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Distributed Automatic Load-Frequency Control with Optimality in
Power Systems
Xin Chen1, Changhong Zhao2, Na Li1
Abstract— With the increasing penetration of renewable en-
ergy resources, power systems face new challenges in balanc-
ing power supply and demand and maintaining the nominal
frequency. This paper studies load control to handle these
challenges. In particular, a fully distributed automatic load
control algorithm, which only needs local measurement and
local communication, is proposed. We prove that the control
algorithm globally converges to an optimal operating point
which minimizes the total disutility of users, restores the
nominal frequency and the scheduled tie-line power flows, and
respects the thermal constraints of transmission lines. It is also
shown that the convergence still holds even when inaccurate
system parameters are used in the control algorithm. Lastly,
the effectiveness, optimality, and robustness of the proposed
algorithm are demonstrated via numerical simulations.
I. INTRODUCTION
In power systems, generation and load are required to be
balanced all the time. Once a mismatch between generation
and load occurs, the system frequency will deviate from the
nominal value, e.g., 50 Hz or 60 Hz, which may undermine
the electric facilities and even cause system collapse. Hence,
it is crucial to maintain the frequency closely around its
nominal value. Traditionally, the generator-side control [1]
plays a dominant role in frequency regulation, where the
generation is managed to follow the time-varying load. How-
ever, with the deepening integration of renewable energy, it
becomes more challenging to maintain the power balance
and the nominal frequency due to increased volatility in non-
dispatchable renewable generation, such as wind and solar.
To address these challenges, load control has received
considerable attention in the recent decade as a promis-
ing complement to generator control, because controllable
loads are ubiquitously distributed in power systems and
can respond fast to regulation signals or changes in fre-
quency. There has been a large amount of research effort
devoted to frequency regulation provided by controllable
loads, including electric vehicles [3], [4], heating, ventilation
and air-conditioning systems [5], battery storage systems
[6], [7], and thermostatically controlled loads [8]. Several
demonstration projects [9]–[11] verified the viability of load-
side participation in frequency regulation. The literature
above focuses on modeling and operating the loads for
1X. Chen and N. Li are with the School of Engineering and Applied
Sciences, Harvard University, USA. chen xin@g.harvard.edu,
nali@seas.harvard.edu.
2C. Zhao is with the National Renewable Energy Laboratory, Golden,
CO, USA. changhong.zhao@nrel.gov.
The work was supported by NSF 1608509, NSF CAREER 1553407 and
ARPA-E through the NODES program.
frequency regulation, and leaves the development of system-
wide optimal load control techniques as an unresolved task.
For load-side frequency control, centralized methods [12],
[13] need to exchange information over remotely connected
control areas, which imposes a heavy communication burden
with expanded computational and capacity complexities [14].
This concern motivates a number of studies on distributed
control methods. In [15]–[17], load control is implemented
by solving a centralized optimization problem using appro-
priate decomposition methods. The decomposition methods
generate optimal control schemes that respect the operational
constraints, but their convergence relies on network param-
eters. In [18], a distributed proportional-integral (PI) load
controller is designed to attenuate constant disturbances and
improve the dynamic performance of the system, whereas
operational constraints, such as load power limits and line
thermal constraints, are not taken into account. Papers [19]–
[21] reversely engineer power system dynamics as primal-
dual algorithms to solve optimization problems for load
control, and prove global asymptotic stability of the closed-
loop system independently of control parameters. However,
the scheme in [19] requires accurate information of power
imbalance or generator’s shaft angular acceleration, which
is hard to obtain in practice. In addition, to implement the
scheme in [19], each boundary bus has to communicate with
all the other boundary buses within the same control area,
which brings heavy remote communication burden if two
boundary buses in the same area are far away from each
other.
In this paper, we develop an automatic load control
(ALC) method for frequency regulation, which can eliminate
power imbalance, restore system frequency to the nominal
value, and maintain scheduled tie-line power flows in a
way that minimizes the total disutility of users for load
adjustment. Power system frequency dynamics is interpreted
as a primal-dual gradient algorithm that solves a properly
formulated optimal load control problem, from which the
load control algorithm is extracted. In particular, the pro-
posed ALC method integrates four significant merits: 1) The
information of aggregate power imbalance is not required in
the control process. 2) With local measurement and local
communication, it operates in a fully distributed manner
while achieving system-wide optimality. 3) It encodes and
satisfies critical operational constraints such as load power
limits and line thermal limits. 4) It is globally asymptotically
stable even when inaccurate system parameters are used in
the controllers. These features overcome the main limitations
in the existing approaches reviewed above and facilitate
practical implementations of the proposed method.
The remainder of this paper is organized as follows:
Section II introduces the power network dynamic model
and formulates the optimal load control problem. Section
III presents the proposed ALC algorithm and analyzes its
convergence to the optimal operating point. Then numerical
tests are carried out in Section IV, and conclusions are drawn
in Section V.
II. SYS TEM MO DEL AND PROB LEM FO RMULATIO N
A. Dynamic Network Model
Consider a power network delineated by a graph G(N,E),
where N={1,· · · , n}denotes the set of buses and E ⊂
N × N denotes the set of transmission lines connecting the
buses. Suppose that G(N,E)is connected and directed, with
arbitrary directions assigned to the transmission lines. Note
that if ij ∈ E, then ji 6∈ E. The buses are divided into two
types: generator buses and load buses, which are denoted
respectively by the sets Gand Lwith N=G∪L. A generator
bus is connected to both generators and loads, while a load
bus is only connected to loads.
For notational simplicity, all the variables in this paper rep-
resent the deviations from their nominal values determined
by the previous execution of economic dispatch. We consider
the standard direct current (DC) power flow model [22], [23]:
Pij =Bij (θi−θj)∀ij ∈ E (1)
where Pij is the active power flow on line ij, and θidenotes
the voltage phase angle of bus i.Bij is a network constant
defined by
Bij := |Vi||Vj|
xij
cos θ0
i−θ0
j
where |Vi|,|Vj|are the voltage magnitudes at buses iand j
(which are assumed to be constant in the DC model) and xij
is the reactance of line ij (which is assumed to be purely
inductive in the DC model). θ0
iis the nominal voltage phase
angle of bus i. See [21] for a detailed derivation.
The dynamic model of the power network is:
Mi˙ωi=−
Diωi+di−Pin
i+X
j:ij∈E
Pij −X
k:ki∈E
Pki
∀i∈ G (2a)
0 = Diωi+di−Pin
i+X
j:ij∈E
Pij −X
k:ki∈E
Pki
∀i∈ L (2b)
˙
Pij =Bij (ωi−ωj)∀ij ∈ E (2c)
where ωidenotes the frequency deviation from the nom-
inal value, Miis the generator inertia constant, and Di
is the damping coefficient, at bus i. The aggregate power
of controllable load at bus iis denoted by di, and the
difference between the generator mechanical power and the
uncontrollable load power at bus iis denoted by Pin
i. For
load buses i∈ L,Pin
irepresents the minus of the aggregate
uncontrollable load power.
Equations (2a) and (2b) describe the frequency dynamics
at generator and load buses, respectively. Actually, they both
indicate power balance at every time instant of the dynamics,
as shown in Figure 1. The damping term Diωi= (Dg
i+
Dl
i)ωicharacterizes the total effect of generator friction and
frequency-sensitive loads, while Pin
i=Pg
i−Pl
icaptures any
change in net uncontrollable power injection. The line flow
dynamics is delineated by (2c). The model (2) essentially
assumes that the frequency deviation wiis small at every
bus i. See [21] for a justification of the model (2).
Fig. 1. Frequency dynamics at bus i, where Pg
iand Pl
idenote generator
mechanical power and uncontrollable load power, respectively; Dg
iand Dl
i
denote the damping coefficients of generators and loads, respectively.
Remark 1. The simplified linear model (2) is for the
purpose of algorithm design and stability analysis. The ALC
algorithm that will be developed later can be applied to real
power systems that have more complex dynamics. In Section
IV, a high-fidelity power system simulator running a realistic
dynamic model is used to test the ALC algorithm.
B. Optimal Load Control Problem
Consider the scenario when step changes occur in Pin =
Pin
ii∈N . The power imbalance and frequency deviations
caused by these step changes will be eliminated through the
adjustment of controllable loads d= (di)i∈N . Our control
goals are therefore threefold:
1) Restore the system frequency to its nominal value.
2) Rebalance the system power while making each control
area absorb its own power change, so that the sched-
uled tie-line power transfers are maintained.
3) Modulate the controllable loads in an economically
efficient way that minimizes the total disutility for
adjusting all the loads, while respecting critical op-
erational constraints including load power limits and
line thermal limits.
The second and third control goals can be formulated as
the following optimal load control (OLC) problem:
Obj.min
d,θ X
i∈N
Ci(di)(3a)
s.t. di=Pin
i−X
j:ij∈Ein
Bij (θi−θj)
+X
k:ki∈Ein
Bki (θk−θj)∀i∈ N (3b)
di≤di≤di∀i∈ N (3c)
Pij ≤Bij (θi−θj)≤Pij ∀ij ∈ E (3d)
where Ein denotes the subset of lines that connect buses
within the same control area. Constants diand diare the
upper and lower load power limits at bus i, respectively,
and Pij and Pij specify the thermal limits of line ij.
The function Ci(di)quantifies the cost, or disutility, for
load adjustment di. To facilitate the subsequent proof of
convergence, we make the following assumption:
Assumption 1. The cost function Ci(·)is strictly convex and
continuously differentiable.
The objective (3a) is to minimize the total cost of load
control. Equation (3b) guarantees that the power imbalance
is eliminated within each control area; this can be shown by
adding (3b) over the buses in the same area A, which leads
to Pi∈A di=Pi∈A Pin
i. Equations (3c) and (3d) impose
the load power constraints and the line thermal constraints,
respectively. A load control scheme is considered to be
optimal if it leads to a steady-state operating point which
is a solution to the OLC problem (3).
III. OPTIMAL AUTOMATIC LOAD CON TROL
In this section, a fully distributed ALC scheme is devel-
oped for frequency regulation (see Algorithm 1). The basic
approach of controller design is reverse engineering, i.e.,
to interpret the system dynamics as a primal-dual gradient
algorithm to solve the OLC problem (3), which has been
used in recent literature [19]–[21].
A. Reformulated Optimal Load Control Problem
To explicitly take into account the first control goal in
Section II-B, i.e., frequency regulation, the OLC problem
(3) is reformulated as follows:
Obj.min
d,ω,P,ψ X
i∈N
Ci(di) + X
i∈N
1
2Diw2
i(4a)
s.t. di=Pin
i−Diωi−X
j:ij∈E
Pij +X
k:ki∈E
Pki
∀i∈ N
(4b)
di≤di≤di∀i∈ N (4c)
di=Pin
i−X
j:ij∈Ein
Bij (ψi−ψj)
+X
k:ki∈Ein
Bki (ψk−ψi)∀i∈ N (4d)
Pij ≤Bij (ψi−ψj)≤Pij ∀ij ∈ E (4e)
where ψiis an auxiliary variable interpreted as the virtual
phase angle of bus i, and ψij =Bij (ψi−ψj)is the virtual
power flow on line ij; see [19] where the concepts of virtual
phase angle and virtual power flow are first proposed. The
vectors ω:= (ωi)i∈N ,d:= (di)i∈N ,P:= (Pij)ij ∈E, and
ψ:= (ψi)i∈N are defined for notational simplicity.
In the reformulated problem (4), the virtual phase angles ψ
and the constraints (4b) and (4d) are introduced so that the
primal-dual algorithm of (4) is exactly the power network
dynamics under proper control. The equivalence between
problems (3) and (4) is established as follows.
Lemma 1. Let (ω∗, d∗, P ∗, ψ ∗)be an optimal solution of
problem (4). Then ω∗
i= 0 for all i∈ N , and d∗is optimal
for problem (3).
Proof. Let (ω∗, d∗, P ∗, ψ∗)be an optimal solution of (4),
and assume that ω∗
i6= 0 for some i∈ N . The optimal
objective value of (4) is therefore:
f∗=X
i∈N
Ci(d∗
i) + X
i∈N
1
2Di(w∗
i)2.
Then consider another solution {ωo, d∗, P o, ψ∗}with ωo
i= 0
for i∈ N ,Po
ij =Bij ψ∗
i−ψ∗
jfor ij ∈ Ein, and Po
ij =
0for ij ∈ E\Ein. Obviously, this solution is feasible for
problem (4), and its corresponding objective value is
fo=X
i∈N
Ci(d∗
i)< f∗
which contradicts the optimality of (ω∗, d∗, P ∗, ψ∗). Hence
ω∗
i= 0 for all i∈ N .
Next, note that the constraints (3b) and (4d) take the same
form, and that when ω= 0 and given (d, ψ), one can always
find Pthat satisfies (4b) by taking Pij =Bij (ψi−ψj)
for ij ∈ Ein and Pij = 0 for ij ∈ E\Ein. Therefore the
feasible set of (4) restricted to ω= 0 and projected onto
the (d, ψ)-space is the same as the feasible set of (3) on the
(d, θ)-space. As a result, for any (ω∗, d∗, P ∗, ψ∗)that is an
optimal solution of (4), d∗is also optimal for (3).
B. Automatic Load Control Algorithm
A partial primal-dual gradient method is applied to solve
the reformulated OLC problem (4). This solution method can
be exactly interpreted as the dynamics of a power network
with load-frequency control. Based on this interpretation, an
optimal ALC algorithm is derived.
The Lagrangian function of problem (4) is
L=X
i∈N
Ci(di) + X
i∈N
1
2Diw2
i
+X
i∈N
λi
−di+Pin
i−Diωi−X
j:ij∈E
Pij +X
k:ki∈E
Pki
+X
i∈N
µi
−di+Pin
i−X
j:ij∈Ein
Bij (ψi−ψj)
+X
k:ki∈Ein
Bki (ψk−ψi)!
+X
ij∈Ein
σ+
ij Bij (ψi−ψj)−Pij
+X
ij∈Ein
σ−
ij −Bij (ψi−ψj) + Pij
+X
i∈N
γ+
idi−di+X
i∈N
γ−
i(−di+di)(5)
where λi, µi∈Rare the dual variables associated with the
equality constraints (4b) and (4d), and γ+
i, γ−
i, σ+
ij , σ−
ij ≥0
are the dual variables associated with the inequality con-
straints (4c) and (4e). Define λG:= (λi)i∈G ,λL:= (λi)i∈L,
σ:= σ+
ij , σ−
ij ij∈Ein
, and γ:= γ+
i, γ−
ii∈N .
A partial primal-dual gradient method is given by the
following two steps:
Step 1): Solve minωLand then maxλLL, which leads to
the following:
wi=λi,∀i∈ N (6a)
0 =di−Pin
i+Diλi+X
j:ij∈E
Pij −X
k:ki∈E
Pki ∀i∈ L (6b)
Step 2): The primal-dual gradient algorithm on the rest of
variables is:
˙
λi=λi
−di+Pin
i−Diλi−X
j:ij∈E
Pij +X
k:ki∈E
Pki
(7a)
˙
Pij =Pij (λi−λj)(7b)
˙
di=di−C0
i(di) + λi+µi−γ+
i+γ−
i(7c)
˙
ψi=ψi
X
j:ij∈Ein µi−µj−σ+
ij +σ−
ij Bij
+X
k:ki∈Ein µi−µk+σ+
ki −σ−
kiBk i#(7d)
˙γ+
i=γ+
idi−di+
γ+
i
(7e)
˙γ−
i=γ−
i[−di+di]+
γ−
i
(7f)
˙µi=µi
−di+Pin
i−X
j:ij∈Ein
Bij (ψi−ψj)
+X
k:ki∈Ein
Bki (ψk−ψi)!(7g)
˙σ+
ij =σ+
ij Bij (ψi−ψj)−Pij +
σ+
ij
(7h)
˙σ−
ij =σ−
ij −Bij (ψi−ψj) + Pij +
σ−
ij
(7i)
where (7a) is for i∈ G, (7b) is for ij ∈ E , (7c)–(7g) are for
i∈ N , and (7h)–(7i) are for ij ∈ Ein . The notations con-
taining represent appropriately selected positive constant
step sizes. The operator [x]+
ymeans positive projection [24],
which equals xif either x > 0or y > 0, and 0 otherwise; it
ensures σ+
ij , σ−
ij , γ+
i, γ−
i≥0.
Since the instant change Pin
iof the uncontrollable power
injection is usually unknown in practice, a new variable ri
is introduced to substitute µi:
ri=Ki
µi
µi−Ki
λi
λi
where Kiis a positive constant. In this way, the necessity to
know Pin
iis circumvented.
Let λi= 1/Miand Pij =Bij. Then the partial primal-
dual gradient algorithm (6)–(7) can be equivalently written as
the ALC algorithm (8) together with the network dynamics
(2). In (8b), µiis the abbreviation of the expression µi=
ωi·µi/λi+ri·µi/Ki.
Fig. 2. The automatic load control (ALC) mechanism.
Algorithm 1 Automatic Load Control Algorithm.
˙
di=di−C0
i(di) + λi+µi
λi
ωi+µi
Ki
ri−γ+
i+γ−
i
(8a)
˙
ψi=ψi
X
j:ij∈Ein µi−µj−σ+
ij +σ−
ij Bij
+X
k:ki∈Ein µi−µk+σ+
ki −σ−
kiBk i#
(8b)
˙γ+
i=γ+
idi−di+
γ+
i
(8c)
˙γ−
i=γ+
i[−di+di]+
γ+
i
(8d)
˙ri=Ki
Diωi+X
j:ij∈E
Pij −X
k:ki∈E
Pki
−X
j:ij∈Ein
Bij (ψi−ψj) + X
k:ki∈Ein
Bki (ψk−ψi)
(8e)
˙σ+
ij =σ+
ij Bij (ψi−ψj)−Pij +
σ+
ij
(8f)
˙σ−
ij =σ−
ij −Bij (ψi−ψj) + Pij +
σ−
ij
(8g)
The implementation of algorithm (8) is illustrated in Fig-
ure 2. In the physical (lower) layer, each bus imeasures its
own frequency deviation ωiand the power flows (Pki, Pij )
on its adjacent lines. In the cyber (upper) layer, each bus
iexchanges the information (µi, ψi)with its neighboring
buses. Then following algorithm (8), each bus iupdates
the variables ψi, γi, σij, riand computes its load adjust-
ment di. Next, the control command diis sent back to
the physical layer and executed by the load modulation
device. Afterwards, the system frequency and power flows
respond to the load adjustment according to the physical law
(2). In this manner, the combination of network dynamics
(2) and the proposed control algorithm (8) forms a closed
loop. In addition, since only local measurement and local
communication are required in this process, the proposed
ALC algorithm is performed in a fully distributed manner.
Furthermore, the proposed algorithm (8) will converge to
a steady-state operating point that is optimal in the sense that
it solves the reformulated OLC problem (4). This claim is
restated formally as the following theorem.
Theorem 1. The proposed ALC algorithm (8) together
with the network dynamics (2) asympotically converges to
a point (d∗, ω∗, P ∗, ψ ∗, γ∗, r∗, σ∗), where (d∗, ω∗, P ∗, ψ ∗)
is an optimal solution of problem (4).
A challenge in implementing (8e) is that the damping
coefficient Diis in general hard to know exactly. It is shown
below that the proposed control (8) is robust to inaccuracy in
Di, in the sense that the closed-loop system still converges
to an optimal solution of (4), if the inaccuracy in Diis small
enough and some additional conditions are met.
Theorem 2. Assume that the following conditions are met:
i) For i∈ N , the cost function Ci(di)is α-strongly convex
and second-order continuously differentiable, i.e., Ci∈C2
with C00
i(di)≥α > 0, in the interior of its domain (di, di),
and Ci(di)→+∞as di→d−
ior d+
i←di.
ii) For i∈ N , the function C0
iis Lipschitz continuous with
Lipschitz constant L > 0.
iii) Infinitely large step sizes diare used for (8a), which is
then reduced to the following algebraic equation:
−C0
i(di) + λi+µi
λi
ωi+µi
Ki
ri−γ+
i+γ−
i= 0.
iv) An inaccurate ˜
Di=Di+δaiis used instead of Diin
(8e), and the inaccuracy δaisatisfies:
δai∈2d0−qd02+d0Dmin , d0+qd02+d0Dmin
where d0:= 1/L and Dmin := mini∈N Di.
Then the closed-loop system (2),(8) converges to a point
(d∗, ω∗, P ∗, ψ ∗, γ∗, r∗, σ∗), where (d∗, ω∗, P ∗, ψ ∗)is an op-
timal solution of problem (4).
Due to the space limit, the proofs of Theorems 1 and 2
are provided in the longer version of this paper [25].
Remark 2. The conditions imposed in Theorem 2 are mostly
for the purpose of theoretical analysis. That means these
conditions are conservative. As shown in the following case
studies, the proposed load control algorithm is effective even
when the inaccuracy in Diis large.
IV. CAS E STUD IES
The effectiveness and robustness of the proposed ALC
algorithm is demonstrated in numerical simulations. In par-
ticular, the performance of the ALC under step and contin-
uous power changes is tested, and the cases with inaccurate
Fig. 3. The 39-bus New England power network.
damping coefficients are demonstrated. The impact of noise
in measurements is also studied numerically.
A. Simulation Setup
The 39-bus New England power network in Figure 3 is
tested. The simulations were run on Power System Toolbox
(PST) [26], and we embedded the proposed ALC algorithm
through modifying the dynamic model functions of PST.
Compared to the analytic model (2), the PST simulation
model is more complicated and realistic, which involves
the classic two-axis subtransient generator model, the IEEE
Type DC1 excitation system model, the alternating current
(AC) power flow model, and different types of load models.
Detailed configuration and parameters of the simulation
model are available online [27].
There are ten generators located at bus-30 to bus-39,
which are the swing buses. To simulate continuous changes
in power supply, four photovoltaic (PV) units are added
to bus-1, bus-6, bus-9, and bus-16. Since PV units are
integrated with controllable power electronic interfaces, we
view them as negative loads rather than swing generators.
Consequently, bus-1 to bus-29 are load buses with a total
active power demand of 6.2 GW. Every load bus has an
aggregate controllable load, and the disutility function for
load control is
Ci(di) = ci·d2
i
where the cost coefficients ciare set to 1 per unit (p.u.) for
bus-1 to bus-5, and 5 p.u. for other load buses. The adjustable
load limits are set as di=−di= 0.4p.u. with the base
power being 100 MVA. In addition, the loads are controlled
every 250 ms, which is a realistic estimate of the time-
resolution for load control [28]. The damping coefficient Di
of each bus is set to 1 p.u. In the proposed controller, the
step sizes and the constants Kiare all set to 0.5 p.u.
B. Step Power Change
At time t= 1 s, step increases of 1 p.u. in load occur
at bus-1, bus-6, bus-9, and bus-16. With or without ALC,
the system frequency is illustrated in Figure 4. It can be
observed that the power network is not capable of bringing
the frequency back to the nominal value without ALC.
In contrast, the proposed ALC mechanism can restore the
system frequency to the nominal value. Figures 5 and 6
present the load adjustments and the total cost of load control
under ALC, respectively. It is observed that loads with lower
cost coefficients citend to make larger adjustments bounded
by their upper limits. This phenomenon indicates that the
load adjustments are calculated for system-wide efficiency
although the calculations are performed in a distributed way.
As a result, the total cost of ALC approaches to the optimal
cost of the OLC problem (3) or (4) in the steady state.
Fig. 4. The frequency dynamics under step power changes.
Fig. 5. The load adjustments under ALC.
C. Continuous Power Change
We next study the performance of ALC under continuous
power changes. To this end, the PV generation profile of a
real power system located within the territory of Southern
California Edison is used as the power supply of each
of the four PV units. The original 6-second data of PV
outputs are linearly interpolated to generate power outputs
every 0.01 second, which is consistent with the resolution of
PST dynamical simulation. The PV power outputs over 10
minutes are shown in Figure 7.
Fig. 6. The trajectory of the total ALC cost and the optimal cost of OLC
problem (3).
Fig. 7. The PV power outputs.
Fig. 8. The frequency dynamics under continuous power changes.
Fig. 9. The dynamics of voltage magnitudes at the PV buses.
Figures 8 and 9 illustrate the dynamics of system fre-
quency and voltage magnitudes, respectively. It can be
observed that ALC can effectively maintain the nominal
frequency under time-varying power imbalance. With real-
time frequency deviation and power mismatch utilized in the
control process, ALC can respond promptly to the lasting
power fluctuations. Also, from Figure 9, it is observed that
the rise in voltage caused by PV generation is alleviated by
ALC. The reason is that the power imbalance is absorbed
by ubiquitously distributed loads to mitigate the effect of
power over-supply on voltage. In summary, the ALC scheme
not only maintains system frequency, but also improves the
dynamics of voltage magnitudes.
D. Impact of Inaccurate Damping Coefficients
This part is devoted to understanding the impact of inac-
curate damping coefficients on the performance of ALC. Let
the damping coefficient ˜
Diused by the controller have the
following relationship with the accurate Di:
˜
Di=k·Di
where kis a uniform scaling factor for all the buses i∈
N. The factor kis tuned to test the performance of ALC
under step power changes. Figure 10 compares the frequency
dynamics under ALC with different k.
Fig. 10. The frequency dynamics under inaccurate damping coefficients.
As shown in Figure 10, the convergence of the system fre-
quency becomes slower when smaller damping coefficients
are used. As the utilized damping coefficients approach zero,
the frequency can be stabilized but cannot be restored to the
nominal value. This observation can be explained as follows.
When Di= 0, problem (4) imposes no restriction on the
system frequency. As a result, only the power imbalance is
eliminated, but the frequency cannot be restored. In contrast,
when larger damping coefficients are used, the convergence
of frequency dynamics becomes faster, at the cost of in-
creased oscillations. As the damping coefficients increase to
30 times the actual values, the system frequency becomes
unstable. In summary, ALC works well under moderate
inaccuracies in the damping coefficients Di.
E. Impact of Measurement Noise
We now study how the noise in local measurement affects
the performance of ALC. Recall that the implementation of
ALC relies on local measurement of frequency deviation ωi
and adjacent power flows (Pki, Pij )at every bus i∈ N .
First, consider the noise ξω
iin the measurement of ωi,
which is assumed to follow the Gaussian distribution, i.e.,
ξω
i∼ N (0, σ2
ω). The measured frequency deviation is thus
˜ωi=ωi+ξω
i. The standard deviation σωis tuned to
test the performance of ALC under step power changes.
In the simulations, the noise ξω
iis generated independently
over time and across buses, with the Gaussian distribution
truncated within the ±3σinterval to avoid the tail effect. The
resultant frequency dynamics is shown in Figure 11.
Fig. 11. The frequency dynamics with noise in frequency measurement.
Next, we inject noise ξP
ij to the measurement of power
flow, which also follows the Gaussian distribution, i.e., ξP
i∼
N(0, σ2
P). The frequency dynamics under different levels of
such noise is shown in Figure 12.
Figures 11 and 12 show that the system frequency can
be restored to the nominal value under measurement noise.
Moreover, the frequency presents oscillations that increase
with the level of noise.
V. CONCLUSION
Based on the reverse engineering approach, we developed
a fully distributed ALC mechanism for frequency regulation
in power systems. The combination of ALC and power
network dynamics was interpreted as a partial primal-dual
gradient algorithm to solve an optimal load control problem.
Fig. 12. The frequency dynamics with noise in power flow measurement.
As a result, relying purely on local measurement and local
communication, ALC can eliminate power imbalance and
restore the nominal frequency with minimum total cost of
load adjustment, while respecting operational constraints
such as load power limits and line thermal limits. Numerical
simulations of the 39-bus New England system showed
that ALC can maintain system frequency under step or
continuous power changes, and is robust to inaccuracy in
damping coefficients as well as measurement noises.
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