Aggregate Modeling of Distribution Systems for
Multi-Period OPF
Evangelos Polymeneas, Student Member, Sakis Meliopoulos, IEEE Fellow
School of Electrical and Computer Engineering
Georgia Institute of Technology
Atlanta, GA, USA
[email protected], [email protected]
Abstract— Distribution systems with active components, such as
responsive load, distributed storage and renewables,
supplemented with thermostatically controlled loads have the
capability to support the transmission grid and provide part of
the required capacity reserve. Including distribution system
resources in transmission level multi-period economic dispatch
is challenging due to the large number of devices. In this paper,
a two-level scheme is proposed for optimally dispatching the
distribution system’s active & reactive support to the grid, over
a look-ahead horizon using an aggregate model. Initially, a
semidefinite programming problem is solved in order to obtain a
maximal dynamic ellipsoidal model of the feasible region for the
power consumption of the distribution system. Subsequently, a
look-ahead AC-OPF problem is solved for the optimal lookahead dispatch of the transmission grid, using the aggregate
distribution model. The distribution dispatch is then
disaggregated to individual devices. The procedure is repeated
in a receding horizon basis. Results on the accuracy and benefits
of the approach are demonstrated on standard IEEE systems.
Index Terms-- Economic Dispatch, Distribution Systems,
Demand Response, Optimal Power Flow, Distributed Resources
NOMENCLATURE
Nd
P[k ], Q[k ]
s[k]
p j [k ] , q j [k ]
Number of controllable devices in
distribution system
Aggregate active & reactive distribution
consumption at step k
Parameter vector, common for entire
distribution system
Active & reactive consumption, device j at
step k
x j [k ] , u j [k ]
State vector & input of device j step k
Aj , B j , D j
State-update LTI model of device j
C j , E j , Fj
Output model of device j
Hj
Constraint model of device j
ΔB +par , ΔB −par ,
ΔB p , ΔBq
Δd +par , Δd −par ,
Δd p , Δd q
Bk , d k
γ
μik
ai( k )
T
c( X, U)
X k , U k , Pk
λ
x1j
2x2 Adjustment matrices for the timevarying ellipsoidal aggregate model
2x1 adjustment vector for the time-varying
ellipsoidal aggregate model
A positive semidefinite matrix and a vector
defining feasible P-Q ellipsoid in step k
Α non-negative penalty factor in relaxed
system-identification SDP
Slack variable for i-th inclusion constraint
at step k
i-th row k-step A matrix that defines
feasible polyhedron in the P-Q domain
Entire system’s single step cost function
Optimal Power Flow state, control and
parameter vectors for step k
Reactive power weighing parameter in
disaggregation problem
Initial state for distributed device j
I. INTRODUCTION
The increased penetration of variable resources in the
Electric Grid is introducing new challenges for power system
operation, such as thermal unit cycling [1], increased reserve
requirements and thermal unit ramping insufficiencies [2].
These situations are detrimental [3], [4] for both power system
reliability and economic operation. New responsive
components, such as large scale and distributed storage as well
as responsive/flexible load [5] can serve as a solution to this
problem [6]. Recent reports indicate that harnessing the
capabilities of distributed controllable resources, such as
responsive demand and distributed storage will be pivotal in
addressing operational issues caused by renewable energy
variability in general [7], [8] and lack of ramping flexibility in
particular [9].
However, each of these distributed resources is associated
with its own set of operational constraints and dynamics. For
example, responsive thermostatic load operation is constrained
by an upper and lower temperature limit and any operation
that violates those limits is unacceptable from the customer’s
perspective. Hence, the question that arises is how
distribution-level responsive devices can be included in the
transmission-level dispatch framework. Naturally, concerns of
scale and tractability arise, if one attempts to include millions
of kilowatt level models in the gigawatt level transmission
optimization formulations.
This paper focuses on detailing a two-level hierarchical
framework for the inclusion of aggregate responsive
distribution system models in transmission level optimal
power flow formulations.
Various efforts to model aggregate flexibility of
distributed resources, while observing customer constraints,
have been recorded in the literature. A first effort was to
model Thermostatically Controlled Load (TCL) uncertainties
using a state queuing model [10]. A refined version of this
approach resulted in the use of Markov models for TCL state
estimation and control [11]. This framework was later
extended to non-homogenous TCL populations [12]. Efforts to
quantify TCL sensitivity using aggregate battery models have
also been recorded [13]. Other approaches focus on an
approximate first order model for TCL loads [14], or even
higher order models for estimating ON/OFF populations [15].
An excellent method to aggregate thermostatically controlled
loads is given in [16], but it is specific to TCL loads and it
requires the solution of an expensive robust optimization
problem in a receding horizon manner. Existing literature on
aggregate modeling of distributed resources is largely focused
on only one specific active resource, such as thermostatically
controlled loads. The topic of integration and utilization of
aggregate models in transmission-level dispatch or unit
commitment algorithms is mostly neglected.
In this paper the extraction of a generalized data-driven
aggregate model for active distribution systems and the use of
this model within the context of a look-ahead optimal power
flow algorithm for the transmission system are discussed.
Distributed resources connected to the distribution grid have
their own dynamic models and their own binding constraints
which correspond to physical device limits as well as
customer inconvenience constraints. In this paper the
aggregate feasible region for active and reactive powers
consumed by a population of active distributed resources is
approximated by an ellipsoid in the P-Q domain, whose
parameters vary in time as a Linear Time Invariant (LTI)
system, with aggregate consumption given as the input.
Unlike existing research, that has mostly focused on
specific resource types (for example only TCL’s or only
batteries), this formulation provides a description for the time
evolution of the feeder’s feasible set for any population of
flexible resources. In addition, compared to existing literature,
the aggregate dynamic model presented here directly
quantifies the feasible set for the aggregate active & reactive
power consumption of the Distribution System, using a model
identification procedure based on system data. The
identification procedure is implemented by Semidefinite
programming, in order to obtain a maximal but conservative
ellipsoidal approximation of the feasible set. Existing
literature is not concerned with reactive power consumption of
the feeder. However, reactive power resources at the
distribution feeder are very important to levelize the voltage
profile along the feeder (modern feeders with distributed
resources especially rooftop PV may experience wild
variations of voltage profile if not controlled) and also provide
voltage control for the transmission system. Our approach is
based on AC formulation of the look-ahead OPF, which fully
considers voltage control. Specifically, we discuss &
implement both the aggregation, as well as the system-wide
optimization and subsequently the disaggregation to each
distribution-connected device, thus obtaining a full two-level
hierarchical optimization scheme, shown in Fig. 1.
Fig. 1. Two-level Hierarchical Distribution System Dispatch
The rest of the paper is organized as follows. Section II
outlines the computational procedure for the data-driven
identification of the aggregate time-varying ellipsoidal model
of the distribution system. Section III briefly describes the
multi-step OPF formulation used for transmission-level
dispatch, including the distribution aggregate model. The
disaggregation of aggregate distribution commands to
individual devices is performed by a direct load control
scheme in Section IV. Results with standard IEEE test cases
are shown in Section V and conclusions are offered in
Section VI.
II. ELLIPSOIDAL AGGREGATION OF DISTRIBUTION FEASIBLE
SETS
A. Aggregate Feasible Region Description
Suppose a population of responsive distributed energy
resources connected to the distribution feeder. For reasons of
tractability, and in accordance with prior work in this area
(e.g. [13]) we shall assume linear modeling for all such energy
resources and ignore the nonlinearities of feeder power flows.
Hence, the following model is applicable regarding the
aggregate active and reactive power consumption of such a
population at step k:
Nd
P[ k ] =  p j [ k ]
j =1
(1a)
Nd
Q[k ] =  q j [k ]
j =1
x j [k + 1] = A j x j [k ] + B j u j [k ] + D j s[k ]
(1b)
(1c)
 p j [k ] 


 q j [k ]  = C j x j [k ] + E j u j [k ] + F j s[k ]


H j x j [k ] ≤ r j
(1d)
(1e)
Given a state vector x j [k ] and s[k] for every device at
step k, (1e) suggests that x j [k + 1] lie in a polyhedral set,
which means that u j [k ] also lie in a polyhedron, since they
are given as the inverse affine image of a polyhedron in (1c).
As such (1d) suggests that the feasible region for p j , q j is a
polyhedron in R2 and, finally, the feasible region for
{P[k ], Q[k ]} is a polyhedron in R2 as a direct summation of
polyhedral sets. Therefore, it is safe to assume that this
polyhedron is bounded. This assumption is consistent with
physical intuition, since an infinite active or reactive power
consumption by any device in a given period should not be
physically realizable. Let the k-step feasible polyhedron for
the aggregate consumptions be given as:
{
F ( k ) = x = (P[k ] Q[k ])T ∈ R 2 : A( k ) x ≤ b ( k )
}
(2)
Since F ( k ) is given as the projection of a polyhedral set
onto R2, A(k) and b(k) can be obtained algorithmically by
Fourier-Motzkin elimination [17].
In this paper, the purpose is to approximate the variations
in the feasible set (2) as a function of feeder consumption. It
should be stressed that the purpose is to derive an aggregate
model of the distribution system, which will not include the
intricate modeling of individual devices (1c)-(1e). For this
purpose, we use a time-moving ellipsoid to model these
variations. An ellipsoid in R2, unlike a polyhedron in the
general case, can be parameterized by a constant number of
five parameters. Accordingly, we assume an LTI model for
those parameters, whose input is the aggregate active and
reactive consumption of the DER population, as well as any
parameter variations. Consider the following approximation
of the feasible set by a time varying ellipsoid:
 P[k ]
k k
k
(3a)
Q[k ] = B v + d


B k = B k −1 + ΔB p P[k − 1] + ΔBq q[k − 1]
np
np
j =1
j =1
+  s j [k − 1]ΔB −par +  s j [k ]ΔB +par
(3b)
np
np
j =1
j =1
 s j [k − 1]Δd −par +
vk
2
≤1
 s j [k ]Δd +par
and ΔBq . Since the parameters for that model are unknown,
we use a system identification procedure to obtain them. The
unknown parameters are the initial ellipsoid ( B1 , d 1 ) , as well
as the adjustment matrices and vectors for the LTI model. The
identification method is described next.
B. Data-Driven System Identification
Suppose the feasible polyhedron in the P-Q domain is
known for N s steps, in the format of (2). The question then
arises, what is a time-varying ellipsoidal model with linear
updates (3) that best fits that data. We adopted the approach
to choose the maximal model that is a conservative
approximation of the feasible polyhedron at each step. This
leads to a model parameter (see Ed. 3) identification
procedure that yields an ellipsoid which is included in the
feasible polyhedron at each step such that the overall volume
of these ellipsoids is maximal.
However, it is a known fact [18], that an ellipsoid
parameterized by ( B, d ) is included in a polyhedron in the
form of (2) if and only if:
(4)
Ba + aT d ≤ b
i
(3c)
(3d)
In (3) the feasible region for the distribution consumption
is given as the image of a unit ball in (3d), under a mapping
that varies according to an unknown LTI model in (3b) and
(3c). Note that B k must be a positive semidefinite matrix, and
i
i
aiT
where
is the i-th row of A. As such, our system
identification problem becomes the following:
 N s

max
Det ( B k )



Bk ,d k ,ΔB p ,ΔBq ,ΔB −par ,ΔB +par 
k =1
subject to
(5)
(k )
( k )T
(k )
∀i = 1,2 & ∀k ∈{1, S}
Bk ai + ai d k ≤ bi
∀k ∈{2, S}
(3b)
(3c)
∀k ∈{2, S}
This formulation yields very conservative estimates, and
occasionally infeasibility, if N s is very large. This issue is
resolved with the introduction of the following slack variable
formulation, to slightly relax the inclusion constraint and
penalize any violations:
Ns
 N s

Det ( B k ) − γ  μ k 1 


−
+

Bk ,d k ,ΔB p ,ΔBq ,ΔB par ,ΔB par 
k =1
k =1
subject to
max
T
d k = d k −1 + Δd p P[k − 1] + Δd q Q[k − 1]
+
the same holds for matrix corrections ΔB +par , ΔB −par , ΔB p
Bk ai( k ) + ai( k ) d k − μik ≤ bi( k )
∀i = 1,2 & ∀k ∈{1, S}
(3b)
∀k ∈{2, S }
(3c)
∀k ∈{2, S }
μik ≥ 0
∀i = 1,2 & ∀k ∈{1, S}
(6)
Any non-negative slack variables μik are penalized by a
penalty factor γ in the objective. The value of γ regulates the
tradeoff between the conservative requirement and the
amount of distribution controllability offered – i.e. the size of
the feasible set.
Note that (5) and (6) both are semidefinite programs, for
which efficient interior point methods exist. We use CVX to
model such problems [19] and SDPT to solve them [20].
It should be mentioned here that this formulation assumes
ai( k )
IV. AGGREGATE COMMAND DISAGGREGATION
defining the polyhedron at
Let us assume that the look-ahead OPF problem has been
step k are known. These parameters are obtained by
performing feeder simulations and obtaining descriptions of
feasible sets using Fourier - Motzkin elimination to obtain the
projection in the P-Q domain. Control inputs for all devices
are chosen at random for the purposes of this simulation. In
the interest of brevity, the simulation process for data
acquisition is not further discussed here.
solved and aggregate optimal commands P*[k ] and Q*[k ]
for the distribution system have been specified. These
commands are committed in the distribution system by
applying individual device controls such that the aggregate
consumption of the feeder is (ideally) equal to the aggregate
optimal commands. This commitment is realized by solving a
disaggregation problem, whereby the aggregate P-Q
commands
are distributed to the multitudes of devices
connected to the distribution system. Because an approximate
aggregate Distribution System P-Q model was used in the
look-ahead OPF, there are no guarantees that these P-Q
targets are actually feasible. For this reason, the
disaggregation problem is cast as an L2 norm minimization
problem, where the normalized distance between the actual
total consumption and the aggregate optimal consumption
over the next M steps is minimized. This problem is
formulated as:
that the parameters
and
bi( k )
disaggregation procedure and the handling of such errors is
discussed in the next section.
III. MULTI-STEP OPTIMAL POWER FLOW FORMULATION
Once derived, the aggregate time-varying ellipsoidal
description (3) is subsequently used within a multi-step OPF
formulation of the bulk transmission system. Because
thousands of kW-level devices have been aggregated, the
scale of the aggregate model is comparable with that of the
usual multi-step OPF problem. Our formulation is a multistep AC-OPF formulation, including transmission constraints
and voltage/reactive power control. The formulation is
detailed in [21]. The multi-step OPF minimizes the cost over
a look-ahead horizon of K steps, subject to the system’s
equality constraints, such as power flow equations, and
inequality constraints, such as ramping constraints and line
ratings. The formulation is given as:
min
K
 M P[k ] − P *[k ]
M Q[ k ] − Q * [ k ]
min  
+λ 
*
k k
x j ,u j k =1
Q *[ k ]
P [k ]
k =0
2

s.t.
P[ k ] =
 c( X k , U k )
X,U k =1
Q[k ] =
subject to
g( X k −1 , U k −1, X k , U k | Pk ) = 0 k = 1,2,, K
h( X k −1 , U k −1 , X k , U k | Pk ) ≤ 0 k = 1,2,..., K
U min ≤ U k ≤ U max
k = 1,2,..., K
X min ≤ X k ≤ X max
k = 1,2,..., K
(7)
X 0 = Xinit
U 0 = U init
Note that we propose an implementation, as suggested in
Fig. 1, where the horizon for which this look-ahead problem
is solved is larger than the dispatch horizon M for which the
optimal controls are actually committed and applied to each
individual device. As applied to the distribution system
control, this means that every M steps the data-driven model
aggregation of Section II is identified, and the entire
transmission multi-step OPF is re-solved. Upon convergence,
the active and reactive dispatch of a smaller number of M
steps (M<K) is taken as the distribution system’s dispatch.
This dispatch is then disaggregated to individual devices in
the distribution system by solving the disaggregation
problem. This results in individual device-level commands
for thousands of devices. One advantage of this approach is
that the model aggregation is repeated as time advances, thus
not allowing an accumulation of errors that could potentially
result from the aggregation modeling process. The
Nd
 p j [k ]
j =1
Nd
 q j [k ]
j =1


2

∀k ∈ {1, , M }
∀k ∈ {1, , M }
(8)
x j [k + 1] = A j x j [k ]
+ B j u j [ k ] + D j s[k ] ∀k ∈ {1, , M − 1}, ∀j ∈ {1, , N d }
 p j [k ] 


 q j [k ]  = C j x j [k ]


+ E j u j [ k ] + F j s[k ]
∀k ∈ {1, , M }, ∀j ∈ {1, , N d }
H j x j [k ] ≤ r j
∀k ∈ {1, , M }, ∀j ∈ {1, , N d }
x1j
=
x1j
∀j ∈ {1,, N d }
A weighting factor λ multiplies the normalized error of
the reactive power in (8). This is used to regulate the relative
importance of reactive power fitting with respect to active
power fitting. Because the real and reactive power could
possibly be conflicting, it is of interest to explore the tradeoff
between them. More importantly, a greater weight should be
given to active power, since errors in active power dispatch
can cause serious mismatches between generation and load in
the system, and this power will need to be provided by a
standby generator, forcing an out-of-market action. Hence,
ensuring perfect tracking of the active power commands
provided by the look-ahead OPF is more important than
Fig. 2. Model identification results: feasible polyhedron vs. ellipsoidal approximation for various values of slack penalty factor – 8 steps shown
tracking reactive power commands. This is the reason for the
introduction of the parameter λ, with selection λ <1.
( )
Upon solution of (8), the actual controls u kj
*
Table I. Sample Feeder Parameters
THERMOSTATICALLY CONTROLLED LOADS
for each
device j and each step k are obtained. These controls
minimize the distance between the desired aggregate
consumption and the actual consumption of the distribution
feeder. The value of the objective is a good metric for the
performance of the ellipsoidal approximation of the aggregate
distribution system. Since this model approximates the
feasible region of the aggregation, the aggregate commands is
close to feasible with the average distance expressed with the
value of the objective function. Thus our model provides a
quantitative description of how well the ellipsoidal
approximation fits into the actual polyhedral feasible region.
A small level of infeasibility of the aggregate commands can
occur. Infeasibilities manifest as nonzero objective values for
(8). The performance of this framework has been evaluated
and demonstrated in the results section.
It should be noted that, under our standing linearity
assumption, the disaggregation problem (8) is a Quadratic
Program, and as such it can be solved by mature interior point
techniques. The solver used for this research is SeDuMi [22].
Class A
TCL
Class B
TCL
Class C
TCL
Class D
TCL
R
(kW)
2.9
±5%
3.4
±5%
2.9
±5%
3.4
±5%
Batteries
nc
Battery
0.9
±1%
C
Prated
(kWs/0C) (kW)
350
8.0
±10%
±10%
350
8.0
±10%
±10%
450
8.0
±10%
±10%
450
8.0
±10%
±10%
BATTERIES
Pmax
nd
(kW)
0.9
4.4
±1%
±1%
Power
Factor
0.91
±10%
0.94
±10%
0.88
±10%
0.91
±10%
Emax
(kWh)
0.1±1%
θmax
(0C)
18.0
24.0
17.0
23.0
18.5
24.5
17.5
23.5
Emin
(kWh)
6
±1%
A test feeder of 1,000 active devices, consisting of 200
TCL’s from each class and 200 batteries was created. The
maximum consumption of the feeder is approximately 7MW.
A scatterplot with R & C values in the feeder considered in
this section is shown in Fig. 3.
V. RESULTS
The active distribution system modeling methodology
was applied to a sample active distribution system. The
distribution system consists of a population of
thermostatically controlled loads and battery storage systems.
To test the generality of results, the parameters of all devices
are drawn from a uniform random distribution around certain
chosen central values. The various classes of devices are
defined as shown in Table I
θmin
(0C)
Fig. 3. Illustration of TCL classes in sample feeder
The aggregation problem for this feeder is addressed by
solving the ellipsoidal feasible region approximation
problem. The data for this problem comes from a simulation
of the feeder, where the initial conditions for the batteries and
TCL’s are selected via a random generator. A sample
execution of this “aggregation phase” is shown in Fig. 2. The
choice of penalty factor γ in (6) substantially affects the
resulting model. A smaller γ yields a larger feasible ellipsoid,
but may overestimate the actual polyhedral feasible region,
while the larger γ will yield a more conservative
approximation and may lead to contraction of the feasible set
if the number of steps increases. We consider erring on the
conservative side, as it is always preferred, and a penalty
factor γ = 200 is chosen for the remaining results, unless
otherwise stated.
As a first numerical experiment, we run a 100-step
random simulation of this feeder and obtain an aggregate
model by solving (6). Subsequently, we connect this feeder to
bus 17 of the IEEE RTS 24 bus system and run the lookahead OPF at 10-minute intervals in a 24-hour dispatch to
schedule this feeder load. The results are shown in Fig. 4.
(a)
Fig. 4b is the schedule provided by the solution of the lookahead OPF with the aggregate model obtained by the
simulation data, while the solid black line is the closest L2
norm fit resulting from the solution of (8). Hence, even
though the model proves useful in scheduling the distribution
system using the look-ahead algorithm, it does not achieve
feasibility of the target schedule, i.e. the objective of (8) is
not zero. This was expected, since our aggregate model is an
approximation to allow for transmission system optimization,
not an exact modeling approach. This suggests the need for
the receding horizon optimization approach of Fig. 1,
whereby the aggregate model is updated periodically.
If the target dispatch is not realizable, this is a significant
problem for the decision-maker (e.g. the balancing authority
of the transmission system), especially if the active power
consumption is different from the scheduled one. Large
errors, such as the one showed in Fig. 4b should be
unacceptable. Use of the receding horizon framework is
suggested as a solution of the modeling accuracy problem.
Renewal of the model at specified intervals and re-solving of
the look-ahead scheduling algorithm allows a reduction of the
inaccuracies, because it allows extraction of renewed state
information for all TCL’s and Batteries in the system and
updating of the aggregate model.
The results for the distribution system scheduling, for the
same net load pattern as Fig. 4a, in the same transmission
system (RTS ’79) and the same feeder composition as in
Table I are shown in Fig. 5. The dashed lines denote the
target consumption schedules given for the rest of the day by
the look-ahead OPF. There are 10 different dashed lines, each
corresponding to a different target schedule, as the lookahead OPF is solved every two hours. The black line
corresponds to the actual consumption, given by the solution
of the L2 norm minimization problem. Note that only two
hours of each target set point are committed.
(b)
(a)
(c)
Fig. 4. Aggregate Distribution Control with Dispatch Horizon 24 hours (a)
Net System Load (b) Feeder P Target vs. Realized (c) Feeder Q
As shown in Fig. 4b, the look-ahead OPF schedules the
distribution system so that consumption is reduced during the
peak load of the system, at around 12 pm. The dashed line at
(b)
Fig. 5. Aggregate Distribution Control with Dispatch Horizon 2 hours (a)
Active Power Target vs. Realized (b) Reactive Power
From Fig. 5a it should be noted that there is no error
between the target active power consumption and the actual
consumption. However, there is some error in the reactive
power consumption, due to the small weight factor λ=0.3
used in (8), to reduce the importance of reactive power
fitting. Active power mismatches can be much more
threatening to power system operation than reactive power
imbalances. It is also of note that, as shown in Fig. 5, the
schedule shows abrupt changes every two hours, as the model
is updated, as expected. Note that the general schedule pattern
is quite similar to the one yielded by the single-pass approach
in Fig. 4.
Fig. 6 shows the result of the L2 norm fit problem when it
comes to the home temperature for four different TCL
classes, in order to obtain the response shown in Fig. 5. Note
the general pattern of pre-cooling the houses in the early offpeak hours and turning-off the air-conditioners in peak times.
Furthermore, note that different TCL classes have different
thermal models, and are scheduled in different patterns by the
L2 norm minimization algorithm.
Fig. 6. In-house temperature for TCL’s of four different classes
We have not reported efficiency (execution times) for the
multi-step OPF because the size and structure of this OPF is
similar to any other OPF that does not consider modeling of
feeder resources. The aggregation/dis-aggregation of the
feeder resources is quite fast and since it is performed for
only one feeder at a time, do not pause any significant
computational burden. Note also the computations for each
feeder are naturally parallelizable and can be assigned to
different cores of a computer. A comparison of our approach
to one where the entire feeder resources are integrated
explicitly into the multi-period OPF yields the following
conclusions: (a) the latter problem is a practical impossibility
due to the huge number of feeders and feeder resources. (b)
the computational requirements of the proposed method are
practically the same as the multi-step OPF without the
proposed aggregate feeder model. Since we are not claiming
of producing a production grade program, we are not
providing execution times.
VI. CONCLUSIONS AND FUTURE WORK
In this paper, a two-level hierarchical control scheme for
including active distribution systems into the multi-step ACOPF problem was discussed. The derivation of a heuristic
aggregate model was based on simulation data & ellipsoidal
approximation of the feasible region of operation of the
distribution system. The entire power system’s operation was
scheduled by a short-term look-ahead AC Optimal Power
Flow and, subsequently, individual distribution device
commands were obtained by a least-squares disaggregation
problem. The results indicate that the model is effective in
scheduling the aggregate operation of distribution systems
without a prohibitive computational burden to the
transmission-level optimization framework. Case studies
showed that the model is effective and quite accurate for
short-term operation horizons, but accuracy declines as
planning horizons increase. Further research may be targeted
towards more conservative models that provide guarantees for
longer horizons or towards theoretical derivations of modeling
accuracy and limitations.
Implementation of the proposed method will require an
infrastructure that will provide customer data in real time as
well as to use historical data for projecting customer use
patterns for 24 hours. Technology is moving towards this
direction with smart meters but also there is significant
research into developing house management systems that will
provide much better information towards this goal [23].
Further discussion of these issues is not discussed in this
paper.
The approach presented in this paper is data based. The
next improvement of the method is to make it distribution
feeder model based. In this approach the feeder circuits are
explicitly modeled together with the demand, TCLs, storage,
etc. The aggregation and dis-aggregation in this case will be
much more complex and the models involved non-convex.
The advantage of this approach will be that the model based
approach will enable coordination of the distributed along the
feeder resources as well as the ability to enable individual
customers to optimize the use of their resources. A distribution
level state estimator can provide the required models and data.
We have developed such a system [24] and we are currently
investigating the aggregation/dis-aggregation approach for
using the feeder model and data to the multi-period OPF. We
expect to report on these extensions in future papers.
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BIOGRAPHIES
Evangelos Polymeneas (S ’12) was born in Athens,
Greece. He received a Diploma in ECE from National
Tchnical University of Athens, Greece in 2010. He
received his Ph.D. from Department of Electrical and
Computer Engineering, Georgia Institute of
Technology, in 2015. His research interests include
large-scale power system estimation, control and
automation, and power markets. He is focusing on
issues of fleexibility and uncertainty in power systems operations and on
operational methods to facilitate grid support through distributed energy
sources.
A. P. Sakis Meliopoulos (M ’76, SM ’83, F ’93)
was born in Katerini, Greece, in 1949. He received
the M.E. and E.E. diploma from the National
Technical University of Athens, Greece, in 1972; the
M.S.E.E. and Ph.D. degrees from the Georgia
Institute of Technology in 1974 and 1976,
respectively. In 1971, he worked for Western
Electric in Atlanta, Georgia. In 1976, he joined the
Faculty of Electrical Engineering, Georgia Institute
of Technology, where he is presently a Georgia Power Distinguished
Professor. He is active in teaching and research in the general areas of
modeling, analysis, and control of power systems. He has made significant
contributions to power system grounding, harmonics, and reliability
assessment of power systems. He is the author of the books, Power Systems
Grounding and Transients, Marcel Dekker, June 1988, Lightning and
Overvoltage Protection, Section 27, Standard Handbook for Electrical
Engineers, McGraw Hill, 1993. He holds three patents and he has published
over 220 technical papers. In 2005 hze received the IEEE Richard Kaufman
Award. Dr. Meliopoulos is the Chairman of the Georgia Tech Protective
Relaying Conference, a Fellow of the IEEE and a member of Sigma Xi.