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A Practical Integration of Automatic GenerationControl and Demand Response
Dylan J. Shiltz and Anuradha M. Annaswamy
Abstract
—For a power grid to operate properly, electricalfrequency must be continuously maintained close to its nominalvalue. Increasing penetration of distributed generation, such assolar and wind generation, introduces ﬂuctuations in active powerwhile also reducing the natural inertial response of the electricitygrid, creating reliability concerns. While frequency regulationhas traditionally been achieved by controlling generators, thecontrol of Demand Response (DR) resources has been recognizedin recent smart grid literature as an efﬁcient means for providingadditional regulation capability. To this end, several controlmethodologies have been proposed recently, but various featuresof these proposals make their practical implementations difﬁcult.In this paper, we propose a new control algorithm that facilitatesoptimal frequency regulation through direct control of bothgenerators and DR, while addressing several issues that preventpractical implementation of other proposals. In particular, i)our algorithm is ideal for control over a large, low-bandwidthnetwork as communication and measurement is only requiredevery 2 seconds, ii) it enables DR resources to recover energylost during system transients, and iii) it allows the market toimmediately respond to disturbances through feedback of thesystem frequency. We demonstrate the viability of our approachthrough dynamic simulations on a 118-bus grid model.
I. I
NTRODUCTION
Maintaining a constant balance between generation andconsumption of power is critical to effective power systemoperations. Several control layers maintain this balance atvarious time scales. Primary frequency control, based ongovernor action, is a decentralized control system that adjuststhe mechanical power of a generator in response to deviationsin local frequency. Following a disturbance, primary controlarrests the decline (or rise) in grid frequency in less than aminute or so, but will not restore it to nominal. Secondary con-trol (also called Automatic Generation Control (AGC)) updatesgenerator set-points every few seconds in a centralized fashionin an attempt to restore nominal frequency and/or inter-areatie line ﬂows to their scheduled values [1] [2]. While effective,the current AGC system does not incorporate DR, and it doesnot in general allocate generation optimally. Finally, tertiarycontrol (also called economic dispatch) calculates optimalsystem set-points, typically every 5 minutes.In recent literature, control of DR resources has beenproposed as a means to improve many aspects of powersystem operation, including frequency regulation (see for
This work was supported by the NSF initiative, Award no. EFRI-1441301.A part of this work was conducted while the ﬁrst author was a graduatesummer intern with the Electric Power Systems Research group at SandiaNational Laboratories, Albuquerque, NM.Both authors are with the Department of Mechanical Engineering, Mas-sachusetts Institute of Technology, Cambridge, MA, 02139 USA e-mail:
{
djshiltz,aanna
}
@mit.edu
example [3]-[19], and [3] for a detailed overview). Unlikemost generation units, loads can respond to control signalsalmost instantaneously [4] and the development of advancedmetering infrastructure (AMI) [5] has made real-time loadcontrol achievable if appropriate control signals are provided.Many control strategies such as [6]-[9] adjust loads based onlocal frequency, enabling loads to mimic the natural responseof synchronous generators. These approaches have been shownto improve the primary frequency response of the grid in bothsimulations and in small-scale ﬁeld tests [10]. However, oneof the challenges with load control is balancing the objectivesof consumers with those of the grid [3], and so the cost (ordisutility) to consumers must also be considered.In [11], an optimal load control (OLC) strategy is de-veloped that minimizes disutility from ﬂexible consumptionand grid frequency deviations, and an analytical guaranteeof stability for such strategies is provided. In [12], thisstrategy is expanded to restore nominal frequency following adisturbance. However, these strategies are not integrated withexisting primary and secondary control on the generation side.Simultaneous optimal control of generation and deferrableloads using Lagrange multiplier methods was explored in [13]-[19]. However, there are several issues in these approaches thatmake their practical implementations difﬁcult.A common feature to many of the approaches in [3]-[19]is to design and analyze the control system assuming rapidor even instantaneous communication is available, which isnot practical for implementation on large networks. As notedin [20], power system SCADA communications and measure-ments can experience delays of several hundred milliseconds,depending on the communication medium. Such large delayscan have a substantial impact on the stability and settling timeof network control systems [21]. In this paper, we proposean algorithm for frequency control with DR-compatible loadsthat is more practical to implement. In particular, our algorithmrequires discrete communications and measurements only onceevery 2 seconds (i.e. a time-scale already achieved by existingAGC implementations).Another issue with load control that is not often addressedis the so-called
recovery peak
[3] in which loads consumeadditional power following a period of deferment such thatnet energy is unchanged. Some types of DR, such as electricvehicles, have requirements on how much energy they need toconsume over a certain horizon [22]. Other types of DR areshiftable (i.e. their required energy is ﬁxed, but their time of consumption is ﬂexible) [23]. Our algorithm addresses theseeffects by ensuring that DR entities eventually recover all of their energy following a system transient.
2016 American Control Conference (ACC)Boston Marriott Copley PlaceJuly 6-8, 2016. Boston, MA, USA
978-1-4673-8682-1/$31.00 ©2016 AACC 6785
Finally, our algorithm allows the market to immediatelyrespond to disturbances - that is, ﬂuctuations in power thatwere not predicted at the real-time market clearing - throughfrequency feedback. This is important for practical implemen-tation in a real power system, in which load ﬂuctuations,line losses, and other disturbances can impact frequency butare difﬁcult to predict or measure at the real-time market.Thus, our algorithm may be viewed as as a Dynamic MarketMechanism [15]-[17] that performs the combined optimizationof generation and ﬂexible consumption as well as frequencycontrol.The outline of this paper is as follows. In Section II wemodel the dynamics of the electric grid and derive a controllerto send the grid state to its optimal set-point. In Section III wepresent the details of our simulations on the IEEE 118-bus testcase and discuss the results of these simulations. Section IVprovides concluding remarks.II. C
ONTROLLER
D
ERIVATION
In this section we describe the proposed network-basedoptimal AGC controller. We start by modeling the dynamicsof the power grid, including frequency dynamics of individualbuses. Then we design an algorithm that iteratively calculatesthe optimal generation and consumption set-points, includingfrequency feedback. These iterations are broadcast over a widearea network and serve as control references for generatorsand ﬂexible consumers. Finally, we discuss modiﬁcations toenable DR resources to recover their energy following a largetransient event (e.g. a rapid drop in renewable generation or agenerator tripping ofﬂine).
A. Power System Dynamics
Our dynamic model for the power system is based on [18].The power grid can be modeled as an undirected graph inwhich buses are modeled as nodes
N
and transmission linesas edges
E
, with sets of generators
G
and ﬂexible consumers
D
. For simplicity, we adopt a linearized DC power ﬂowmodel [24] which includes the following assumptions: i)voltage magnitudes of buses remain ﬁxed, ii) resistances of electrical lines are negligible, iii) voltage angle differencesare small, iv) reactive power ﬂows are neglected. The powernetwork dynamics, at the fast primary level time-scale, canthen be written as
M
i
˙
ω
i
+
D
i
ω
i
=
j
∈G
i
P
M
j
−
j
∈D
i
P
kD
j
−
P
L
i
−
(
i,j
)
∈E
T
ij
(
δ
i
−
δ
j
)
, i
∈ N
(1)
˙
δ
i
=
ω
i
, i
∈ N
(2)
˙
P
M
i
=
τ
−
1
i
(
P
kG
i
−
P
M
i
−
R
−
1
i
ω
i
)
,i
∈ G
.
(3)In the above equations,
M
i
and
D
i
are the inertia anddamping coefﬁcients of a bus,
P
M
i
is mechanical powergeneration, and
P
D
i
is ﬂexible power consumption. Each busalso has conventional (i.e. non-ﬂexible) consumption
P
L
i
.In (3) we model the turbine-governor dynamics as a ﬁrstorder system with time constant
τ
i
and droop coefﬁcient
R
i
.Finally, we deﬁne
ω
i
as the deviation in frequency of bus
i
from the nominal frequency, and
δ
i
as the correspondingphase angle. Transmission coefﬁcients
T
ij
=
1
X
ij
V
i,
0
V
j,
0
areconstant, where
X
ij
is the line reactance (inductive) and
V
i,
0
denotes the ﬁxed voltage magnitude at bus
i
.The terms
P
kG
and
P
kD
are setpoints for generators andﬂexible consumers. These setpoints are updated periodicallyat discrete time increments
t
k
,k
∈
Z
+
, with
t
k
−
t
k
−
1
= ∆
k
.In the next subsections we discuss how these setpoints aredetermined, as well as the period
∆
k
.To simplify further analysis, we write the power griddynamics (1)-(3) compactly as
˙
y
=
A
y
y
+
B
x
x
+
B
L
P
L
(4)where
y
=
ω
T
δ
T
P
T M
T
,
x
is a vector of control inputsspeciﬁed in Section II-B, and
A
y
,
B
x
, and
B
L
are constantmatrices containing parameters of system (1)-(3).
B. Optimal Power Flow
For our controller to operate efﬁciently, it is desired toupdate
P
kG
and
P
kD
iteratively at each
t
k
, such that thecontrolled system satisﬁes Optimal Power Flow (OPF) atequilibrium. The purpose of OPF is to determine the most cost-effective way to meet power demands, subject to constraints ongenerators, consumers, and transmission lines. In this paper,we use the DC-OPF formulation given in [16], which includesﬂexible consumption as decision variables. An independentsystem operator (ISO) attempts to solve this problem bymaximizing Social Welfare, denoted by
S
W
and deﬁned as
S
W
=
i
∈D
U
D
i
(
P
D
i
)
−
i
∈G
C
G
i
(
P
G
i
)
(5)where quadratic utility curves of ﬂexible consumers andquadratic cost curves of generators are given in (6) and (7)respectively.
U
D
i
(
P
D
i
) =
b
D
i
P
D
i
+
c
D
i
2
P
2
D
i
(6)
C
G
i
(
P
G
i
) =
b
G
i
P
G
i
+
c
G
i
2
P
2
G
i
(7)The overall DC-OPF can then be formulated as the follow-ing optimization problem, written in simpliﬁed matrix-vectornotation.min
−
S
W
(8)subject to
h
(
x
) =
A
D
P
D
−
A
G
P
G
+
T
bus,R
θ
+ ˆ
P
L
+
βρ
= 0
(9)
P
minG
≤
P
G
≤
P
maxG
(10)
P
minD
≤
P
D
≤
P
maxD
(11)
−
P
max
≤
T
line,R
θ
≤
P
max
(12)Here our decision vectors are voltage angles
θ
(with theslack bus removed), ﬂexible consumption
P
D
, and generation
6786
P
G
. We refer to the entire decision vector as
x
= [
θ
T
P
T D
P
T G
]
T
.Matrices
T
bus,R
and
T
line,R
give the net power ﬂow out of a bus and through a transmission line, respectively, whenmultiplied by
θ
, and matrices
A
D
and
A
G
are incidencematrices that map ﬂexible consumers and generators to theirrespective buses. The equality constraints
h
(
x
) = 0
representpower balance at each bus in the grid, where the term
βρ
cor-responds to feedback control from grid frequency, which willbe discussed in Section II-D. Since the inﬂexible consumption
P
L
is not known precisely, the OPF uses a prediction
ˆ
P
L
.Equations (10)-(12) correspond to inequality constraints,written compactly as
g
(
x
)
≤
0
. The components of
g
(
x
)
aregiven by
g
1
(
x
) =
P
minG
−
P
G
,
g
2
(
x
) =
P
G
−
P
maxG
,
g
3
(
x
) =
P
minD
−
P
D
,
g
4
(
x
) =
P
D
−
P
maxD
,
g
5
(
x
) =
−
P
max
−
T
line,R
θ
,and
g
6
(
x
) =
T
line,R
θ
−
P
max
, where
g
n
(
x
)
,n
= 1
,...,
6
arethemselves vectors.
C. An Iterative Solution of OPF
To solve this optimization problem (8)-(12), ﬁrst we formits Lagrangian with penalty vectors
λ
and
µ
for the equalityand inequality constraints, respectively.
L
(
x,λ,µ
) =
−
S
W
(
x
) +
λ
T
h
(
x
) +
µ
T
g
(
x
)
(13)To simplify further notation, we re-write the Lagrangian as
L
(
x,λ,µ
) =
f
(
x,µ
) +
λ
T
h
(
x
)
(14)We utilize a solution method similar to the one foundin [17], which is a Newton-like primal dual interior pointmethod. We deﬁne the Hessian matrix as
H
=
∇
2
xx
f
(
x,µ
) =
0 0 00
−
c
D
00 0
c
G
.
(15)We now construct an augmented Hessian matrix by choosing apositive parameter
γ
such that the following matrix is positivedeﬁnite
¯
H
=
H
+
γNN
T
(16)where
N
=
∇
x
h
(
x
)
is a constant matrix. We then deﬁne theupdate equations for
x
and
µ
as
x
k
+1
=
x
k
−
α
¯
H
−
1
∇
x
L
(
x
k
,
ˆ
λ
k
,µ
k
)
(17)
µ
k
+1
= [
µ
k
+
K
µ
g
(
x
k
)]
+
(18)where
ˆ
λ
k
= (
N
T
¯
H
−
1
N
)
−
1
(
h
(
x
k
)
−
N
T
¯
H
−
1
∇
x
f
(
x
k
,µ
k
))
(19)
∇
x
L
(
x
k
,
ˆ
λ
k
,µ
k
) =
∇
x
f
(
x
k
,µ
k
) +
N
ˆ
λ
k
(20)and
α
and
K
µ
are positive parameters chosen at the designstage such that iterates (17)-(18) converge. The operation
[
·
]
+
= max(0
,
·
)
. At convergence, the solution satisﬁes theKarush-Kuhn-Tucker (KKT) optimality conditions which aresufﬁcient for global optimality (see Theorem 1).
D. Feedback Control using ACE
If the prediction of inﬂexible demand
ˆ
P
L
is not exact,the market would not schedule the appropriate amount of generation to balance demand, resulting in a deviation infrequency from its nominal value. Large electricity grids areusually divided into balancing areas, which are connected toone another via tie lines. Each balancing area is managedby a corresponding balancing authority, whose goal is tomaintain system frequency within acceptable limits. TypicallyArea Control Error (ACE) is used to determine each balancingauthority’s obligation to support frequency control by addingor removing generation. Imbalances arise due to discrepanciesin supply and demand within the balancing area, as well asdiscrepancies in tie line ﬂows between areas [1]. In this paper,we assume a single area with no tie lines to adjacent areas. Amulti-area version of this controller is a topic of future work.The total frequency response bias of the balancing area,
β
area
, relates the magnitude of a power imbalance to thecorresponding deviation in frequency (typically measured inMW/0.1Hz). We deﬁne the bias of a single bus as
β
i
=
D
i
+
j
∈G
i
R
−
1
j
and
β
area
=
i
∈N
β
i
. We deﬁne
¯
ω
(
t
)
asthe average of
ω
i
(
t
)
,i
∈ N
. The Area Control Error can beexpressed as
ACE
(
t
) =
β
area
¯
ω
(
t
)
.
(21)We can equivalently express ACE in terms of the state vari-ables introduced in (4) as
ACE
(
t
) =
β
area
Qy
(
t
)
(22)where
Q
is a constant matrix used to compute the averagefrequency
¯
ω
(
t
)
. Typical practice is to measure ACE at discretetime invervals
t
k
. To model this procedure, we express ACEas a discrete time signal given by
ACE
k
=
β
area
Qy
k
(23)where
y
k
is a discrete sample of continuous variable
y
in (4).We feed the ACE signal into the algorithm through the scalar
ρ
in the power balance constraints (9). Recall that
h
(
x
k
)
isused to update iterates (17)-(18). We update the scalar
ρ
k
as
ρ
k
+1
=
ρ
k
−
K
f
1
β
area
ACE
k
(24)where
K
f
is a suitably chosen feedback gain (see Theorem1 for stability analysis). The purpose of
ρ
k
is to distributepower imbalance measured through ACE to individual buses(weighted by their bias factors
β
i
), such that these imbalancescan be met optimally by the market. Unlike existing AGC, thisenables our control algorithm to deploy ﬂexible consumptionas well as generation to stabilize grid frequency, and to do soin a way that maximizes Social Welfare.We note that (24) can be viewed as an aggregation scheme,in an effort to incorporate unmeasured disturbances in gen-eration and load. As ACE is an indirect measure of thesedisturbances, the use of
ρ
can be viewed as an aggregate,based on the frequency bias factors of buses. This may lead tosuboptimality, which may be reduced by using more advancedalgorithms in lieu of (24).
6787
E. Energy Recovery for DR Resources
Another feature of our control algorithm is the inclusionof DR resources
P
D
in a way that allows them to recovertheir energy following a large excursion in system frequency.Our goal is to modify the control algorithm to ensure that theadditional net energy consumed or deferred by a DR resourceconverges to zero. To this end, we introduce a new set of statevariables
E
D
which are the net energy consumed or deferredby the DR resources. This energy is updated as
E
k
+1
D
=
E
kD
+
K
E
P
kD
(25)where
K
E
is a diagonal matrix of positive scaling valueswhose magnitudes depend on how quickly energy payback is needed. Some DR resources (such as HVAC units) mayneed their energy back very quickly, while others (such asPHEV’s) can defer for longer periods. We then replace theDR resources’ inequality constraints in (18) with
g
3
(
x
k
,E
kD
) =
−
E
kD
−
P
kD
(26)
g
4
(
x
k
,E
kD
) =
P
kD
+
E
kD
(27)At equilibrium, we have
E
∗
D
=
P
∗
D
= 0
, which ensures that thetotal energy consumed converges to zero, and that the powerof each DR resource returns to its nominal value.
F. Stability Analysis
To summarize, the primary and secondary level dynamics of the grid together with our proposed controller can be expressedas the following hybrid dynamic system.
˙
y
=
A
y
y
+
B
x
x
k
+
B
L
P
L
(28)
µ
k
+1
= [
µ
k
+
K
µ
g
(
x
k
,E
kD
)]
+
(29)
x
k
+1
=
x
k
−
α
¯
H
−
1
∇
x
L
(
x
k
,
ˆ
λ
k
,µ
k
)
(30)
ρ
k
+1
=
ρ
k
−
K
f
Qy
k
(31)
E
k
+1
D
=
E
kD
+
K
E
0
I
0
x
k
(32)This includes both the existing primary control system as partof (28) as well as the new secondary control system givenby (29)-(32). We analyze stability and convergence propertiesof this combined primary-secondary system with the followingTheorem:
Theorem 1:
For properly chosen control parameters
α
,
K
µ
,
K
f
, and
K
E
, system (28)-(32) is stable and converges to theglobal optimum of Problem (8)-(12).
To prove Theorem 1 we make the following assumptions.
Assumption 1:
Problem (8)-(12) is feasible. We assume thatsufﬁcient generation has been scheduled in advance to meetpower demands, and that the transmission system is capableof handling the necessary power ﬂows. This involves a unitcommitment problem which is solved at a slower time-scale.
Assumption 2:
At equilibrium, a subset of the elementsof
µ
are projected. As Problem (8)-(12) includes bounds ondecision variables, it is not possible for a decision variableto simultaneously equal its minimum and maximum valueat equilibrium. Thus, some of the corresponding Lagrangemultipliers
µ
must be projected.See Appendix for the proof of Theorem 1.
G. Network Implementation
Next we discuss how this control algorithm might be imple-mented over a large network. An independent system operator(ISO) broadcasts set-points
x
k
to each of the buses every
∆
k
seconds (in this paper
∆
k
= 2
seconds). Upon receiving theirset-points, each generator and DR consumer responds to theISO with its entry of
∇
x
f
(
x
k
,µ
k
)
. These quantities can bethought of as the marginal cost or marginal utility for eachparticipant at the current set-point. Each generator and DRconsumer is responsible for updating its own value of
µ
, givenby (29), and DR consumers update their own value of
E
D
,given by (32). The ISO is responsible for measuring systemfrequency, calculating ACE, and updating
ρ
using (31). At thispoint, the ISO has everything it needs to compute the nextset of set-points
x
k
+1
(consisting of a simple matrix-vectorproduct) and the process repeats. We note that the existingAGC/SCADA system updates set-points every 4-6 seconds [2],which includes frequency measurement, processing, and com-munication. Thus, the communication requirements of ourcontrol algorithm should be within the capability of existingcommunication and measurement infrastructure with minimalmodiﬁcation, making the algorithm feasible to implement.As it is not practical for thousands or millions of devicesto interface directly with an ISO or other central authority,we envision a hierarchy in which demand response aggre-gators [3][25][26] communicate with the ISO and distributecontrol actions for their respective DR resources. These ag-gregators would gather information from their constituentsand determine characteristics of the group (such as the utilitycurves in (6)). The design and operation of such aggregatorsis an open research question and beyond the scope of thispaper. Notable work in this area includes [26] which discussesspeciﬁc modeling methodologies as well as case studies withsupermarket HVAC systems.III. S
IMULATION
S
TUDIES ON
IEEE 118 B
US
G
RID
In this section we simulate the combined primary-secondarycontrol system on a 118 bus grid following a sudden increasein load
P
L
. This may simulate a generator trip, or perhapsa rapid drop in renewable generation or a sudden surge indemand. We say that the trip was
detected
if
ˆ
P
L
=
P
L
following the trip, and we say that the trip was
undetected
if
ˆ
P
L
does not correct for the trip.To analyze the performance of the algorithm, we run atotal of 3 tests. In test 1, we model the system with nofeedback control (i.e. all control gains are zero). This modelsthe primary control response with no AGC. In test 2, weenable feedback control but the trip is undetected. In test 3, weenable feedback control and the trip is detected. In all tests, thegrid experiences 20 seconds of nominal operation, then
P
L
isincreased by
1%
of total base load with deviations randomlydistributed among the buses.To evaluate system performance, we analyze both ACE andthe achieved Social Welfare. We deﬁne the achieved SocialWelfare identical to (5) but with the commanded generation
P
G
replaced by the actual mechanical generation
P
M
.
6788

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