transient performance of power systems with distributed
TRANSCRIPT
Transient Performance of Power Systemswith Distributed Power-Imbalance Allocation Control
Kaihua Xia,โ, 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
โ 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.
โCorresponding authorEmail addresses: [email protected] (Kaihua Xi), [email protected]
(Hai Xiang Lin), [email protected] (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 Hโ 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 โ V ร V, where a node represents a bus and edge (๐, ๐)represents the direct transmission line between node ๐ and ๐ .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 ๏ฟฝ๏ฟฝ๐ ๐ for (๐, ๐) โ E.The set of the buses of the synchronous machines, of fre-quency dependent power sources, of passive loads are denotedby V๐ ,V๐น ,V๐ respectively, thus V = V๐ โช V๐น โช V๐ . Thedynamics of the system can be modelled by the following Dif-ferential Algebraic Equations (DAEs), e.g., [3, 7],
ยค\๐ = ๐๐ , ๐ โ V๐ โชV๐น , (1a)
๐๐ ยค๐๐ = ๐๐ โ ๐ท๐๐๐ โโ๐โV
๐พ๐ ๐ sin (\๐ โ \ ๐ ) + ๐ข๐ , ๐ โ V๐ ,
(1b)
0 = ๐๐ โ ๐ท๐๐๐ โโ๐โV
๐พ๐ ๐ sin (\๐ โ \ ๐ ) + ๐ข๐ , ๐ โ V๐น ,
(1c)
0 = ๐๐ โโ๐โV
๐พ๐ ๐ sin (\๐ โ \ ๐ ), ๐ โ V๐ , (1d)
where \๐ is the phase angle at node ๐, ๐๐ is the frequency devi-ation from the nominal value (e.g., 50 or 60 Hz), ๐๐ > 0 is themoment of inertia of the synchronous machine, ๐ท๐ > 0 is thedroop control coefficient, ๐๐ is the power supply (or demand),๐พ๐ ๐ = ๏ฟฝ๏ฟฝ๐ ๐๐๐๐ ๐ is the effective susceptance of the transmissionline between node ๐ and ๐ , ๐๐ is the voltage, ๐ข๐ is the input forthe secondary control. The nodes in V๐ and V๐น are assumedto be equipped with secondary frequency controllers, denotedby V๐พ = V๐ โช V๐น . 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
๐๐ ๐ฆ๐ =
โ๐โV ๐๐ +
โ๐โV๐พ ๐ข๐โ
๐โV๐โชV๐น ๐ท๐. (2)
The condition ๐๐ ๐ฆ๐ = 0 at a steady state can be satisfied bysolving the economic power dispatch problem in the secondaryfrequency control [2],
min๐ข๐ โ๐
โ๐โV๐พ
๐ฝ๐ (๐ข๐) (3)
๐ .๐ก.โ๐โV
๐๐ +โ๐โV๐พ
๐ข๐ = 0,
where ๐ฝ๐ (๐ข๐) = 12๐ผ๐๐ข
2๐
represents the control cost at node ๐. 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 ๐๐ฝ๐ (๐ข๐)/๐๐ข๐ of the nodesare all identical, i.e.,
๐ผ๐๐ข๐ = ๐ผ ๐๐ข ๐ , โ ๐, ๐ โ V๐พ .
For a secondary control law with the objective of (3), it isrequired that the total control input ๐ข๐ (๐ก) =
โ๐โV๐พ ๐ข๐ con-
verges to the unknown โ๐๐ = โโ๐โV ๐๐ 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 V๐พare 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]
ยค[๐ =โ
๐โV๐โชV๐น๐ท๐๐๐ , (4a)
ยคb๐ = โ๐1 (โ๐โV๐
๐๐๐๐ + [๐ ) โ ๐2b๐ , (4b)
๐ข๐ =๐ผ๐
๐ผ๐๐2b๐ , ๐ โ V๐พ , (4c)
where [๐ โ R, b๐ โ R are state variables of the central con-troller, ๐1, ๐2 are positive control gain coefficients, ๐ผ๐ is thecontrol price at node ๐ as defined in the optimization problem(3), ๐ผ๐ = (โ๐โV๐พ 1/๐ผ๐)โ1 is a constant.
Definition 2.3 (DPIAC). Consider the power system (1), de-fine the Distributed Power-Imbalance Allocation Control(DPIAC) law as,
ยค[๐ = ๐ท๐๐๐ + ๐3
โ๐โV๐พ
๐๐ ๐ (๐2๐ผ๐b๐ โ ๐2๐ผ ๐b ๐ ), (5a)
ยคb๐ = โ๐1 (๐๐๐๐ + [๐) โ ๐2b๐ , (5b)๐ข๐ = ๐2b๐ , (5c)
for node ๐ โ V๐พ , where [๐ โ R and b๐ โ R are state variablesof the local controller at node ๐, ๐1, ๐2 and ๐3 are positive gain
coefficients, (๐๐ ๐ ) defines a weighted undirected communicationnetwork with Laplacian matrix (๐ฟ๐ ๐ )
๐ฟ๐ ๐ =
{โ๐๐ ๐ , ๐ โ ๐ ,โ๐โ ๐ ๐๐๐ , ๐ = ๐ ,
and ๐๐ ๐ โ [0,โ) is the weight of the communication line con-necting node ๐ and ๐ . The marginal cost at node ๐ is ๐ผ๐๐ข๐ =
๐2๐ผ๐b๐ .
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,
ยค[๐ = ๐ท๐๐๐ , (6a)ยคb๐ = โ๐1 (๐๐๐๐ + [๐) โ ๐2b๐ , (6b)๐ข๐ = ๐2b๐ , (6c)
for node ๐ โ V๐พ , where ๐1 and ๐2 are positive gain coefficients.
For a fast recovery from an imbalance while avoiding theovershoot problem, the control gain coefficient ๐2 should sat-isfy ๐2 โฅ 4๐1. For details of the configuration of ๐1 and ๐2, werefer to [15]. In this paper, we set ๐2 = 4๐1 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 ๐2 = 4๐1, there are two control gain coefficients ๐1 and๐3 in DPIAC. We focus on the following problem.
Problem 3.1. How do the coefficients ๐1 and ๐3 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,
ยค๐ = ๐จ๐ + ๐ฉ๐, (7a)๐ = ๐ช๐, (7b)
where ๐ โ R๐, ๐จ โ R๐ร๐ is Hurwitz, ๐ฉ โ R๐ร๐, ๐ช โ R๐งร๐,the input is denoted by ๐ โ R๐ and the output of the system isdenoted by ๐ โ R๐ง . The squared H2 norm of the transfer matrix๐ฎ of the mapping (๐จ, ๐ฉ,๐ช) from the input ๐ to the output ๐ isdefined as
| |๐ฎ | |22 = tr(๐ฉ๐๐ธ๐๐ฉ) = tr(๐ช๐ธ๐๐ช๐ ), (8a)
๐ธ๐๐จ + ๐จ๐๐ธ๐ + ๐ช๐๐ช = 0, (8b)
๐จ๐ธ๐ + ๐ธ๐๐จ๐ + ๐ฉ๐ฉ๐ = 0, (8c)
3
where tr(ยท) denotes the trace of a matrix, ๐ธ๐,๐ธ๐ โ R๐ร๐ are theobservability Grammian of (๐ช, ๐จ) and controllability Gram-mian of (๐จ, ๐ฉ) respectively [23],[24, chapter 2].
The H2 norm can be interpreted as follows. When theinput ๐ is modeled as the Gaussian white noise such that๐ค๐ โผ ๐ (0, 1) for all ๐ = 1, ยท ยท ยท , ๐ and for all ๐ โ ๐ , thescalar Brownian motions ๐ค๐ and ๐ค ๐ are independent, the ma-trix ๐ธ๐ฃ = ๐ช๐ธ๐๐ช
๐ is the variance matrix of the output at thesteady state [25, Theorem 1.53], i.e.,
๐ธ๐ฃ = lim๐กโโ
๐ธ [๐(๐ก)๐๐ (๐ก)]
where ๐ธ [ยท] denotes the expectation. Thus
โ๐ฎโ22 = tr(๐ธ๐ฃ ) = lim
๐กโโ๐ธ [๐(๐ก)๐ ๐(๐ก)] . (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 V๐น =
โ , V๐ = โ and for all ๐ โ V๐ , ๐๐ = ๐ > 0, ๐ท๐ = ๐ > 0, ๐ผ๐ =1. For DPIAC, assume that the topology of the communicationnetwork is the same as the one of the power system such that๐๐ ๐ = ๐พ๐ ๐ for all (๐, ๐) โ 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 \๐ ๐ = \๐ โ \ ๐ is usually small for relatively small๐๐ compared to the line capacity in practice, we approximatesin \๐ ๐ by \๐ ๐ 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 \๐ ๐ by \๐ ๐ , we obtain
ยค๐ฝ = ๐ (10a)๐ด ยค๐ = โ๐ณ๐ฝ โ ๐ซ๐ + ๐ฉ๐ + ๐, (10b)
where ๐ฝ = col(\๐) โ R๐, ๐ denotes the number of nodes in thenetwork, ๐ = col(๐๐) โ R๐, ๐ด = diag(๐๐) โ R๐ร๐, ๐ณ โ R๐ร๐is the Laplacian matrix of the network, ๐ซ = diag(๐ท๐) โ R๐ร๐,๐ = col(๐ข๐) โ R๐ร๐. The disturbances of ๐ท = col(๐๐) โ R๐have been modeled by ๐ฉ๐ as the inputs with ๐ฉ โ R๐ร๐ and๐ โ R๐ as in Definition 3.2. Here, col(ยท) denotes the columnvector of the indicated elements and diag(๐ฝ๐) denotes a diago-nal matrix ๐ท = diag({๐ฝ๐ , ๐ ยท ยท ยท ๐}) โ R๐ร๐ with ๐ฝ๐ โ R. Denotethe identity matrix by ๐ฐ๐ โ R๐ร๐ and the ๐ dimensional vectorwith all elements equal to one by 1๐.
The transient performance of ๐(๐ก) and ๐(๐ก) are measured bythe H2 norm of the corresponding transfer functions with input๐ and output ๐ = ๐ and ๐ = ๐ respectively. The squared H2norms are denoted by | |๐ฎ๐ (๐, ๐) | |22 and | |๐ฎ๐ (๐, ๐) | |22 where thesub-index ๐ = ๐ or ๐ 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 ๐ณ โ R๐ร๐,there exist an invertible matrix ๐ธ โ R๐ร๐ such that
๐ธโ1 = ๐ธ๐ , (11a)
๐ธโ1๐ณ๐ธ = ๐ฒ, (11b)
๐ธ1 =1โ๐
1๐, (11c)
where ๐ธ = [๐ธ1, ยท ยท ยท ,๐ธ๐], ๐ฒ = diag(_๐) โ R๐ร๐, ๐ธ๐ โ R๐ isthe normalized eigenvector of ๐ณ corresponding to eigenvalue_๐ , thus ๐ธ๐
๐๐ธ ๐ = 0 for ๐ โ ๐ . Because ๐ณ1๐ = 0, _1 = 0 is one
of the eigenvalues with normalized eigenvector ๐ธ1.
We study the transient performance of ๐ and ๐ 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 ๐ข๐ = 1๐๐1b๐ at
node ๐ as in (4). With the notations of section 3 and Assumption3.3, and ๐2 = 4๐1, we obtain from (10) and (4) the closed-loopsystem of GBPIAC in a vector form as follows.
ยค๐ฝ = ๐, (12a)
๐๐ฐ๐ ยค๐ = โ๐ณ๐ฝ โ ๐๐ฐ๐๐ + 4๐1b๐
๐1๐ + ๐ฉ๐, (12b)
ยค[๐ = ๐1๐๐๐, (12c)ยคb๐ = โ๐1๐1๐๐๐ โ ๐1[๐ โ 4๐1b๐ , (12d)
where [๐ โ R and b๐ โ R.For the transient performance of ๐(๐ก), ๐(๐ก) in GBPIAC, the
following theorem can be proved.
Theorem 4.2. Consider the closed-loop system (12) ofGBPIAC with ๐ฉ = ๐ฐ๐. The squared H2 norm of the frequencydeviation ๐ and of the control inputs ๐ are,
| |๐ฎ๐ (๐, ๐) | |22 =๐ โ 12๐๐
+ ๐ + 5๐๐1
2๐(2๐1๐ + ๐)2 , (13a)
| |๐ฎ๐ (๐, ๐) | |22 =๐1
2. (13b)
4
Proof: With the linear transform ๐1 = ๐ธโ1๐ฝ , ๐2 = ๐ธโ1๐ where๐ธ is defined in Lemma 4.1, we derive from (12) that
ยค๐1 = ๐2,
ยค๐2 = โ 1๐๐ฒ๐1 โ
๐
๐๐ฐ๐๐2 +
4๐1b๐
๐๐๐ธโ11๐ +
1๐๐ธโ1๐,
ยค[๐ = ๐1๐๐๐ธ๐2,
ยคb๐ = โ๐1๐1๐๐๐ธ๐2 โ ๐1[๐ โ 4๐1b๐ ,
where ๐ฒ is the diagonal matrix defined in Lemma 4.1. Since1๐ is an eigenvector of ๐ณ corresponding to _1 = 0, we obtain๐ธโ11๐ = [
โ๐, 0, ยท ยท ยท , 0]๐ . Thus the components of ๐1 and ๐2
can be decoupled as
ยค๐ฅ11 = ๐ฅ21, (15a)
ยค๐ฅ21 = โ ๐๐๐ฅ21 +
4๐1
๐โ๐b๐ +
1๐๐ธ๐1 ๐, (15b)
ยค[๐ = ๐โ๐๐ฅ21, (15c)
ยคb๐ = โ๐1๐โ๐๐ฅ21 โ ๐1[๐ โ 4๐1b๐ (15d)
and for ๐ = 2, ยท ยท ยท , ๐,
ยค๐ฅ1๐ = ๐ฅ2๐ , (16a)
ยค๐ฅ2๐ = โ_๐๐๐ฅ1๐ โ
๐
๐๐ฅ2๐ +
1๐๐ธ๐๐ ๐, (16b)
We rewrite the decoupled systems of (15) and (16) in the gen-eral form as (7) with
๐ =
๐1๐2[๐ b๐
, ๐จ =
0 ๐ฐ๐ 0 0โ ๐ฒ๐
โ ๐๐๐ฐ๐ 0 4๐1
๐โ๐๐
0 ๐โ๐๐๐ 0 0
0 โ๐1๐โ๐๐๐ โ๐1 โ4๐1
, ๏ฟฝ๏ฟฝ =
0
๐ธโ1
๐
00
,where ๐๐ = [1, 0, ยท ยท ยท , 0] โ R๐. The H2 norm of a state vari-able e.g., the frequency deviation and the control cost, can bedetermined by setting the output ๐ฆ as that state variable. Be-cause the closed-loop system (12) is asymptotically stable, ๐จ isHurwitz regardless the rotations of the phase angle ๐ฝ .
For the transient performance of ๐(๐ก), setting ๐ = ๐ = ๐ธ๐2and ๐ช = [0,๐ธ, 0, 0], we obtain the observability Grammian ๐ธ๐of (๐ช, ๐จ) (8b) in the form,
๐ธ๐ =
๐ธ๐11 ๐ธ๐12 ๐ธ๐13 ๐ธ๐14๐ธ๐๐12 ๐ธ๐22 ๐ธ๐23 ๐ธ๐24
๐ธ๐๐13 ๐ธ๐
๐23 ๐ธ๐33 ๐ธ๐34๐ธ๐๐14 ๐ธ๐
๐24 ๐ธ๐๐34 ๐๐44
.Thus,
โ๐ฎ๐ (๐, ๐)โ22 = tr(๏ฟฝ๏ฟฝ๐๐ธ๐ ๏ฟฝ๏ฟฝ) =
tr(๐ธ๐ธ๐22๐ธ๐ )
๐2 =tr(๐ธ๐22)๐2 . (17)
Because
๐ช๐๐ช =
0 0 0 00 ๐ฐ๐ 0 00 0 0 00 0 0 0
,
the diagonal elements ๐ธ๐22 (๐, ๐) of ๐ธ๐22 can be calculated bysolving the observability Gramian ๏ฟฝ๏ฟฝ๐ of (๐ช๐ , ๐จ๐) which satis-fies
๏ฟฝ๏ฟฝ1๐จ1 + ๐จ๐1 ๏ฟฝ๏ฟฝ1 + ๐ช๐1 ๐ช1 = 0
where
๐จ1 =
0 1 0 00 โ ๐
๐0 4๐1
๐โ๐
0 ๐โ๐ 0 0
0 โ๐1๐โ๐ โ๐1 โ4๐1
,๐ช๐1 =
0100
and
๏ฟฝ๏ฟฝ๐๐จ๐ + ๐จ๐๐ ๏ฟฝ๏ฟฝ๐ + ๐ช๐๐ ๐ช๐ = 0, ๐ = 2, ยท ยท ยท , ๐,
where
๐จ๐ =
[0 1
โ_๐๐
โ ๐๐
],๐ช๐๐ =
[01
].
In this case, the diagonal elements of ๐ธ๐22 satisfy ๐ธ๐22 (๐, ๐) =๏ฟฝ๏ฟฝ๐ (2, 2) for ๐ = 1, ยท ยท ยท , ๐. We thus derive from the observabilityGramian ๏ฟฝ๏ฟฝ๐ that
tr(๏ฟฝ๏ฟฝ๐๐ธ๐ ๏ฟฝ๏ฟฝ) =๐ โ 12๐๐
+ ๐ + 5๐๐1
2๐(2๐1๐ + ๐)2 , (18)
which yields (13a) directly. Similarly by setting ๐ฆ = ๐ข๐ (๐ก) =
4๐1b๐ (๐ก) and ๐ช = [0, 0, 0, 4๐1], we derive the norm of ๐ข๐ (๐ก) as
โ๐ฎ๐ (๐ข๐ , ๐)โ22 =
๐1๐
2. (19)
With ๐ข๐ =๐ข๐ ๐
for ๐ = 1, ยท ยท ยท , ๐ and โ๐(๐ก)โ2 = ๐(๐ก)๐ ๐(๐ก) =โ๐๐ ๐ข
2๐(๐ก), we further derive(13b) for the control cost. ๏ฟฝ
The norm of ๐(๐ก) 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 ๐(๐ก) and๐(๐ก). This is because of Assumption 3.3 and the identicalstrength of the disturbances at all the nodes with ๐ฉ = ๐ฐ๐.
Remark 4.4. The overall frequency deviation depends on ๐1while the relative frequency deviation is independent of ๐1.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 ๐1 can accelerate the restorationof the frequency. This will be further described in Section 5.However, a large ๐1 leads to a high control cost. Hence there isa trade-off between the overall frequency deviation suppressionand control cost, which is determined by ๐1.
5
Theorem 4.2 with ๐ฉ = ๐ฐ๐ 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 ๐ฉ where the disturbances are correlatedwith non-identical strength.
Theorem 4.5. Consider the closed-loop system (12) ofGBPIAC with a positive definite ๐ฉ โ R๐ร๐, the squared H2norm of the frequency deviation ๐ and of the control inputs ๐satisfy
๐พ2min๐บ๐ โค ||๐ฎ๐ (๐, ๐) | |22 โค ๐พ2
max๐บ๐ , (20a)๐1
2๐พ2
min โค ||๐ฎ๐ (๐, ๐) | |22 โค ๐1
2๐พ2
max, (20b)
where ๐พmin and ๐พmax are the smallest and the largest eigenvalueof ๐ฉ, and
๐บ๐ =(๐ โ 1
2๐๐+ ๐ + 5๐๐1
2๐(2๐1๐ + ๐)2
).
Proof: Since ๐ฉ > 0, then there exist ๐ผ and ๐ช such that ๐ฉ =
๐ผ๐ ๐ช๐ผ where ๐ผ is an orthogonal matrix and ๐ช is a diagonalmatrix with the diagonal elements being the eigenvalues of ๐ฉ,which are all strictly positive. With the decomposition of ๐ฉ,from (17) we derive
trac(๐ฉ๐๐ธ๐ธ๐22๐๐ฉ
)= trac
(๐ผ๐ ๐ช๐ผ๐ธ๐ธ๐22๐ธ
๐๐ผ๐ ๐ช๐ผ)
= trac(๐ช๐ผ๐ธ๐ธ๐22๐ธ
๐๐ผ๐ ๐ช)
โค ๐พ2maxtrac
(๐ผ๐ธ๐ธ๐22๐ธ
๐๐ผ๐)
= ๐พ2maxtrac
(๐ธ๐22
),
Similarly, we derive
๐พ2mintrac
(๐ธ๐22
)โค trac
(๐ฉ๐๐ธ๐ธ๐22๐ธ
๐ ๐ฉ).
From the above inequalities and (17), we can easily follow theprocedure as in the proof of Theorem 4.2 to obtain the traceof ๐๐22 and further derive the inequalities in (20a). A similarprocedure is conducted to obtain the inequalities in (20b). ๏ฟฝ
It is demonstrated by Theorem 4.5 that when the disturbancesare correlated with non-identical strength, the impact of ๐1 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 ๐2 = 4๐1, we derive the closed-loop system of DPIAC from (10) and (5) as
ยค๐ฝ = ๐, (23a)๐๐ฐ๐ ยค๐ = โ๐ณ๐ฝ โ ๐๐ฐ๐๐ + 4๐1๐ + ๐ฉ๐ (23b)
ยค๐ผ = ๐ซ๐ + 4๐1๐3๐ณ๐, (23c)ยค๐ = โ๐1๐ด๐ โ ๐1๐ผ โ 4๐1๐, (23d)
where ๐ผ = col([๐) โ R๐ and ๐ = col(b๐) โ R๐. Note that๐ณ = (๐ฟ๐ ๐ ) โ R๐ร๐ 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 4๐1๐ณ๐ (๐ก), we use the squared norm of (4๐1๐ณ๐ (๐ก)) tomeasure the coherence of the marginal costs in DPIAC. Denotethe squared H2 norm of the transfer matrix of (23) with output๐ = 4๐1๐ณ๐ by | |๐บ๐ (4๐1๐ฟb, ๐) | |2. In this subsection, we alsocalculate the squared H2 norm of 4๐1๐ณ๐ as an additional metricfor the influence of ๐3 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 ๐ฉ = ๐ฐ๐, the squared H2 norm of ๐(๐ก), ๐(๐ก) and 4๐1๐ณ๐are
โ๐ฎ๐ (๐, ๐)โ22 =
12๐
๐โ๐=2
๐1๐
๐๐+ ๐ + 5๐๐1
2๐(2๐1๐ + ๐)2 ,
(24a)
โ๐ฎ๐ (๐, ๐)โ22 =
๐1
2+
๐โ๐=2
๐2๐
๐๐, (24b)
โ๐ฎ๐ (4๐1๐ณ๐, ๐)โ22 =
๐โ๐=2
_2๐๐2๐
๐2๐๐, (24c)
where
๐1๐ = _2๐ (4๐2
1๐3๐ โ 1)2 + 4๐๐๐31
+ ๐1 (๐ + 4๐1๐) (4๐_๐๐1๐3 + 5_๐ + 4๐๐1),๐2๐ = 2๐๐3
1 (๐ + 2๐1๐)2 + 2_๐๐41๐
2 (4๐1๐3๐ + 4),๐๐ = ๐_
2๐ (4๐2
1๐3๐ โ 1)2 + 16๐_๐๐41๐3๐
2 + ๐2_๐๐1
+ 4๐1 (๐ + 2๐1๐)2 (๐๐1 + _๐ + ๐_๐๐1๐3).
Proof: Let ๐ธ โ R๐ร๐ be defined as in Lemma 4.1 and let๐1 = ๐ธโ1๐ฝ , ๐2 = ๐ธโ1๐, ๐3 = ๐ธโ1๐ผ, ๐4 = ๐ธโ1๐, we obtainthe closed-loop system in the general form as (7) with
๐ =
๐1๐2๐3๐4
, ๐จ =
0 ๐ฐ๐ 0 0
โ 1๐๐ฒ โ ๐๐ ๐ฐ๐ 0 4๐1
๐ ๐ฐ๐0 ๐๐ฐ๐ 0 4๐1๐3๐ฒ0 โ๐1๐๐ฐ๐ โ๐1๐ฐ๐ โ4๐1๐ฐ๐
, ๐ฉ =
0
๐ธโ1
๐00
,where ๐ฒ is the diagonal matrix defined in Lemma 4.1. Each ofthe block matrices in the matrix ๐จ is either the zero matrix or adiagonal matrix, so the components of the vector ๐1 โ R๐, ๐2 โR๐, ๐3 โ R๐, ๐4 โ R๐ can be decoupled.
With the same method for obtaining (18) in the proof ofTheorem 4.2, setting ๐ = ๐ = ๐๐ฅ2, ๐ช = [0,๐ธ, 0, 0], we de-rive (24a) for ๐(๐ก). Then, setting ๐(๐ก) = ๐(๐ก) = 4๐1๐ (๐ก) and๐ช = [0, 0, 0, 4๐1๐ธ], we derive (24b) for the norm of ๐(๐ก). Fi-nally for the coherence measurement of the marginal cost, set-ting ๐ = 4๐1๐ณ๐ and ๐ช = [0, 0, 0, 4๐1๐ณ๐ธ], we derive (24c).๏ฟฝ
Similar as Theorem 4.5, for a positive definite ๐ฉ โ R๐ร๐, weobtain the following theorem.
6
Theorem 4.7. Consider the closed-loop system (23) of DPIACwith a positive definite ๐ฉ โ R๐ร๐, the squared H2 norm of ๐(๐ก),๐(๐ก) and 4๐1๐ณ๐ satisfy
๐พ2min๐บ๐ โค โ๐ฎ๐ (๐, ๐)โ2
2 โค ๐พ2max๐บ๐ ,
๐พ2min (
๐1
2+
๐โ๐=2
๐2๐
๐๐) โค โ๐ฎ๐ (๐, ๐)โ2
2 โค ๐พ2max (
๐1
2+
๐โ๐=2
๐2๐
๐๐)
๐พ2min
๐โ๐=2
_2๐๐2๐
๐2๐๐โค โ๐ฎ๐ (4๐1๐ณ๐, ๐)โ2
2 โค ๐พ2max
๐โ๐=2
_2๐๐2๐
๐2๐๐
where ๐พmin and ๐พmax are the smallest and the largest eigenvalueof ๐ฉ, ๐1๐ , ๐2๐ and ๐๐ are defined in Theorem 4.6 and
๐บ๐ =1
2๐
๐โ๐=2
๐1๐
๐๐+ ๐ + 5๐๐1
2๐(2๐1๐ + ๐)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 ๐1 and ๐3 on thenorms are similar as in the case with identical strength of dis-turbances.
Based on Theorem 4.6, we analyze the impact of ๐1 and ๐3on 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 ๐1 when ๐3 is fixed.
The norm of ๐ 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 ๐1 increases. Hence,when the overall frequency deviation dominates the relative fre-quency deviation, the convergence can also be accelerated by alarge ๐1 as analyzed in Remark 4.4 for GBPIAC. From (24a),we derive
lim๐1โโ
| |๐ฎ๐ (๐, ๐) | |22 =1
2๐
๐โ๐=2
_2๐๐2
3
๐_2๐๐2
3 + ๐ (1 + 2_๐๐3), (27)
which indicates that even with a large ๐1, the frequency devi-ations cannot be decreased anymore when ๐3 is nonzero. Sosimilar to GBPIAC, the relative frequency deviation cannot besuppressed to an arbitrary small positive value in DPIAC. How-ever, when ๐3 = 0, DPIAC reduces to DecPIAC (6), thus
โ๐ฎ๐ (๐, ๐)โ22 โผ ๐ (๐โ1
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 ๐1 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 ๐1 is fixed, it can be easily observed from (24a) thatthe order of ๐3 in the term ๐1๐ is 2 which is the same as in theterm ๐๐ , thus ๐3 has little influence on the frequency deviation.
4.2.2. The control costWe first analyze the influence of ๐1 on the cost and then the
influence of ๐3. For any ๐3 โฅ 0, we derive from (24b) that
โ๐ฎ๐ (๐, ๐)โ22 โผ ๐ (๐1),
which indicates that the control cost increases as ๐1 increases.Recalling the impact of ๐1 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 ๐3 influences the control cost. From(24b), we obtain that
โ๐ฎ๐ (๐, ๐)โ22 โผ ๐1
2+๐ (๐1๐
โ13 ), (29)
where the second term is positive. It shows that the controlcost decreases as ๐3 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 ๐3 has little influence on the frequency deviation. Hencethe control cost can be decreased by ๐3 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 โ๐ฎ๐ (4๐1๐ณ๐, ๐) | |2. From (24c), we obtain
โ๐ฎ๐ (4๐1๐ณ๐, ๐)โ22 = ๐ (๐โ1
3 ),
which indicates that the difference of the marginal costs de-creases as ๐3 increases. Hence, this analytically confirms thatthe consensus speed can be increased by increasing ๐3.
Remark 4.10. In practice, similar to ๐1, the configuration of๐3 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 ๐1, we can easily obtain from (13b, 24b) that
โ๐ฎ๐ (๐, ๐)โ < โ๐ฎ๐ (๐, ๐) | |, (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
lim๐3โโ
โ๐ฎ๐ (๐, ๐)โ22 =
๐ โ 12๐๐
+ ๐ + 5๐๐1
2๐(2๐1๐ + ๐)2 = | |๐ฎ๐ (๐, ๐) | |2,
lim๐3โโ
โ๐ฎ๐ (๐, ๐)โ22 =
๐1
2= โ๐ฎ๐ (๐, ๐)โ2.
Hence, as ๐3 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. HenceV๐ = {๐บ1, ๐บ2, ๐บ3, ๐บ4, ๐บ5, ๐บ6, ๐บ7, ๐บ8, ๐บ9, ๐บ10}, V๐ =
{30, 31, 32, 33, 34, 35, 36, 37, 38, 39} and the other nodes are inset V๐น . The nodes in V๐ โชV๐น are all equipped with secondaryfrequency controllers such that V๐พ = V๐ โช V๐น . Because thevoltages are constants, the angles of the synchronous machineand the bus have the same dynamics [29]. Except the controlgain coefficients ๐1 and ๐3, 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 ๐๐ ๐ = 1 for the communication ifnode ๐ and ๐ 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 ๐1 and ๐3 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 ๐ก = 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 ๐1 and ๐3 monotonically. Wecalculate the following two metrics
๐ =
โซ ๐0
0๐๐ (๐ก)๐(๐ก)๐๐ก, and ๐ถ =
12
โซ ๐0
0๐๐ (๐ก)๐ถ๐(๐ก)๐๐ก,
to measure the performance of ๐(๐ก) and ๐(๐ก) during the tran-sient phase, where ๐0 = 40, ๐ = col(๐๐) for ๐ โ V๐ โช V๐น ,๐ = col(๐ข๐) for ๐ โ V๐ โชV๐น and ๐ถ = diag(๐ผ๐).
From Fig.2 (๐1-๐2), it can be observed that the frequencyrestoration is accelerated by a larger ๐1 with an accelerated con-vergence of the control input as shown in Fig. 2 (๐1). FromFig.2 (๐2-๐3), it can be seen that the consensus of the marginalcosts is accelerated by a larger ๐3 with little influences on thefrequency deviation as shown in Fig.2 (๐2-๐3). It can be easilyimagined that the marginal costs converge to that of GBPIAC asshown in Fig. 2 (b4) as ๐3 further increases. Hence, by increas-ing ๐1 and ๐3, the convergence of the state of the closed-sytstemto the optimal state can be accelerated, and by increasing ๐3,the performance of the distributed control method DPIAC ap-proaches to that of the centralized control method GBPIAC.
Fig.2 (๐2 โ ๐3) show the trends of ๐ and ๐ถ with respect to๐1 and ๐3. It can be observed from Fig.2 (๐2) that as ๐1 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, ๐converges to a non-zero value as ๐1 increases which is con-sistent with the anlsysis in (27). However, the control cost isbounded as ๐1 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 (๐3), it can seen that the control cost decreases inverselyto a non-zero value as ๐3 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
ร106
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
ร106
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
ร105
0
1
2
3
4
DPIAC (k3 = 10): EC and ES
Fre
qu
en
cy d
evia
tio
n
ร10-8
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
ร105
0
1
2
3
4
DPIAC (k1 = 2.4): EC and ES
Fre
qu
en
cy d
evia
tio
n
ร10-8
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 ๐ค๐ โผ ๐ (0, ๐2
๐) with
๐๐ = 0.002 for ๐ โ V๐ . 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 (๐1-๐2) that the variance ofthe frequency deviation can be suppressed with a large ๐1 whichhowever increases the variances of the marginal costs and thetotal control cost as shown in Fig. 3 (๐2) and (๐1) respectively.These observations are consistent with the analysis of Theo-rem 4.6 when ๐3 is fixed. From Fig. 3 (๐2-๐3), it can be seenthat increasing ๐3 can effectively suppress the variances of themarginal costs.
We calculate the following metrics to study the impact of๐1 and ๐3 on the variance of the frequency deviation and theexpected control cost,
๐ธ๐ = ๐ธ [๐๐ (๐ก)๐(๐ก)], and ๐ธ๐ถ =12๐ธ [๐๐ (๐ก)๐ถ๐(๐ก)] .
Fig.3 (๐2-๐3) show the trend of ๐ธ๐ and ๐ธ๐ถ as ๐1 and ๐3 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 (๐2). The difference is that the controlcost increases linearly as ๐1 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 (๐3) illustrates the trend of the expected control costand the variance of the frequency deviations with respect to ๐3.It can be observed that the expected control cost decreases as ๐3increases, 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.
References
[1] P. Kundur, Power system stability and control, McGraw-Hill, New York,1994.
[2] A. J. Wood, B. F. Wollenberg, Power generation, operation, and control,2nd ed., John Wiley & Sons, New York, 1996.
[3] F. Dorfler, S. Grammatico, Gather-and-broadcast frequency control inpower systems, Automatica 79 (2017) 296 โ 305. doi:http://dx.doi.org/10.1016/j.automatica.2017.02.003.
[4] C. Zhao, E. Mallada, F. Dorfler, Distributed frequency control for sta-bility and economic dispatch in power networks, in: Proc. 2015 Amer.Control Conf., 2015, pp. 2359โ2364. doi:https://doi.org/10.1109/ACC.2015.7171085.
[5] N. Li, C. Zhao, L. Chen, Connecting automatic generation control andeconomic dispatch from an optimization view, IEEE Trans. ControlNetw Syst. 3 (2016) 254โ263. doi:https://doi.org/10.1109/TCNS.2015.2459451.
[6] C. Zhao, E. Mallada, S. Low, J. Bialek, A unified framework for fre-quency control and congestion management, in: Proc. 2016 Power Sys.Comp. Conf., 2016, pp. 1โ7. doi:https://doi.org/10.1109/PSCC.2016.7541028.
[7] K. Xi, J. L. A. Dubbeldam, H. X. Lin, J. H. van Schuppen, Power im-balance allocation control of power systems-secondary frequency con-trol, Automatica 92 (2018) 72โ85. doi:https://doi.org/10.1016/j.automatica.2018.02.019.
[8] J. W. Simpson-Porco, B. K. Poolla, N. Monshizadeh, F. Dorfler,Quadratic performance of primal-dual methods with application to sec-ondary frequency control of power systems, in: Proc. 2016 IEEE Conf.on Decis. Control, 2016, pp. 1840โ1845. doi:https://doi.org/10.1109/CDC.2016.7798532.
[9] K. Porter, What caused the UKโs power blackout and will it happen again.,2019. (accessed 12 August 2019). URL: http://watt-logic.com/2019/08/12/august-2019-blackout/.
[10] H. Bevrani, Robust Power System Frequency Control, 2nd ed., Springer,New York, 2014.
[11] D. Rerkpreedapong, A. Hasanovic, A. Feliachi, Robust load frequencycontrol using genetic algorithms and linear matrix inequalities, IEEETrans. Power Syst. 18 (2003) 855โ861. doi:https://doi.org/10.1109/TPWRS.2003.811005.
[12] Y. Mi, Y. Fu, C. Wang, P. Wang, Decentralized sliding mode load fre-quency control for multi-area power systems, IEEE Trans. Power Syst.28 (2013) 4301โ4309. doi:https://doi.org/10.1109/TPWRS.2013.2277131.
[13] K. Vrdoljak, N. Peric, I. Petrovic, Sliding mode based load-frequencycontrol in power systems, Electric Power Syst. Res. 80 (2010) 514 โ 527.doi:https://doi.org/10.1016/j.epsr.2009.10.026.
[14] E. Cam, Ilhan Kocaarslan, Load frequency control in two area powersystems using fuzzy logic controller, Energy Convers. Manag. 46 (2005)233 โ 243. doi:https://doi.org/10.1016/j.enconman.2004.02.022.
[15] K. Xi, H. X. Lin, C. Shen, J. H. van Schuppen, Multi-level power-imbalance allocation control for secondary frequency control of powersystems, IEEE Trans. Autom. Control (2019) 1โ16. doi:https://doi.org/10.1109/TAC.2019.2934014.
[16] M. Andreasson, E. Tegling, H. Sandberg, K. H. Johansson, Coherence insynchronizing power networks with distributed integral control, in: Proc.2017 IEEE Conf. on Decis. Control, 2017, pp. 6327โ6333. doi:https://doi.org/10.1109/CDC.2017.8264613.
[17] X. Wu, F. Dorfler, M. R. Jovanovic, Input-Output analysis and decen-tralized optimal control of inter-area oscillations in power systems, IEEETrans. Power Syst. 31 (2016) 2434โ2444. doi:https://doi.org/10.1109/TPWRS.2015.2451592.
[18] B. K. Poolla, J. W. Simpson-Porco, N. Monshizadeh, F. Dorfler,Quadratic performance analysis of secondary frequency controllers,in: Proc. 2019 IEEE Conf. on Decis. Control, 2019, pp. 7492โ7497.doi:https://doi.org/10.1109/CDC40024.2019.9029647.
[19] B. K. Poolla, S. Bolognani, F. Dorfler, Optimal placement of virtual in-ertia in power grids, IEEE Trans. Autom. Control 62 (2017) 6209โ6220.doi:https://doi.org/10.1109/TAC.2017.2703302.
[20] L. Guo, C. Zhao, S. H. Low, Cyber network design for secondary fre-quency regulation: A spectral approach, in: 2018 Power Systems Compu-tation Conference (PSCC), 2018, pp. 1โ7. doi:https://doi.org/10.23919/PSCC.2018.8442814.
[21] J. W. Simpson-Porco, F. Dorfler, F. Bullo, Voltage collapse in complexpower grids, Nat. Commun. 7 (2016) 10790. doi:https://doi.org/10.1038/ncomms10790.
[22] Y. Khayat et al, On the secondary control architectures of AC micro-
10
grids: An overview, IEEE Trans. Power Electron. 35 (2020) 6482โ6500.doi:https://doi.org/10.1109/TPEL.2019.2951694.
[23] J. C. Doyle, K. Glover, P. P. Khargonekar, B. A. Francis, State-spacesolutions to standard H2 and H infinty control problems, IEEE Trans.Autom. Control 34 (1989) 831โ847. doi:https://doi.org/10.1109/9.29425.
[24] R. Toscano, Structured controllers for uncertain systems, Springer-verlag,London, 2013.
[25] I. Karatzas, S. Shreve, Brownian motion and stochastic calculus,Springer-Verlag, Berlin, 1987.
[26] E. Tegling, B. Bamieh, D. F. Gayme, The price of synchrony: Evaluatingthe resistive losses in synchronizing power networks, IEEE Trans. ControlNetw. Syst. 2 (2015) 254โ266. doi:https://doi.org/10.1109/TCNS.2015.2399193.
[27] P. M. Anderson, A. A. Fouad, Power System Control and Stability, 1sted., The Iowa State University Press, Iowa, 1977.
[28] F. Milano, Power systems analysis toolbox, University of Castilla,Castilla-La Mancha, Spain, 2008.
[29] M. Ilic, J. Zaborszky, Dynamics and control of large electric power sys-tems, John Wiley & Sons, New York, 2000.
[30] K. Wang, M. L. Crow, Numerical simulation of stochastic differen-tial algebraic equations for power system transient stability with randomloads, in: 2011 IEEE PES General Meeting, 2011, pp. 1โ8. doi:https://doi.org/10.1109/PES.2011.6039188.
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