Grid-forming frequency shaping control
GGrid-forming frequency shaping control
Yan Jiang , Andrey Bernstein , Petr Vorobev , and Enrique Mallada Abstract — As power systems transit to a state of highrenewable penetration, little or no presence of synchronousgenerators makes the prerequisite of well-regulated frequencyfor grid-following inverters unrealistic. Thus, there is a trendto resort to grid-forming inverters which set frequency directly.We propose a novel grid-forming frequency shaping control thatis able to shape the aggregate system frequency dynamics into afirst-order one with the desired steady-state frequency deviationand Rate of Change of Frequency (RoCoF) after a suddenpower imbalance. The no overshoot property resulting from thefirst-order dynamics allows the system frequency to monoton-ically move towards its new steady-state without experiencingfrequency Nadir, which largely improves frequency security. Weprove that our grid-forming frequency-shaping control rendersthe system internally stable under mild assumptions. Theperformance of the proposed control is verified via numericalsimulations on a modified Icelandic Power Network test case.
I. I
NTRODUCTION
Power system frequency control by storage units has beena topic of extensive research over the last decade, especiallyunder the circumstances of the increasing penetration of re-newable generation. Compared to conventional synchronousgenerators, storage units have outstanding ramping capa-bilities, which makes them an ideal choice for provisionof various types of frequency control services. At present,special policies for storage participation in frequency controlservices are being developed by system operators aroundthe world [1], [2]. For instance, the existing rules of theEnhanced Frequency Response –program introduced by Na-tional Grid in Great Britain– already assume the power-frequency response with a gain of up to p.u. [3], farexceeding typical capabilities of synchronous generators ( - p.u.). Thus, with the fall of the power system inertia andprimary frequency reserves due to the increased penetrationof renewables, energy storage systems have a potential tobecome the major providers of frequency control services inthe future power systems.So far, synthetic inertia and droop response by storagedominate the scientific literature. These two services aresupposed to compensate for the falling system inertia andprimary reserves, and seem to be a logical solution underexisting grid codes. Typically, the storage units are supposedto realize the power-frequency type of response while beingin the so-called grid-following mode. That is, inverters of the This work was supported by any organization Y. Jiang and E. Mallada are with the Johns Hopkins University,Baltimore, MD 21218, USA. Emails: { yjiang,mallada } @jhu.edu A. Bernstein is with the National Renewable Energy Laboratory, Golden,CO 80401, USA. Email: [email protected] P. Vorobev is with the Skolkovo Institute of Science and Technology,Moscow 143026, Russia. Email:
[email protected] storage units measure the grid frequency and then inject (orconsume) power based on a particular control strategy. Suchan approach seems to be effective, yet the fact that there arecertain delays associated with inverter control systems posesa threat to the frequency security. These delays are originatedfrom the frequency measurement system – typically a phase-locked-loop (PLL), and also from inverter current control andpulse-width modulation (PWM) systems. It is foreseeablethat, in the future low-inertia grid, these delays (from severaldecades of milliseconds to hundreds of milliseconds) canbecome fatal to frequency security. As an example, duringthe already famous South Australian blackout of 2016, theRate of Change of Frequency (RoCoF) has hit the valuesas high as − [4]. Clearly, it becomes vital to developnew methods for storage participation in frequency controlso as to minimize any possible response delays.Grid-forming inverters [5] have recently attracted a lotof attention from the research community, mainly in thecontext of autonomous microgrids. Beneficially, this type ofinverters bring a broad range of new options for frequencycontrol. First, they naturally adjust power almost with no de-lays (apart from some electro-magnetic transients in filters).Second, new control options become available. For instance,inertial response can be realized without any low-pass filters(hence, even less delays), since in the grid-forming modethis type of control becomes strictly causal. Third, invertersin the grid-forming mode are much less susceptible to gridvoltage variations that often accompany frequency transients,which provides more reliability to the system. In the presentmanuscript, we explore a new approach for frequency controlrealized by grid-forming inverters – a topic that is not yetstudied sufficiently by both power and control communities.We propose a novel grid-forming frequency shaping con-trol that is inspired by its grid-following counterpart pro-posed in [6]. We first show that the proposed control isable to fashion the aggregate system frequency dynamics,a.k.a. Center of Inertia (CoI) Frequency, into a first-order onewith the desired steady-state frequency deviation and RoCoF(following a sudden power imbalance). Notably, a first-ordersystem frequency evolution naturally avoids overshoot sothat the frequency deviation moves towards its steady-stateincrementally without experiencing frequency Nadir, whichis what we mean by “Nadir elimination” hereafter. Nadirelimination largely improves the frequency security since itreduces the risk of under-frequency load shedding. We thenshow that the proposed control ensures the internal stabilityof the overall system under mild conditions by using thedecentralized stability criterion developed in [7], where thecrux of the matter is to check a positive realness (PR) [8] a r X i v : . [ ee ss . S Y ] S e p h ( s ) ˆ h i ( s ) ˆ h n ( s ) u P L B s Network Dynamics p e p in − + ω Bus Dynamics
Fig. 1. Block diagram of power network. requirement. We finally confirm the good performance ofthe proposed controller through numerical simulations on themodified Icelandic Power Network test case [9].II. P
OWER S YSTEM M ODEL
We consider a power network composed of n busesindexed by i ∈ N := { , . . . , n } and transmission linesdenoted by unordered pairs { i, j } ∈ E ⊂ {{ i, j } : i, j ∈N , i (cid:54) = j } . As illustrated by the block diagram in Fig. 1,the system dynamics are modeled as a feedback intercon-nection of bus dynamics and network dynamics. The inputsignals p in := ( p in ,i , i ∈ N ) ∈ R n represent power injectionchanges and the output signals ω := ( ω i , i ∈ N ) ∈ R n represent the bus frequency deviations from its nominalvalue. We now discuss the dynamic elements in more detail.
1) Bus Dynamics:
The set of buses N is a disjoint unionof the set of generator buses G and the set of inverter buses I , i.e., N = G (cid:93) I . The bus dynamics that map net powerbus imbalances u P := ( u P ,i , i ∈ N ) ∈ R n to frequencydeviations ω can be described by the transfer function matrix ˆ H ( s ) := diag(ˆ h i ( s ) , i ∈ N ) , where ˆ h i ( s ) is the transferfunction of either generator or inverter depending on whether i ∈ G or i ∈ I . a) Generator Dynamics: We consider generator dy-namics that are composed of the standard swing dynamicswith turbine droop, i.e., ˆ h i ( s ) = (cid:32) m i s + d i + r − ,i τ i s + 1 (cid:33) − , ∀ i ∈ G , (1)where m i > denotes the aggregate generator inertia, d i > the aggregate generator damping, τ i > the turbine timeconstant, and r t ,i > the turbine droop coefficient. b) Inverter Dynamics: We consider grid-forming in-verters, which set local grid frequency deviations ω i directlyas a function of their power output variation q r ,i = − u P ,i .The detailed function depends on the control law ˆ h i ( s ) employed to map u P ,i to ω i for buses with i ∈ I .
2) Network Dynamics:
The network power fluctuations p e := ( p e ,i , i ∈ N ) ∈ R n are given by a linearized model ofthe power flow equations [10]: ˆ p e ( s ) = L B s ˆ ω ( s ) , (2)where ˆ p e ( s ) and ˆ ω ( s ) denote the Laplace transforms of p e and ω , respectively. The matrix L B is an undirectedweighted Laplacian matrix of the network with elements L B ,ij = ∂ θ j n (cid:88) k =1 | V i || V k | b ik sin( θ i − θ k ) (cid:12)(cid:12)(cid:12) θ = θ . Here, θ := ( θ i , i ∈ N ) ∈ R n denotes the angle deviationfrom its nominal, θ := ( θ ,i , i ∈ N ) ∈ R n are the equilib-rium angles, | V i | is the (constant) voltage magnitude at bus i , and b ij is the line { i, j } susceptance.
3) Closed-Loop Dynamics:
We are interested in theclosed-loop response of the system in Fig. 1 from the powerinjection changes p in to frequency deviations ω , which canbe described by the transfer function matrix ˆ T ω p ( s ) := ˆ ω ( s )ˆ p in ( s ) = (cid:18) I n + ˆ H ( s ) L B s (cid:19) − ˆ H ( s ) . (3)It is in general tough to analyze or tune the perfor-mance of ˆ T ω p ( s ) . Nevertheless, when the system is tightly-connected [11], all buses exhibit a coherent response approx-imated by ˆ T ω p ( s ) ≈ ˆ h c ( s ) n Tn , (4)where n ∈ R n is the vector of all ones and ˆ h c ( s ) := (cid:32)(cid:88) i ∈G ˆ h − i ( s ) + (cid:88) i ∈I ˆ h − i ( s ) (cid:33) − . (5)Henceforth, we refer to ˆ h c ( s ) in (5) as the coherent dynamics of the network.III. G RID - FORMING F REQUENCY S HAPING C ONTROL
Motivated by (4), we focus in this paper on shapingthe response ˆ h c ( s ) , instead of (3). Thus, given generatordynamics ˆ h i ( s ) for buses with i ∈ G , our goal is todesign inverter dynamics ˆ h i ( s ) for buses with i ∈ I suchthat the coherent dynamics ˆ h c ( s ) is a first-order transferfunction with two degrees of freedom. Such a coherentdynamics actually naturally ensures Nadir elimination as wellas tunable steady-state frequency deviation and RoCoF, asthe following theorem formally states. Theorem 1 (Grid-forming frequency shaping control).
Consider generator dynamics ˆ h i ( s ) , i ∈ G , as in (1) . Then,the grid-forming inverter control law ˆ h i ( s ) = 1 m I ,i s + d I ,i − ˆ g I ,i ( s ) , ∀ i ∈ I , (6) We use hat to distinguish the Laplace transform from its time domaincounterpart. ith m I ,i , d I ,i > , renders a first-order coherent dynamics ˆ h c ( s ) = 1 as + b , (7) with a, b > given by a := (cid:88) i ∈I m I ,i + (cid:88) i ∈G m i , (8a) b := (cid:88) i ∈I d I ,i + (cid:88) i ∈G d i , (8b) if and only if (cid:88) i ∈I ˆ g I ,i ( s ) = (cid:88) i ∈G r − ,i τ i s + 1 . (9) In this case, the frequency deviations will experience noNadir and the steady-state frequency deviations ω ( ∞ ) andthe RoCoF | ˙ ω | ∞ will be determined by ω ( ∞ ) ≈ (cid:80) ni =1 u ,i b n and | ˙ ω | ∞ ≈ (cid:80) ni =1 u ,i a n , (10) when the system undergoes step power injection changes,i.e., p in = u t ≥ ∈ R n with u ∈ R n being any arbitraryvector direction and t ≥ being the unit-step function.Proof. Applying the desired coherent dynamics given by(7) and the generator transfer function given by (1) to thedefinition of coherent dynamics given by (5) yields as + b = (cid:88) i ∈G (cid:32) m i s + d i + r − ,i τ i s + 1 (cid:33) + (cid:88) i ∈I ˆ h − i ( s ) . Thus, the desired inverter control law should satisfy (cid:88) i ∈I ˆ h − i ( s ) = (cid:32) a − (cid:88) i ∈G m i (cid:33) s + (cid:32) b − (cid:88) i ∈G d i (cid:33) − (cid:88) i ∈G r − ,i τ i s + 1 . It is straightforward that the control law determined by (6),(8), and (9) guarantees that the above condition hold. Thisconcludes the proof of the first statement.Next, combining (3) and (4), we can see that the frequencydeviations ˆ ω ( s ) of the system ˆ T ω p in response to step powerinjection changes ˆ p in ( s ) = u /s is given by ˆ ω ( s ) = ˆ T ω p ( s )ˆ p in ( s ) ≈ ˆ h c ( s ) n Tn u s = n (cid:88) i =1 u ,i ˆ h c ( s ) s n , (11)which can be interpreted as that the frequency deviation oneach bus reacts to the aggregate step power injection changeof size (cid:80) ni =1 u ,i with the coherent dynamics ˆ h c ( s ) . Now,applying initial and final value theorems to (11) with ˆ h c ( s ) given by (7), we find that a and b satisfy the followingrelations: | ˙ ω | ∞ = lim s →∞ s ˆ ω ( s ) ≈ lim s →∞ s (cid:80) ni =1 u ,i s ( as + b ) n = (cid:80) ni =1 u ,i a n ,ω ( ∞ ) = lim s → s ˆ ω ( s ) ≈ lim s → s (cid:80) ni =1 u ,i s ( as + b ) n = (cid:80) ni =1 u ,i b n , which concludes the proof of (10).Clearly, given specific requirements on steady-state fre-quency and RoCoF, there are infinite ways of choosing m I ,i and d I ,i to satisfy (8). A straightforward choice is to set m I ,i = a − (cid:80) i ∈G m i |I| and d I ,i = b − (cid:80) i ∈G d i |I| , ∀ i ∈ I , (13)where |I| denotes the cardinality of I . Similarly, we proposethe following two strategies to meet (9). • Matching individual turbine dynamics by individualinverters:
Assume the cardinality of I is no less thanthat of G , i.e., |I| ≥ |G| . Let I t ⊂ I such that there isa bijection between I t and G that maps each j ∈ G todistinct i ∈ I t by the following relation ˆ g I ,i ( s ) = r − ,j τ j s + 1 . ∀ i ∈ I \ I t , simply set ˆ g I ,i ( s ) = 0 . • Distributing the first-order reduced order model of theaggregate turbine dynamics [12] over inverters:
Let z i ≥ , ∀ i ∈ I , be weighting parameters satisfying (cid:80) i ∈I z i = 1 . Set ˆ g I ,i ( s ) = z i ˜ r − (˜ τ s + 1) , ∀ i ∈ I , with ˜ r t and ˜ τ being the turbine droop coefficient andtime constant, respectively, of a first-order reduced ordermodel of (cid:88) i ∈G r − ,i τ i s + 1 . Tuning ˆ g I ,i ( s ) by distributing the first-order reduced ordermodel of the aggregate turbine dynamics over inverters seemsto be a more practical choice for two reasons. First, it getsrid of the need to accurately estimate droop coefficients andtime constants of all individual turbines. Second, it relaxesthe cardinality assumption |I| ≥ |G| . Remark 1 (Meeting frequency specifications (10) ). Choos-ing a and b to meet frequency specifications (10) naturallyasks for knowledge of the current network composition via (8) and (9) . The estimation of dynamic parameters, includingbut not limited to inertia, is currently an active researcharea [12]–[14]. This endorses our utilization of (10) forsafety specification. Arguably, whether (8) holds rigorouslyfor chosen a and b is not of major concern. We highlight thatthe proposed control always improve RoCoF for any positive m I ,i and steady-state for large enough d I ,i , ∀ i ∈ I . Remark 2 (Steady-state power output from grid-formingfrequency shaping control inverters).
It is easy to showfrom (1) , (5) , (6) , and (8) that the steady-state power outputfrom the proposed inverters depends on the relation between d i for i ∈ I and r − ,i for i ∈ G . Note that, if I = ∅ , then ˆ h c (0) = 1 / (cid:80) i ∈G (cid:0) d i + r − ,i (cid:1) ; otherwise ˆ h c (0) =1 / (cid:0)(cid:80) i ∈G d i + (cid:80) i ∈I d I ,i (cid:1) . Hence, as long as (cid:80) i ∈I d I ,i > (cid:80) i ∈G r − ,i , the collection of inverters will provide power inteady-state since the steady-state frequency deviation willbe reduced. Remark 3 (Freedom of resources allocation).
The coherentdynamics ˆ h c ( s ) depends merely on the summation of theinverse of grid-forming frequency shaping control transferfunctions ˆ h i ( s ) over i ∈ I , but not on the way of howthese control resources are distributed across the network.Although, in our discussion above, control resources aremainly equally distributed over inverters, there are actuallymany other possibilities. Thus, a promising future researchdirection will be the exploration of how to optimally allocatecontrol resources based on additional performance metricsthat may be of interest. Considering the two choices of ˆ g I ,i ( s ) suggested before,we make the following assumption on the form of ˆ g I ,i ( s ) . Assumption 1 (The form of ˆ g I ,i ( s ) ). ∀ i ∈ I , ˆ g I ,i ( s ) is inone of the two forms below, i.e., ˆ g I ,i ( s ) = 0 or ˆ g I ,i ( s ) = ρ i σ i s + 1 , (14) where ρ i , σ > . IV. S
TABILITY A NALYSIS
In this section, we show that the grid-forming frequencyshaping control given by (6) and (14) ensures internalstability of the overall system in Fig. 1 under mild conditionscompatible with (9). To this end, we first review somestandard concepts that play a role in our stability analysis.
Definition 1 ( H ∞ space [15]). H ∞ is the Hardy spaceof functions ˆ F ( s ) that are analytic in the open right-half complex plane C + with a bounded norm (cid:107) ˆ F (cid:107) ∞ :=sup s ∈ C + | ˆ F ( s ) | . Definition 2 (Positive real [8]).
A proper rational transferfunction matrix ˆ F ( s ) is called positive real (PR) if: • Poles of all elements of ˆ F ( s ) are in the closed left-halfcomplex plane C − . • For any ν ∈ R such that j ν is not a pole of any elementof ˆ F ( s ) , the matrix ˆ F ( j ν ) + ˆ F T ( − j ν ) is positivesemidefinite. • For any ν ∈ R such that j ν is a pole of someelement of ˆ F ( s ) , the pole j ν is simple and the residuematrix lim s → j ν ( s − j ν ) ˆ F ( s ) is positive semidefiniteHermitian.Here, j represents the imaginary unit that satisfies j = − . Remark 4 (Real rational subspace of H ∞ ). The realrational subspace of H ∞ consists of all proper real rationalstable transfer matrices. Thus, in order to check whether aproper real rational transfer function belongs to H ∞ or not,it is sufficient to check whether it is stable or not. Remark 5 (Applications of positive realness).
The positiverealness was originally introduced in electrical networksynthesis [16] and recently extended to mechanical networksynthesis [17]. Moreover, it has been applied a lot to stabilityanalysis for both linear and nonlinear systems.
We are now ready to conduct a stability analysis.
Theorem 2 (Internal stability under grid-forming fre-quency shaping control).
Let Assumption 1 hold. The sys-tem ˆ T ω p with (1) and (6) is internally stable if d I ,i > ρ i , ∀ i ∈ I with nonzero ˆ g I ,i ( s ) .Proof. According to the decentralized stability criterion pro-posed in [7], the system ˆ T ω p is internally stable if ∃ τ α , (cid:15) > such that γ i ˆ h i ( s ) ∈ Q , ∀ i ∈ N , (15)with Q := (cid:26) ˆ q ( s ) ∈ H ∞ (cid:12)(cid:12)(cid:12)(cid:12) ˆ q (0) (cid:54) = 0 , ss + τ α (cid:18)
1+ ˆ q ( s ) s (cid:19) − (cid:15) ∈ PR (cid:27) ,γ i := 2 n (cid:88) j =1 V i V j b ij , where V i and V j denote the maximum allowable voltagemagnitudes at endpoints of the line { i, j } . Thus, the keyis to check whether the condition in (15) holds for ˆ h i ( s ) , ∀ i ∈ N .Combining (6) and (14), we know that ∀ i ∈ I , ˆ h i ( s ) = 1 m I ,i s + d I ,i or ˆ h i ( s ) = (cid:18) m I ,i s + d I ,i − ρ i σ i s + 1 (cid:19) − . We begin with the later case, from which we get γ i ˆ h i ( s ) = γ i ( σ i s + 1) m I ,i σ i s + ( m I ,i + d I ,i σ i ) s + d I ,i − ρ i . (16)First, it is well-known that a second-order transfer functionis stable if all coefficients of its denominator have the samesign. Thus, m I ,i , d I ,i , σ i > , and d I ,i > ρ i , ∀ i ∈ I ,guarantee the stability of (16), i.e., γ i ˆ h i ( s ) ∈ H ∞ . Second,it is trivial to check that γ i ˆ h i (0) = γ i / ( d I ,i − ρ i ) (cid:54) = 0 . Lastbut not least, we need to show that ∃ τ α , (cid:15) > such that s + τ α (cid:20) s + γ i ( σ i s + 1) m I ,i σ i s +( m I ,i + d I ,i σ i ) s + d I ,i − ρ i (cid:21) − (cid:15) ∈ PR , which is equivalent to ξ ,i s + ξ ,i s + ξ ,i s + ξ ,i η ,i s + η ,i s + η ,i s + η ,i ∈ PR (17)with ξ ,i := γ i − ( d I ,i − ρ i ) τ α (cid:15) , (18a) ξ ,i := ( d I ,i − ρ i ) (1 − (cid:15) ) + γ i σ i − ( m I ,i + d I ,i σ i ) τ α (cid:15) , (18b) ξ ,i := ( m I ,i + d I ,i σ i ) (1 − (cid:15) ) − m I ,i σ i τ α (cid:15) , (18c) ξ ,i := m I ,i σ i (1 − (cid:15) ) , (18d) η ,i := ( d I ,i − ρ i ) τ α , (18e) η ,i := ( d I ,i − ρ i ) + ( m I ,i + d I ,i σ i ) τ α , (18f) η ,i := m I ,i + d I ,i σ i + m I ,i σ i τ α , (18g) η ,i := m I ,i σ i . (18h)We now show that (17) holds by performing the algebraictest for positive realness proposed in [18]. That is, for theondegenerate case, i.e., ( ξ ,i , ξ ,i , ξ ,i , ξ ,i ) T ∈ R ≥ and ( η ,i , η ,i , η ,i , η ,i ) T ∈ R ≥ \ with being the zerovector of size , the condition (17) holds if and only if ( ξ ,i + η ,i ) ( ξ ,i + η ,i ) ≥ ( ξ ,i + η ,i ) ( ξ ,i + η ,i ) . (19)We check the nonnegativity of all coefficients in (17) first.Suppose τ α > and < (cid:15) < . Clearly, it followsdirectly from m I ,i , d I ,i , σ i > , and d I ,i > ρ i , ∀ i ∈ I , that ξ ,i , η ,i , η ,i , η ,i , η ,i > . Also, for any given τ α > , ξ ,i , ξ ,i , ξ ,i > if (cid:15) is sufficiently small. Now we are readyto check whether (19) holds or not. Applying (18) to the lefthand side of (19) yields ( ξ ,i + η ,i ) ( ξ ,i + η ,i ) (20) = [( d I ,i − ρ i ) (2 − (cid:15) ) + γ i σ i + ( m I ,i + d I ,i σ i ) τ α (1 − (cid:15) )][( m I ,i + d I ,i σ i ) (2 − (cid:15) ) + m I ,i σ i τ α (1 − (cid:15) )] . Applying (18) to the right hand side of (19) yields ( ξ ,i + η ,i ) ( ξ ,i + η ,i ) (21) = [ γ i + ( d I ,i − ρ i ) τ α (1 − (cid:15) )] m I ,i σ i (2 − (cid:15) ) . Through standard algebra, using (20) and (21), we get ( ξ ,i + η ,i ) ( ξ ,i + η ,i ) − ( ξ ,i + η ,i ) ( ξ ,i + η ,i )= ( d I ,i − ρ i ) ( m I ,i + d I ,i σ i ) (2 − (cid:15) ) + ( m I ,i + d I ,i σ i ) τ α (2 − (cid:15) ) (1 − (cid:15) ) + γ i d I ,i σ i (2 − (cid:15) )+ [ γ i σ i + ( m I ,i + d I ,i σ i ) τ α (1 − (cid:15) )] m I ,i σ i τ α (1 − (cid:15) ) ≥ , for any sufficiently small (cid:15) , which means (19) holds. Thus,the required positive realness in (17) has been proved.Therefore, γ i ˆ h i ( s ) ∈ Q in this case.We then turn to the simple case where γ i ˆ h i ( s ) = γ i m I ,i s + d I ,i . (22)First, the stability of (22), i.e., γ i ˆ h i ( s ) ∈ H ∞ , follows fromthe fact that the only pole of it is − d I ,i /m I ,i < . Second, γ i ˆ h i (0) = γ i /d I ,i (cid:54) = 0 . As for the required positive realness,(22) can be considered as a special case of (16) with ρ i = 0 and σ i = 0 . Plugging ρ i = 0 and σ i = 0 into (18) gives ξ ,i , ξ ,i , ξ ,i , η ,i , η ,i , η ,i > , ξ ,i = η ,i = 0 , and ( ξ ,i + η ,i ) ( ξ ,i + η ,i ) − ( ξ ,i + η ,i ) ( ξ ,i + η ,i )= d I ,i m I ,i (2 − (cid:15) ) + m ,i τ α (2 − (cid:15) ) (1 − (cid:15) ) ≥ , for any sufficiently small (cid:15) , which lead to the requiredpositive realness. Therefore, γ i ˆ h i ( s ) ∈ Q in this case.Finally, from (1), we know that ∀ i ∈ G , γ i ˆ h i ( s ) = γ i ( τ i s + 1) m i τ i s + ( m i + d i τ i ) s + d i + r − ,i . (23)Observe that (23) and (16) have the same form except forsome minor sign differences. Thus, the proof of γ i ˆ h i ( s ) ∈ Q follows from a similar argument on (16). This concludes theproof that the system ˆ T ω p is internally stable. V. N UMERICAL I LLUSTRATIONS
In this section, we present simulation results that com-pare the novel grid-forming frequency shaping control withthe popular grid-forming virtual inertia control [19]. Thesimulations are conducted on the Icelandic Power Networkavailable in the Power Systems Test Case Archive [9].Instead of the linearized network model used in the analysis,the simulations are built upon a nonlinear setup includingnonlinear power flows and line losses. The original dynamicmodel contains generator buses and load buses, whoseunion is denoted as N . To mimic a low-inertia scenario, weonly keep generator buses that are equipped with turbinesout of original generator buses. Each of above generatorbuses is distinctly indexed by some i ∈ { , . . . , } := G here.We then randomly pick buses from the set N \G as inverterbuses. Each of above inverter buses is distinctly indexedby some i ∈ { , . . . , } := I here. The remaining busesare left as load buses denoted by L := N \ ( G ∪ I ) .For every generator bus i ∈ G , the aggregate generatorinertia m i , the turbine time constant τ i , and the turbine droopcoefficient r t ,i are directly obtained from the dataset. Inaddition, turbine governor deadbands are taken into accountsuch that turbines are only responsive to frequency deviationsexceeding ± .
036 Hz [20]. Given that the values of generatordamping coefficients are not provided by the dataset, weset d i = 1 p.u.. For every load buses i ∈ L , the dampingcoefficient is chosen as / of the mean of all generatordamping coefficients, i.e., ¯ d := ( (cid:80) i ∈G d i ) / |G| .The inverter control law on buses i ∈ I is either grid-forming virtual inertia (GF-VI) or grid-forming frequencyshaping (GF-FS). The GF-VI is modelled as ˆ h i ( s ) = 1 m v ,i s + d v ,i , ∀ i ∈ I , where m v ,i > is the virtual inertia constant and d v ,i > is the virtual damping constant. ∀ i ∈ I , we set m v ,i =¯ m := ( (cid:80) i ∈G m i ) / |G| and d v ,i = ¯ d . As for the GF-FS in(6), we only test the more practical tuning method suggestedin Section III, where ˆ g I ,i ( s ) is obtained by distributing thefirst-order reduced model of the aggregate turbine dynamicsover inverters. Thus, ∀ i ∈ I , we set m I ,i = ¯ m , d I ,i = ¯ d + ˜ r − and ˆ g I ,i ( s ) = ˜ r − τ s + 1) , which ensures that the RoCoF and steady-state frequencydeviations under GF-VI and GF-FS are the same so as toprovide a fair comparison. Note that, with this setting, thestability condition required in Theorem 2 is satisfied since d I ,i = ¯ d + ˜ r − / > ˜ r − / ρ i , ∀ i ∈ I .For the purpose of comparison, the frequency deviationof the system without inverters when there is a step changeof − . p.u. in power injection at a randomly picked busat time t = 1 s is provided in Fig. 2(a). The performancesof the system under the two inverter control laws are givenin Fig. 2(b) and Fig. 2(c). Some observations can be made.First, the system under GF-FS almost exhibits a first-ordercoherent dynamics as predicted by Theorem 1, while the (a) System without inverters, where inverters on buses i ∈ I are replaced byloads with damping coefficients given by ¯ d/ and the generator dampingis increased so as to exactly compensate the lost inverter damping (b) System with GF-VI inverters (c) System with GF-FS invertersFig. 2. Performance of the system when a − . p.u. step change in powerinjection is introduced to a randomly picked bus. system under GF-VI experiences a deep Nadir. Second, Nadirelimination via GF-FS only requires an acceptable amountof control effort.VI. C ONCLUSIONS AND F UTURE W ORK
A novel grid-forming frequency shaping control has beenproposed for inverter-based frequency control in low-inertiapower systems. The proposed control is able to force thesystem frequency to exhibit first-order coherent dynamicswith specified steady-state frequency deviations and RoCoFin response to sudden power injection changes. The key ben-efit of a first-order frequency response is that the frequencydeviations gradually evolve towards the final equilibriumwithout experiencing Nadir so as to improve frequencysecurity. The internal stability of the system is guaranteedby the proposed control under mild conditions. The perfor-mance of the proposed control is verified through numericalsimulations.Future work include: (i) developing a more advancedcontrol to achieve a second-order coherent dynamics with de-sired steady-state frequency deviations, RoCoF, and tunableNadir; (ii) investigating the problem of optimal allocationof the proposed control resources over the network; (iii) considering a more detailed inverter model to throw lightto device-level execution of the proposed control.R
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