bi

, ✏i

) = (1.37, 1, 0.08)shows that they are satisfied for any �

i

� 0.18, whichis clearly su�cient to conclude stability for this network(here [L

B

]ii

= 1). Note that if they were failed, we wouldthen just have to go back and retune the design of ouriDroop controller to obtain more favourable parameters(or test (20) instead). To stress the scalability of thisprocedure, observe that we did not use the fact that thiswas a two bus example at any stage. This same localmethod can be used for designing iDroop controllers forheterogeneous buses in networks of any size, and the aboveshows that this specific bus can be connected into anypossible transmission network provided the local aggregatenetwork susceptance is less than 1/0.18.

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