Transient Performance of Power Systemswith Distributed Power-Imbalance Allocation Control

Kaihua Xia,∗, Hai Xiang Linb, Jan H. van Schuppenb

aSchool of Mathematics, Shandong University, Jinan, 250100, Shandong, ChinabDelft Institute of Applied Mathematics, Delft University of Technology, Van Mourik Broekmanweg 6, 2628 XE Delft, The Netherlands.

Abstract

We investigate the sensitivity of the transient performance of power systems controlled by Distributed Power-Imbalance AllocationControl (DPIAC) on the coefficients of the control law. We measure the transient performance of the frequency deviation and thecontrol cost by the H2 norm. Analytic formulas are derived for the H2 norm of the transient performance of a power system withhomogeneous parameters and with a communication network of the same topology as the power network. It is shown that thetransient performance of the frequency can be greatly improved by accelerating the convergence to the optimal steady state throughthe control gain coefficients, which however requires a higher control cost. Hence, in DPIAC, there is a trade-off between thefrequency deviation and the control cost which is determined by the control gain coefficients. In addition, by increasing one of thecontrol gain coefficients, the behavior of the state approaches that of a centralized control law. These analytical results are validatedthrough a numerical simulation of the IEEE 39-bus system in which the system parameters are heterogeneous.

Keywords: Secondary frequency control, Transfer matrix, H2 norm, Control gain coefficients, Overshoot.

1. Introduction

The power system is expected to keep the frequency withina small range around the nominal value so as to avoid damagesto electrical devices. This is accomplished by regulating theactive power injection of sources. Three forms of frequencycontrol can be distinguished from fast to slow timescales, i.e.,primary control, secondary control, and tertiary control [1, 2].Primary frequency control has a control objective to maintainthe synchronization of the frequency based on local feedbackat each power generator. However, the synchronized frequencyof the entire power system with primary controllers may stilldeviate from its nominal value. Secondary frequency controlrestores the synchronized frequency to its nominal value and isoperated on a slower time scale than primary control. Basedon the predicted power demand, tertiary control determines theset points for both primary and secondary control over a longerperiod than used in secondary control. The operating point isusually the solution of an optimal power flow problem.

The focus of this paper is on the secondary frequency whichhas been traditionally actuated by the passivity-based methodAutomatic Genertaion Control (AGC) for half a century. Re-cently, considering the on-line economic power dispatch in the

★This work is supported in part by the research project of the FundamentalResearch Funds of Shandong University under Grant 2018HW028 and in partby the Foundation for Innovative Research Groups of National Natural ScienceFoundation of China under Grant 61821004.

∗Corresponding authorEmail addresses: kxi@sdu.edu.cn (Kaihua Xi), H.X.Lin@tudelft.nl

(Hai Xiang Lin), jan.h.van.schuppen@tudelft.nl (Jan H. vanSchuppen)

secondary frequency control between all the controllers in thepower system [2], various control methods are proposed forthe secondary frequency control. These include passivity-basedcentralized control methods such as Gather-Broadcast Control[3], and distributed control methods such as the DistributedAverage Integral (DAI)[4], primal-dual algorithm based dis-tributed method such as the Economic Automatic GenerationControl (EAGC) [5], Unified Control [6] etc.. However, theprimary design objectives of these methods focus on the steadystate only. As investigated in our previous study in [7], thecorresponding closed-loop system may have a poor transientperformance even though the control objective of reaching thesteady state is achieved, e.g. [8, 7]. For example, from theglobal perspective of the entire power system, the passivitybased methods, e.g., AGC, GB and DAI, are actually a formof integral control. A drawback of integral control is that largeintegral-gain coefficients may result in extra oscillations dueto the overshoot of the control input while small gain coeffi-cients result in a slow convergence speed towards a steady state.For instance, an overshoot problem occurred after the blackoutwhich happened in UK in August 2019 [9].

The continuous increase of integration of renewable energyinto the power systems, which may bring serious fluctuations,asks for more attentions on the transient performance of thepower systems. For the secondary frequency control, the way toimprove the transient performance of the traditional methods isto tune the control gain coefficients either by obtaining satisfac-tory eigenvalues of the linearized closed-loop system or by us-ing a control law based on H2 or H∞ control synthesis [10, 11].However, besides the complicated computations, the improve-ment of the transient performance is still limited because it also

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depends on the structure of the control laws. In order to improvethe transient performance, sliding-mode-based control laws,e.g.,[12, 13] and fuzzy control-based control laws,e.g., [14]are proposed, which are all able to shorten the transient phasewithout the overshoot. However, those control laws use eithercentralized or decentralized control structure without consider-ing economic power dispatch. Concerning the transient perfor-mance and the balance of the advantages of the centralized anddistributed control structure, the authors have proposed a multi-level control method, Multi-Level Power-Imbalance AllocationControl (MLPIAC) [15] for the secondary frequency control,which is suitable for large scale power systems. There are twospecial cases of MLPIAC, a centralized control called Gather-Broadcast Power-Imbalance Allocation Control (GBPIAC),and a distributed control called Distributed Power-ImbalanceAllocation Control (DPIAC). Numerical study with comparisonto the existing methods show that the overshoot problem canbe avoided by MLPIAC, thus the transient performance can beimproved by accelerating the convergence of the state [7, 15].However, the analysis is incomplete, which lacks of quantifyingthe impact of these control coefficients on the transient perfor-mance.

The H2 norm of a time-invariant linear input-output systemreflects the response of the output to the input, which has beenwidely used to study the response of power systems to distur-bances, e.g., the performance analysis of secondary frequencycontrol methods in [16, 8, 17, 18], the optimal virtual inertiaplacement in Micro-Grids [19], and the cyber network designfor secondary frequency control [20].

In this paper, we focus on the distributed control law DPIAC,and analyze the impact of the control coefficients on the tran-sient performance of the frequency deviation and the controlcost after a disturbance. For comparison with DPIAC, we alsoinvestigate the transient performance of the centralized controlmethod GBPIAC. We measure the transient performance by theH2 norm. We will show analytically and numerically that (1)the transient performance can be improved by tuning the controlgain coefficients monotonically; (2) there is a trade-off betweenthe transient performance of frequency deviation and the con-trol cost, which is determined by the control coefficients; and(3) the performance of the distributed control approaches to thatof the centralized control as a gain coefficient is increased. Themain contributions of this paper are,

(i) analytic formulas for how the transient performance of thefrequency deviation and the control cost depends on thecontrol gain coefficients;

(ii) a numerical study of the transient performance and its de-pendence on the control gain coefficients.

The paper is organized as follows. We first introduce themodel model of the power system, GBPIAC and DPIAC in sec-tion 2, then formulate the problem of this paper with introduc-tion of the H2 norm in section 3. We calculate the correspond-ing H2 norms and analyze the impact of the control coefficientson the transient performance of the frequency deviation and thecontrol cost in section 4 and verify the analysis by simulationsin section 5. Finally, we conclude with remarks in section 6.

2. The secondary frequency control laws

The transmission network of a power system can be de-scribed by a graph G = (V, E) with nodes V and edgesE ⊆ V × V, where a node represents a bus and edge (𝑖, 𝑗)represents the direct transmission line between node 𝑖 and 𝑗 .The buses can connect to synchronous machines, frequency de-pendent power sources (or loads), or passive loads. We focuson a power system with loss-less transmission lines and denotethe susceptance of the transmission line by ��𝑖 𝑗 for (𝑖, 𝑗) ∈ E.The set of the buses of the synchronous machines, of fre-quency dependent power sources, of passive loads are denotedby V𝑀 ,V𝐹 ,V𝑃 respectively, thus V = V𝑀 ∪ V𝐹 ∪ V𝑃 . Thedynamics of the system can be modelled by the following Dif-ferential Algebraic Equations (DAEs), e.g., [3, 7],

¤\𝑖 = 𝜔𝑖 , 𝑖 ∈ V𝑀 ∪V𝐹 , (1a)

𝑀𝑖 ¤𝜔𝑖 = 𝑃𝑖 − 𝐷𝑖𝜔𝑖 −∑𝑗∈V

𝐾𝑖 𝑗 sin (\𝑖 − \ 𝑗 ) + 𝑢𝑖 , 𝑖 ∈ V𝑀 ,

(1b)

0 = 𝑃𝑖 − 𝐷𝑖𝜔𝑖 −∑𝑗∈V

𝐾𝑖 𝑗 sin (\𝑖 − \ 𝑗 ) + 𝑢𝑖 , 𝑖 ∈ V𝐹 ,

(1c)

0 = 𝑃𝑖 −∑𝑗∈V

𝐾𝑖 𝑗 sin (\𝑖 − \ 𝑗 ), 𝑖 ∈ V𝑃 , (1d)

where \𝑖 is the phase angle at node 𝑖, 𝜔𝑖 is the frequency devi-ation from the nominal value (e.g., 50 or 60 Hz), 𝑀𝑖 > 0 is themoment of inertia of the synchronous machine, 𝐷𝑖 > 0 is thedroop control coefficient, 𝑃𝑖 is the power supply (or demand),𝐾𝑖 𝑗 = ��𝑖 𝑗𝑉𝑖𝑉 𝑗 is the effective susceptance of the transmissionline between node 𝑖 and 𝑗 , 𝑉𝑖 is the voltage, 𝑢𝑖 is the input forthe secondary control. The nodes in V𝑀 and V𝐹 are assumedto be equipped with secondary frequency controllers, denotedby V𝐾 = V𝑀 ∪ V𝐹 . Since the control of the voltage andthe frequency can be decoupled when the transmission lines arelossless [21], we do not model the dynamics of the voltages andassume the voltages are constant which can be derived from apower flow calculation [1].

The synchronized frequency deviation can be expressed as

𝜔𝑠𝑦𝑛 =

∑𝑖∈V 𝑃𝑖 +

∑𝑖∈V𝐾 𝑢𝑖∑

𝑖∈V𝑀∪V𝐹 𝐷𝑖. (2)

The condition 𝜔𝑠𝑦𝑛 = 0 at a steady state can be satisfied bysolving the economic power dispatch problem in the secondaryfrequency control [2],

min𝑢𝑖 ∈𝑅

∑𝑖∈V𝐾

𝐽𝑖 (𝑢𝑖) (3)

𝑠.𝑡.∑𝑖∈V

𝑃𝑖 +∑𝑖∈V𝐾

𝑢𝑖 = 0,

where 𝐽𝑖 (𝑢𝑖) = 12𝛼𝑖𝑢

2𝑖

represents the control cost at node 𝑖. Theconstraints on the control input at each node is not included into(3) as in [2, 15] based on the assumption that the constraint of

2

the capacity is not triggered by the disturbances in the time-scale of the secondary frequency control. This assumption al-lows the use of the H2 norm in the transient performance anal-ysis.

A necessary condition for the solution of the optimizationproblem (3) is that the marginal costs 𝑑𝐽𝑖 (𝑢𝑖)/𝑑𝑢𝑖 of the nodesare all identical, i.e.,

𝛼𝑖𝑢𝑖 = 𝛼 𝑗𝑢 𝑗 , ∀ 𝑖, 𝑗 ∈ V𝐾 .

For a secondary control law with the objective of (3), it isrequired that the total control input 𝑢𝑠 (𝑡) =

∑𝑖∈V𝐾 𝑢𝑖 con-

verges to the unknown −𝑃𝑠 = −∑𝑖∈V 𝑃𝑖 and the marginal

costs achieve a consensus at the steady state. For the overviewof the secondary frequency control laws, see [22]. To obtaina good transient performance, a fast convergence of the con-trol inputs to the optimal solution of (3) is critical, which mayintroduce the overshoot problem. To avoid the overshoot prob-lem, MLPIAC has been proposed in [15] with two special cases,GBPIAC and DPIAC. In this paper, we focus on the impact ofthe control gain coefficients of DPIAC on the transient perfor-mance and compare with that of GBPIAC. The following as-sumption on the connectivity of the communication network isrequired to realize the coordination control.

Assumption 2.1. For the power system (1), there exists a undi-rected communication network such that all the nodes in V𝐾are connected.

The definition of the distributed method DPIAC and the cen-tralized method GBPIAC follow.

Definition 2.2 (GBPIAC). Consider the power system (1),the Gather-Broadcast Power-Imbalance Allocation Control(GBPIAC) law is defined as [15]

¤[𝑠 =∑

𝑖∈V𝑀∪V𝐹𝐷𝑖𝜔𝑖 , (4a)

¤b𝑠 = −𝑘1 (∑𝑖∈V𝑀

𝑀𝑖𝜔𝑖 + [𝑠) − 𝑘2b𝑠 , (4b)

𝑢𝑖 =𝛼𝑠

𝛼𝑖𝑘2b𝑠 , 𝑖 ∈ V𝐾 , (4c)

where [𝑠 ∈ R, b𝑠 ∈ R are state variables of the central con-troller, 𝑘1, 𝑘2 are positive control gain coefficients, 𝛼𝑖 is thecontrol price at node 𝑖 as defined in the optimization problem(3), 𝛼𝑠 = (∑𝑖∈V𝐾 1/𝛼𝑖)−1 is a constant.

Definition 2.3 (DPIAC). Consider the power system (1), de-fine the Distributed Power-Imbalance Allocation Control(DPIAC) law as,

¤[𝑖 = 𝐷𝑖𝜔𝑖 + 𝑘3

∑𝑗∈V𝐾

𝑙𝑖 𝑗 (𝑘2𝛼𝑖b𝑖 − 𝑘2𝛼 𝑗b 𝑗 ), (5a)

¤b𝑖 = −𝑘1 (𝑀𝑖𝜔𝑖 + [𝑖) − 𝑘2b𝑖 , (5b)𝑢𝑖 = 𝑘2b𝑖 , (5c)

for node 𝑖 ∈ V𝐾 , where [𝑖 ∈ R and b𝑖 ∈ R are state variablesof the local controller at node 𝑖, 𝑘1, 𝑘2 and 𝑘3 are positive gain

coefficients, (𝑙𝑖 𝑗 ) defines a weighted undirected communicationnetwork with Laplacian matrix (𝐿𝑖 𝑗 )

𝐿𝑖 𝑗 =

{−𝑙𝑖 𝑗 , 𝑖 ≠ 𝑗 ,∑𝑘≠𝑖 𝑙𝑖𝑘 , 𝑖 = 𝑗 ,

and 𝑙𝑖 𝑗 ∈ [0,∞) is the weight of the communication line con-necting node 𝑖 and 𝑗 . The marginal cost at node 𝑖 is 𝛼𝑖𝑢𝑖 =

𝑘2𝛼𝑖b𝑖 .

Without the coordination on the marginal costs, DPIAC re-duces to a decentralized control method as follows.

Definition 2.4 (DecPIAC). Consider the power system (1), theDecentralized Power-Imbalance Allocation Control (DecPIAC)is defined as,

¤[𝑖 = 𝐷𝑖𝜔𝑖 , (6a)¤b𝑖 = −𝑘1 (𝑀𝑖𝜔𝑖 + [𝑖) − 𝑘2b𝑖 , (6b)𝑢𝑖 = 𝑘2b𝑖 , (6c)

for node 𝑖 ∈ V𝐾 , where 𝑘1 and 𝑘2 are positive gain coefficients.

For a fast recovery from an imbalance while avoiding theovershoot problem, the control gain coefficient 𝑘2 should sat-isfy 𝑘2 ≥ 4𝑘1. For details of the configuration of 𝑘1 and 𝑘2, werefer to [15]. In this paper, we set 𝑘2 = 4𝑘1 in the followinganalysis so as to simplify the deduction of the explicit formulawhich shows the impact of the control coefficients on the tran-sient performance. For the control procedure and the asymp-totic stability of GBPIAC, DPIAC, see [15]. Fo the control lawMLPIAC, see [15].

3. Problem formulation

With 𝑘2 = 4𝑘1, there are two control gain coefficients 𝑘1 and𝑘3 in DPIAC. We focus on the following problem.

Problem 3.1. How do the coefficients 𝑘1 and 𝑘3 influence thetransient performance of the frequency deviation and controlcost in the system (1) controlled by DPIAC?

To address Problem 3.1, we introduce the H2 norm to mea-sure the transient performance, which is defined as follows.

Definition 3.2. Consider a linear time-invariant system,

¤𝒙 = 𝑨𝒙 + 𝑩𝒘, (7a)𝒚 = 𝑪𝒙, (7b)

where 𝒙 ∈ R𝑛, 𝑨 ∈ R𝑛×𝑛 is Hurwitz, 𝑩 ∈ R𝑛×𝑚, 𝑪 ∈ R𝑧×𝑛,the input is denoted by 𝒘 ∈ R𝑚 and the output of the system isdenoted by 𝒚 ∈ R𝑧 . The squared H2 norm of the transfer matrix𝑮 of the mapping (𝑨, 𝑩,𝑪) from the input 𝒘 to the output 𝒚 isdefined as

| |𝑮 | |22 = tr(𝑩𝑇𝑸𝑜𝑩) = tr(𝑪𝑸𝑐𝑪𝑇 ), (8a)

𝑸𝑜𝑨 + 𝑨𝑇𝑸𝑜 + 𝑪𝑇𝑪 = 0, (8b)

𝑨𝑸𝑐 + 𝑸𝑐𝑨𝑇 + 𝑩𝑩𝑇 = 0, (8c)

3

where tr(·) denotes the trace of a matrix, 𝑸𝑜,𝑸𝑐 ∈ R𝑛×𝑛 are theobservability Grammian of (𝑪, 𝑨) and controllability Gram-mian of (𝑨, 𝑩) respectively [23],[24, chapter 2].

The H2 norm can be interpreted as follows. When theinput 𝒘 is modeled as the Gaussian white noise such that𝑤𝑖 ∼ 𝑁 (0, 1) for all 𝑖 = 1, · · · , 𝑚 and for all 𝑖 ≠ 𝑗 , thescalar Brownian motions 𝑤𝑖 and 𝑤 𝑗 are independent, the ma-trix 𝑸𝑣 = 𝑪𝑸𝑐𝑪

𝑇 is the variance matrix of the output at thesteady state [25, Theorem 1.53], i.e.,

𝑸𝑣 = lim𝑡→∞

𝐸 [𝒚(𝑡)𝒚𝑇 (𝑡)]

where 𝐸 [·] denotes the expectation. Thus

‖𝑮‖22 = tr(𝑸𝑣 ) = lim

𝑡→∞𝐸 [𝒚(𝑡)𝑇 𝒚(𝑡)] . (9)

For other interpretations, see [26].There are so many parameters which influence the transient

performance of the system that it is hard to deduce an explicitformula of the H2 norm for the closed-loop system when theparameters are heterogeneous. To simplify the analysis and fo-cus on the impact of the control gain coefficients, we make thefollowing assumption.

Assumption 3.3. For GBPIAC and DPIAC, assume that V𝐹 =

∅, V𝑃 = ∅ and for all 𝑖 ∈ V𝑀 , 𝑀𝑖 = 𝑚 > 0, 𝐷𝑖 = 𝑑 > 0, 𝛼𝑖 =1. For DPIAC, assume that the topology of the communicationnetwork is the same as the one of the power system such that𝑙𝑖 𝑗 = 𝐾𝑖 𝑗 for all (𝑖, 𝑗) ∈ E.

The frequency dependent nodes are excluded from the modelby this assumption while the nodes of the passive power loadscan be involved into the model in this assumption by Kron Re-duction [27]. From the practical point of view, the analysis withAssumption 3.3 is valuable because it provides us the insight onhow to improve the transient behavior by tuning the control co-efficients. For the general case without the restriction of thisassumption, in which the model includes the frequency depen-dent nodes, we resort to simulations in Section 5.

Since \𝑖 𝑗 = \𝑖 − \ 𝑗 is usually small for relatively small𝑃𝑖 compared to the line capacity in practice, we approximatesin \𝑖 𝑗 by \𝑖 𝑗 as in e.g., [4, 5] to focus on the transient perfor-mance. With Assumption 3.3, rewriting (1) into a vector formby replacing sin \𝑖 𝑗 by \𝑖 𝑗 , we obtain

¤𝜽 = 𝝎 (10a)𝑴 ¤𝝎 = −𝑳𝜽 − 𝑫𝝎 + 𝑩𝒘 + 𝒖, (10b)

where 𝜽 = col(\𝑖) ∈ R𝑛, 𝑛 denotes the number of nodes in thenetwork, 𝝎 = col(𝜔𝑖) ∈ R𝑛, 𝑴 = diag(𝑀𝑖) ∈ R𝑛×𝑛, 𝑳 ∈ R𝑛×𝑛is the Laplacian matrix of the network, 𝑫 = diag(𝐷𝑖) ∈ R𝑛×𝑛,𝒖 = col(𝑢𝑖) ∈ R𝑛×𝑛. The disturbances of 𝑷 = col(𝑃𝑖) ∈ R𝑛have been modeled by 𝑩𝒘 as the inputs with 𝑩 ∈ R𝑛×𝑛 and𝒘 ∈ R𝑛 as in Definition 3.2. Here, col(·) denotes the columnvector of the indicated elements and diag(𝛽𝑖) denotes a diago-nal matrix 𝜷 = diag({𝛽𝑖 , 𝑖 · · · 𝑛}) ∈ R𝑛×𝑛 with 𝛽𝑖 ∈ R. Denotethe identity matrix by 𝑰𝑛 ∈ R𝑛×𝑛 and the 𝑛 dimensional vectorwith all elements equal to one by 1𝑛.

The transient performance of 𝝎(𝑡) and 𝒖(𝑡) are measured bythe H2 norm of the corresponding transfer functions with input𝒘 and output 𝒚 = 𝝎 and 𝒚 = 𝒖 respectively. The squared H2norms are denoted by | |𝑮𝑖 (𝝎, 𝒘) | |22 and | |𝑮𝑖 (𝒖, 𝒘) | |22 where thesub-index 𝑖 = 𝑐 or 𝑑 which refers to the centralized methodGBPIAC or the distributed method DPIAC.

4. The transient performance analysis

In this section, we calculate the H2 norms of the frequencydeviation and of the control cost for GBPIAC and DPIAC.

Lemma 4.1. For a symmetric Laplacian matrix 𝑳 ∈ R𝑛×𝑛,there exist an invertible matrix 𝑸 ∈ R𝑛×𝑛 such that

𝑸−1 = 𝑸𝑇 , (11a)

𝑸−1𝑳𝑸 = 𝚲, (11b)

𝑸1 =1√𝑛

1𝑛, (11c)

where 𝑸 = [𝑸1, · · · ,𝑸𝑛], 𝚲 = diag(_𝑖) ∈ R𝑛×𝑛, 𝑸𝑖 ∈ R𝑛 isthe normalized eigenvector of 𝑳 corresponding to eigenvalue_𝑖 , thus 𝑸𝑇

𝑖𝑸 𝑗 = 0 for 𝑖 ≠ 𝑗 . Because 𝑳1𝑛 = 0, _1 = 0 is one

of the eigenvalues with normalized eigenvector 𝑸1.

We study the transient performance of 𝝎 and 𝒖 of GBPIACand DPIAC in subsection 4.1 and 4.2 respectively by calculat-ing the corresponding H2 norm. In addition, for DPIAC, wealso calculate a H2 norm which measures the coherence of themarginal costs. The performance of GBPIAC and DPIAC willbe compared in subsection 4.3.

4.1. Transient performance analysis for GBPIAC

By Assumption 3.3, we derive the control input 𝑢𝑖 = 1𝑛𝑘1b𝑠 at

node 𝑖 as in (4). With the notations of section 3 and Assumption3.3, and 𝑘2 = 4𝑘1, we obtain from (10) and (4) the closed-loopsystem of GBPIAC in a vector form as follows.

¤𝜽 = 𝝎, (12a)

𝑚𝑰𝑛 ¤𝜔 = −𝑳𝜽 − 𝑑𝑰𝑛𝜔 + 4𝑘1b𝑠

𝑛1𝑛 + 𝑩𝒘, (12b)

¤[𝑠 = 𝑑1𝑇𝑛𝝎, (12c)¤b𝑠 = −𝑘1𝑚1𝑇𝑛𝝎 − 𝑘1[𝑠 − 4𝑘1b𝑠 , (12d)

where [𝑠 ∈ R and b𝑠 ∈ R.For the transient performance of 𝝎(𝑡), 𝒖(𝑡) in GBPIAC, the

following theorem can be proved.

Theorem 4.2. Consider the closed-loop system (12) ofGBPIAC with 𝑩 = 𝑰𝑛. The squared H2 norm of the frequencydeviation 𝝎 and of the control inputs 𝒖 are,

| |𝑮𝑐 (𝝎, 𝒘) | |22 =𝑛 − 12𝑚𝑑

+ 𝑑 + 5𝑚𝑘1

2𝑚(2𝑘1𝑚 + 𝑑)2 , (13a)

| |𝑮𝑐 (𝒖, 𝒘) | |22 =𝑘1

2. (13b)

4

Proof: With the linear transform 𝒙1 = 𝑸−1𝜽 , 𝒙2 = 𝑸−1𝝎 where𝑸 is defined in Lemma 4.1, we derive from (12) that

¤𝒙1 = 𝒙2,

¤𝒙2 = − 1𝑚𝚲𝒙1 −

𝑑

𝑚𝑰𝑛𝒙2 +

4𝑘1b𝑠

𝑚𝑛𝑸−11𝑛 +

1𝑚𝑸−1𝒘,

¤[𝑠 = 𝑑1𝑇𝑛𝑸𝒙2,

¤b𝑠 = −𝑘1𝑚1𝑇𝑛𝑸𝒙2 − 𝑘1[𝑠 − 4𝑘1b𝑠 ,

where 𝚲 is the diagonal matrix defined in Lemma 4.1. Since1𝑛 is an eigenvector of 𝑳 corresponding to _1 = 0, we obtain𝑸−11𝑛 = [

√𝑛, 0, · · · , 0]𝑇 . Thus the components of 𝒙1 and 𝒙2

can be decoupled as

¤𝑥11 = 𝑥21, (15a)

¤𝑥21 = − 𝑑𝑚𝑥21 +

4𝑘1

𝑚√𝑛b𝑠 +

1𝑚𝑸𝑇1 𝒘, (15b)

¤[𝑠 = 𝑑√𝑛𝑥21, (15c)

¤b𝑠 = −𝑘1𝑚√𝑛𝑥21 − 𝑘1[𝑠 − 4𝑘1b𝑠 (15d)

and for 𝑖 = 2, · · · , 𝑛,

¤𝑥1𝑖 = 𝑥2𝑖 , (16a)

¤𝑥2𝑖 = −_𝑖𝑚𝑥1𝑖 −

𝑑

𝑚𝑥2𝑖 +

1𝑚𝑸𝑇𝑖 𝒘, (16b)

We rewrite the decoupled systems of (15) and (16) in the gen-eral form as (7) with

𝒙 =

𝒙1𝒙2[𝑠b𝑠

, 𝑨 =

0 𝑰𝑛 0 0− 𝚲𝑚

− 𝑑𝑚𝑰𝑛 0 4𝑘1

𝑚√𝑛𝒗

0 𝑑√𝑛𝒗𝑇 0 0

0 −𝑘1𝑚√𝑛𝒗𝑇 −𝑘1 −4𝑘1

, �� =

0

𝑸−1

𝑚

00

,where 𝒗𝑇 = [1, 0, · · · , 0] ∈ R𝑛. The H2 norm of a state vari-able e.g., the frequency deviation and the control cost, can bedetermined by setting the output 𝑦 as that state variable. Be-cause the closed-loop system (12) is asymptotically stable, 𝑨 isHurwitz regardless the rotations of the phase angle 𝜽 .

For the transient performance of 𝝎(𝑡), setting 𝒚 = 𝝎 = 𝑸𝒙2and 𝑪 = [0,𝑸, 0, 0], we obtain the observability Grammian 𝑸𝑜of (𝑪, 𝑨) (8b) in the form,

𝑸𝑜 =

𝑸𝑜11 𝑸𝑜12 𝑸𝑜13 𝑸𝑜14𝑸𝑇𝑜12 𝑸𝑜22 𝑸𝑜23 𝑸𝑜24

𝑸𝑇𝑜13 𝑸𝑇

𝑜23 𝑸𝑜33 𝑸𝑜34𝑸𝑇𝑜14 𝑸𝑇

𝑜24 𝑸𝑇𝑜34 𝑄𝑜44

.Thus,

‖𝑮𝑐 (𝝎, 𝒘)‖22 = tr(��𝑇𝑸𝑜 ��) =

tr(𝑸𝑸𝑜22𝑸𝑇 )

𝑚2 =tr(𝑸𝑜22)𝑚2 . (17)

Because

𝑪𝑇𝑪 =

0 0 0 00 𝑰𝑛 0 00 0 0 00 0 0 0

,

the diagonal elements 𝑸𝑜22 (𝑖, 𝑖) of 𝑸𝑜22 can be calculated bysolving the observability Gramian ��𝑖 of (𝑪𝑖 , 𝑨𝑖) which satis-fies

��1𝑨1 + 𝑨𝑇1 ��1 + 𝑪𝑇1 𝑪1 = 0

where

𝑨1 =

0 1 0 00 − 𝑑

𝑚0 4𝑘1

𝑚√𝑛

0 𝑑√𝑛 0 0

0 −𝑘1𝑚√𝑛 −𝑘1 −4𝑘1

,𝑪𝑇1 =

0100

and

��𝑖𝑨𝑖 + 𝑨𝑇𝑖 ��𝑖 + 𝑪𝑇𝑖 𝑪𝑖 = 0, 𝑖 = 2, · · · , 𝑛,

where

𝑨𝑖 =

[0 1

−_𝑖𝑚

− 𝑑𝑚

],𝑪𝑇𝑖 =

[01

].

In this case, the diagonal elements of 𝑸𝑜22 satisfy 𝑸𝑜22 (𝑖, 𝑖) =��𝑖 (2, 2) for 𝑖 = 1, · · · , 𝑛. We thus derive from the observabilityGramian ��𝑖 that

tr(��𝑇𝑸𝑜 ��) =𝑛 − 12𝑚𝑑

+ 𝑑 + 5𝑚𝑘1

2𝑚(2𝑘1𝑚 + 𝑑)2 , (18)

which yields (13a) directly. Similarly by setting 𝑦 = 𝑢𝑠 (𝑡) =

4𝑘1b𝑠 (𝑡) and 𝑪 = [0, 0, 0, 4𝑘1], we derive the norm of 𝑢𝑠 (𝑡) as

‖𝑮𝑐 (𝑢𝑠 , 𝒘)‖22 =

𝑘1𝑛

2. (19)

With 𝑢𝑖 =𝑢𝑠𝑛

for 𝑖 = 1, · · · , 𝑛 and ‖𝒖(𝑡)‖2 = 𝒖(𝑡)𝑇 𝒖(𝑡) =∑𝑛𝑖 𝑢

2𝑖(𝑡), we further derive(13b) for the control cost. �

The norm of 𝝎(𝑡) in (13a) includes two terms. The first onedescribes the relative deviations which depend on the primarycontrol, and the second one describes the overall frequency de-viation of which the suppression is the task of the secondarycontrol. From the proof of Theorem 4.2, it can be observed thatthe relative frequency deviations are derived from the eigen-direction of nonzero eigenvalues, and the overall frequency de-viation from the eigen-direction of the zero eigenvalue.

Remark 4.3. It is demonstrated by Theorem 4.2 that the topol-ogy of the network has no influence on the norm of 𝝎(𝑡) and𝒖(𝑡). This is because of Assumption 3.3 and the identicalstrength of the disturbances at all the nodes with 𝑩 = 𝑰𝑛.

Remark 4.4. The overall frequency deviation depends on 𝑘1while the relative frequency deviation is independent of 𝑘1.Hence, the frequency deviation cannot be suppressed to an ar-bitrary small positive value. When the overall frequency de-viation caused by the power imbalance dominates the relativefrequency deviation, a large 𝑘1 can accelerate the restorationof the frequency. This will be further described in Section 5.However, a large 𝑘1 leads to a high control cost. Hence there isa trade-off between the overall frequency deviation suppressionand control cost, which is determined by 𝑘1.

5

Theorem 4.2 with 𝑩 = 𝑰𝑛 includes the assumption that allthe disturbances are independent and of identical strength. Thefollowing theorem describes the impact of the control coeffi-cients with a general 𝑩 where the disturbances are correlatedwith non-identical strength.

Theorem 4.5. Consider the closed-loop system (12) ofGBPIAC with a positive definite 𝑩 ∈ R𝑛×𝑛, the squared H2norm of the frequency deviation 𝝎 and of the control inputs 𝒖satisfy

𝛾2min𝐺𝑐 ≤ ||𝑮𝑐 (𝝎, 𝒘) | |22 ≤ 𝛾2

max𝐺𝑐 , (20a)𝑘1

2𝛾2

min ≤ ||𝑮𝑐 (𝒖, 𝒘) | |22 ≤ 𝑘1

2𝛾2

max, (20b)

where 𝛾min and 𝛾max are the smallest and the largest eigenvalueof 𝑩, and

𝐺𝑐 =(𝑛 − 1

2𝑚𝑑+ 𝑑 + 5𝑚𝑘1

2𝑚(2𝑘1𝑚 + 𝑑)2

).

Proof: Since 𝑩 > 0, then there exist 𝑼 and 𝚪 such that 𝑩 =

𝑼𝑇 𝚪𝑼 where 𝑼 is an orthogonal matrix and 𝚪 is a diagonalmatrix with the diagonal elements being the eigenvalues of 𝑩,which are all strictly positive. With the decomposition of 𝑩,from (17) we derive

trac(𝑩𝑇𝑸𝑸𝑜22𝑄𝑩

)= trac

(𝑼𝑇 𝚪𝑼𝑸𝑸𝑜22𝑸

𝑇𝑼𝑇 𝚪𝑼)

= trac(𝚪𝑼𝑸𝑸𝑜22𝑸

𝑇𝑼𝑇 𝚪)

≤ 𝛾2maxtrac

(𝑼𝑸𝑸𝑜22𝑸

𝑇𝑼𝑇)

= 𝛾2maxtrac

(𝑸𝑜22

),

Similarly, we derive

𝛾2mintrac

(𝑸𝑜22

)≤ trac

(𝑩𝑇𝑸𝑸𝑜22𝑸

𝑇 𝑩).

From the above inequalities and (17), we can easily follow theprocedure as in the proof of Theorem 4.2 to obtain the traceof 𝑄𝑜22 and further derive the inequalities in (20a). A similarprocedure is conducted to obtain the inequalities in (20b). �

It is demonstrated by Theorem 4.5 that when the disturbancesare correlated with non-identical strength, the impact of 𝑘1 onthe norm is similar as in the case with identical strength of dis-turbances.

4.2. Transient performance analysis for DPIAC

With Assumption 3.3 and 𝑘2 = 4𝑘1, we derive the closed-loop system of DPIAC from (10) and (5) as

¤𝜽 = 𝝎, (23a)𝑚𝑰𝑛 ¤𝝎 = −𝑳𝜽 − 𝑑𝑰𝑛𝝎 + 4𝑘1𝝃 + 𝑩𝒘 (23b)

¤𝜼 = 𝑫𝝎 + 4𝑘1𝑘3𝑳𝝃, (23c)¤𝝃 = −𝑘1𝑴𝝎 − 𝑘1𝜼 − 4𝑘1𝝃, (23d)

where 𝜼 = col([𝑖) ∈ R𝑛 and 𝝃 = col(b𝑖) ∈ R𝑛. Note that𝑳 = (𝐿𝑖 𝑗 ) ∈ R𝑛×𝑛 is the weighted Laplacian matrix of the

power network and also of the communication network. Be-cause the differences of the marginal costs can be fully repre-sented by 4𝑘1𝑳𝝃 (𝑡), we use the squared norm of (4𝑘1𝑳𝝃 (𝑡)) tomeasure the coherence of the marginal costs in DPIAC. Denotethe squared H2 norm of the transfer matrix of (23) with output𝒚 = 4𝑘1𝑳𝝃 by | |𝐺𝑑 (4𝑘1𝐿b, 𝒘) | |2. In this subsection, we alsocalculate the squared H2 norm of 4𝑘1𝑳𝝃 as an additional metricfor the influence of 𝑘3 on the control cost.

The following theorem states the H2 norms of the frequencydeviation, the control cost and the coherence of the marginalcosts in DPIAC.

Theorem 4.6. Consider the closed-loop system (23) of DPIACwith 𝑩 = 𝑰𝑛, the squared H2 norm of 𝝎(𝑡), 𝒖(𝑡) and 4𝑘1𝑳𝝃are

‖𝑮𝑑 (𝝎, 𝒘)‖22 =

12𝑚

𝑛∑𝑖=2

𝑏1𝑖

𝑒𝑖+ 𝑑 + 5𝑚𝑘1

2𝑚(2𝑘1𝑚 + 𝑑)2 ,

(24a)

‖𝑮𝑑 (𝒖, 𝒘)‖22 =

𝑘1

2+

𝑛∑𝑖=2

𝑏2𝑖

𝑒𝑖, (24b)

‖𝑮𝑑 (4𝑘1𝑳𝝃, 𝒘)‖22 =

𝑛∑𝑖=2

_2𝑖𝑏2𝑖

𝑚2𝑒𝑖, (24c)

where

𝑏1𝑖 = _2𝑖 (4𝑘2

1𝑘3𝑚 − 1)2 + 4𝑑𝑚𝑘31

+ 𝑘1 (𝑑 + 4𝑘1𝑚) (4𝑑_𝑖𝑘1𝑘3 + 5_𝑖 + 4𝑑𝑘1),𝑏2𝑖 = 2𝑑𝑘3

1 (𝑑 + 2𝑘1𝑚)2 + 2_𝑖𝑘41𝑚

2 (4𝑘1𝑘3𝑑 + 4),𝑒𝑖 = 𝑑_

2𝑖 (4𝑘2

1𝑘3𝑚 − 1)2 + 16𝑑_𝑖𝑘41𝑘3𝑚

2 + 𝑑2_𝑖𝑘1

+ 4𝑘1 (𝑑 + 2𝑘1𝑚)2 (𝑑𝑘1 + _𝑖 + 𝑑_𝑖𝑘1𝑘3).

Proof: Let 𝑸 ∈ R𝑛×𝑛 be defined as in Lemma 4.1 and let𝒙1 = 𝑸−1𝜽 , 𝒙2 = 𝑸−1𝝎, 𝒙3 = 𝑸−1𝜼, 𝒙4 = 𝑸−1𝝃, we obtainthe closed-loop system in the general form as (7) with

𝒙 =

𝒙1𝒙2𝒙3𝒙4

, 𝑨 =

0 𝑰𝑛 0 0

− 1𝑚𝚲 − 𝑑𝑚 𝑰𝑛 0 4𝑘1

𝑚 𝑰𝑛0 𝑑𝑰𝑛 0 4𝑘1𝑘3𝚲0 −𝑘1𝑚𝑰𝑛 −𝑘1𝑰𝑛 −4𝑘1𝑰𝑛

, 𝑩 =

0

𝑸−1

𝑚00

,where 𝚲 is the diagonal matrix defined in Lemma 4.1. Each ofthe block matrices in the matrix 𝑨 is either the zero matrix or adiagonal matrix, so the components of the vector 𝒙1 ∈ R𝑛, 𝒙2 ∈R𝑛, 𝒙3 ∈ R𝑛, 𝒙4 ∈ R𝑛 can be decoupled.

With the same method for obtaining (18) in the proof ofTheorem 4.2, setting 𝒚 = 𝝎 = 𝑄𝑥2, 𝑪 = [0,𝑸, 0, 0], we de-rive (24a) for 𝝎(𝑡). Then, setting 𝒚(𝑡) = 𝒖(𝑡) = 4𝑘1𝝃 (𝑡) and𝑪 = [0, 0, 0, 4𝑘1𝑸], we derive (24b) for the norm of 𝒖(𝑡). Fi-nally for the coherence measurement of the marginal cost, set-ting 𝒚 = 4𝑘1𝑳𝝃 and 𝑪 = [0, 0, 0, 4𝑘1𝑳𝑸], we derive (24c).�

Similar as Theorem 4.5, for a positive definite 𝑩 ∈ R𝑛×𝑛, weobtain the following theorem.

6

Theorem 4.7. Consider the closed-loop system (23) of DPIACwith a positive definite 𝑩 ∈ R𝑛×𝑛, the squared H2 norm of 𝝎(𝑡),𝒖(𝑡) and 4𝑘1𝑳𝝃 satisfy

𝛾2min𝐺𝑑 ≤ ‖𝑮𝑑 (𝝎, 𝒘)‖2

2 ≤ 𝛾2max𝐺𝑑 ,

𝛾2min (

𝑘1

2+

𝑛∑𝑖=2

𝑏2𝑖

𝑒𝑖) ≤ ‖𝑮𝑑 (𝒖, 𝒘)‖2

2 ≤ 𝛾2max (

𝑘1

2+

𝑛∑𝑖=2

𝑏2𝑖

𝑒𝑖)

𝛾2min

𝑛∑𝑖=2

_2𝑖𝑏2𝑖

𝑚2𝑒𝑖≤ ‖𝑮𝑑 (4𝑘1𝑳𝝃, 𝒘)‖2

2 ≤ 𝛾2max

𝑛∑𝑖=2

_2𝑖𝑏2𝑖

𝑚2𝑒𝑖

where 𝛾min and 𝛾max are the smallest and the largest eigenvalueof 𝑩, 𝑏1𝑖 , 𝑏2𝑖 and 𝑒𝑖 are defined in Theorem 4.6 and

𝐺𝑑 =1

2𝑚

𝑛∑𝑖=2

𝑏1𝑖

𝑒𝑖+ 𝑑 + 5𝑚𝑘1

2𝑚(2𝑘1𝑚 + 𝑑)2 .

The proof of this theorem follows that of Theorem 4.5. Hence,similar as in GBPIAC, when the disturbances are correlatedwith non-identical strength, the impact of 𝑘1 and 𝑘3 on thenorms are similar as in the case with identical strength of dis-turbances.

Based on Theorem 4.6, we analyze the impact of 𝑘1 and 𝑘3on the norms by focusing on 1) the frequency deviation, 2) con-trol cost and 3) coherence of the marginal costs.

4.2.1. The frequency deviationWe first pay attention to the influence of 𝑘1 when 𝑘3 is fixed.

The norm of 𝝎 also includes two terms in (24a) where the firstone describes the relative frequency oscillation and the secondone describes the overall frequency deviation. The overall fre-quency deviation decreases inversely as 𝑘1 increases. Hence,when the overall frequency deviation dominates the relative fre-quency deviation, the convergence can also be accelerated by alarge 𝑘1 as analyzed in Remark 4.4 for GBPIAC. From (24a),we derive

lim𝑘1→∞

| |𝑮𝑑 (𝝎, 𝒘) | |22 =1

2𝑚

𝑛∑𝑖=2

_2𝑖𝑘2

3

𝑑_2𝑖𝑘2

3 + 𝑑 (1 + 2_𝑖𝑘3), (27)

which indicates that even with a large 𝑘1, the frequency devi-ations cannot be decreased anymore when 𝑘3 is nonzero. Sosimilar to GBPIAC, the relative frequency deviation cannot besuppressed to an arbitrary small positive value in DPIAC. How-ever, when 𝑘3 = 0, DPIAC reduces to DecPIAC (6), thus

‖𝑮𝑑 (𝝎, 𝒘)‖22 ∼ 𝑂 (𝑘−1

1 ). (28)

Remark 4.8. By DecPIAC, it follows from (28) that if all thenodes are equipped with the secondary frequency controllers,the frequency deviation can be controlled to any prespecifiedrange. However, it results in a high control cost for the entirenetwork. In addition, the configuration of 𝑘1 is limited by theresponse time of the actuators.

Remark 4.9. This analysis is based on Assumption 3.3 whichrequires that each node in the network is equipped with a sec-ondary frequency controller. However, for the power systemswithout all the nodes equipped with the controllers, the distur-bance from the node without a controller must be compensatedby the other nodes with controllers. In that case, the equilib-rium of the system is changed and the oscillation can never beavoided even when the controllers are sufficiently sensitive tothe disturbances.

When 𝑘1 is fixed, it can be easily observed from (24a) thatthe order of 𝑘3 in the term 𝑏1𝑖 is 2 which is the same as in theterm 𝑒𝑖 , thus 𝑘3 has little influence on the frequency deviation.

4.2.2. The control costWe first analyze the influence of 𝑘1 on the cost and then the

influence of 𝑘3. For any 𝑘3 ≥ 0, we derive from (24b) that

‖𝑮𝑑 (𝒖, 𝒘)‖22 ∼ 𝑂 (𝑘1),

which indicates that the control cost increases as 𝑘1 increases.Recalling the impact of 𝑘1 on the overall frequency deviationin (24a), we conclude that minimizing the control cost alwaysconflicts with minimizing the frequency deviation. Hence, atrade-off should be determined to obtain the desired frequencydeviation with an acceptable control cost.

Next, we analyze how 𝑘3 influences the control cost. From(24b), we obtain that

‖𝑮𝑑 (𝒖, 𝒘)‖22 ∼ 𝑘1

2+𝑂 (𝑘1𝑘

−13 ), (29)

where the second term is positive. It shows that the controlcost decreases as 𝑘3 increases due to the accelerated consensusspeed of the marginal costs. This will be further discussed in thenext subsubsection on the coherence of the marginal costs. Notethat 𝑘3 has little influence on the frequency deviation. Hencethe control cost can be decreased by 𝑘3 without increasing thefrequency deviation much.

4.2.3. The coherence of the marginal costs in DPIACWe measure the coherence of the marginal costs by the norm

of ‖𝑮𝑑 (4𝑘1𝑳𝝃, 𝒘) | |2. From (24c), we obtain

‖𝑮𝑑 (4𝑘1𝑳𝝃, 𝒘)‖22 = 𝑂 (𝑘−1

3 ),

which indicates that the difference of the marginal costs de-creases as 𝑘3 increases. Hence, this analytically confirms thatthe consensus speed can be increased by increasing 𝑘3.

Remark 4.10. In practice, similar to 𝑘1, the configuration of𝑘3 depends on the communication devices and cannot be arbi-trarily large. In addition, the communication delay and noisealso influence the transient performance, which still needs fur-ther investigation.

7

4.3. Comparison of the GBPIAC and DPIAC control laws

With a positive 𝑘1, we can easily obtain from (13b, 24b) that

‖𝑮𝑐 (𝒖, 𝒘)‖ < ‖𝑮𝑑 (𝒖, 𝒘) | |, (30)

which is due to the differences of the marginal costs. The differ-ence in the control cost between these two control laws can bedecreased by accelerating the consensus of the marginal costsas explained in the previous subsection. From (24a) and (24b)we derive that

lim𝑘3→∞

‖𝑮𝑑 (𝝎, 𝒘)‖22 =

𝑛 − 12𝑚𝑑

+ 𝑑 + 5𝑚𝑘1

2𝑚(2𝑘1𝑚 + 𝑑)2 = | |𝑮𝑐 (𝝎, 𝒘) | |2,

lim𝑘3→∞

‖𝑮𝑑 (𝒖, 𝒘)‖22 =

𝑘1

2= ‖𝑮𝑐 (𝒖, 𝒘)‖2.

Hence, as 𝑘3 goes to infinity, the transient performance ofDPIAC converges to that of GBPIAC.

5. Simulations

In this section, we numerically verify the analysis of thetransient performance of DPIAC using the IEEE 39-bus sys-tem as shown in Fig. 1 with the Power System Analy-sis Toolbox (PSAT) [28]. We compare the performance ofDPIAC with that of GBPIAC. For a comparison of DPIACwith the traditional control laws, see [7, 15]. The sys-tem consists of 10 generators, 39 buses, which serves a to-tal load of about 6 GW. As in [15], we change the buseswhich are neither connected to synchronous machines norto power loads into frequency dependent buses. HenceV𝑀 = {𝐺1, 𝐺2, 𝐺3, 𝐺4, 𝐺5, 𝐺6, 𝐺7, 𝐺8, 𝐺9, 𝐺10}, V𝑃 =

{30, 31, 32, 33, 34, 35, 36, 37, 38, 39} and the other nodes are inset V𝐹 . The nodes in V𝑀 ∪V𝐹 are all equipped with secondaryfrequency controllers such that V𝐾 = V𝑀 ∪ V𝐹 . Because thevoltages are constants, the angles of the synchronous machineand the bus have the same dynamics [29]. Except the controlgain coefficients 𝑘1 and 𝑘3, all the parameters of the power sys-tem, including the control prices, damping coefficients and con-stant voltages are identical to those in the simulations in [15].The communication topology are the same as the one of thepower network and we set 𝑙𝑖 𝑗 = 1 for the communication ifnode 𝑖 and 𝑗 are connected. We remark that with these con-figurations of the parameters, Assumption 3.3 is not satisfiedin the simulations. We first verify the impact of 𝑘1 and 𝑘3 onthe transient performance in the deterministic system where thedisturbance is modeled by a step-wise increase of load, then ina stochastic system with the interpretation of the H2 norm asthe limit of the variance of the output.

5.1. In the deterministic system

We analyze the impact of the control gain coefficients onthe performance on the deterministic system where the distur-bances are step-wise increased power loads by 66 MW at nodes4, 12 and 20 at time 𝑡 = 5 seconds. This step-wise disturbancelead the overall frequency to dominate the relative frequenciesas described in Remark 4.4, which illustrates the function of

Figure 1: IEEE 39-bus test power system

the secondary frequency control. The system behavior follow-ing the disturbance also show us how the convergence of thestate can be accelerated by tuning 𝑘1 and 𝑘3 monotonically. Wecalculate the following two metrics

𝑆 =

∫ 𝑇0

0𝝎𝑇 (𝑡)𝝎(𝑡)𝑑𝑡, and 𝐶 =

12

∫ 𝑇0

0𝒖𝑇 (𝑡)𝜶𝒖(𝑡)𝑑𝑡,

to measure the performance of 𝝎(𝑡) and 𝒖(𝑡) during the tran-sient phase, where 𝑇0 = 40, 𝝎 = col(𝜔𝑖) for 𝑖 ∈ V𝑀 ∪ V𝐹 ,𝒖 = col(𝑢𝑖) for 𝑖 ∈ V𝑀 ∪V𝐹 and 𝜶 = diag(𝛼𝑖).

From Fig.2 (𝑎1-𝑎2), it can be observed that the frequencyrestoration is accelerated by a larger 𝑘1 with an accelerated con-vergence of the control input as shown in Fig. 2 (𝑐1). FromFig.2 (𝑏2-𝑏3), it can be seen that the consensus of the marginalcosts is accelerated by a larger 𝑘3 with little influences on thefrequency deviation as shown in Fig.2 (𝑎2-𝑎3). It can be easilyimagined that the marginal costs converge to that of GBPIAC asshown in Fig. 2 (b4) as 𝑘3 further increases. Hence, by increas-ing 𝑘1 and 𝑘3, the convergence of the state of the closed-sytstemto the optimal state can be accelerated, and by increasing 𝑘3,the performance of the distributed control method DPIAC ap-proaches to that of the centralized control method GBPIAC.

Fig.2 (𝑐2 − 𝑐3) show the trends of 𝑆 and 𝐶 with respect to𝑘1 and 𝑘3. It can be observed from Fig.2 (𝑐2) that as 𝑘1 in-creases, the frequency deviation decreases while the controlcost increases. Hence, to obtain a better performance of thefrequencies, a higher control cost is needed. In addition, 𝑆converges to a non-zero value as 𝑘1 increases which is con-sistent with the anlsysis in (27). However, the control cost isbounded as 𝑘1 increases due to the bounded disturbance, whichis different from the conclusion from Theorem 4.6. When thedisturbance is unbounded, the control cost is also unbounded,which will be further discussed in the next subsection. FromFig.2 (𝑐3), it can seen that the control cost decreases inverselyto a non-zero value as 𝑘3 increases, which is also consistentwith the analysis in (29).

8

Time (sec)

0 10 20 30 40

Fre

qu

en

cy

(H

z)

59.9

59.92

59.94

59.96

59.98

60

60.02DPIAC(k1 = 0.4, k3 = 10): Frequency

Time (sec)

0 10 20 30 40

Fre

qu

en

cy

(H

z)

59.9

59.92

59.94

59.96

59.98

60

60.02DPIAC(k1 = 0.8, k3 = 10): Frequency

Time (sec)

0 10 20 30 40

Fre

qu

en

cy

(H

z)

59.9

59.92

59.94

59.96

59.98

60

60.02DPIAC(k1 = 0.8, k3 = 20): Frequency

Time (sec)

0 10 20 30 40

Fre

qu

en

cy

(H

z)

59.9

59.92

59.94

59.96

59.98

60

60.02GBPIAC(k1 = 0.8): Frequency

Time (sec)

0 10 20 30 40

Ma

rgin

al

Co

st

0

5

10

15

20

25DPIAC(k1 = 0.4, k3 = 10): Marginal Cost

Time (sec)

0 10 20 30 40

Ma

rgin

al

Co

st

0

5

10

15

20

25DPIAC(k1 = 0.8, k3 = 10): Marginal Cost

Time (sec)

0 10 20 30 40

Ma

rgin

al

Co

st

0

5

10

15

20

25DPIAC(k1 = 0.8, k3 = 20): Marginal Cost

Time (sec)

0 10 20 30 40

Ma

rgin

al

Co

st

0

5

10

15

20

25GBPIAC(k1 = 0.4): Marginal Cost

Time (sec)

0 10 20 30 40

Po

wer

(M

W)

0

50

100

150

200

250DPIAC: Sum of inputs

k1 = 0.4, k3 = 10

k1 = 0.8, k3 = 10

k1 = 0.8, k3 = 20

Ps

Gain coefficient k1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Co

ntr

ol c

ost

×106

4.5

5

DPIAC(k3 = 10): C and S

Fre

qu

en

cy d

ev

iati

on

0

1

2

C S

Gain coefficient k3

2 4 6 8 10 12 14 16 18 20

Co

ntr

ol c

ost

×106

4.5

5

DPIAC(k1 = 0.4): C and S

Fre

qu

en

cy d

ev

iati

on

0

1

2

CS

(a1) (a2) (a3) (a4)

(b1) (b2) (b3) (b4)

(c1) (c2) (c3)

Figure 2: The simulation result of the determistic system.

Time (sec)

0 10 20 30 40

Fre

qu

en

cy

(H

z)

59.98

59.99

60

60.01

60.02

DPIAC (k1 = 0.4, k3 = 10): Frequency

Time (sec)

0 10 20 30 40

Fre

qu

en

cy

(H

z)

59.98

59.99

60

60.01

60.02

DPIAC (k1 = 1.6, k3 = 10): Frequency

Time (sec)

0 10 20 30 40

Fre

qu

en

cy

(H

z)

59.98

59.99

60

60.01

60.02

DPIAC (k1 = 1.6, k3 = 20): Frequency

Time (sec)

0 10 20 30 40

Fre

qu

en

cy

(H

z)

59.98

59.99

60

60.01

60.02

GBPIAC (k1 = 1.6): Frequency

Time (sec)

0 10 20 30 40

Ma

rgin

al

Co

st

-8

-4

0

4

8DPIAC(k1 = 0.4, k3 = 10): Marginal Cost

Time (sec)

0 10 20 30 40

Ma

rgin

al

Co

st

-8

-4

0

4

8DPIAC(k1 = 1.6, k3 = 10): Marginal Cost

Time (sec)

0 10 20 30 40

Ma

rgin

al

Co

st

-8

-4

0

4

8DPIAC(k1 = 1.6, k3 = 20): Marginal Cost

Time (sec)

0 10 20 30 40

Ma

rgin

al

Co

st

-8

-4

0

4

8GBPIAC(k1 = 1.6): Marginal Cost

Time (sec)

0 10 20 30 40

Po

wer

(M

W)

-20

-10

0

10

20

DPIAC: Sum of inputs

k1 = 0.4, k3 = 10

k1 = 1.6, k3 = 10

k1 = 1.6, k3 = 20

Gain coefficient k1

0.4 0.8 1.2 1.6 2 2.4 2.8 3.2

Co

ntr

ol co

st

×105

0

1

2

3

4

DPIAC (k3 = 10): EC and ES

Fre

qu

en

cy d

evia

tio

n

×10-8

2.5

3.5

4.5

5.5EC ES

Gain coefficient k3

6 8 10 12 14 16 18 20 22 24 26 28 30

Co

ntr

ol co

st

×105

0

1

2

3

4

DPIAC (k1 = 2.4): EC and ES

Fre

qu

en

cy d

evia

tio

n

×10-8

2.5

3.5

4.5

5.5ECES

(a1) (a2) (a3) (a4)

(b1) (b2) (b3) (b4)

(c1) (c2) (c3)

Figure 3: The simulation result of the stochastic system.

9

5.2. In the stochastic system

With the interpretation of the H2 norm where the distur-bances are modelled by white noise, we assume the distur-bances are from the nodes of loads and 𝑤𝑖 ∼ 𝑁 (0, 𝜎2

𝑖) with

𝜎𝑖 = 0.002 for 𝑖 ∈ V𝑃 . With these noise signal, the distur-bances are unbounded and the closed-loop system becomes astochastic algebraic differential system. We refer to [30] for anumerical algorithm to solve this stochastic system.

It can be observed from Fig. 3 (𝑎1-𝑎2) that the variance ofthe frequency deviation can be suppressed with a large 𝑘1 whichhowever increases the variances of the marginal costs and thetotal control cost as shown in Fig. 3 (𝑏2) and (𝑐1) respectively.These observations are consistent with the analysis of Theo-rem 4.6 when 𝑘3 is fixed. From Fig. 3 (𝑎2-𝑎3), it can be seenthat increasing 𝑘3 can effectively suppress the variances of themarginal costs.

We calculate the following metrics to study the impact of𝑘1 and 𝑘3 on the variance of the frequency deviation and theexpected control cost,

𝐸𝑆 = 𝐸 [𝝎𝑇 (𝑡)𝝎(𝑡)], and 𝐸𝐶 =12𝐸 [𝒖𝑇 (𝑡)𝜶𝒖(𝑡)] .

Fig.3 (𝑐2-𝑐3) show the trend of 𝐸𝑆 and 𝐸𝐶 as 𝑘1 and 𝑘3 in-crease. Similar to the discussion in the previous subsection,a trade-off can be found between the frequency deviation andthe control cost in Fig.3 (𝑐2). The difference is that the controlcost increases linearly as 𝑘1 increases unboundly because ofthe unbounded disturbances. This is consistent with the resultin Theorem 4.6, which further confirms that a better frequencyresponse requires a higher control cost.

Fig. 3 (𝑐3) illustrates the trend of the expected control costand the variance of the frequency deviations with respect to 𝑘3.It can be observed that the expected control cost decreases as 𝑘3increases, which is consistent as in (29). However, the varianceof the frequency deviation is slightly increased, which is alsoconsistent with our analysis in (27).

6. Conclusion

For the power system controlled by DPIAC, it has beendemonstrated analytically and numerically that the transientperformance of the frequency can be improved by tuning thecoefficients monotonically, and a trade-off between the controlcost and frequency deviations has to be resolved to obtain a de-sired frequency response with acceptable control cost.

There usually are noises and delays in the state measurementand communications in practice, which are neglected in this pa-per. How these factors influence the transient behaviors of thestate requires further investigation.

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