constrained load/frequency control problems in networked ... · a hierarchical control structure,...

Journal of the Franklin Institute 348 (2011) 832–852

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Constrained load/frequency control problems innetworked multi-area power systems

G. Franz�e�, F. Tedesco

DEIS - Universit �a degli Studi della Calabria, Via P. Bucci, Cubo 42 C, 87036 Rende (CS), Italy

Received 18 February 2010; received in revised form 18 November 2010; accepted 19 February 2011

Available online 4 March 2011

Abstract

In this paper, we present a supervisory discrete-time predictive control strategy for load/frequency

control problems in networked multi-area power systems subject to coordination constraints.

Coordination between the control center and the spatially distributed areas is accomplished via data

networks subject to communication latency modeled by time-varying time-delay. The aim here is

finding supervising strategies able to reconfigure, whenever necessary in response to unexpected load

changes and/or faults, the nominal set-points on frequency and generated power to the generators of

each area so that viable evolutions would arise for the overall power system and a new sustainable

equilibrium is reached. In order to demonstrate the effectiveness of the strategy, examples on a

four-area power system are presented.

& 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction

In a restructured system, power systems consist of several interconnected control areaswhere each one is responsible for its native load and scheduled interchanges withneighboring areas. A hierarchical control structure, with many local and some wide-areacontrollers is required to effectively handle all adverse phenomena, such as imbalances,fluctuations, disturbances, etc.Typical local control objectives include: power system protection [1], power production

stabilization [2], local voltage control and power flow control. As a common feature,

2.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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nding author.

dresses: [email protected] (G. Franz�e), [email protected] (F. Tedesco).

dx.doi.org/10.1016/j.jfranklin.2011.02.009

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G. Franz�e, F. Tedesco / Journal of the Franklin Institute 348 (2011) 832–852 833

a regional communication infrastructure is required to be effective because their actionsdepend on data spread out in the power grid at far away locations.

Load frequency control (LFC) is the mechanism by which a balance between powergeneration and demand is satisfied. Its basic objective is the global matching between powergeneration and demand, which must be maintained as close as possible regardless of loadfluctuations. If an imbalance results, large frequency deviations may occur with seriousimpacts on system operations. Power imbalances are caused by unusual load profiles(see [3,4]), fault/failure events on generating units and/or breakdowns of the tie-linesconnecting the utilities amongst the areas. All above phenomena are hard to be anticipatedand, if not timely detected and corrected, may cause massive power shortages or excesseswhich result in unacceptable frequency deviations exceeding minimum and maximumlimits [5]. This usually leads to protection relays activation with the correspondingdisconnection of many components of the power grid and the breakup of the entire systeminto relatively small disjoined islands.

Typical LFC actions are finalized to maintain the frequency deviations within acceptableand secure limits during these events. However, it is fair commenting that the LFC scope islimited to the stabilization of small imbalances around the nominal equilibrium. In fact,these controllers are not usually capable, for example, of taking the system from a pre-faultequilibrium and steering it into another stable post-fault equilibrium, outside the stabilityregion of the former [6]. Current LFC architectures become ineffective during extreme events(e.g. huge power demand or tie-lines breakdown) and, as a consequence, undesirable transientphenomena could compromise the system integrity if the action of the tripping devices wouldnot take place.

Details on LFC can be found e.g. in [5–14]. The recent survey [15] offers an update andcomplete overview of LFC strategies. Moreover, a detailed discussion on technicalancillary services and economic features strictly related to frequency and voltage controlproblems in power systems around the world is given in the two surveys [16,17]. In essence,the LFC requirements are satisfied by the so-called ALFC (automatic load frequencycontrol) stage, which is implemented in the primary and secondary control layers. Theprimary loop provides a basic small-signal compensation and is computed on the basisof local information only. While, the secondary LFC stage is implemented mainly forthat purpose and is in charge to regulate frequency deviations by collecting, via dedicateddata links, information on frequencies and power flows at relevant points of thegrid and generate, usually in a centralized fashion (in a central station), appropriatecounter-actions.

An important aspect in networked applications, rarely considered in the load/frequencycontrol literature, is how to take care of the communication latency affecting the data linksconnecting each area to the control center. In this paper, the network latency is consideredand it is abstractly modeled by a time-varying time-delay. If the network’s protocol isequipped with a time-stamping mechanism, the linear time-invariant discrete-time (LTI-TD)paradigm can be used for system description. See [18] and references therein for furtherdiscussions and theoretical justifications on this point.

The necessity to face with the problem of delayed communication is receiving increasingimportance from the researcher community and it is now well understood that time-delaycan compromise the system integrity by driving the power system toward the instability orother unacceptable behaviors when authoritative wide-area control actions are imple-mented. Currently, only few recent works (see [6,7,9,14]) have analyzed the presence of the

G. Franz�e, F. Tedesco / Journal of the Franklin Institute 348 (2011) 832–852834

communication time-delay in power system networks, and limitedly to normal operatingscenarios.In this paper we propose an on-line tertiary LFC supervisory approach based on

constrained control ideas, initially developed for telecontrol problems [19], aimed atsupervising remotely located generation units, connected by distribution lines to remotelydistributed loads, under coordination requirements consisting of enforcing pointwise-in-timeconstraints on the evolutions of relevant system variables. In particular, a LFC scheme ableto directly enforce constraints on maximum allowable frequency and power deviations fromnominal values is proposed, and it will be shown that the scheme is also effective in thepresence of time-varying communication time-delays and possible data-loss.We focus on the master/slaves scenario depicted in Fig. 1, where the ALFC master

represents a supervisor in charge of taking decisions based on delayed information comingfrom the areas via data networks, namely suitable state vectors xiðt�tÞ, with t being thetime-delay representing the network latency. At the remote sides, the ALFC slave units arethose parts of the strategy which require only local, though updated, information. Then,they are implemented distributively in the proximity of each generating unit (for simplicitywe will assume hereafter that a single generating unit per area exists). Primary andsecondary standard control structures are still present and are not modified by the additionof this proposed tertiary control level.In Fig. 1, the signals ri, di, xi, yi and ci represent respectively: nominal set-points, loads,

states, performance-related and coordination-related outputs for the i-th area. In such acontext, the supervisory task can be expressed as the requirement of satisfying sometracking performance, viz. yiðtÞ � riðtÞ, whereas the coordination task consists of enforcingpointwise-in-time constraints of the form ciðtÞ 2 Ci on each area and/or f ðc1ðtÞ; c2ðtÞ;. . . ; cNðtÞÞ 2 C on the overall evolution of the power-grid. These constraints can be used toimpose all safety, operative and coordination requirements that ensure feasible evolutionsto the power-grid so as to prevent the actions of the tripping devices from occurring.For each area, the ALFC master action consists of a vector zi :¼ ½g0i y

0i�0, where gi

represents the best approximation of ri compatible with the prescribed constraints. Themodification of ri in gi takes place if the equilibrium corresponding to ri is no longersustainable given the actual changed conditions. As a result, if gi is applied instead of ri, itdoes not produce constraints violation and allows the overall network to reach a newfeasible and sustainable equilibrium. The vector yi is an additional optional command.

Fig. 1. A distributed tertiary LFC structure for a two-area power system.


If used, it represents an offset sequence to be added at the input terminal of the generationunit in order to enlarge the set of feasible evolutions for that area and for the overall powergrid. It is worth commenting that, because of time-delay, the computed vectors ziðt; tþ tÞare double indexed. Such a notation indicates that they are computed at time t for beingapplied exactly at time tþ t or nevermore. It has been shown [22] that their exactapplication at time tþ t is mandatory to ensure constraints fulfilment and one cannotdepart from. Finally, because of possible data-loss, if the vector ziðt�t; tÞ is not available attime t at the remote side, the ALFC slave logic is typically instructed to replace it with thelast applied vector ziðt�1Þ. Then, we denote with zi(t) the vector of commands reallyapplied at time t and allow it to be different from ziðt�t; tÞ in general.

The above ALFC scheme makes use of a LTI-TD dynamic model of each area forcomputing the future state predictions of the power grid. Moreover, unlike most LFC schemesreported in the literature, here the mathematical models of the power grid are defined in termsof absolute variables. The rationale of such a choice hinges upon the possibility of takingadvantage of set-point adjustments w.r.t. their nominal values in order to enlarge the set offeasible transients. This is of paramount importance in the presence of faults and/or unexpectedlarge load changes for avoiding, as long as possible, the intervention of tripping devices.

It is expected that the above described ALFC strategy, in response to critical events, iscapable to reconfigure the set-points and the additional offsets for the local primary/secondary control laws in such a way to maintain the overall system evolutions alwayscoordinated, within the prescribed safety and operative constraints.

The core of the proposed supervisory strategy is based on a predictive control approachused recently to synthesize reference governor (RG) [20–24] and parameter governor (PG)units [25]. The idea here is to combine in a single unit both strategies referred hereafter toas reference-offset governor (ROG). Moreover, in order to take care of the time-varyingcommunication latency possibly existing from the control center to each area, an extensionof this strategy is presented, hereafter referred to as distributed reference offset governor

(DROG), by resorting to the master/slaves ideas of [22].

2. Power system model and operative constraints

In this section, a two-area power system and its load/frequency control scheme(see Fig. 2) is detailed.

By referring to a single area, a small-signal dynamical model of the two-area powersystem is as follows:

_f iðtÞ ¼ �1

TPi

ðfiðtÞ�firef Þ þKPi

TPi

PTiðtÞ�

KPi

TPi

PtieðtÞ�KPi

TPi

PDiðtÞ�

firef

KPi

� �ð1Þ

_PTiðtÞ ¼ �

1

TTi

PTiðtÞ þ

1

TTi

PviðtÞ ð2Þ

_PviðtÞ ¼ �

1

RiTGi

ðfiðtÞ�firef Þ�1

TGi

PviðtÞ�

1

TGi

PciðtÞ þ

1

TGi

yiðtÞ ð3Þ

_PciðtÞ ¼ �KiðPtieðtÞ þ Bsi

ðfiðtÞ�firef ÞÞ ð4Þ

_PtieðtÞ ¼ T12ððf1ðtÞ�f1ref Þ�ðf2ðtÞ�f2ref ÞÞ ð5Þ

Kp11+sTp1

1R1

11+sTG1

-K1s

Bs1

11+sTT1

T12s

Kp21+sTp2

11+sTG2

-K2s

11+sTT2

a12 a12

1Bs2

+

+

+

+

+

+

+

+

+

+

+ +

+

_

_

_

_

_

_

_

_

_

f1

f2

f1ref

f2ref

Pc2

Pv1

Pv2

PD1

PD2

θ1

θ2

PT1

PT2

Ptie

Pc1

primary control levelsecondary

control level

Kp2

f2ref

+

+

Kp1

f1ref

+

+

R2

Fig. 2. Block diagram of a two-area power system.


where firef and yi represent respectively the frequency set-point and the additional offset tobe added to the nominal control input. The constant term firef =KPi

in Eq. (1) accounts forthe representation of the LTI model of the power system in terms of absolute variables(unlike the standard incremental models used more often in the literature [13,26]).Moreover, PTi

ðtÞ is the active power produced by the generator unit, PDiðtÞ the local load

demand, PviðtÞ the change in the valve position and Pci

ðtÞ the control action, related to theACEi(t) error index described below, provided by the secondary control layer forcompensating frequency deviations. All the variables are standard and their descriptioncan be found in [13].The task of the primary/secondary control layers is to maintain the power system within

acceptable operating limits by achieving the load balance PTiðtÞ � PDi

ðtÞ while maintainingthe area frequency fi(t) at their nominal value firef. When the power balance is no longerpreserved, because e.g. of a change in the local load PDi

or in the tie-line power of Ptie(t),frequency deviations arise (the frequency fi(t) shifts w.r.t. its set-point). The effect forcustomers is a lower quality of the delivered electrical energy and damaging vibrations inthe turbines (at frequencies lower than � 57258 Hz) may occur.The load/frequency control requirements are traditionally satisfied by the ALFC stage

(primary and secondary loops), which typically takes the form of a PI local controller. Thepurpose of the primary loop is to achieve fast adjustments on the produced power PTi

ðtÞ inresponse to local frequency deviations. To this end, by means of the regulation parameterRi, the following control action:

PciðtÞ�

1

Ri

ðfiðtÞ�firef Þ ð6Þ


is fed into the governor unit, here represented by a first order lag 1=ð1þ sTGiÞ: Such a

command causes a position change PviðtÞ of the control valve which, in turn, translates into

a power increment of PTiðtÞ:

The secondary ALFC action is based on an error signal, referred to as area control error

(ACE), which in addition to local frequency deviations also contains information on globalfrequency deviations via the power exchanged between the areas via the tie-line [26]. Foreach i-th area, the ACE index is defined as

ACEiðtÞ ¼ PtieðtÞ þ BsiðfiðtÞ�firef Þ ð7Þ

and the command PciðtÞ is computed as shown in Eq. (4). Finally, the block Kpi

=ð1þ sTpiÞ

is used to model the linear power system dynamics (inertia constants and dampingcoefficients).

The above ALFC loops perform in a satisfactory way under small power imbalancesarising during normal operating conditions. On the contrary, when the power system is inan emergency state due to a sudden generator loss or to a tie-line disconnection, the ALFCstage is ineffective because turbines have slow responses. In practice, for handling suchsituations, the power system is equipped with the so-called emergency control whose task isto recover the normal operating state. However, such efforts could fail and the systemwould end up in a total blackout.

2.1. Operative constraints

Our approach in trying to mitigate the occurrence of the above abnormal scenarios isthat of defining a set of operative and safety constraints, which characterize more preciselythe normal conditions one would like to impose, to be fulfilled as longer as possible by thesystem evolutions under the action of the proposed tertiary supervisory scheme. To thisend, the following constraints will be considered:

fmin;irfiðtÞrfmax;i; i ¼ 1; 2 ð8Þ

jPtieðtÞjrbmax ð9Þ

gmin;irPTiðtÞrgmax;i; i ¼ 1; 2 ð10Þ

Specifically, constraint (8) prescribes bounds on maximum frequency deviations for thei-th area, Eq. (9) imposes bounds on the power flow exchanged between the areas via thetie-line, while Eq. (10) limits the maximum generable power in each area. The latter springsfrom the existing limitations on each generation unit, which can be identified as asaturation on the power production.

3. Distributed reference-offset governor (DROG) device

In this section, the main ideas and properties of the proposed master/slaves DROGapproach will be presented. This solution is a direct generalization of the GC schemepresented in [29]. For the sake of clarity, the approach is presented for a single slave system(a single area). The extension to the multi-slaves case is straightforward and follows similarideas of [22]. Consider the master/slave scenario depicted in Fig. 3.

x(t)

z(t,t+τ)

x(t-τ)

r(t+τ)

z(t)ROGSlave

ROGMaster

PrimalController

θ(t)

xc(t)

c(t)y(t)

Plant

xp(t)

g(t)u(t)

++

d(t)z(t-τ,t),

Fig. 3. The distributed reference-offset governor (DROG).


The typical system structure for the slave side consists of a primal compensated plant(dashed box) described by the following state-space representation:

xðtþ 1Þ ¼ FxðtÞ þ GzðtÞ þ GddðtÞ

yðtÞ ¼ HyxðtÞ

cðtÞ ¼ HcxðtÞ þ LzðtÞ þ LddðtÞ

8><>: ð11Þ

with t 2 Zþ :¼ f0; 1; . . .g; xðtÞ 2 Rn the state vector (which includes the controller states ifany); zðtÞ ¼ ½gðtÞyðtÞ� 2 R

2m the ROG output, where gðtÞ 2 Rm is the manipulable referencewhich, if no constraints were present, it would essentially coincide with the referencerðtÞ 2 Rm. Moreover, yðtÞ 2 Rm is an adjustable offset on the nominal control inputwhich we assume to be selected from a given convex and compact set Y, with 0m 2 int Y;dðtÞ 2 Rnd an exogenous bounded disturbance satisfying dðtÞ 2 D;8t 2 Zþ with D aspecified convex and compact set such that 0nd

2 D; yðtÞ 2 Rm the output, viz. aperformance related signal; and finally cðtÞ 2 Rnc the constrained vector

cðtÞ 2 C; 8t 2 Zþ ð12Þ

with C � Rnc a prescribed convex and compact set. Moreover, the following matrices aredefined

G ¼ Gg Gy

h i;L ¼ Lg Ly

h iIt is assumed that

ðA1Þ

ð1Þ F is an asymptotically stable matrix;

ð2Þ System ð11Þ is offset� free w:r:t: gðtÞ i:e:

HyðIn�FÞ�1Gg ¼ Im

8><>:

At the master side, the system representation (11) is used as a model for computing the

ROG master action zðt; tþ tÞ ¼ ½gTðt; tþ tÞy

Tðt; tþ tÞ�T 2 R2m (see Fig. 3), which has to

be read as: commands generated at time t for being applied at time tþ t. Typically it wouldbe zðtÞ ¼ zðt�t; tÞ under normal conditions. However, if zðt�t; tÞ would not be available atthe remote side at time t for congestions or data-loss, the ROG slave logic is instructed toapply the previous applied command, viz. z(t)=z(t�1).Moreover, it is required that: (1) gðt; tþ tÞ-r whenever rðtþ tÞ-r, with r the best

admissible approximation of r and yðt; tþ tÞ-0m; and (2) the DROG have a finite settlingtime, viz. gðt; tþ tÞ ¼ r and yðt; tþ tÞ ¼ 0m for a possibly large but finite t whenever thereference stays constant after a finite time.


In order to make our discussion more general, we allow hereafter the time-delay to bepossibly time-varying. To this end, let tf ðtÞ and tbðtÞ be the forward and, respectively,backward time-delays at each time instant t (expressed in sampling units), viz. tf ðtÞ is thedelay from the master to the slave unit whereas tbðtÞ is the delay in the opposite direction.We assume further that the following upper-bounds tf and tb

tf ðtÞrtf and tbðtÞrtb ð13Þ

apply.At each sampling time t, we indicate with tbrt the most recent sampling instant at which

the area has sent a piece of information which is received by the ROG master at timeinstant t and with tf rt be the most recent sampling instant at which the ROG master hassent a command that is received by the area at time instant t. Then, under Eq. (13) it resultsthat t�tbrtb and t�tf rtf , which means that all the delivered signals are timely received.The situation is clarified in Fig. 4.

The idea here is that the DROG master logic device acts as if the time-delay would not bepresent by modifying, whenever necessary, the reference rðtþ tf Þ into gðt; tþ tf Þ and by addingan offset yðt; tþ tf Þ on the nominal control action. In so doing, at each time t, an admissiblecommand zðt; tþ tf Þ is generated. The computation of the master DROG action relies on thefuture state and constrained vector predictions from tþ tf onward, computed under theassumption that a constant vector z is applied. To this end, let tbrt be the most recent samplinginstant at which the master has received a piece of information from the slave and x(tb) its stateat that time. Notice that the following inequality tþ tf�tbrtb þ tf holds true, whichrepresents the maximum round-trip delay on the network.

By linearity, one is allowed to separate the effects of the initial conditions and inputsfrom those of disturbances, e.g. xðtÞ ¼ xðtÞ þ ~xðtÞ; where xðtÞ is the disturbance-freecomponent and ~xðtÞ depends only on the disturbances. Then, the remote disturbance-freestate at tþ tf can easily be computed as

xðtþ tf jtbÞ :¼ Ftþt f�tb xðtbÞ þXtþt f�1

i¼tb

Ftþt f�i�1Gzði�tf ; iÞ ð14Þ

under the assumption that all commands z delivered from tb to tþ tf�1 have beenreceived and timely applied.

In the sequel, we adopt the following notations:

xz :¼ ðIn�FÞ�1Gz

yz :¼ HyðIn�FÞ�1Gz

Fig. 4. The arrows indicate the delivering directions.

x

z

.................................................

.....................

CC0

C∞

C1

Ck

x

z

δ

W�

C�

cz

z

xz

C∞

Fig. 5. (Left) Nested set family Ck. (Right) Sets Cd, Wd and VðxÞ. The set cz denotes the equilibrium points

corresponding to constant commands z 2Wd.


cz :¼ HcðIn�FÞ�1Gzþ Lz ð15Þ

for the disturbance-free equilibrium solutions of Eq. (11) to a constant command zðtÞ � z;with zðtÞ � ½gy�; and g; y 2 Rm constant vectors. Consider next the following set of recursion(see Fig. 5 (Left)):

C0 :¼ C�LdDCk :¼ Ck�1�HcFk�1GdD^

C1 :¼\1k¼0

Ck ð16Þ

where, for given sets A; E � Rn, A�E � A is a proper restriction of A denoted in theliterature as the P-difference between sets [27]:

A�E :¼ fx 2 Rn : xþ e 2 A; 8e 2 Eg ð17Þ

In [23,28] it has been proved that the sets Ck are non-conservative restrictions of C such thatcðtÞ 2 C1;8t 2 Zþ; implies that cðtÞ 2 C;8t 2 Zþ: In the above references, it has also beenshown that the sets Ck are the largest restrictions of C such that cðtÞ 2 Ck impliescðtþ iÞ 2 C, any dðtþ iÞ 2 D;8i 2 f0; 1; . . . ; k�1g. In other words, if the disturbance-freeconstrained vector cðtÞ is contained in Ck at a certain time instant t, then one is ensured thatno constraint violations can occur for the next k time instants due to disturbances. Inparticular, cðtÞ 2 C1;8t 2 Zþ; implies that cðtÞ 2 C;8t 2 Zþ: If C1 is empty the problemhas no solution. If non-empty, all Ck’ s, 8k 2 Zþ, are non-empty, convex, compact andsatisfy the nesting condition Ck � Ck�1; 8k 2 Zþ: Thus, one is allowed to consider thedisturbance-free system evolutions only and adopt a ‘‘worst case’’ approach. For reasonswhich have been clarified in [20], it is convenient to introduce the following sets for agiven d40:

Cd :¼ C1�Bd ð18Þ

Wd :¼ fz 2 R2m : cz 2 Cdg ð19Þ

where Bd is a ball of radius d centered at the origin (see Fig. 5 (Right)). Let us assume thatthere exists a sufficiently small d40 such thatWd is non-empty. In particular,Wd is the setof all commands whose corresponding steady-state solution cz satisfies the constraints with


margin d: From the foregoing definitions and assumptions, it follows thatWd is closed andconvex. Again, emptiness of Wd means that the problem has no solution.

Then, the approach in selecting at each time t the DROG action z(t) will consist ofrestricting the choice amongst all vectors of a suitable x(t)-depending subset of Wd, eachvector of which, if constantly applied as a command to the system from the time instant t

onwards, gives rise to system evolutions which do not produce constraint violations.Notice that the choice zðtÞ 2Wd only ensures constraints fulfillment in steady-stateconditions, viz. when czðtÞ 2 Cd, but nothing can be said about the transient from x(t) toxzðtÞ: If many choices exist, a selection index is used to determine the best feasibleapproximation to r. Such a ROG command is applied, a new state is measured and theprocedure is repeated at next time instant tþ1 on the basis of the new state x(tþ1).

In this respect, we consider the following family of constant virtual command sequences:

zð�Þ ¼ fzðkÞ � z 2Wd;8k 2 Zþg ð20Þ

and, as a consequence, the disturbance-free state and c-vector future predictions emanatingfrom xðtþ tf jtbÞ under a whatever constant command z are given by

xðk; xðtþ tf jtbÞ; zÞ :¼ Fkxðtþ tf jtbÞ þXk�1i¼0

Fk�i�1Gz

cðk; xðtþ tf jtbÞ; zÞ :¼ Hcxðk;xðtþ tf jtbÞ; zÞ þ Lz ð21Þ

where cðk;xðtþ tf jtbÞ; zÞ has to be understood as the disturbance-free virtual evolution atvirtual time instant k (opposite to the real time t) of the constrained vector c, from theinitial condition x(t) (applied at virtual time zero) under the constant command zð�Þ � z:Finally, we define the set Vðxðtþ tf jtbÞÞ as

Vðxðtþ tf jtbÞÞ ¼ fz 2Wd : cðk;xðtþ tf jtbÞ; zÞ � Ckþtbþt f; 8k 2 Zþg ð22Þ

collects all constant virtual commands inWd whose corresponding c-evolutions starting attime tþ tf from the state xðtþ tf jtbÞ satisfy the constraints for all k 2 Zþ: This is requiredfor feasibility reasons because we have used the disturbances-free state prediction xðtþ

tf jtbÞ instead of xðtþ tf Þ, that does not contain information on the disturbances, as initialstate in Eq. (21) at each time t. However, because of linearity, we can equivalently take careof the overall effect of disturbances on the state evolution from tb to tþ tf in Eq. (14) byrestricting the feasibility conditions in Eq. (22) by an amount corresponding to effect of thedisturbances acting for a time equal to tb þ tf , which is an upper-bound of tþ tf�tb:Notice that Vðxðtþ tf jtbÞÞ is finitely determined as proved in [22]. Moreover, because oftime-invariance, the above machinery (20)–(22) can be used at each time instant t. As aconsequence Vðxðtþ tf jtbÞÞ �Wd; which, if non-empty, represents the set of all constantvirtual sequences in Wd whose evolutions starting from x(t) satisfy the constraints alsoduring transients. In [24], it has been proved that such a set is finitely determined, viz. thereexists a positive integer k0 such that Eq. (22) is identically characterizable by restrictingk 2 f0; . . . ; k0g, with k0 computable off-line.

Then, provided that Vðxðtþ tf jtbÞÞ is non-empty, closed and convex at each time t 2 Zþ;the DROG master output is chosen according to the solution of the following constrainedoptimization problem:

zðt; tþ tf Þ ¼ arg minz2Vðxðtþt f jtbÞÞ

Jg�rðtþ tf ÞJ2Cgþ JyJ2Cy

ð23Þ


where Cg ¼ CTg 40m, Cy ¼ CT

y 40m and JvJC :¼ vTCv: It represents the best feasible appro-ximation of rðtÞ; which, if constantly applied from t onwards to the system, would never produceconstraints violation. The slave part of the DROG logic is far more simpler and reduces to

zðtÞ ¼ zðt�tf ; tÞ ð24Þ

This strategy is justified by the fact that Vðxðtf j0bÞÞ non-empty implies that Vðxðtþ tf jtbÞÞ isnon-empty along system evolutions generated by the DROG strategy (23)–(24). The mainproperties of the DROG strategy may be summarized in Theorem 1.

Theorem 1. Consider system (11) along with the DROG logic (23)–(24). Let assumptions

(A1) be fulfilled and Vðxðt0 þ tf jt0bÞÞ be non-empty, where t0b

is the time instant at which the

master receives the first piece of information from the slave, generated at time t0brt0: Then

1.
The minimizer in Eq. (23) uniquely exists at each t 2 Zþ and can be obtained by solving a
convex constrained optimization problem, viz. Vðxðtf j0ÞÞ non-empty implies Vðxðtþtf jtbÞÞ non-empty along the trajectories generated by the DROG logic (23)–(24).

2.
The set Vðxðtþ tf jtbÞÞ, 8xðtÞ 2 Rn, is finitely determined, viz. there exists an integer k0
such that if cðk; xðtþ tf jtbÞ; zÞ 2 Ck, k 2 f0; 1; . . . k0g, then cðk;xðtþ tf jtbÞ; zÞ 2Ck8k 2 Zþ. Such a constraint horizon k0 can be determined off-line.

3.
The constraints are fulfilled for all t 2 Zþ. 4. The overall system is asymptotically stable; in particular, whenever rðtÞ � r, limt-1yðtÞ ¼ 0m
and gðtÞ converges either to r or to its best steady-state admissible approximation r; with

z :¼r

0m

" #:¼ argmin

z2Wd

Jðz; rÞ ð25Þ

where

Jðz; rÞ :¼ Jg�rJ2Cgþ JyJ2Cy

ð26Þ

Consequently, by the offset-free condition ðA1Þð2Þ, limt-þ1yðtÞ ¼ r, where y is the

disturbance-free component of y.
5. The strategy achieves a complete time-delay compensation, in that all commands generated
at time t at the master side will be applied to the slave with exactly tf sampling steps of

delay.

Proof. The proof is straightforward by noting that the DROG logic (23)–(24) essentiallyimplements the ROG scheme of [29] replaced by

zðtþ tf Þ ¼ zðt; tþ tf Þ :¼ zðxðtþ tf Þ; rðtþ tf ÞÞ:

Therefore, the properties 1–4 trivially follow. Moreover, because zðtþ tf Þ ¼ zðt; tþ tf Þ;8t;the stated time-delay compensation property also follows. &

Remark 1. It is worth commenting that the previous DROG strategy has been presented for asingle area only for the sake of clarity. However, the scheme can directly be applied to anymulti-area scenario depicted in Fig. 6. In fact, a four-area system is considered in the finalexample. In order to give a brief sketch of how generalizing the scheme to an N-area power

AREA

AREA

ROGMaster

w1

wN

w1

wN

x1

xN

x1

xN

y1

c1

yN

cN

r 1

r N...

.

.

.

.

.

. ...

Fig. 6. A multi-area power system scenario.


system let the overall vector r ¼ ½rT1 ; . . . ; r

TN �

T collect all references ri to each i-th area and so on

for the other overall vectors of interest, i.e. the command vector w ¼ ½wT1 ; . . . ; wT

N �T ; the states

vector x ¼ ½xT1 ; . . . ;x

TN �

T ; the performance output vector y ¼ ½yT1 ; . . . ; y

TN �

T and the constraints

vector c ¼ ½cT1 ; . . . ; c

TN �

T : Then, at the ROG master side, all N-area models and their primal

controllers are used for computing the future state predictions of the overall power system. Inparticular, all future state predictions are updated as soon as a new piece of information isreceived from a whatever single area. Therefore, the computation of the setsWd and Vðxðtþtf jtbÞÞ follows exactly the same lines as before with the obvious noticeable difference that the

computational complexity increases with N. Details can be found in [22]. &

Remark 2. Finally, it is worth to point out that the ideas developed here could be arrangedfor implementation in a distributed way. Along this direction, a preliminary contribution isgiven in [33], where a distributed supervision strategy for multi-agent linear systemsconnected via data networks and subject to coordination constraints is proposed.Applications to distributed LFC problems have been reported in [34]. &

4. Simulations

The aim of this section is to analyze the behavior of the supervised networked powersystem and to verify the capabilities of the proposed tertiary LFC scheme to reconfigurethe frequency set-points and control offsets despite large load deviations from nominalconditions and/or the occurrence of faults.

In the numerical simulations, the linear turbine model has been enriched by consideringa saturation effect (70.1 limits) on the rate of change of the produced power (see Fig. 9).Such a static nonlinear device has been introduced to take into account practicallimitations in the turbine response. Moreover, the model has been also equipped withsaturation devices on the produced powers in order to explicitly take into account theirphysical limits (10).

0 20 40 60 80 100 120 140 1600

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time [s]

τ f

Fig. 7. Communication latency.


In all forthcoming simulations we have used identical forward and backward randomtime-varying delay bounded by tf ¼ tb ¼ 2 (sampling steps), depicted in Fig. 7. Thoughthe delay changes significantly in the figure before and after time instant t=110 s, it willresult that the performance is quite independent on the actual value of the delay anddepends only on the assumed upper-bounds, the lower the better.For a detailed discussion on the constraints (8)–(10), the interested reader is referred

to [29].The full nonlinear model depicted in Fig. 9 is considered in the simulations for

assessments. Notice however that the system evolution under the DROG action essentiallycoincides with that obtained by using a LTI model of the power system. This hinges uponthe fact that the power system remains within its linear regime when all constraints arefulfilled and the time-delay is bounded.Coordination of a four-area power system will be considered in the simulations.

Comparisons with traditional strategies not based on the use of DROG units will be alsoconsidered. All simulations have been carried out with Matlab 7.0 and Simulink 6.0.In the sequel, an interconnected four-area power system has been used to confirm the

effectiveness of the proposed approach in a more realistic power system scenario. Asimplified representation for an interconnected system in a general form, including bothring [8] and bus [10] interconnections, is shown in Fig. 8. In particular, only Area 1 isdepicted in details in Fig. 9, with its interconnections toward the other areas.The state space model for this system has been described in [13]. For the sake of

comprehension, we report the expression for the power exchanged via the tie-lines with thei-th area Ptie i

Ptie i ¼Xjai

Tij

sððfi�fi ref Þ�ðfj�fj ref ÞÞ ¼

Xjai

Ptie;ij ð27Þ

Then, Ptie i represents the total power exchanged by the i-th area via the tie-lines whereasPtie,ij the specific contribution between the i-th and j-th areas. It is important to note that

AREA 3

AREA 1

AREA 2

AREA 4

Fig. 8. A four-area power system.


the above expression is a linear combination, under the weights Tij, of the frequencydifferences amongst the areas. Clearly, the following power balance is always satisfied:X

i

Ptie i ¼ 0

In the simulations, we have assumed that the nominal operating frequency for each area isfiref ðtÞ ¼ 60 Hz; i ¼ 1; . . . ; 4 8t: Again, the power demand has the following form PDi

ðtÞ ¼

PDiþ ~PDi

ðtÞ; i ¼ 1; . . . ; 4 where the same nominal load PDi¼ 3 MW; has been set for each

area, while ~PDiðtÞ identifies possible different load variations. Moreover, under nominal

conditions, each area has the capability to autonomously balance its own nominal load. Inthe simulations, we have supposed that power demands could vary up to 33% with respectto PDi

; i ¼ 1; . . . ; 4. As a consequence, the following convex and compact region

D ~PD:¼ f ~PD 2 R4 : U ~PDrhg ð28Þ

where U ¼ ½ I4�I4� and h ¼ ½1 1 1 1 1 1 1 1�T , characterizes the set of admissible load

disturbances (in MW). Observe that such a bound represents the largest disturbance thatthe DROG may handle. The following set of constraints (expressed in Hz and MW)

58:5rfiðtÞr61:5

jPtie iðtÞjr0:7

2:2rPTiðtÞr3:8; i ¼ 1; . . . ; 4 ð29Þ

to be fulfilled pointwise-in-time are considered. Finally, the following initial condition:

xð0Þ ¼ ½60 3 3 3 0 60 3 3 3 0 60 3 3 3 0 60 3 3 3 0�T

has been chosen and the initial admissible DROG command vector has been determined asgið0Þ ¼ 60; i ¼ 1; . . . ; 4 and yið0Þ ¼ 0; i ¼ 1; . . . ; 4.

We assume for Area 1 the load variations depicted in Fig. 10. It consists first of a stepincrement of 0.6 MW on the power demand occurring at t=2 s. Then, a further incrementof 0.25 MW is simulated at t=12 s (which is higher than the maximum generable powerfor the Area 1) and finally, after t=32 s, the load demand settles down to 3.4 MW. Here,we focus on a transmission line failure, which seems to be the most interesting example inorder to show the benefits coming out from the interconnection among many areas. We

Kp11+sTp1

1R1

11+sTG1

-K1s

Bs1

1s

+

+ +

+

+ +

_

_

_

_

f1

f1ref

Pc1Pv1

PD1

θ1

PT1

+_f2

f2ref

++

+_f3

f3ref

+

+_f4

f4ref

+

_

_

_

T14

s1

T12+

+T13 +

Ptie1

1TT1

PT1.

+ _

Kp1

f1ref

+

+

Fig. 9. Block diagram of Area 1.

G.

Fra

nz �e,

F.

Ted

esco/

Jo

urn

al

of

the

Fra

nk

linIn

stitute

34

8(

20

11

)8

32

–8

52

846

0 10 20 30 40 50 60 70 80 90 1002.9

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

PD [MW]

Time [s]

AR

EA

13.85 MW

Fig. 10. Load demand of Area 1.

AREA 3

AREA 1

AREA 2

AREA 4

Fig. 11. The four-area power scheme under transmission line failures.


impose breakdowns on the transmission lines between Area 1 and Areas 2 and 4 during allthe simulation (see Fig. 11). It is evident that Area 4 is disconnected from the others. As aconsequence, it is not influenced from the remaining network and its exchanged power viathe tie-line is always zero (see Fig. 13).

In order to clarify the power system response, we restrict our attention to the highestload request (3.85 MW), which overcomes the maximum power production of Area 1 (seeconstraints (29)). The corresponding system evolutions are reported in Figs. 12–14.

In the faulty network topology of Fig. 11, Area 1 is directly connected only to Area 3.Area 1 is not capable to produce the necessary power to balance its local demand and thepower fraction required to accomplish the request is furnished via the tie-line Ptie 1: Inparticular PT1

¼ 3:575 MW and Ptie 1 ¼ 0:275 MW; while Ptie 2 ¼ 0:13 MW; and Ptie 3 ¼

0:145 MW: Recalling that Ptie 3 ¼ Ptie 31 þ Ptie 32 and observing that Ptie 2 ¼ Ptie 23 ¼

�Ptie 32; and Ptie 1 ¼ Ptie 13 ¼ �Ptie 31; one can conclude that the overall power balance is

0 10 20 30 40 50 60 70 80 90 1002

3

4PT [MW]

AR

EA

1

0 10 20 30 40 50 60 70 80 90 1002

3

4

AR

EA

2

0 10 20 30 40 50 60 70 80 90 1002

3

4

AR

EA

3

0 10 20 30 40 50 60 70 80 90 1002

3

4

AR

EA

4

Time [s]

3.575 MW

3.013 MW

3.0145 MW

Fig. 12. Generated powers: with DROG.

0 10 20 30 40 50 60 70 80 90 100-1

0

1Ptie [MW]

AR

EA

1

0 10 20 30 40 50 60 70 80 90 100-1

0

1

AR

EA

2

0 10 20 30 40 50 60 70 80 90 100-1

0

1

AR

EA

3

0 10 20 30 40 50 60 70 80 90 100-1

0

1

AR

EA

4

Time [s]

-0.275 MW

0.13 MW

0.145 MW

Fig. 13. Tie-line power flows with DROG.


0 10 20 30 40 50 60 70 80 90 10058

60

62g [Hz]

AR

EA

1

0 10 20 30 40 50 60 70 80 90 10058

60

62

AR

EA

2

0 10 20 30 40 50 60 70 80 90 10058

60

62

AR

EA

3

0 10 20 30 40 50 60 70 80 90 10058

60

62

AR

EA

4

Time [s]

^

Fig. 15. Frequency commands.

0 10 20 30 40 50 60 70 80 90 10058

60

62f [Hz]

AR

EA

1

0 10 20 30 40 50 60 70 80 90 10058

60

62

AR

EA

2

0 10 20 30 40 50 60 70 80 90 10058

60

62

AR

EA

3

0 10 20 30 40 50 60 70 80 90 10058

60

62

AR

EA

4

Time [s]

59.9 Hz

59.9 Hz

Fig. 14. Frequencies: with DROG.


0 10 20 30 40 50 60 70 80 90 100-4

-2

0

2θ

AR

EA

1

0 10 20 30 40 50 60 70 80 90 100-1

0

1

2

AR

EA

2

0 10 20 30 40 50 60 70 80 90 100-1

0

1

2

AR

EA

3

0 10 20 30 40 50 60 70 80 90 100-1

0

1

2

AR

EA

4

Time [s]

^

Fig. 16. Offset commands.


satisfied. In fact, as evident from Fig. 12, both Areas 2 and 3 produce more power thantheir local needs and the surplus contributes to match the demand of Area 1 via the pathArea 2-Area 3-Area 1: This coincides with a new power routing which can beinfluenced by the choice of the weighting matrices Cg and Cy in the cost function (23).For the sake of completeness, also the frequency evolutions are shown in Fig. 14, where

it clearly results that the constraints are never violated. However, some coupling amongstareas is still present as it result from unexpected frequency adjustments in the 4-th Area(see Fig. 14), not interested during the faulty condition to any power exchange. In Figs. 15and 16 the DROG commands (frequency and offset respectively) are also depicted.Finally, the relationships between the computed commands w and the applied commandsw are depicted in Fig. 17. Also in this case, because the upper-bounds (13) on the time-delay are always fulfilled, the applied commands w are just a translated version of w; that iswðtÞ ¼ wðt�tf ; tÞ:

5. Conclusions

In this paper the problem of supervising multi-area power systems subject to LFCrequirements and bounded time-delays on the communication channels has beenconsidered. Detailed investigations have been undertaken on the effect of communicationtime-delay of the coordination performance. Centralized master/slaves strategies have beenpresented and discussed and their relevant properties summarized. Special efforts have

12 12.5 13 13.5 1459.9

59.95

60A

RE

A 1

12 12.5 13 13.5 1459.97

59.98

59.99

60

AR

EA

2

12 12.5 13 13.5 14

60

60.02

60.04

AR

EA

3

12 12.5 13 13.5 1460

60.0260.0460.06

Time [s]

AR

EA

4

12 12.5 13 13.5 14-0.8-0.6-0.4-0.2

12 12.5 13 13.5 140

0.1

0.2

12 12.5 13 13.5 140

0.1

0.2

12 12.5 13 13.5 140

0.1

0.2

Time [s]

gg^ θ

θ

^

Fig. 17. Computed w (continuous line) and applied w (dashed line) commands. Frequency (left side) and power

(right side): zoom on the time interval [12,15] s.


been devoted at investigating how the proposed strategy behaves under critical events asfaults, failure or abrupt changes in the load demand.

The simulations have shown that the proposed strategy is capable to handle hard-to-predict adverse events whereas standard ALFC schemes result to be ineffective. Inparticular, the proposed approaches ensure viable evolutions to the overall networkedsystem with respect to safety and operative constraints, despite remarkable changes in theload demand, in the presence of faulty situations and communication time-delays.

References

[1] K. Tomsovic, D.E. Bakken, V. Venkatasubramanian, A. Bose, Designing the next generation of real-time

control, communication and computations for large power systems, Proceedings of the IEEE 93 (5) (2005)

965–979.

[2] H. Liao, Y. Tang, Mathematical aspects of static and dynamic stability problems in power systems, Journal

of the Franklin Institute 341 (2004) 443–463.

[3] E.A. Cano, H.E. Tacca, Power load identification, Journal of the Franklin Institute 342 (2005) 97–113.

[4] A.M. Gaouda, Power system disturbance modeling under deregulated environment, Journal of the Franklin

Institute 344 (2007) 507–519.

[5] G. Andersson, Modelling and analysis of electrical power systems, Lecture 227-0528-00, EEH-Power Systems

Laboratory, 2004, ETH Zurich.


[6] M.D. Ilic, H. Allen, W. Chapman, C.A. King, J.H. Lang, E. Litvinov, Preventing future blackouts by means

of enhanced electric power systems control: from complexity to order, Proceedings of the IEEE 93 (11) (2005)

1920–1941.

[7] S. Bhowmik, K. Tomsovic, A. Bose, Communication models for third party load frequency control, IEEE

Transactions on Power Systems 19 (1) (2004) 543–548.

[8] N. Cohn, Control of Generation and Power Flow on Interconnected Systems, Wiley, New York, 1959.

[9] C.H. Hauser, D.E. Bakken, A. Bose, A failure to communicate: next generation communication

requirements, technologies, and architecture for the electric power grid, Power and Energy Magazine 3 (2)

(2005) 47–55.

[10] T. Hiyana, Design of decentralised load frequency regulators for interconnected power systems, IEE

Proceedings C 129 (1982) 17–23.

[11] I.P. Kumar, D.P. Kothari, Recent philosophies of automatic generation control strategies in power systems,

IEEE Transaction on Power Systems 20 (1) (2005) 346–357.

[12] H. Shayeghi, H.A. Shayeghi, O.P. Malik, Robust decentralized neural networks based LFC in a deregulated

power system, Electric Power Systems Research 77 (2007) 241–251.

[13] T.C. Yang, Z.T. Ding, H. Yu, Decentralised power system load frequency control beyond the limit of

diagonal dominance, Electrical Power and Energy Systems 24 (2002) 173–184.

[14] X. Yu, K. Tomsovic, Application of linear matrix inequalities for load frequency control with

communication delays, IEEE Transactions on Power Systems 19 (3) (2004) 1508–1515.

[15] H. Shayeghi, H.A. Shayeghi, A. Jalili, Load frequency control strategies: a state-of-the-art survey for the

researcher, Energy Conversion and Management 50 (2009) 344–353.

[16] Y.G. Rebours, D.S. Kirschen, M. Trotignon, S. Rossignol, A survey of frequency and voltage control

ancillary services—Part I: technical features, IEEE Transactions on Power Systems 22 (1) (2007) 350–357.

[17] Y.G. Rebours, D.S. Kirschen, M. Trotignon, S. Rossignol, A survey of frequency and voltage control

ancillary services—Part II: economic features, IEEE Transactions on Power Systems 22 (1) (2007) 358–366.

[18] G.C. Walsh, H. Ye, Scheduling of networked control systems, IEEE Control Systems Magazine 21 (2001)

57–65.

[19] A. Casavola, E. Mosca, M. Papini, Predictive teleoperation of constrained dynamical systems via internet-

like channels, IEEE Transactions on Control Systems Technology 12 (1) (2004) 193–204.

[20] A. Bemporad, A. Casavola, E. Mosca, Nonlinear control of constrained linear systems via predictive

reference management, IEEE Transactions on Automatic Control 42 (1997) 340–349.

[21] A. Casavola, E. Mosca, D. Angeli, Robust command governors for constrained linear systems, IEEE

Transactions on Automatic Control 45 (2000) 2071–2077.

[22] A. Casavola, M. Papini, G. Franz�e, Constrained supervision of dynamic systems in spatial networks, IEEE

Transaction on Automatic Control 51 (3) (2006) 421–437.

[23] E.G. Gilbert, I.V. Kolmanovsky, K. Tin Tan, Discrete-time reference governors and the nonlinear control of

systems with state and control constraints, International Journal on Robust and Nonlinear Control 5 (1995)

487–504.

[24] E.G. Gilbert, K. Tin Tan, Linear systems with state and control constraints: the theory and applications of

maximal output admissible sets, IEEE Transactions on Automatic Control 36 (1991) 1008–1020.

[25] I.V. Kolmanovsky, J. Sun, Parameter governors for discrete-time nonlinear systems with pointwise-in-time

state and control constraints, Automatica 42 (5) (2006) 841–848.

[26] O.I. Elgerd, Control of electric power systems, IEEE Control Systems Magazine 1 (2) (1981) 4–16.

[27] A. Kurzhanski, I. Valyi, Ellipsoidal Calculus for Estimation and Control, Birkhauser, Berlin, 1997.

[28] I. Kolmanovsky, E.G. Gilbert, Theory and computation of disturbance invariant sets for discrete-time linear

systems, Mathematical Methods in Engineering: Theory, Methods and Applications 4 (1998) 317–367.

[29] A. Casavola, D. Famularo, G. Franz�e, E. Garone, Set-points reconfiguration in networked multi-area

electrical power systems, International Journal of Adaptive Control Signal Processing 23 (2009) 808–832.

[33] E. Garone, F. Tedesco, A. Casavola, Distributed coordination strategies for interconnected multi-agent

systems, in: 8th IFAC Symposium on Nonlinear Control Systems, September 1–3, Bologna, Italy, 2010.

[34] A. Casavola, G. Franz�e, E. Garone, F. Tedesco, Distributed coordination-by-constraint strategies in

networked multi-area power Systems, in: Proceedings of the IEEE ISIE 2011 — IEEE International

Symposium on Industrial Electronics, Gdansk Poland, June 2011.

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