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1
Abstract—This paper proposes a data-driven method based on
distributionally robust optimization to determine the maximum
penetration level of distributed generation (DG) for active
distribution networks (ADNs). In our method, the uncertain DG
outputs and load demands are formulated as stochastic variables
following some ambiguous distributions. In addition to the given
expectations and variances, the polyhedral uncertainty intervals
are employed for the construction of the probability distribution
set to restrict possible distributions. Then we decide the optimal
sizes and locations of DG to maximize the total DG hosting
capacity under the worst-case probability distributions among
this set. Since more information is utilized, our proposed model is
expected to be less conservative than the robust optimization
method and the traditional distributionally robust method. Using
the CVaR (Conditional Value at Risk) reformulation technique
and strong duality, we transform the proposed model into an
equivalent bilinear matrix inequality problem, and a sequential
convex optimization algorithm is applied for solution. Our model
guarantees that the probability of security constraints being
violated will not exceed a given risk threshold. Besides, the
predefined risk level can be tuned to control the conservativeness
of our model in a physically meaningful way. The effectiveness
and robustness of this proposed method are demonstrated
numerically on the two modified IEEE test systems.
Index Terms—Active distribution network, DG capacity
assessment, distributionally robust optimization.
NOMENCLATURE
A. Sets
b
Set of all branches.
n
Set of all buses.
sub
Set of substation buses.
dg
Set of candidate buses potentially connected
to distributed generators.
()i
Set of buses directly connected to bus i.
Set of all time periods for DG capacity
assessment.
B. Parameters
i,t
w
Predicted efficiency factor of DG output at
bus i at period t.
,it
The power factor angle of DG output at bus
i at period t.
Manuscript received XX, 2016. This work was supported in part by the
National Key Basic Research Program of China (Grant 2013CB228203), the
National Science Foundation of China (Grant 51477083). The authors are
with the Department of Electrical Engineering, Tsinghua University, Beijing
100084, China (email: wuwench@tsinghua.edu.cn, Correspondig author:
Wenchuan Wu).
LL
i,t i,t
P ,Q
Predicted active, reactive load demand at
bus i at period t.
ref
U
Reference of squared voltage magnitude
controlled by substation.
min maxi, i,
U ,U
Lower and upper limits of squared voltage
magnitude at bus i.
maxij,
s
Apparent power capacity of branch ij.
ij ij
r , x
Resistance and reactance of branch ij.
C. Variables
i
S
Installed DG capacity at bus i.
i,t
w
Actual efficiency factor of DG output at bus
i at period t.
dg dg
i,t i,t
P ,Q
Active, reactive DG output at bus i at period
t.
LL
i,t i,t
P ,Q
Actual active, reactive load demand at bus i
at period t.
i,t
V
Voltage magnitude at bus i at period t.
i,t
U
Squared voltage magnitude at bus i at
period t.
ij,t ij,t
p , q
Active, reactive power flow of branch ij
from bus i to bus j at period t.
,,
,
dg L
i t i t
Uncertainty variables of the DG output and
load demand.
k
Scaling variable.
D. Vectors and Matrices
ξ
Vector of uncertainty variables whose
elements are
,,
,
dg L
i t i t
.
,
dg dg
pq
Vector of active, reactive DG outputs whose
elements are
dg dg
i,t i,t
P ,Q
.
x
Decision vector of installed DG capacity
whose elements are
i
S
.
,
LL
pq
Vector of actual active, reactive load demands
whose elements are
LL
i,t i,t
P ,Q
.
,
bb
pq
Vector of branch active, reactive power flow
whose elements are
ij,t ij,t
p , q
.
b
u
Vector of branch squared voltage magnitude.
n
u
Vector of bus squared voltage magnitude
whose elements are
i,t
U
.
ref
u
Vector of the reference of squared voltage
magnitude whose elements are
ref
U
.
λ
Vector of scaling variables whose elements are
k
.
1
Unit vector whose elements are 1.
Xin Chen, Student Member, IEEE, Wenchuan Wu, Senior Member, IEEE, Boming Zhang, Fellow, IEEE,
Chenhui Lin, Student Member, IEEE
Data-driven DG Capacity Assessment Method for
Active Distribution Networks
2
W
Matrix of actual DG efficiency factor whose
elements are
i,t
w
.
T
Diagonal matrix of
,
tan it
.
A
Node-branch incidence matrix of the
distribution networks.
,
bb
RX
Diagonal matrix of branch resistance,
reactance.
E. Acronyms
SO
Stochastic optimization
RO
Robust optimization.
DRO
Distributionally robust optimization.
D-CAM
Deterministic DG capacity assessment model.
DR-CAM
Distributionally robust DG capacity
assessment model.
DRJCC
Distributionally robust joint chance constraint.
DRICC
Distributionally robust individual chance
constraint.
BMI
Bilinear matrix inequality.
LMI
Linear matrix inequality.
SDP
Semidefinite programming.
CVaR
Conditional Value at Risk.
I. INTRODUCTION
ith a large scale integration of distributed generation
(DG), active distribution networks (ADNs) are formed
to facilitate the control of distributed energy resources and
enhance system reliability. The proliferation of DG, especially
photovoltaic (PV) generators benefits ADNs in various way,
involving voltage profiles improvement, amelioration of
power quality, losses reduction, and deferring network
reinforcements [1]. Hence, there are intrinsic motivations for
fully exploitation of renewable energy based DG. In light of
the advent of DG, the techniques for planning, designing and
operating of distribution networks have changed profoundly
[2].
However, a series of technical problems arise when a high
penetration level of DG is reached, such as reverse power flow
and overvoltage. These issues have adverse impacts on system
operation and deteriorate the ability of ADNs to accommodate
DG. Therefore, the DG capacity assessment task is required to
be completed ahead. Through this assessment process, system
operators can gain a better understanding about the maximum
DG capacity that can be absorbed by present ADNs, and make
the optimal DG sitting and size decisions. Until now, there
have been considerable researches conducted on the issue. In
[3, 4], analytical methods were proposed based on sensitivity
analysis of the voltage and branch power flow, but only one
DG location can be evaluated at a time. The repetitive power
flow solutions [5] and heuristic approaches [6, 7] were applied
to obtain maximum DG penetration as well. In [8, 9], the
alternating current optimal power flow (ACOPF) technique
was used, where the variability of loads and DG outputs was
depicted with multiple time periods. Literature reviews and
comprehensive analysis of DG planning and optimization
methods can be found in [10, 11].
Although forecasting techniques have been investigated in
many studies, accurately predicting long-term DG outputs and
load demands remains a challenge. As a result, the obvious
forecasting errors introduce significant uncertainty risks to the
DG capacity assessment process. However, the uncertainties
associated with outputs of renewable energy sources based
DG and load demands have not been recognized in most
previous studies. This may lead to erroneous capacity
evaluation results and inappropriate DG installation schemes.
Hitherto, the two predominant techniques used to model
uncertainty in DG capacity evaluation have been stochastic
optimization (SO) [12, 13] and robust optimization (RO) [14,
15]. In the SO approach, the uncertain forecasting errors are
treated as random variables with given probability distribution
functions (PDFs). While obtaining the exact PDFs of uncertain
variables is intractable in practice, and performing the
optimization itself is extremely computationally intensive. In
the worst-case oriented RO method, decisions are made based
on the worst scenarios from the predefined uncertainty
intervals. The PDF information is not required in the RO
method, but it usually comes up with a two-stage nonconvex
optimization model and results in over conservative strategies.
Generally, only partial statistical information, such as the
first- and second-order moments, is available regarding the
uncertainty variables. To cope with such cases, a novel
data-driven optimization tool, distributionally robust
optimization (DRO), was developed to tackle the uncertainty.
This method fully utilizes existing knowledge and inherits the
advantages of both SO and RO [16]. In the DRO approach,
uncertain parameters are treated as random variables, while
detailed PDFs are ambiguous. Based on the known moment
information, a probability distribution set is constructed to
appropriately restrict the possible PDFs. Then, as in the RO
method, robustness is ensured by making decisions under the
worst-case PDFs. The DRO method has recently been
successfully applied in power system optimization, including
transmission congestion management [17], unit commitment
[18], and reserve scheduling [19]. A conceptual framework
for organizing the data analytics functions and supporting
data-driven decision making in an electrical distribution
network is proposed in [20]. However, the ranges of
uncertainty variables are neglected in these traditional DRO
applications, which will extend the optimization region to the
entire space and obtain much more conservative solutions.
Furthermore, native range constraints are imposed on the
uncertainty variables in many realistic problems; for instances,
the outputs of PV unit cannot exceed its capacity. In [21], an
ellipsoid region is used to restrict the holistic uncertainty
range, while this form cannot ensure that the native range
constraints are satisfied.
In this paper, we propose a data-driven DG hosting capacity
assessment methodology for ADNs, based on the DRO
algorithm, to capture the uncertainties of both DG outputs and
load demands. The main procedures and contributions of this
paper are summarized below.
Based on a deterministic DG capacity assessment model
(D-CAM), we developed a distributionally robust capacity
assessment model (DR-CAM). In our DR-CAM, the uncertain
DG outputs and load demands are described as random
variables following ambiguous distributions. The probability
distribution set containing all the possible PDFs is constructed
consistent with the given moment information, where the
polyhedral uncertainty intervals are incorporated as well. Then
W
3
the voltage security constraints and branch thermal constraints
are reformulated as a distributionally robust joint chance
constraint (DRJCC). In this manner, DR-CAM guarantees that
the system security requirements can be satisfied under any
possible PDFs in the distribution set. To solve this model,
DRJCC-based DR-CAM is transformed equivalently into a
bilinear matrix inequality (BMI) problem, and a sequential
optimization algorithm is applied for solution.
Via DR-CAM, we can evaluate the maximum DG capacity
that can be installed in ADNs and determine the robust
optimal placement and size of the DG units. The specific
merits of DR-CAM are as follows:
1) The polyhedral uncertainty intervals generally employed
in the RO method are incorporated to this model. Since the
statistic information of the uncertainty variables, including the
first-, second-order moments and the ranges, is fully exploited
in the assessment process, the results of DR-CAM are
practical and less conservative than the RO method in theory.
To the best of our knowledge, there is no such DRO
application on power system yet that utilizes both the moment
information and the polyhedral uncertainty intervals
simultaneously.
2) Detailed PDFs of the uncertainty variables are not
required in DR-CAM. As the installation decisions are made
based on the worst-case PDFs, they are robust and immune to
forecasting errors.
3) The acceptable risk level in DRJCC can be tuned to
control the conservativeness of DR-CAM, enabling a tradeoff
between conservativeness and robustness. This is superior to
the RO method, as the tuning has explicit physical meaning
and can guarantee that the probability of a security constraint
violation does not exceed a given risk threshold.
The remainder of this paper is organized as follows: In
Section II, we present the mathematical formulation of
DR-CAM. In Section III, DR-CAM is transformed into a BMI
problem, and a corresponding solution algorithm is
introduced. Numerical tests to assess the performance of our
approach are discussed in Section IV. Conclusions are drawn
in Section V, and the proof of the transformation mentioned
above is provided in Section VI.
II. MATHEMATICAL MODEL BUILDING
A. Deterministic DG Capacity Assessment Model
Essentially, the capacity assessment issue is an optimization
problem to maximize the hosting DG capacity with a number
of system operation constraints. In ACOPF approach, we built
a multi-period deterministic DG capacity assessment model
(D-CAM), which is presented as follows.
1) Objective Function:
0Ψ
max
idg
i
Si
Obj. S
(1)
The objective function (1) aims at maximizing the total DG
hosting capacity.
2) DG Output Constraints:
,,
, , ,
. . tan
Ψ,
dg
i t i t i
dg dg
i t i t i t
dg
P w S
st Q P
it
(2)
In equation(2), the DG units are assumed to operate in the
maximum power point tracking (MPPT) mode for making full
use of the renewable energy.
3) Power Balance Constraints at Buses:
, , ,
Ψ()
, , ,
Ψ()
Ψ,
dg L
i t i t ij t
ji
dg L
i t i t ij t
ji
n
P P p
Q Q q
it
(3)
Equation (3) depicts the active and reactive power balance
constraints of each bus in ADNs. The left terms of these
equalities are the net power injection to bus i, and the right
denote the total power flow moving from bus i to its directly
connected buses.
4) Power Flow Equality Constraints:
The power flow equations in radial ADNs can be described
by the Distflow model (4) (5) [22].
2
,, Ψ,
i t i t n
U = V i t
(4)
22
22
,,
2
Φ,
ij,t ij,t
i t j t ij ij,t ij ij,t ij ij i,t
b
pq
U U r p x q r x
i
U
jt
(5)
Equation (5) is nonconvex due to the quadratic term of
network power losses, while this term can be abandoned for
relatively small values. It results in the linear power flow
constraints (6), which are used in this paper.
, , , ,
2
Φ,
i t j t ij ij t ij ij t
b
i
U U r p x q
jt
(6)
5) Voltage Reference Constraints:
,,
i t ref sub
U =U i t
(7)
6) Branch Thermal Constraints:
2 2 2
, , ,max
,
ij t ij t ij
b
p q s
tij
(8)
Equation (8) is a quadratic circular constraint, which needs
to be linearized to facilitate the subsequent formula derivation.
With the quadratic constraint linearization method introduced
in [23], several square constraints can be used to approximate
the circular constraint. Here, two square constraints (9) is
employed to substitute for equation (8), shown as Fig 1, which
is sufficiently accurate for engineering applications.
P
Q
S
Figure 1. Circular constraint linearization method.
4
,max , ,max
,max , ,max
,max , , ,max
,max , , ,max
22
22
Φ,
ij ij t ij
ij ij t ij
ij ij t ij t ij
ij ij t ij t ij
b
i
s p s
s q s
s p q s
s p q s
jt
(9)
7) Voltage Security Constraints:
min max
Ψ
i, i,t i,
n
U U U
i ,t
(10)
Comprising equation (1)-(3), (6)-(7) and (9)-(10), D-CAM
is in the formulation of a linear programming (LP). It should
be noted that D-CAM handles multiple candidate DG buses
simultaneously, and we can set different DG buses to be
evaluated through changing
Ψdg
. Since the issue which we
focus on is assessing the overall hosting DG capacity of ADNs,
after carefully investigation, any buses potentially connected
to DG should be added to
Ψdg
. If the system operators have no
prior knowledge about the locations of DG units in some
extreme cases, all the possible buses in ADNs could be set as
the candidate DG buses for D-CAM to deal with the sitting
uncertainty problem.
Since the DG outputs and load demands are modeled as
deterministic parameters with forecasting values, D-CAM
does not take the uncertainty into account, which needs further
improvements.
B. Modeling of DG and Load Uncertainty
Firstly, we model the actual DG efficiency factors
,it
w
and
load demands
,,
,
LL
i t i t
PQ
as equation (11), and the random
variables
,,
=,
dg L
i t i t
ξ
are used to describe their
uncertainties.
, , ,
,
, , , , ,
,
= 1+
1+ ,
dg
i t i t i t
L
it
L L L L L
i t i t i t i t i t
L
it
ww
Q
P P Q P
P
(11)
Although exact PDFs of the uncertainty variables
ξ
N
Rξ
are unavailable, we can easily calculate the empirical mean
vector
μ
and covariance matrix
Σ
based on the profiles of
historical data. To make full exploitation of the known
moment information, we establish the probability distribution
set
Π,Σ,Sμ
as expression (12), which is a family of all
possible PDFs that have equal means and variances to
μ
and
Σ
respectively.
S
denotes the support sets of
ξ
.
=1
Π, = =
=
dF
F dF
dF
S
S
S
Σ,S
Σ
ξ
ξ
ξ
ξ
μ ξ ξ μ
ξξ ξ μμ
TT
(12)
To further mitigate the conservativism of solutions, we
incorporate the polyhedral uncertainty intervals (13) generally
used in the RO method to the modelling of uncertainty
variables.
1,2, ,
i i i ξ
ξ Δ ,ΔiN
(13)
Here,
i
Δ
and
i
Δ
denote the corresponding lower and
upper bounds, respectively;
ξ
N
is the number of elements of
uncertainty vector
ξ
. It should be noted that when we define
the polyhedral uncertainty intervals, some native range
requirements should be satisfied to avoid the contradictions to
the reality. For example, the output of PV unit cannot exceed
its capacity or be negative. In other words, constraint (14)
must be met.
,,
0 1, 0 L
i t i t
wP
(14)
Then, the uncertainty intervals (13) can be rewritten in a
quadratic form (15).
0 1,2, ,
i i i i ξ
ξ Δ ξ Δ iN
(15)
Accordingly, we establish the support sets
S
as expression
(16) to limit the ranges of uncertainty variables, which is a
compact formulation of expression (15) indeed. Here,
i
I
is a
diagonal matrix; its diagonal elements are all zero except for
the ith value which equals one.
Δ
and
Δ
are the vectors of
lower and upper bounds, respectively.
0 1,2, ,
iξ
iN S=ξ ξ Δ Iξ Δ
T
(16)
The probability distribution set
Π,Σ,Sμ
combines the
information of moments and supports of the uncertainty
variables. Comparing with the RO method and the traditional
DRO method, the solutions of DR-CAM are expected to be
less conservative due to utilization of more information.
Meanwhile, it is accessible to procure the characteristic
parameters required for the probability distribution set in
reality.
C. Distributionally Robust DG Capacity Assessment Model
To explain explicitly, we express the linear power equality
constraints (2), (3) and (6)-(7) in matrix forms as equation (17),
(18) and (19), respectively. Here,
-1
A
denotes the inverse of
matrix
A
, and
A-T
represents the inverse and transpose of
matrix
A
.
dg
dg dg
p =W x
q = T p
(17)
-1
-1
b dg L
b dg L
p = A p - p
q = A q - q
(18)
-1 -1
22
2 , 2
A
AA
b b b b b
n ref b ref dg L dg L
bb
u R p + X q
u u u u R p - p + X q -q
R R A X X A
-T
-T -T
(19)
According to equation (17)-(19) and (11), we can formulate
the actual power flow state variables
,,
b b n
p q u
as bilinear
functions of the decision vector
x
and the uncertainty vector
ξ
. Since the remaining constraints of D-CAM, namely branch
thermal constraints (9) and voltage security constraints (10),
are just linear constraints on the power flow state variables,
they can be expressed as bilinear equation (21) and substitute
5
for all the constraints of D-CAM.
0()
k
zx
and
()
k
zx
are the
corresponding linear function and linear function vector of
x
,
respectively; K is the number of the remaining constraints (9)
(10).
Hence, the uncertain DG capacity assessment problem is
presented as follows.
0
. maxObj x1x
T
(20)
0( ) ( ) 0
.. 1,2, ,K
kk
z
st k
x z x ξ
T
(21)
To capture the uncertainty and ensure robustness of the
solutions, we transform equation (21) into a distributionally
robust joint chance constraint (DRJCC) (22). It means that the
probability of branch overloading or voltage security violation
will not exceed the predefined risk level
δ
, even in the
worst-case PDFs among the probability distribution set
Π,Σ,Sμ
.
0
Π,
( ) ( ) 0
inf Pr 1
1, ,K
kk
FF
zδ
k
μ ξ
x z x ξ
～Σ,S
T
(22)
As a consequence, the distributionally robust DG capacity
assessment model (DR-CAM) is formulated as:
0
. max
Subject to Equations 12 22
Obj
DR-CAM x1x
T
The DG installation strategies generated via DR-CAM are
robust to the uncertainty in forecasting errors of DG outputs
and load demands, and it guarantees that the security
requirements can be met under any PDFs in the probability
distribution set. Besides, the risk level
δ
can be adjusted to
control the conservatism of DR-CAM in a clear physically
meaningful way.
III. SOLUTION METHOD FOR DISTRIBUTIONALLY ROBUST DG
CAPACITY ASSESSMENT MODEL
The DRJCC of DR-CAM is nonconvex and difficult to
apply in practice. In this section, the DRJCC is converted into
a BMI problem using CVaR formulation and dualization.
Then the sequential convex optimization algorithm [24] is
employed to solve this problem iteratively.
A. CVaR-Based Reformulation of DR-CAM
We approach the problem of solving the DRJCC (22) by
decomposing it into a number of distributionally robust
individual chance constraints (DRICCs), shown as equation
(23). Then each DRICC can be reformulated equivalently as a
worst-case CVaR [25] constraint (24):
0
Π,
inf Pr ( ) ( ) 0 1 ; 1, ,K
k k k
FF
zεk
μ ξ x z x ξ
～Σ,S
T
(23)
+
0
Π,
+
0
Π,
1
sup inf ( ) ( ) 0
1
inf sup ( ) ( ) 0
1, ,K
kk
Fk
kk
F
k
z
ε
z
ε
k
F
R
Σ,S
F
RΣ,S
Ε
Ε
μ
μ
x z x ξ
x z x ξ
T
T
(24)
Here,
max 0,xx
,
R
is an intermediate variable,
and
x
F
Ε
denotes the expected value of
x
.
According to duality theory [26], equation (24) combined
with the probability distribution set (12) can be converted into
equation (25) using strong duality, whose proof is provided in
the Appendix of this paper.
12
n 1 +
1Tr , 0
0, 0
1, ,K
; ; ,
k
k
k
ii
ε
k
ab
YQ
Q Q+
R Q Z R
DD
(25)
In equation (25), the definitions of matrix
Y
,
1
D
and
2
k
D
are shown as expression (26).
n1
QZ
is the
symmetric matrix associated with the dual variables, and the
space of symmetric matrixes of dimension n is denoted by
n
Z
.
Tr ,YQ
is the trace of the scalar product of
Y
and
Q
.
1
20
1
-
-
1()
2
1( ) ( )
2
i i i i
ii
i i i i
ii
kk
i i k i i
ii
k
kk
k i i k i i
ii
aa
aa
bb
b z b
Σ
Yμμ μ
μ
IIΔ
DΔIΔIΔ
I - z x I Δ
D- z x ΔI - x ΔIΔ
T
TT
T T T
(26)
In this way, the uncertainty variables
ξ
are eliminated,
and each DRICC is transformed into a linear matrix inequity
(LMI) in the form of a semidefinite programming (SDP).
However, the issue how to allocate the risk level
δ
in the
DRJCC (22) to each
k
ε
in the DRICC (23) appropriately to
make this decomposition equivalent must be addressed.
Bonferroni approximation [27], where the risk level is equally
divided among the DRICCs by setting
=K
k
ε δ
, is overly
conservative and cannot be used here. In this paper, DR-CAM
is converted into the following equivalent optimization
problem, which is denoted as DR-CAM*. In light of Theorem
3.6 in [24], this conversion is exact if the scaling vector
1K
,,
λ
is introduced and treated as decision
variables.
*
0, 0
max
Subject to , 0
DR-CAM xλ1x
Gxλ
T
where
,Gxλ
is defined as equation (27), and the definition
of
2,
k
λ
D
is presented as equation (28).
6
1 2,
n 1 +
1
, = Tr ,
0; 0
1, ,K
; ; ,
k
λ
k
ii
k
ab
YQ
QQ
R Q Z R
Gxλ
DD
(27)
0
2,
1()
2
()
1()
2
kk
i i k k i i
ii
k
λkk
kk
k k i i ii
ii
bb
z
bb
I - z x I Δ
D-x
- z x ΔIΔIΔ
TT T
(28)
DR-CAM* is a SDP-based BMI problem, which can be
solved using the following sequential convex optimization
algorithm.
B. Sequential Convex Optimization Algorithm
Inspired by the idea that the BMI problem is convex and
tractable when the values of the scaling variables
λ
are fixed,
we decompose DR-CAM* into a master problem and a sub-
problem, and then optimize
λ
and
x
alternately.
max
Subject to , 0
f
Master Problem x0
*
1x
Gxλ
T
In the master problem,
*
λ
is the given scaling parameters.
We optimize
x
to obtain the optimal strategies of DG sitting
and sizing and the maximum DG installation capacity
f
.
max ,
Sub Problem *
λ0Gx λ
In the sub-problem,
*
x
is fixed. We solve it to attain the
optimal scaling parameters
*
λ
corresponding to
*
x
.
By solving the master problem and sub-problem alternately,
we can gain a sequence of monotonically increasing objective
values. The detailed sequential convex optimization procedure
is presented as follows.
1) STEP 1 Initialization: Set a feasible solution
0
x
for the
master problem and the corresponding objective value is
denoted as
0
f
; initialize the iteration counter
1l
and set a
small tolerance
γ
.
2) STEP 2 Scaling Parameter Optimization: Solve the
sub-problem with input
1l
x
and obtain the optimal scaling
parameters
*
λ
; set
l*
λ λ
.
3) STEP 3 Decision Optimization: Solve the master
problem with input
l
λ
and obtain the optimal decisions
*
x
with the corresponding objective value
l
f
; set
l*
xx
.
4) STEP 4 Termination: If
11l l l
f f f γ
, terminate and
output the final decisions
l
x
and objective value
l
f
;
otherwise set
1ll
and go back to STEP 2.
In the first step, an initial feasible solution of the master
problem can be obtained using the Bonferroni approximation
technique mentioned above. Theorem 3.8 in [24] implies that
since
l
x
is bounded, the sequence of objective values
l
f
will increase monotonically and converge to a finite limit. The
master problem and sub-problem are LMI issues in the
formulation of SDP, which can be solved efficiently via many
available commercial optimizers.
IV. NUMERICAL TESTS
In section IV, numerical tests are carried out to demonstrate
the characteristics of DR-CAM. Firstly, we give a brief
introduction about the modified IEEE 33-bus distribution
system. Secondly, the assessment results of DR-CAM with a
variety of input parameters are investigated. Then, we make
comparisons between DR-CAM and the other methods in
optimality and robustness using Monte Carlo simulations.
Next, additional case studies are implemented based on the
IEEE 123-bus distribution system. Finally, the simulation
platform and computational efficiency are provided.
A. Introduction to IEEE 33-bus System
The modified IEEE 33-bus distribution system, shown as
Fig. 2, was used as the test system. There are four candidate
sites for the PV generators located at bus-14, bus-21, bus-24,
and bus-31, so we let
Ψ14,21,24,31
dg
. We set the
voltage magnitude of the substation bus to 10.5 kV as a
reference, and the lower and upper bounds of voltage for each
bus were 10 kV and 11 kV, respectively. The detailed
description of the parameters and configurations of this test
system is available online [28].
01 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
18 19 20 21 PV-1
PV-2
22 23 24
PV-3
25 26 27 28 29 30 31 32
PV-4
:Residential loads : Industrial loads :Commercial loads
Figure 2. The modified IEEE 33-bus distribution system.
To simulate the real situation, we considered three common
load modes, including residential, industrial, and commercial
loads, and the distribution network was divided into three
areas accordingly. A typical day separated to 24h was chosen
as the multiple assessment time periods. The PV output curve
and three types of daily load curves are presented in Fig. 3.
The time-varying shape coefficients are normalized
multipliers, whose basic values are the nominal load demands
and the PV installation capacities.
2 4 6 8 10 12 14 16 18 20 22 24
0.0
0.2
0.4
0.6
0.8
1.0
Load/PV Output (p.u.)
Time/h
PV output
Residential load
Industrial load
Commercial load
Figure 3. The daily PV output and load demand curves.
B. Tests on IEEE 33-bus Distribution System
In this part, we assumed that the uncertain PV outputs and
load demands follow a particular multivariate PDF consistent
with the given statistic information. Considering the
correlation of PV generation and each type of load located in
7
neighboring areas, we specified four independent uncertainty
variables
= , , ,
pv res ind com
ξ ξ ξ ξξ
to delineate the forecasting
errors of PV outputs and residential, industrial and commercial
loads sequentially. Their corresponding expectation values
μ
and variances
Σ
were given by:
pv res ind com
μ,μ,μ,μ
μ=
(29)
2 2 2 2
pv res ind com
diag σ,σ,σ,σΣ=
(30)
where diag(x) denotes the diagonal matrix with element x.
The uncertainty intervals were defined as:
- , , ,
i i i
ξ Δ ,Δi pv res ind com
(31)
where the symmetrical ranges were employed.
Since the assessment results of DR-CAM heavily depend on
the probability distribution set
Π,Σ,Sμ
and the predefined
acceptable risk level
δ
, the following numerical tests were
carried out to investigate the effects of these crucial
parameters.
(1) Case 1: Effect of the second-order moments.
In the first case, we assumed that
= = = = 0
pv res ind com
μ μ μ μ
,
0.15
i
Δ
, and
= = = =
pv res ind com
σ σ σ σ σ
increases by 0.02 from
0.05 to 0.09. The maximum total PV hosting capacity and the
corresponding PV installation strategies suggested by
DR-CAM are presented in Fig.4 and Fig.5, respectively.
0.00 0.05 0.10 0.15 0.20 0.25 0.30
3.66
3.72
3.78
3.84
3.90
3.96
4.02
4.08
Total Capacity (MW)
Risk Level δ
σ=0.09
σ=0.07
σ=0.05
Figure 4. The maximum total PV hosting capacity obtained via DR-CAM
with different risk levels and standard deviations when the expectations are
fixed as 0 and the ranges are fixed as 0.15.
As shown in Fig.4, the maximum total PV hosting capacity
declined with the decreasing of the risk level
δ
, when the
standard deviation
σ
was fixed. This trend is consistent with
the intuition that stricter system security requirements lead to a
lower PV penetration level. Besides, as the risk level
δ
was
specified, the total PV hosting capacity had a monotonically
decreasing relationship with the standard deviation
σ
. In fact,
a larger
σ
means that the extent of the probability deviation
from the basic scenarios is greater, and a more worst-case PDF
will be selected from the probability distribution set for
decision-making. As a consequence, more conservative results
of capacity assessment are obtained.
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.52
0.54
0.56
0.58
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.90
0.92
0.94
0.96
0.00 0.05 0.10 0.15 0.20 0.25 0.30
1.36
1.40
1.44
1.48
1.52
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.92
0.96
1.00
1.04
Installation Capacity (MW)
Risk Level δ
σ=0.09
σ=0.07
σ=0.05
Bus-14
Installation Capacity (MW)
Risk Level δ
σ=0.09
σ=0.07
σ=0.05
Bus-21
Installation Capacity (MW)
Risk Level δ
σ=0.09
σ=0.07
σ=0.05
Bus-24
Installation Capacity (MW)
Risk Level δ
σ=0.09
σ=0.07
σ=0.05
Bus-31
Figure 5. Detailed optimal PV installation strategies obtained via DR-CAM
with different risk levels and standard deviations when the expectations are
fixed as 0 and the ranges are fixed as 0.15.
Fig.5 illustrates the optimal PV installation capacity for
each candidate bus, which contributes to the total hosting
capacity discussed above. It is seen that the detailed PV
installation capacities with distinct parameters exhibited the
similar trends to their aggregation shown in Fig.4.
From Fig.4 and Fig.5, a significant phenomenon is observed:
when the risk level became smaller, the total PV capacity and
the optimal one for each bus decreased monotonically, and
finally converged to particular values. Furthermore, these
results of convergence turned out to be exactly the solutions
obtained via the RO method, which are presented in Table I.
TABLE I
CAPACITY ASSESSMENT RESULTS OBTAINED VIA THE RO METHOD
bus-14
bus-21
bus-24
bus-31
Total
Capacity/MW
0.516
0.897
1.347
0.911
3.671
As a matter of fact, DRO is a general optimization method
for SO and RO. If we employ the probability distribution set
that contains distributions that put all their weight at a single
point within the support set, then DRO reduces to the RO
method [29]. When the risk level is set to zero, which means
that there is no possibility to violate the security constraints,
the moment information representing stochasticity becomes
useless. Therefore, only the polyhedral uncertainty intervals
are utilized in decision-making, and DR-CAM reduces to the
RO method. It is also a cogent evidence to prove that
DR-CAM is superior in optimality, for its most conservative
schemes are still the optimal strategies generated by the RO
method.
(2) Case 2: Effect of the first-order moments.
In the second case, we set
= = = = =0.07
pv res ind com
σ σ σ σ σ
,
0.2
i
Δ
, and defined
L res ind com
μ=μ=μ=μ
. DR-CAM with
different expectations was applied repeatedly in the following
three test scenarios:
0.05
1: 0.05
pv
L
μ
Scen μ
0
2: 0
pv
L
μ
Scen μ
0.05
3: 0.05
pv
L
μ
Scen μ
It should be mentioned that a larger
L
μ
means unpredicted
higher loads, and a smaller
pv
μ
reflects lower transformation
8
efficiency of solar energy or less sunlight radiations than the
prediction. The maximum total PV hosting capacity yielded by
DR-CAM under these above scenarios are presented as Fig.6.
0.00 0.05 0.10 0.15 0.20 0.25 0.30
3.4
3.6
3.8
4.0
4.2
Total Capacity (MW)
Risk Level δ
Scen3
Scen2
Scen1
Figure 6. The maximum total PV hosting capacity obtained via DR-CAM
with different risk levels and expectations when the standard deviations are
fixed as 0.07 and the ranges are fixed as 0.2.
The results shown in Fig.6 indicates that the scenarios with
bigger
L
μ
and smaller
pv
μ
were capable to accommodate a
larger scale of PV integration. Because less solar radiations
lead to less power injection when PV capacity is fixed, and
higher load demands enable more outputs to be absorbed on
the spot. As a conclusion, the level of local load demands and
the generation potential of renewable energy sources are the
fundamental factors that determine the maximum installation
capacity of DG.
(3) Case 3: Effect of the ranges.
In this case, we set
= 0.07σ
,
= = 0
pv L
μ μ
, and increased
i
Δ
by 0.05 from 0.15 to 0.25. The maximum total PV hosting
capacity obtained via DR-CAM under different uncertainty
intervals are provided in Fig.7.
0.00 0.05 0.10 0.15 0.20 0.25 0.30
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
Total Capacity (MW)
Risk Level δ
Δ=0.25
Δ=0.2
Δ=0.15
Figure 7. The maximum total PV hosting capacity obtained via DR-CAM
with different risk levels and ranges when the standard deviations are fixed as
0.07 and the expectations are fixed as 0.
It is observed that the curves with different range values in
Fig.7 ascend from distinct starting points and then
approximate each other, as the risk level increases. As
mentioned above, the solutions of DR-CAM will converge to
those of the RO method when a sufficiently small risk level is
achieved. Hence, these starting points are corresponding to the
maximum total PV hosting capacity obtained via the RO
method. Since the wider uncertainty intervals generally lead to
more conservative results, the curve with larger range values
starts from a lower point with the increment of the risk level.
According to Fig.4-Fig.7, we can draw the conclusions that
when the risk level is quite small, the capacity assessment
results of DR-CAM are mainly effected by the polyhedral
uncertainty intervals; while the moment information plays a
predominant role as the risk level becomes larger.
C. Comparisons against the Other Methods
In this part, Monte Carlo simulations were implemented to
compare the optimality and robustness among D-CAM, the
RO method, the T-DRO method and DR-CAM. Here, the
T-DRO method is the DG capacity assessment method based
on the traditional DRO algorithm that only utilizes the
moment parameters without considering the ranges of
uncertainty variables.
As for the simulation tests, we set the expectations
==
pv L
μ μ μ
to 0 and defined the risk level as 0.1. The actual
DG outputs and loads were assumed to be independent
random variables following Gaussian distributions, which
were in accord with the given expectations and variances.
Several test cases were carried out under distinct ranges and
standard deviations. For each test case, up to 10000 sampling
scenarios were generated randomly and independently within
the polyhedral uncertainty intervals. The PV installation
strategies obtained via the above four models were tested in
these scenarios, and we calculated their average PV outputs.
To facilitate the calculation, we only used the time period t=14
hours for test. Once the violation of voltage security occurred
on a certain PV bus during the simulation process, we
triggered the protection mechanism and shed the connected
PV units. The simulation results are summarized in Table II.
TABLE II
COMPARISON RESULTS IN MONTE CARLO SIMULATIONS FOR IEEE 33-BUS
SYSTEM
Range
Δ
Standard
deviation
σ
Average PV Power Outputs/MW
D-CAM
Robust
T-DRO
DR-CA
M
0.15
0.05
2.2177
3.6689
3.6966
3.78
0.07
2.157
3.6694
3.4091
3.6987
0.09
2.1384
3.6686
2.796
3.6718
0.2
0.05
2.1853
3.429
3.6935
3.7436
0.07
2.1478
3.4308
3.4084
3.56
0.09
2.1371
3.4293
2.7941
3.4824
0.25
0.05
2.1667
3.208
3.6927
3.7206
0.07
2.1469
3.2064
3.4081
3.5075
0.09
2.1197
3.2039
2.7944
3.327
From Table II, it is seen that DR-CAM outperformed the
other models with greatest average PV outputs in all cases.
The RO method and the T-DRO method provided overly
conservative assessment results and installation strategies due
to lacking of the range or moment information; this led to
lower PV outputs than DR-CAM and caused a waste of solar
energy resource. On the contrary, too radical PV installation
9
decisions were made by D-CAM for ignoring the impact of
uncertainty, which exacerbated the violation of security
constraints. As a result, the severe voltage security problems
limited the amount of power generated by PV unit in return.
While DR-CAM ensures the robustness and optimality of its
strategies simultaneously, and has apparent superiors over the
other three methods.
D. Tests on IEEE 123-bus Distribution System
To explore the scalability of DR-CAM, a larger distribution
system, shown as Fig.8, was applied for numerical tests. This
modified IEEE 123-bus system comprises 123 buses and 122
branches, which is comparable to a real distribution network
of a district. There are six PV candidate sites that locate at
bus-34, bus-67, bus-85, bus-97, bus-111 and bus-123,
respectively. The detailed parameters and configurations of
this test system are provided online [30]. All the load demands
were assumed to share the same load pattern as the residential
loads for simplification.
35
18
22
24
26
29
31 121
49 48 50 51 52
88
89
90
91
92
93
56
57
55
5859
60 61
63
64
65
66 67
5453
119
95
94
97
96
45 46 47
43 44
41 42
120 36
38 37 39 40
62
118
1
123 117 112
111
113 114 115
110
109
106
102
98
122
68
73
77
87
78 79 80
81
82
83
84
85 86
74 75 76
69 70 71 72
99 100 101 116
107 108
103 104 105
15
14
1617
19
2021
23
25
27
32
30
33
28
34
2
46
75
3
89
10
12
11
13
PV-1
PV-3
PV-4
PV-5
PV-6
PV-2
Figure 8. The modified IEEE 123-bus distribution networks.
In the first case, we set
= = 0
pv L
μ μ
and
0.2
i
Δ
, then
increased the standard deviations from 0.05 to 0.09 by 0.02.
The maximum total PV hosting capacity and the
corresponding PV installation strategies generated by
DR-CAM are illustrated in Fig.9 and Fig.10, respectively.
0.00 0.05 0.10 0.15 0.20 0.25 0.30
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
Total Capacity (MW)
Risk Level δ
σ=0.09
σ=0.07
σ=0.05
Figure 9. The maximum total PV hosting capacity obtained via DR-CAM
with different risk levels and standard deviations when the expectations are
fixed as 0 and the ranges are fixed as 0.2.
0.00 0.05 0.10 0.15 0.20 0.25 0.30
1.16
1.20
1.24
1.28
1.32
1.36
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.60
0.64
0.68
0.72
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.40
0.42
0.44
0.46
0.48
0.50
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.34
0.36
0.38
0.40
0.42
0.44
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.64
0.68
0.72
0.76
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.92
0.96
1.00
1.04
1.08
1.12
Installation Capacity (MW)
Risk Level δ
σ=0.09
σ=0.07
σ=0.05
Bus-34
Installation Capacity (MW)
Risk Level δ
σ=0.09
σ=0.07
σ=0.05
Bus-67
Installation Capacity (MW)
Risk Level δ
σ=0.09
σ=0.07
σ=0.05
Bus-85
Installation Capacity (MW)
Risk Level δ
σ=0.09
σ=0.07
σ=0.05
Bus-97
Installation Capacity (MW)
Risk Level δ
σ=0.09
σ=0.07
σ=0.05
Bus-111
Installation Capacity (MW)
Risk Level δ
σ=0.09
σ=0.07
σ=0.05
Bus-123
Figure 10. Detailed optimal PV installation strategies obtained via DR-CAM
with different risk levels and standard deviations when the expectations are
fixed as 0 and the ranges are fixed as 0.2.
Similar to the test results shown in Fig.4 and Fig.5, the
optimal PV capacities suggested by DR-CAM have
monotonically increasing and decreasing relationships with
the risk level
δ
and the standard deviation
σ
, respectively.
In addition, as extreme small
δ
was reached, the total PV
hosting capacity and the optimal capacity for each bus
declined to the particular values, which proved to be exactly
the solutions obtained by the robust method.
In the second case, Monte Carlo simulations were
conducted to compare the performances among D-CAM, the
RO method, the traditional DRO method and DR-CAM.
Parameter setting and simulation process were identical to
those in Part C. Each case was tested for 10000 sampling
scenarios, which were generated randomly and independently
in Gaussian distributions within the uncertainty intervals. The
average PV outputs of different models are presented in Tab
III.
TABLE III
COMPARISON RESULTS IN MONTE CARLO SIMULATIONS FOR IEEE 123-BUS
SYSTEM
Range
Δ
Standard
deviation
σ
Average PV Power Outputs/MW
D-CAM
Robust
T-DRO
DR-CA
M
0.15
0.05
2.6077
4.3301
4.5144
4.5401
0.07
2.534
4.3307
4.2481
4.336
0.09
2.5142
4.3306
3.9986
4.3306
0.2
0.05
2.5915
4.0606
4.5098
4.5213
0.07
2.5162
4.059
4.2478
4.2851
0.09
2.4906
4.0602
4.0077
4.0942
0.25
0.05
2.5582
3.8059
4.5091
4.5091
0.07
2.5232
3.8059
4.2463
4.2589
0.09
2.4507
3.8075
4.0064
4.0506
10
Tab III shows that DR-CAM has the highest average PV
outputs at all times for the fully exploitation of existing
knowledge. While overly pessimistic or optimistic decisions
were made under the other three methods, leading to less
power generation in realistic applications.
E. Computational Platform and Efficiency
DR-CAM was programmed in Matlab 2014a and solved by
an embedded Mosek solver with the YALMIP interface. We
implemented all the above numerical tests in a computational
environment with Intel(R) Core(TM) i7-4510U CPUs running
at 2.60 GHz with 8 GB RAM. Regarding the modified IEEE
33-bus system, it takes about one minutes on average to
accomplish a DR-CAM process. The CPU time for solving the
master problem and sub-problem are around 13.4 seconds and
3.1 seconds respectively, and it usually iterates three to four
times to reach convergence. In terms of the 123-bus system,
the average CPU time for the solution of DR-CAM is 6.7
minutes, and it spends 100.5s and 20.1s on average to solve
the master problem and sub-problem, respectively. Since the
DG capacity assessment process is an off-line task, it does not
put much pressure on the computational efficiency in practical
applications.
V. CONCLUSION
In this paper, we propose a distributionally robust
optimization based data-driven DG capacity assessment model
(DR-CAM) for active distribution networks, to deal with the
uncertain forecasting errors of DG outputs and load demands.
Distinguished from the traditional DRO method, the
polyhedral uncertainty intervals are also incorporated to the
construction of the probability distribution set, in addition to
the moment parameters. As more information is utilized,
DR-CAM is deemed to be less conservative than the RO
method and the traditional DRO method. Besides, the
predefined risk level can be adjusted to control the
conservativeness of DR-CAM with explicit physical meaning,
making a tradeoff between conservativeness and robustness.
Numerical tests implemented on the modified IEEE 33-bus
system and an IEEE 123-bus system demonstrated the
characteristics of DR-CAM, and comparisons against the other
methods verified its robustness and optimality. In future
works, the uncertainty of moment information will be
addressed.
VI. APPENDIX
As for each of the worst-case CVaR constraints (24), we
define the worst-case expectation as
+
0
Π,
0
sup ( ) ( )
=1
=
.. =
0 1, 2, ,
k k k
F
iξ
z dF
dF g
dF
st dF
iN
Σ,S S
S
S
S
Σ
S
μξ
ξ
ξ
ξ
x x z x ξ-ξ
ξ
ξ ξ μ g
ξξ ξ μμ G
=ξ ξ Δ Iξ Δ
T
TT
T
(32)
Dual variables
0
gR
,
n
Rg
, and
n
ZG
are assigned
to the equality constraints in (32), and its corresponding dual
problem (33) is presented as follows.
0
0
0
inf ,
. . max 0, ( ) ( )
k
kk
g
st g z
Σ
S
xgμGμμ
gξ ξ Gξx z x ξ ξ
TT
T T T
(33)
Since the moment matrix
ΣμμT
is positive definite,
strong duality holds between problem (32) and (33). The
semi-definite constraint in (33) can be expanded as two
equivalent semi-definite constraints (34) and (35).
00g Sgξ ξ Gξ ξ
TT
(34)
0
0( ) ( )
kk
gz
Sgξ ξ Gξx z x ξ-ξ
T T T
(35)
To remove the limits of the support
Sξ
, we apply the
idea of positivstellensatz [21]. The positive variables
,k
ii
ab
are introduced to convert constraints (34) and (35) to (36) and
(37), respectively. This conversion is equivalent according to
the S-lemma [31].
0ii
i
ga
gξ ξ Gξ ξ Δ Iξ Δ
T
TT
(36)
0
0( ) ( ) +
kk
k
ii
i
gz
b
gξ ξ Gξx z x ξ
ξ Δ Iξ Δ
T T T
T
(37)
Constraints (36) and (37) can be expressed in compact
forms as below:
10
11
Q+
ξ ξ
D
T
(38)
20
11
k
Q+
ξ ξ
D
T
(39)
where
1
-
-
i i i i
ii
i i i i
ii
aa
aa
IIΔ
DΔIΔIΔ
TT
,
20
1()
2
1( ) ( )
2
kk
i i k i i
ii
k
kk
k i i k i i
ii
bb
b z b
I - z x I Δ
D- z x ΔI - x ΔIΔ
T T T
,
and
n+1
0
g
Q = Z
Gg
gT
is the matrix of dual variables.
n
Ζ
denotes the space of symmetric matrixes of dimension n .
Consequently, the worst-case CVaR constraints (24) are
transformed into the following SDP constraints.
11
12
n 1 +
1Tr , 0
0, 0
1, ,K
; ; ,
k
k
k
ii
ε
k
ab
YQ
Q Q+
R Q Z R
DD
(25)
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