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Operational Planning of Active Distribution Grids under Uncertainty

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Modern distribution system operators are facing constantly changing operating conditions caused by the increased penetration of intermittent renewable generators and other distributed energy resources. Under these conditions, the distribution system operators are required to operate their networks with increased uncertainty, while ensuring optimal, cost-effective, and secure operation. This paper proposes a centralized scheme for the operational planning of active distribution networks under uncertainty. A multi-period optimal power flow algorithm is used to compute optimal set-points of the controllable distributed energy resources located in the system and ensure its security. Computational tractability of the algorithm and feasibility of the resulting flows are ensured with the use of an iterative power flow method. The system uncertainty, caused by forecasting errors of renewables, is handled through the incorporation of chance constraints, which limit the probability of insecure operation. The resulting operational planning scheme is tested on a low-voltage distribution network model using real forecasting data for the renewable energy sources. We observe that the proposed method prevents insecure operation through efficient use of system controls.
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Operational Planning of Active Distribution Grids
under Uncertainty
Stavros Karagiannopoulos, Line Roald, Petros Aristidou, Gabriela Hug
EEH - Power Systems Laboratory, ETH Zurich, Physikstrasse 3, 8092 Zurich, Switzerland
Los Alamos National Laboratory, Los Alamos, NM 87545
School of Electronic and Electrical Engineering, University of Leeds, Leeds LS2 9JT, UK
Emails: {karagiannopoulos, hug}@eeh.ee.ethz.ch, roald@lanl.gov, p.aristidou@leeds.ac.uk
Abstract—Modern distribution system operators are facing
constantly changing operating conditions caused by the increased
penetration of intermittent renewable generators and other
distributed energy resources. Under these conditions, the distri-
bution system operators are required to operate their networks
with increased uncertainty, while ensuring optimal, cost-effective,
and secure operation. This paper proposes a centralized scheme
for the operational planning of active distribution networks under
uncertainty. A multi-period optimal power flow algorithm is used
to compute optimal set-points of the controllable distributed
energy resources located in the system and ensure its security.
Computational tractability of the algorithm and feasibility of the
resulting flows are ensured with the use of an iterative power flow
method. The system uncertainty, caused by forecasting errors
of renewables, is handled through the incorporation of chance
constraints, which limit the probability of insecure operation.
The resulting operational planning scheme is tested on a low-
voltage distribution network model using real forecasting data
for the renewable energy sources. We observe that the proposed
method prevents insecure operation through efficient use of
system controls.
Index Terms—active distribution network, chance constrained
multi-period optimal power flow, backward forward sweep power
flow, distributed energy resources
I. INTRODUCTION
The increasing penetration of Distributed Energy Resources
(DERs) in medium and Low Voltage (LV) distribution net-
works is posing new challenges to system operation, while
at the same time creating exciting new opportunities. On
the one hand, large shares of renewables, such as wind
and photovoltaic generation (PVs), lead to increased power
flow variability due to fluctuations of the injections. This is
forcing Distribution Network Operators (DNOs) to operate
under increased uncertainty and at higher risks. On the other
hand, controllable DERs, such as Battery Energy Storage
Systems (BESS), flexible loads, and dispatchable inverter-
based Distributed Generators (DGs), provide new sources
of flexibility. In active distribution networks, the DERs are
requested by the DNO to provide ancillary services, making
the distribution network operation more efficient and providing
the tools to handle increased uncertainty and increased load.
Ancillary services provided by DERs include among oth-
ers reactive power control [1], active power curtailment [2],
network upgrade deferral by means of BESS [3], and load
shifting [4]. To tackle voltage or congestion problems in dis-
tribution networks, reactive power control is usually preferred.
By adjusting the reactive power injection of DERs, DNOs can
perform voltage control or decrease the flow over congested
lines. In certain cases, for example when DERs are obliged to
operate at unity power factor or the system has a high R/X
ratio, reactive power control might be less effective for voltage
regulation. In these situations, active power curtailment can be
used as an additional means of alleviating the aforementioned
problems. However, curtailing active power typically incurs a
higher compensation cost to be paid by the DNO. Finally, a
combination of reactive and active power control can be used,
taking into account both the DER characteristics [3], [5] and
the grid rules, to achieve an optimal or near optimal operation
of distribution networks.
Based on the communication infrastructure available for
controlling the DERs, operation schemes can be generally
characterized as centralized or decentralized. In decentralized
schemes, e.g. [1], [2], local control strategies are employed to
tackle power quality issues, without the use of communication
to coordinate the units. Thus, only local information is used
to modify the DER behaviour. Centralized distribution grid
optimization, used for active distribution grid operation, has
lately attracted significant attention thanks to advances in
computational power and new theoretical developments in
approximations of the nonlinear AC power flow equations. In
centralized schemes, e.g. [6], [7], a central entity makes use
of communication infrastructure to gather information from
local DERs and employ network-level optimization to provide
system-wide optimal settings for the controlled DERs. Lately,
hybrid schemes have been proposed, e.g. [8], that apply off-
line, centralized optimization to derive optimized local control
schemes that can be applied when little or no communication
infrastructure is available.
In this paper, we investigate a centralized scheme for
operating a distribution network with large shares of renew-
ables under uncertainty. The method employs optimization
techniques that exploit system-wide information about the
network and consider the impact of uncertainty to derive
optimal DER setpoints by running a multi-period, chance-
constrained Optimal Power Flow (OPF).
II. RE LATE D WO RK A ND CONTRIBUTIONS
In this section, we review related work in distribution
network optimization. We consider three different aspects: a)
the available active measures for distribution network control,
b) the choice of power flow representation and c) the methods
to account for uncertainty. We then describe the distinguishing
contributions of our work.
a) Active Measures Considered: A variety of measures
for distribution network control are considered in the literature.
Many references use only one active measure, such as active
power curtailment [2], reactive power control [9], controllable
loads [4], etc., while others use a combination of control
measures, e.g. active power curtailment and reactive power
control [7], [8], [10], [11], active power curtailment, reactive
power control and BESS [12], [13] or active power curtail-
ment, reactive power control, BESS and controllable loads [3].
In this work, we consider a combination of active power
curtailment, reactive power control, BESS and controllable
loads. In addition, we include controllable On Load Tap
Changing (OLTC) transformers that can control the voltage
magnitude of the transformer.
b) Power Flow Equations: The DC power flow approx-
imation, which is used widely in transmission grid studies, is
not suitable for distribution networks since the line resistances
are not negligible and the voltage magnitudes are typically
not close to nominal. At the same time, the use of the non-
convex non-linear AC power flows in an OPF framework, as
in [7], can easily become computationally complex. Convex
relaxations, based on e.g. semidefinite relaxations [14], find
solutions that are globally optimal for the original problem in
many practical cases. However, using a relaxation might lead
to an optimization outcome which is not a physically valid
solution to the original AC power flow equations. Moreover,
the computational complexity of the convex relaxations is still
high [10], [11]. By exploiting the radial or weakly meshed
distribution network topology, it is possible to solve the power
flow problem using the iterative backward and/or forward
sweep (BFS) power flow method [15], as shown in [3], [6].
In this paper, we apply a modified version of the BFS-OPF to
ensure computational tractability and feasibility of the multi-
period OPF problem.
c) Uncertainty modeling: The increasing penetration of
renewable, distributed, generation has increased the level of
uncertainty in distribution networks. The safe integration of
uncertain resources has hence gained a lot of attention, both in
the transmission and distribution grids. Recent papers use tools
from risk-aware portfolio optimization [11] or rely on sin-
gle [11], [13], [16]–[19] or joint [20], [21] chance constraints
to ensure that the corresponding limits will be enforced with
a pre-described probability to limit possible adverse impacts
of uncertainty. To reformulate the chance constraints into
a tractable representation, some methods assume a certain
(Gaussian) distribution of the forecast error and provide an
analytical reformulation [11], [13], [16], [18], while others
are distribution agnostic [19]–[21]. In this work, we consider
single chance constraints, and use an iterative solution scheme
[18], [22] to enforce these constraints. The iterative scheme is
based on the observation that the chance constrained problem
can be interpreted as a deterministic problem with tightened
constraints, where the optimal tightening, which strikes the
best trade-off between cost and system security, is a function
of the optimization variables. The iterative scheme alternates
between solving the deterministic problem with a given set of
tightenings, and evaluating the optimal constraint tightening
based on the solution of the deterministic problem. A feasible
solution is found when the tightenings do not change between
iterations. The benefit of the iterative approach is that the
evaluation of the constraint tightenings is done outside of
the optimization problem. This enables the use of accurate,
but computationally heavy evaluation methods, such as Monte
Carlo simulations [22]. Using a Monte Carlo simulation based
on samples of the uncertain variables allows us to obtain
accurate values for the tightenings, without making limiting
assumptions about the system equations or the distribution of
the generation from the DERs.
This paper proposes a multi-period OPF formulation for
the operational planning of active distribution grids under
uncertainty. The contributions of this paper are twofold:
1) We extend previous modelling of active distribution net-
work operations:
We include a wider range of measures for active con-
trol, considering not only active power curtailment,
reactive power control, BESS and controllable loads,
but also OLTC transformers.
We apply the BFS power flow algorithm to ensure
tractability, but make some modifications to previous
schemes to improve feasibility.
We mitigate adverse impacts of forecast error uncer-
tainty by formulating a chance constrained problem,
which is reformulated based on a Monte Carlo ap-
proach and solved using an iterative solution scheme.
2) We investigate the added benefit of the control through a
case study. We compare the behavior of different sets of
available active measures and we investigate the impact
of considering uncertainties on constraint violations and
on costs.
The remainder of the paper is organized as follows: Sec-
tion III presents the general mathematical formulation of the
deterministic multi-period AC-OPF considering the modeling
of active measures. Section IV describes the principles of the
iterative BFS power flow method and the incorporation of BFS
within the multi-period OPF, while Section Vexplains the
modeling of PV uncertainty, as well as the formulation using
chance constraints. Section VI summarizes the overall solution
algorithm, while Section VII introduces the considered case
study and the simulation results. Finally, conclusions are
drawn in Section VIII.
III. OPE RATI ONAL PLANNIN G OF DISTRIBUTION GR ID S
In this section, we detail the modeling of the deterministic
multi-period OPF problem formulation including the full, non-
linear AC power flow equations. We describe the objective, as
well as the constraints on power balance, on power quality
and on active measures.
A. Formulation of the Deterministic Multi-period Optimal
Power Flow
We consider a distribution network with a set of nodes
J:= 1,2, ..., Nb(denoted by index j) and a set of lines
I:= 1,2, ..., Nbr (denoted by index i). In order to account
for the inter-temporal constraints of DERs and the OLTC
transformer, we need to solve the following multi-period AC
OPF over the time horizon T:= 1, ..., Nhor (with each
timestep denoted by index t):
min
u
c(x,u)(1a)
s.t. f(x,u,y)=0 j, t J ,T,(1b)
hV(x,u,y)0j, t J ,T,(1c)
hI(x,u,y)0i, t ∈ I,T,(1d)
hDER(x,u,y)0j, t J ,T,(1e)
gDER(x,u,y)=0 j, t J ,T.(1f)
Here, urepresents the control vector, e.g. the DER active and
reactive power setpoints, the position of the transformer taps,
etc.; xcorresponds to the state vector, i.e. the bus voltage
magnitudes and angles (except for the slack bus, where the
angle is set to 0 degrees and the magnitude is fixed); and
ydefines the constant parameters vector, comprising of the
network topology, physical characteristics of the grid, and the
thermal and voltage constraint limits.
The DNO optimizes the control vector uover the objective
function (1a), where the function c(x,u)represents the com-
bined cost of the required control measures and the cost of
covering the losses.
Equation (1b) corresponds to the standard AC power flow
equations enforcing active and reactive power balances at
each node. It is a function of the states |V|,Θ, denoting the
voltage magnitudes and angles, and Pinj,Qinj denoting the
nodal injections of active and reactive power. Equations (1c)-
(1d) ensure that the voltage and current magnitudes remain
within acceptable limits, and (1e)-(1f) refer to DER models
and constraints.
In the following sections, we will elaborate on the objective
function and the modeling of all available active measures used
in this work, i.e. the DERs and OLTC transformer.
1) Objective function-(1a): The objective of the DNO is
to minimize its operating costs, associated with the cost of
DER control to guarantee a safe grid operation, and network
losses. In this work, the cost of DER control is determined
based on the curtailment of active energy and provision of
reactive power support by DGs. We assume that the other
active measures (such as BESS, OLTC transformer and load
control) do not incur operational cost to the DNO. In the
general case, their cost can be easily included in the objective
function. The objective function is evaluated by summing the
cost of DER control over all network nodes Nb, branches Nbr
and the entire time horizon Nhor,
min
u
Nhor
X
t=1
Nb
X
j=1
(cT
P·Pcurt,j,t +cT
Q·Qctrl,j,t) +
Nbr
X
i=1
cT
P·Ploss,i,t
| {z }
losses
t,
(2)
where tis the length of each time period. The curtailed
power of the DG at node jand time tis given by Pcurt,j,t =
Pmax
g,j,t Pf
g,j,t, where Pmax
g,j,t is the maximum available active
power and Pf
g,j,t is the actual infeed. The use of the reactive
power support Qf
g,j,t for each DG at node jand time tis
minimized by including the term Qctrl,j,t =|Qf
g,j,t|in the
objective function. The losses in each branch iat time t
are calculated by Ploss,i,t =|Ibr,i,t|2·Rbr,i, where |Ibr,i,t|is the
magnitude of the current flow and Rbr,i its resistance. Finally,
the coefficients cT
Pand cT
Qrepresent, respectively, the DG
cost of curtailing active power and providing reactive power
support. Selecting cT
QcT
Pprioritizes the use of reactive
power control over active power curtailment.
2) Power balance constraints-(1b): The power injection
equations at every node jand time step tare given by
Pf
inj,j,t =Pf
g,j,t Pf
lflex,j,t (Pch
B,j,t Pdis
B,j,t),(3a)
Qf
inj,j,t =Qf
g,j,t Pf
lflex,j,t ·tan(φload).(3b)
For each node jand time step t,Pf
inj,j,t and Qf
inj,j,t are the
net active and reactive power injections of the nodes. Pf
g,j,t
and Qf
g,j,t are the active and reactive power infeeds of the
DGs; Pf
lflex,j,t and Pf
lflex,j,t ·tan(φload)are the active and reactive
node demands (after control), with cos(φload)being the power
factor of the load; Pch
B,j,t and Pdis
B,j,t are respectively the charging
and discharging power of the BESS. The nodal power balance
equations using the full, non-linear AC power flow are given
by
Pf
inj,j,t =|Vbus,k,t|
Nb
X
m=1
|Vbus,m,t|(Gkm cosθkm,t +Bkmsinθkm,t ),
(4a)
Qf
inj,j,t =|Vbus,k,t|
Nb
X
m=1
|Vbus,m,t|(Gkm sinθkm,t Bkmcosθkm,t ).
(4b)
Here, Gkm +jBkm =Ykm form the nodal admittance matrix,
|Vbus,k,t|,|Vbus,m,t|are the voltage magnitudes at buses kand
mrespectively, and θkm,t =θk,t θm,t is the voltage angle
difference between these buses, both at time t.
3) Power quality constraints-(1c,1d): The voltage con-
straints for each bus jand the current constraints for each
line i, for each time step t, are given by
Vmin ≤ |Vbus,j,t| ≤ Vmax ,(5a)
|Vslack|= 1, θ1= 0,(5b)
|Ibr,i,t| ≤ Ii,max,(6)
where |Vslack|and |Vbus,j,t |are the voltage magnitudes at the
slack and all other buses respectively, and Vmax,Vmin the upper
and lower acceptable voltage limits; the slack bus voltage
angle, i.e. θ1is set to zero degrees; and Ii,max is its maximum
thermal limit.
Since this work focuses on distribution networks, the slack
bus is the bus on the high voltage side of the distribution
transformer, which can be equipped with OLTC capabilities.
The operation and constraints of the OLTC transformer are
modelled as
|VLV-trfo|=|Vslack Vtap ·ρt|,(7a)
24
X
t=2
(|ρtρt-1|)2,(7b)
ρmin ρtρmax,(7c)
where |Vtap|is the voltage magnitude change caused by
one tap switching action of the OLTC transformer (assumed
constant for simplicity), ρtis an integer value defining the
position of the tap, and VLV-trfo is the voltage on the low-
voltage side of the transformer. Constraint (7b) assures that
there is a maximum of two tap switching actions within a day
to avoid wear and tear on the transformer, and the parameters
(ρmin, ρmax ) in (7c) define the minimum and maximum tap
positions of the OLTC transformer.
4) Active measures constraints-(1e,1f):
a) DG limits: In this work, we consider inverter-based
DGs such as PVs. The limits are given by
Pmin
g,j,t Pf
g,j,t Pmax
g,j,t ,(8a)
tan(φmax)·Pf
g,j,t Qf
g,j,t tan(φmax)·Pf
g,j,t,(8b)
where Pmin
g,j,t and Pmax
g,j,t are the upper and lower limits for
active DG power at each node jand time t. These limits
vary depending on the type of the DG and the control
schemes implemented. Usually, small DGs have technical or
regulatory [23] limitations on the power factor they can operate
at. Here, we use the reactive power limit given in (8b), which
limits the reactive power output as a function of the maximum
power factor cos(φmax).
b) Controllable loads: We consider flexible loads with
an on/off controllable nature, i.e loads which can shift a fixed
amount of power over some time. The behavior of such loads
at each controllable node jis given by
Plflex,j,t =Pl,j,t +zj,t ·Pshift,j,1zj,t 1,(9a)
24
X
t=1
zj,t = 0,(9b)
where Plflex,j,t is the final controlled active demand at node j
and time t,Pshift,j is the constant shiftable load at node jand
zj,t ∈ {−1,0,1}is an integer variable indicating an increase
or a decrease of the load when shifted from the known initial
demand Pl,j,t. Constraint (9b) assures that the final total daily
energy demand is maintained.
c) Battery Energy Storage Systems: Finally, the con-
straints related to the BESS are given as
SoC bat
min ·Ebat
inv,j Ebat
j,t SoC bat
max ·Ebat
inv,j,(10a)
Ebat
j,1 =Estart,(10b)
Ebat
j,t =Ebat
j,t-1 + (ηbat ·Pch
B,j,t Pdis
B,j,t
ηbat
)·t, (10c)
Pch
B,j,t 0, P dis
B,j,t 0,(10d)
Pch
B,j,t ·Pdis
B,j,t ˆη. (10e)
Here, Ebat
inv,j is the installed BESS capacity at node j,SoCbat
min,
SoC bat
max are the fixed minimum and maximum per unit limits
for the battery state of charge, and Ebat
j,t is the available energy
capacity at node jand time t. The initial energy content
of the BESS in time period 1 is given by Estart, and (10c)
updates the energy capacity at each time step tbased on the
BESS efficiency ηbat, time interval tand the charging and
discharging power of the BESS Pch
B,j,t and Pdis
B,j,t. The charging
and discharging power are defined as positive according to
(10d). Equation (10e) ensures that the BESS is not charging
and discharging at the same time, by using an arbitrarily small
value ˆη= 105.
In order to avoid the bi-linearity of (10e), we replace the
constraint with
Pch
B,j,t ·(Pl,j,t Pmax
g,j,t )ˆη, (10f)
Pdis
B,j,t ·(Pl,j,t Pmax
g,j,t )ˆη. (10g)
Here, we make the assumption that when excess (deficit) of
generation is expected locally, the BESS at that node is not
allowed to discharge (charge) [24].
IV. BACK WARD /FO RWARD S WE EP P OWER FLOW ME TH OD
Distribution networks differ from transmission grids in that
they are typically radially operated and have high R/X ratio.
Furthermore, the loading at the three phases is unbalanced and
the lines/cables are not transposed. Due to these differences,
some conventional power flow methods may be inefficient for
the distribution networks. However, other solution methods,
such as the BFS power flow method considered in this work,
exploit the radial or weakly meshed distribution grid topology
to increase efficiency.
By considering the full AC power balance equations in the
OPF problem, the inter-temporal constraints of many active
measures (OLTC transformer, BESS, controllable loads, etc.),
and the integer variables of the controllable loads and the
tap position, the problem can easily solve slowly or become
computationally complex. To tackle this problem, we replace
the full power flow equations in the OPF formulation with a
single iteration of the BFS method. After the OPF solution,
we perform an exact BFS power flow computation. In this
way, we obtain a solution to the full, non-linear set of AC
power flow equations for the chosen set of controls. In the
next iteration, we again solve the OPF problem, starting from
the AC feasible solution for the previous set of controls. In
this section, we present the BFS power flow technique and its
incorporation into the OPF framework.
A. BFS power flow solution
The basic formulation of the BFS used in this work is taken
from [15] and is shown in Algorithm 1. The solution of the
power flow problem is achieved by iteratively ”sweeping” the
distribution network and updating the network variables at
each iteration. The structure of the grid is captured by two
matrices: the Bus Injection to Branch Current (BIBC), and
the Branch Current to Bus Voltage (BCBV) matrices. BIBC
is a matrix with real elements composed of ones and zeros,
capturing the topology of a given network, whereas BCBV is
a matrix with the complex impedance of the lines as elements.
One iteration of the algorithm consists of two sweeps. First,
in the backward sweep step of the kth BFS iteration, the
current injections at all buses are calculated (11a) and the
corresponding branch currents are computed using the BIBC
matrix (11b), i.e.
Ik
inj =(Pinj +jQinj )
Vk
bus ,(11a)
Ik
br =BI BC ·Ik
inj.(11b)
where Ik
inj and Ik
br are the complex current injections at all
buses and flows at all branches respectively. Then, in the
forward sweep step, the currents are used to calculate the
voltage drop over all branches using BCBV (12a), and the bus
voltages are updated for the next iteration (12b) as follows
Vk+1 =BCBV ·Ik
br,(12a)
Vk+1
bus =Vslack Vtap ·ρt+ ∆Vk+1.(12b)
In LV grids, it is the voltage magnitude differences that
dictate the flows in the lines/cables and the voltage angles are
typically small, due to the high R/X ratio. Thus, a reasonable
approximation for LV grids would be to consider only the real
part of the voltage drop. In this case, (12a) can be substituted
by
Vk+1 Re BC BV ·Ik
br.(13)
This assumption, i.e. assuming zero voltage angles and thus
considering only the real voltage drop, is reasonable for LV
networks [6], where the angles are typically below 10. The
algorithm converges when the maximum voltage magnitude
Algorithm 1 Main steps of BFS power flow based on [15]
Input: BIBC, BCBV, Pinj, Qinj,Vslack
Output: Ik
br,Vk+1
bus
1: initialize: k= 1,Vk
bus = 10
2: do
3: Backward sweep: Ik
inj =(Pinj+j Qinj)
Vk
bus
4: Ik
br =BI BC ·Ik
inj
5: Forward sweep: Vk+1 =BCBV ·Ik
br
6: Vk+1
bus =Vslack Vtap ·ρt+ ∆Vk+1
7: Update iteration: k+=1
8: while max|(|Vk
bus|−|Vk1
bus |)| ≥ ¯η
9: return Ik
br,Vk
bus
difference between two subsequent iterations is smaller than
a predetermined threshold ¯η.
B. BFS-OPF implementation
To incorporate the BFS-based power balance equations
into our OPF framework, (4a)-(4b) are replaced by a single
sweeping iteration of the BFS method. Thus, using (11), the
constraint for the current magnitude for all branches i, at each
iteration kand time step t, is given by
|Ik
br,i,t| ≤ Ii,max.(14a)
Furthermore, considering also the OLTC capabilities (7a),
the equations for the voltage magnitudes at all nodes are
now given by (12b). According to (13), we can consider only
the real part of the voltage drop, approximating the voltages
with Vk
bus,j,t Vslack Vtap ·ρt+Re Vk+1, where all
elements have zero imaginary part. Therefore, considering the
magnitude |Vk
bus,j,t| ≈ VslackVtap ·ρt+Re Vk+1, we end
up with a linear voltage constraint inside the OPF formulation,
given by
Vmin Vslack Vtap ·ρt+Re Vk+1Vmax .(15)
By considering this approximation, we avoid the use of the
non-convex exact AC power flow, and we solve the OPF with
one BFS iteration. After the optimal setpoints of the OPF
problem are obtained, the exact BFS power flow is performed
to update the operating point and project it into the feasible
domain of the exact power flow equations. This new operating
point will be used as input to the subsequent iteration of
the BFS-OPF problem, and this loop will be repeated until
convergence in terms of voltage magnitude mismatch. This
procedure is sketched in Fig. 5, labeled as multi-period BFS-
OPF.
This approach is in contrast to [6] where only one BFS
iteration is performed after the OPF solution. The formulation
used in this work fully computes the voltages and currents
after each OPF solution and can improve (or facilitate) the
convergence of the BFS-OPF, especially when the current
operating point is far from the optimum or close to the stability
limits and a single BFS iteration would not give a good
approximate of the values.
V. ACCOUNTING FOR UNCERTAIN TY T HRO UG H CHANCE
CONSTRAINTS
With the above formulation, we have a tractable optimiza-
tion problem for the multi-period OPF. What remains is to
account for the effect of uncertainty, and formulate a problem
that limits possible adverse effects.
Uncertainty is becoming an important issue for distribution
grids due to increasing installations with variable output, e.g.
wind farms in medium voltage and PV panels in LV grids,
respectively. This work focuses on LV networks and thus,
assumes that the power injection from the PV units is the
only uncertainty source. However, load uncertainty could be
modelled and included in the optimization problem in a similar
way, without requiring any extensions to the method.
A. Formulation of the Chance Constraints
In order to mitigate the effect of uncertainty on system
operation, is is necessary to account for uncertainty within
the optimization framework. In our model, the branch current
flows and the voltage magnitudes are functions of the power
injections and are hence influenced by PV power uncertainty.
To limit the risk of constraint violations, we model the corre-
sponding voltage and current constraints as chance constraints.
Chance constraints are probabilistic constraints which ensure
that the limits will hold with a pre-described probability 1 - ε,
where εis the acceptable violation probability. With this defi-
nition, the voltage constraints (5a) and current constraints (6)
become
P{|Vbus,j,t| ≤ Vmax } ≥ 1ε, (16)
P{|Vbus,j,t| ≥ Vmin } ≥ 1ε, (17)
P{|Ibr,i,t| ≤ Ii,max} ≥ 1ε. (18)
The probabilistic constraints (16) - (18) are not tractable in
their current form, and require reformulation into deterministic
constraints. This can be done by interpreting them as tightened
versions of the original constraints [16], [22], where the
tightening represents a security margin against uncertainty, i.e.,
an uncertainty margin. Using this observation, it is possible
to express (16), (18) as
Vmin + Ωlower
V j,t ≤ |Vk
bus,j,t| ≤ Vmax upper
V j,t ,(19)
|Ik
br,i,t| ≤ Ii,max Ibr,i ,(20)
where lower
V,upper
Vare the tightenings for the lower and upper
voltage magnitude constraints and Ibr are the tightenings of
the current magnitude constraints. Figures 1and 2schemati-
cally show the uncertainty margins on the voltage and current
flow constraints due to the introduction of the uncertainty sets.
B. Iterative Solution Algorithm
Since the effect of the uncertainty is captured in the uncer-
tainty margins and does not occur elsewhere in the problem,
it is possible to solve the problem using an iterative algorithm
Feasible area for |Vk
bus|
Uncertainty margins
Vmin Vmin +Ωlower
VVmax
Vmax upper
V
|V|
lower
V
lower
Vupper
V
upper
V
Fig. 1. Voltage constraint tightening due to the uncertainty margins.
Feasible area for |Ibr,i|
Uncertainty margin
Ii,max
Ii,max I
|Ibr|
0
I
I
Fig. 2. Maximum current tightening due to the uncertainty margin.
[18], [22]. The iterative algorithm for AC chance-constrained
OPF alternates between solving a deterministic OPF with
tightened constraints, and calculating the uncertainty margins
lower
V,upper
V,Ibr based on the obtained solution. If the
maximum changes in the tightenings between two subsequent
iterations are below certain thresholds η
V, η
I, the algorithm
has converged and a feasible solution has been found. This
procedure is sketched in Fig. 5in the green box labeled
Uncertainty tightenings.
C. Evaluation based on Monte Carlo Simulations
The iterative algorithm, which evaluates the uncertainty
margins in an outer iteration and not within the OPF problem
itself, enables several ways of obtaining the uncertainty mar-
gins lower
V,upper
V,Ibr . One approach is to use the analytical
reformulation approach based on linear sensitivity factors and
the assumption of a Gaussian distribution [17], or related
analytical methods that are distribution agnostic [19], [25].
Another approach, which is possible due to the iterative nature
of the solution, is to use a Monte Carlo simulation [22]. This
method has the benefit that it accounts for the full non-linearity
of the AC power flow equations and requires no restrictive (or
conservative) assumptions about the uncertainty distribution.
If a sufficient number of representative samples is available,
we are able to get a very accurate estimate of the necessary
uncertainty margins.
The calculation procedure for the Monte Carlo based tight-
enings is summarized in Fig. 3and Fig. 4. Given the optimal
set-points from the BFS-OPF, the Monte Carlo simulations
are used to calculate an empirical distribution function for
the voltages and currents at each time step. Since we con-
sider separate chance constraints, each constraint has its own
empirical distribution as depicted schematically for a voltage
constraint in Fig. 4. Enforcing a chance constraint with 1
probability is equivalent to ensuring that the 1quantile of
the distribution remains within bounds. Hence, the tightening
corresponds to the difference between the forecasted voltage
magnitude at zero forecast error |Vk,0
bus,j,t|and the 1quantile
Vk,1-%
bus,j,t .
The upper 1quantile Vk,1-%
bus,j,t and the lower quantile
Vk,%
bus,j,t of the voltage magnitudes, as well as the 1quantile
of the current magnitudes are evaluated based on the empirical
distribution. The empirical uncertainty margins are then given
by
upper
V j,t =|Vk,1-
bus,j,t|−|Vk,0
bus,j,t|,(21a)
lower
V j,t =|Vk,0
bus,j,t|−|Vk,
bus,j,t|,(21b)
upper
Ibr,i =|Ik,1-
br,i,t |−|Ik,0
br,i,t|,(21c)
where superscript 0indicates the current or voltage magnitude
at the operating point with zero forecast error.
VI. SOLUTION ALGORITHM
In this section, we summarize the proposed method for
optimal operational planning of active distribution networks,
sketched in Fig. 5.
Deterministic
forecast
Multi - period
BFS-OPF |V0
bus,j|,|I0
br,i|
Monte Carlo
sampling
Single - period
BFS-PF
|V1-
bus,j|,|V
bus,j|,
|I1-
br,i|
Empirical
tightenings
upper
Vj,t ,
lower
Vj,t ,
Ibr,i
Optimal set-points
Fig. 3. Procedure to derive the empirical uncertainties sets.
0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12
0
50
100
150
lower
Vupper
V
= 5%
probability
= 5%
probability
Voltage magnitude (p.u.)
Number of occurrences (-)
Empirical distribution
and (1 )quantiles
No forecast error
Fig. 4. Example of an empirical distribution for a voltage constraint, with
upper and lower tightenings.
At the core of the proposed methodology lies the formula-
tion of the multi-period centralized OPF for active distribution
networks described in Section IV. The uncertainty is treated
in the outer loop as described in Section V. The initialization
stage sets the uncertainty margins to zero and initializes the
voltage levels for the multi-period OPF to a flat voltage profile.
Then, the BFS-OPF calculates the optimal setpoints of the
available active measures based on the first step of the BFS
algorithm. The BFS power flow algorithm then runs until
convergence for the solution point, and we check whether the
converged point is similar to the voltage profile assumed by
the OPF. After the multi-period BFS-OPF has converged, the
uncertainty margins are evaluated based on the MC approach.
The iteration index of the BFS-OPF loop is denoted by k
and the iteration of the uncertainty loop by m. The iterative
procedure continues until all parts of the algorithm have
reached convergence.
VII. CAS E STU DY - RE SU LTS
A. Network description - Case study setup
In order to demonstrate the proposed method, we use the
data from the benchmark radial LV grid presented in [26]
and shown in Fig. 6. The installed PV capacity is expressed
as a percentage of the total maximum load as follows: PV
nodes = [12,16,18,19], PV share (%) = [45,40,30,45].
The dimensions of the remaining units used were selected
following the planning approach of [3]. More specifically,
we consider at node 16 a BESS of 26 kWh, and a flexible
load of 5kW, whose total daily energy consumption needs
to be maintained constant. In this work, we only consider
balanced, single-phase system operation, but the framework
can be extended to three-phase unbalanced networks.
The operational costs are assumed to be cP= 0.3CHF
kWh and
cQ= 0.01 ·cP. The BESS cost is considered in the planning
stage [3] and thus, the use of the BESS does not incur any
operational cost to the DNO.
Regarding the uncertainty modeling, we use historical fore-
cast error distributions and we enforce the chance constraints
with an ε= 5% violation probability. We assume a maximum
acceptable voltage of 1.04 p.u and cable current magnitude of
1p.u. (on the cable base). The minimum acceptable voltage
is set to 0.92 p.u..
Using this system, we investigate the effectiveness of dif-
ferent active control measures and the impact of uncertainty
on the network constraint violations and computational bur-
den. To achieve this, we consider different scenarios with
incrementally more active control measures provided to the
DNO. That is, the first scenario includes only active power
curtailment, and the last, all of the control measures described
Initialize:
k= 0,Vk
bus = 10
m= 1,m1
ibr = Ωm1
V i = 0
Run multi-period OPF
with one BFS iteration
Run BFS power flow until
convergence (Algorithm 1)
max|(|Vk
bus|−|VPF
bus |)| ≤ ˜η
Evaluate m
V i,m
Ibr
and check tightenings
max|m
V i m1
V i | ≤ η
V
&
max|m
ibr m1
ibr | ≤ η
I
Stop
Yes
No
Yes
No
Multi - period BFS-OPF Uncertainty tightenings
VP F
bus
Vk
bus
m
V i
m
ibr
Fig. 5. Proposed solution algorithm.
Energy Storage Applications
MV ring
1 2 3 4 5 6 7 8 9 10 11
12 17 19
13
14 15 16 18
LV
HV
Fig. 6. Cigre LV grid.
in Section III, i.e. active power curtailment, reactive power
control, control of BESS, OLTC transformer, and flexible
loads. Finally, for each of the above scenarios, we consider
a deterministic operational planning approach, the impact
of forecasting error, and the benefit of explicitly treating
uncertainty through the chance constraints.
B. Forecast error distributions
Reference [27] provides forecasts every 3 hours for 10 PV
stations in Switzerland. The goal of this section is to capture
forecast error distributions which will serve as inputs to our
approach.
Figure 7shows the histograms of the forecast error distri-
butions of the daily production hours 06:00-20:00 for 9,6and
3-hour ahead forecasts for different seasons of the year 2013.
As expected, the 3-hour ahead forecasts are more accurate
in all cases due to the shorter lead time. These distributions
will be used in combination with a new PV injection forecast,
to account for uncertainty. The evaluation of the uncertainty
margins is carried out based on these samples.
For our case study, we use 1000 samples from the 9-hour
ahead forecast error distribution of the summer power profiles
as seen in Fig. 7c. We assume a perfect spatial correlation,
implying that all PVs follow the same distribution.
C. Optimization results
In this part, we present the results for the operational
planning optimization of active distribution networks, for a
summer day with high PV infeed. Initially, we show the results
without any measures, by running power flow calculations with
the forecasted PV injections. Then, we apply the proposed
methodology, and we investigate:
the operational cost and total active power curtailment for
different sets of active measures
the impact of considering uncertainty in the operational
cost and violation probabilities
the computational time and convergence characteristics
of the proposed method
1) Base Case - No control measures: In absence of any
control action and assuming perfect forecasts, the grid will
experience overvoltage and thermal congestion issues, as can
be observed by the power flow calculations. Figure 8shows the
0.50 0.5
0
200
400
600
Occurences (-)
Winter
power profiles
(a)
0.50 0.5
Spring / Autumn
power profiles
9 hours - ahead
(b)
0.50 0.5
Summer
power profiles
(c)
0.50 0.5
0
200
400
600
Occurences (-)
(d)
0.50 0.5
6 hours - ahead
(e)
0.50 0.5
(f)
0.50 0.5
0
200
400
600
800
Occurences (-)
(g)
0.50 0.5
3 hours - ahead
(h)
0.50 0.5
(i)
Fig. 7. Histograms of PV forecast errors in p.u. of the installed capacity.
daily voltage magnitude distribution at all nodes. Considering
a maximum acceptable voltage magnitude of 1.04 p.u., several
nodes face overvoltage issues at noon hours with high solar
radiation. Similarly, the boxplots of the thermal loading for
all cables shown in Fig. 9, indicate that the cables 23and
34will be overloaded. Thus, active control measures are
needed to bring the voltages and the currents under acceptable
thresholds, leading to a safe grid operation.
2) Optimal scheduling of active distribution grids with
different available active measures: The operational flexibility
of active measures is used in order to relieve the grid from the
aforementioned violations. This section aims at investigating
the use of different sets of active measures. More specifically,
we quantify the needed control activation and the change of
the operational cost as more active measures become available
to the DNO.
Figure 10 summarizes the objective function value with and
without the consideration of uncertainty. As can be observed,
in both cases, the expected cost decreases with more available
measures. This occurs due to the decrease of total active power
curtailment when other control measures with less operational
costs are used, as shown in Table I. At the same time, it can
2 4 6 8 10 12 14 16 18 20 22 24
2
4
6
8
10
12
14
16
18
Time (h)
Node (-)
0.92
0.94
0.96
0.98
1
1.02
1.04
1.06
1.08
Fig. 8. Daily voltage magnitude distribution at all nodes without any control.
1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 4-12 5-13 13-14 14-15 15-16 7-17 10-18 11-19
0
50
100
Cable (-)
Loading (%)
Fig. 9. Boxplots for the daily thermal loading of the cables. Each boxplot
shows the minimum, the first quartile, the median, the third quartile, and the
maximum value over the time considered.
be seen that the introduction of uncertainty leads to higher
curtailment necessary to satisfy the network constraints. In
the deterministic case, the total curtailment using only active
power curtailment is more than 8times higher compared to the
case with all measures available. A similar pattern is observed
in the chance-constrained case, with the use of reactive power
control appearing to be the most efficient measure.
3) Impact of considering uncertainty: To highlight the
importance of considering the PV injection uncertainty, we
provide insights into the distribution of voltage and current
magnitudes at specific nodes and cables. We consider the
TABLE I
TOTAL CU RTAIL ME NT NE ED ED FO R DI FFER EN T SET S OF AVAILA BL E
ACTIVE MEASURES
Total Active Power Curtailment (%p.u.)
UncertaintyAvailable Measures
Without With
Active Power Curtailment 10.18 24.83
+ Reactive Power Control 5.28 17.90
+ BESS 4.68 17.27
+ OLTC transformer 2.17 16.48
+ Controllable Load 1.22 15.14
APC + RPC + BESS + OLTC + CL
0
1
2
3
Increasing control availability
Available measures (-)
Operational Cost (kCHF)
Deterministic
Chance-constrained
Fig. 10. Operational costs for different sets of active measures.
APC = Active Power Curtailment, RPC = Reactive Power Control,
BESS = Battery Energy Storage System, OLTC = On Load Tap Changing
transformer, CL = Controllable Load
TABLE II
MAX IMU M PROBA BI LIT Y OF VO LTAGE AN D THE RM AL LO ADI NG
VIOLATION.
max(P{|Vbus| ≤ Vmax})(%) max(P{|Ibr | ≤ Ii,max})(%)
Uncertainty UncertaintyAvailable Measures
Without With Without With
Active Power Curtailment 51 5 20 5
+ Reactive Power Control 35.6 4 51.7 5
+ BESS 35.6 4 51.7 5
+ OLTC transformer 48.9 4.8 52.8 5
+ Controllable Load 46.2 5 60.2 5
case where active power curtailment, reactive power control
and BESS are the available active measures, and enforce the
chance constraints with violation probability = 5%. We
evaluate the behavior of the grid using 1000 samples from
the forecast error distribution of Fig. 7c. The same samples
are used for both optimization and evaluation to demonstrate
that the method achieves the prescribed violation probability.
Out-of-sample testing has been performed in earlier work [22].
Table II shows the maximum probability of overvoltage
and thermal violation among all nodes and cables. Based on
the examined 9-hours ahead forecast error distribution, the
distribution network which does not consider uncertainties will
face overvoltage and overload issues with a high probability
(higher than 35% in most cases). The chance constraints limit
this probability to 5% as prescribed by our choice of .
Figures 11 and 12 show the histograms of the voltage and
current magnitudes at node 16 and cable 23, respectively.
Note that the consideration of the PV uncertainty through
the chance constraints shifts the distribution to lower values,
reducing both the probability of overvoltage and thermal
overload to below the prescribed value.
4) Computational time and convergence characteristics of
the proposed method: In this part, we elaborate on the
convergence features and computational tractability of the
iterative BFS-OPF. The proposed methodology is implemented
in MATLAB using YALMIP [28] as the modeling layer, and
0.96 0.98 1 1.02 1.04 1.06 1.08
0
50
100
Overvoltage
Voltage magnitude (p.u.)
Number of occurrences (-)
Deterministic
Chance-constrained
Fig. 11. Histograms of the voltage magnitudes at 14:00 at Node 16, evaluated
based on Monte Carlo samples for both the deterministic (blue) and chance-
constrained (green) solutions.
0.2 0.4 0.6 0.811.2 1.4 1.6
0
50
100
Thermal Overload
Current magnitude (p.u.)
Number of occurrences (-)
Deterministic
Chance-constrained
Fig. 12. Histogram of the current magnitudes at 14:00 at Cable 23,
evaluated based on Monte Carlo samples for both the deterministic (blue)
and chance-constrained (green) solutions.
solved with Gurobi [29], using an Intel Core i7-2600 CPU and
16 GB of RAM.
Figure 13 shows the computational time needed for different
available active measures. We observe a fairly constant burden
for the deterministic case, where for all cases, almost the same
number of iterations is needed until convergence, leading to
similar computation times (see flowchart of the BFS-OPF). In
the chance-constrained case, the more the available measures,
the longer the time needed for the multi-period BFS-OPF. This
can be explained by the consideration of more variables, in
particular integer variables for the OLTC transformer and the
controllable load. More iterations are needed in the chance-
constrained case due to the second iteration loop which
updates the constraints tightenings.
Finally, Figs. 14 and 15 provide insights in terms of
the convergence characteristics of the proposed methodology.
Without considering uncertainty, only the multi-period BFS-
OPF loop is applied and, as can be observed in Fig. 14
for m= 1 (only this part is relevant for the deterministic
case), convergence is reached after 4 iterations. The maximum
voltage magnitude deviation of the first iteration for k= 1 is
APC + RPC + BESS + OLTC + CL
0
200
400
600
+ integer variables
Increasing control availability
Available measures (-)
Computational Time (sec)
Deterministic
Chance-constrained
Fig. 13. Computational time needed for different sets of active measures.
APC = Active Power Curtailment, RPC = Reactive Power Control,
BESS = Battery Energy Storage System, OLTC = On Load Tap Changing
transformer, CL = Controllable Load
large due to the initial flat voltage profile. From the second
iteration on, i.e. from k= 2, the maximum voltage magnitude
deviation is rather small, and the BFS-OPF converges after 4
iterations, when the maximum deviation is below the threshold
of ˜η= 104.
In the chance-constrained case, after the voltage magnitudes
have converged, the voltage and current uncertainty tight-
enings are checked for convergence. Once these tightenings
are updated, the BFS-OPF iteration loop is run again, which
increases the computational time compared to the deterministic
case. Figure 14 shows the maximum voltage magnitude devia-
tion for all iterations of the BFS-OPF loop. Note that the itera-
tions until the first BFS-OPF convergence, i.e. k∈ {1,2,3,4}
and m= 1 are identical with the deterministic case, because of
the same voltage initialization. Subsequently, the voltage and
current uncertainty tightenings are updated, since they deviate
more than the predefined thresholds of η
V=η
I= 103, as
can be observed in Fig. 15. The second run of the BFS-OPF
loop for m= 2 converges faster, namely after 3 iterations,
due to a better initial voltage profile. In total 4 iterations of
the uncertainty tightening loop are needed in order to reach
convergence of both loops.
VIII. CONCLUSION
In this paper, we propose a chance constrained multi-
period optimal power flow algorithm, which can be used
in the operational planning procedure of active distribution
networks under uncertainty. The tractability is achieved by
using an iterative power flow approach and the consideration
of uncertainty is based on uncertainty margins calculated by
Monte Carlo simulations.
In the case study considered, we provide results concerning
the utilization of various sets of active measures, and the
importance of accounting for uncertainty in terms of constraint
violation and cost increase. The more the available measures,
the lower the operational costs, with reactive power control
0 2 4 6 8 10 12 14
0
1
2
3
4
·102
m= 1 m= 2 m= 3 m= 4
Iteration k(-)
Maximum Deviation (p.u.)
max|(|Vk
bus|−|VP F
bus |)|
Fig. 14. Voltage magnitude convergence of the BFS-OPF loop. The determin-
istic case corresponds to the first iteration m= 1, plotted in blue. The chance-
constrained optimization requires several outer iterations m= 1,2,3,4to
achieve convergence of the tightenings.
1 2 3 4
0
0.05
0.1
Iteration m(-)
Maximum Deviation (p.u.)
max|m
V i m1
V i |
max|m
ibr m1
ibr |
Fig. 15. Convergence of the voltage and current tightenings in the chance-
constrained optimization. Maximum change in the voltage tightenings (plotted
in orange) and current tightenings (in blue) between subsequent outer itera-
tions m.
being the most efficient measure. The consideration of uncer-
tainty more than doubles the operational cost, but limits the
constraint violation probability to less than a predetermined
value. Finally, we elaborate on the algorithmic performance
of the proposed iterative method. Future work will focus on
evaluating the convergence and optimality of the obtained
solutions compared to full AC OPF and common convex
relaxation techniques.
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... As a result, there is a range of stochastic OPF models that have been developed specifically for distribution grids. This includes robust and distributionally robust methods [15]- [17], stochastic approximation techniques [18], and chanceconstrained formulations [19]. These models focus primarily on managing or reducing voltage magnitude violations [15], [17] while minimizing objectives such as cost of energy [16], [17], [19], deviation from a desired power withdrawal at the substation [17], or losses [15], [18], [19]. ...
... This includes robust and distributionally robust methods [15]- [17], stochastic approximation techniques [18], and chanceconstrained formulations [19]. These models focus primarily on managing or reducing voltage magnitude violations [15], [17] while minimizing objectives such as cost of energy [16], [17], [19], deviation from a desired power withdrawal at the substation [17], or losses [15], [18], [19]. They often leverage power flow formulations that rely on a radial network topology [20]- [24], include approximate representations of unbalance [25]- [28], and sometimes take advantage of iterative solution algorithms such as forward-backward sweep [6], [29], [30]. ...
... This includes robust and distributionally robust methods [15]- [17], stochastic approximation techniques [18], and chanceconstrained formulations [19]. These models focus primarily on managing or reducing voltage magnitude violations [15], [17] while minimizing objectives such as cost of energy [16], [17], [19], deviation from a desired power withdrawal at the substation [17], or losses [15], [18], [19]. They often leverage power flow formulations that rely on a radial network topology [20]- [24], include approximate representations of unbalance [25]- [28], and sometimes take advantage of iterative solution algorithms such as forward-backward sweep [6], [29], [30]. ...
Preprint
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The growing penetration of distributed energy resources (DERs) is leading to continually changing operating conditions, which need to be managed efficiently by distribution grid operators. The intermittent nature of DERs such as solar photovoltaic (PV) systems as well as load forecasting errors not only increase uncertainty in the grid, but also pose significant power quality challenges such as voltage unbalance and voltage magnitude violations. This paper leverages a chance-constrained optimization approach to reduce the impact of uncertainty on distribution grid operation. We first present the chance-constrained optimal power flow (CC-OPF) problem for distribution grids and discuss a reformulation based on constraint tightening that does not require any approximations or relaxations of the three-phase AC power flow equations. We then propose two iterative solution algorithms capable of efficiently solving the reformulation. In the case studies, the performance of both algorithms is analyzed by running simulations on the IEEE 13-bus test feeder using real PV and load measurement data. The simulation results indicate that both methods are able to enforce the chance constraints in in- and out-of-sample evaluations.
... However, in this paper, we focus on demonstrating the ability of DERs, which are expected to play a key role in active distribution grids, to assist the DSOs not just in active power balancing but also in voltage control. Tractability is achieved by formulating the controller as a Backward/Forward Sweep (BFS) OPF, extending our previous works [20]- [22]. In contrast to previous work, we investigate the technical potential of the inverters under various existing grid codes and standards, and we highlight the need of harmonization and modernization of new guidelines. ...
... Since the OPF will be used to process several scenarios in a multi-period framework, it is necessary to use some approximations to increase its computational performance. For this reason, the iterative BFS power flow [27] method is used in this work, extending the formulation presented by the authors in [20]- [22], [28] for a single and three-phase systems. Following our previous work, a single iteration of the BFS power-flow method is used to replace the AC power-flow constraints in the OPF formulation. ...
... The parameters (ρ min , ρ max ) are respectively the minimum and maximum tap positions of the OLTC transformer. This convex formulation provides a good approximation of the nonlinear AC OPF [29], is computationally tractable even in a three-phase model [21], [22], and results in AC feasible solutions which can account for uncertainties, see [20]. More specifically, [29] which is based on a purely linear formulation of the BFS-OPF reports a 2% difference in terms of the objective value compared to solving the exact AC OPF using interior-point methods. ...
Preprint
The increasing installation of controllable Distributed Energy Resources (DERs) in Distribution Networks (DNs) opens up new opportunities for the provision of ancillary services to the Transmission Network (TN) level. As the penetration of smart meter devices and communication infrastructure in DNs increases, they become more observable and controllable with centralized optimization-based control schemes becoming efficient and practical. In this paper, we propose a centralized tractable Optimal Power Flow (OPF)-based control scheme that optimizes the real-time operation of active DNs, while also considering the provision of voltage support as an ancillary service to the TN. We embed in the form of various constraints the current voltage support requirements of Switzerland and investigate the potential benefit of ancillary services provision assuming different operational modes of the DER inverters. We demonstrate the performance of the proposed scheme using a combined HV-MV-LV test system.
... Indeed, as more and more stochastic resources are being integrated in the distribution grid, the role of uncertainty is becoming increasingly significant. There are several works that model AC OPF in a stochastic setting, e.g., through chanceconstraints in centralized control [29] and in data-driven approaches [30] that emphasize on the operational aspect. A chance-constrained AC OPF formulation is also employed in [31], which accounts for small-scale generators as controllable DERs, while treating all behind-the-meter DER as uncontrollable, and includes pricing considerations with chanceconstrained generation and voltage limits. ...
Preprint
Full-text available
In this paper, we consider the day-ahead operational planning problem of a radial distribution network hosting Distributed Energy Resources (DERs) including rooftop solar and storage-like loads, such as electric vehicles. We present a novel decomposition method that is based on a centralized AC Optimal Power Flow (AC OPF) problem interacting iteratively with self-dispatching DER problems adapting to real and reactive power Distribution Locational Marginal Costs. We illustrate the applicability and tractability of the proposed method on an actual distribution feeder, while modeling the full complexity of spatiotemporal DER capabilities and preferences, and accounting for instances of non-exact AC OPF convex relaxations. We show that the proposed method achieves optimal Grid-DER coordination, by successively improving feasible AC OPF solutions, and discovers spatiotemporally varying marginal costs in distribution networks that are key to optimal DER scheduling by modeling losses, ampacity and voltage congestion, and, most importantly, dynamic asset degradation.
... whereV * j,t is the voltage magnitude at node j at time t, * indicates the complex conjugate and the hat indicates that the value from the previous iteration is used (the interested reader is referred to [12,13] for more details in terms of the use of BFS in an OPF framework); I inj t = [I inj j,t , ∀ j] and I br t = [I br i,t , ∀ i] represent the vectors of bus injection and branch flow currents, respectively (I br i,t is the i-th branch current); BIBC (Bus Injection to Branch Current) is a matrix with ones and zeros, capturing the radial topology of the DN; the entries in ∆V t correspond to the voltage drops over all branches; BCBV (Branch Current to Bus Voltage) is a matrix with the complex impedances of the lines as elements; V slack is the voltage in per unit at the slack bus (here assumed to be 1 < 0 • ). Thus, the constraint for the current magnitude for all branches i at time t is given by ...
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Lately, data-driven algorithms have been proposed to design local controls for Distributed Generators (DGs) that can emulate the optimal behaviour without any need for communication or centralised control. The design is based on historical data, advanced off-line optimization techniques and machine learning methods, and has shown great potential when the operating conditions are similar to the training data. However, safety issues arise when the real-time conditions start to drift away from the training set, leading to the need for online self-adapting algorithms and experimental verification of data-driven controllers. In this paper, we propose an online self-adapting algorithm that adjusts the DG controls to tackle local power quality issues. Furthermore, we provide experimental verification of the data-driven controllers through power Hardware-in-the-Loop experiments using an industrial inverter. The results presented for a low-voltage distribution network show that data-driven schemes can emulate the optimal behaviour and the online modification scheme can mitigate local power quality issues.
... This work considers EVs of both types. The battery model is adopted from [24] and is given by: ...
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In recent years, renewable energy resources have been increasingly embedded in the distribution grids, raising new issues such as reverse power flows, and challenging the traditional distribution system operation. In order to mitigate these issues, it has been proposed to operate the distribution system more flexibly. For instance, residential buildings are ideal candidates to offer energy flexibility locally and defer avoidable and expensive system expansions. Due to advances in smart meter technologies and trends towards digitalization, it becomes more and more common that electrical appliances in residential buildings are equipped with remote communication and control capabilities. This paper aims at quantifying a flexibility envelope of actively controlled flexible buildings and at analyzing the sensitivity of flexibility levels with respect to system configuration, control strategy and objective function settings. We consider rooftop photovoltaic units, air-sourced heat pumps for space heating and domestic hot water, thermal energy storage and electric vehicles. The results show that an optimal control aiming at minimizing energy costs while limiting peak power can lead to savings of up to 25% compared to the existing rule-based control. When carbon emissions are considered in the cost function, the optimized controller leads to an emission reduction of up to 21%. The time-dependent quantification of the flexibility envelope further reveals that high and low power levels can only be sustained for a limited period, whereas medium power levels can be sustained the longest.
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In this paper, we consider the day-ahead operational planning problem of a radial distribution network hosting Distributed Energy Resources (DERs) including rooftop solar and storage-like loads, such as electric vehicles. We present a novel decomposition method that is based on a centralized AC Optimal Power Flow (AC OPF) problem interacting iteratively with self-dispatching DER problems adapting to real and reactive power Distribution Locational Marginal Costs. We illustrate the applicability and tractability of the proposed method on an actual distribution feeder, while modeling the full complexity of spatiotemporal DER capabilities and preferences, and accounting for instances of non-exact AC OPF convex relaxations. We show that the proposed method achieves optimal Grid-DER coordination, by successively improving feasible AC OPF solutions, and discovers spatiotemporally varying marginal costs in distribution networks that are key to optimal DER scheduling by modeling losses, ampacity and voltage congestion, and, most importantly, dynamic asset degradation.
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With higher shares of fluctuating electricity generation from renewables, new operational planning methods to handle uncertainty from forecast errors and short-term fluctuations are required. In this paper, we formulate a probabilistic AC optimal power flow where the uncertainties are accounted for using chance constraints on line currents and voltage magnitudes. The chance constraints ensure that the probability of limit violations remain small, but require a tractable reformulation. To achieve this, an approximate, analytical reformulation of the chance constraints is developed based on linearization around the expected operating point and the assumption of normally distributed deviations. Further, an iterative solution approach is suggested, which allows for a straightforward adaption of the method based on any existing AC OPF implementation. We evaluate the performance of our method in a case study on the 24-bus IEEE RTS96 system. The proposed algorithm is found to converge fast and substantially reduce constraint violations.
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