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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)≤0∀j, t ∈ J ,T,(1c)

hI(x,u,y)≤0∀i, t ∈ I,T,(1d)

hDER(x,u,y)≤0∀j, 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

(|ρt−ρt-1|)≤2,(7b)

ρmin ≤ρt≤ρmax,(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,−1≤zj,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= 10−5.

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 = 1∠0◦

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|−|Vk−1

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| ≈ Vslack−∆Vtap ·ρt+Re ∆Vk+1, we end

up with a linear voltage constraint inside the OPF formulation,

given by

Vmin ≤Vslack −∆Vtap ·ρt+Re ∆Vk+1≤Vmax .(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 } ≥ 1−ε, (16)

P{|Vbus,j,t| ≥ Vmin } ≥ 1−ε, (17)

P{|Ibr,i,t| ≤ Ii,max} ≥ 1−ε. (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 −Ωupper

V j,t ,(19)

|Ik

br,i,t| ≤ Ii,max −ΩIbr,i ,(20)

where Ωlower

V,Ωupper

Vare the tightenings for the lower and upper

voltage magnitude constraints and ΩIbr 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 −Ωupper

V

|V|

Ωlower

V

Ωlower

VΩupper

V

Ωupper

V

Fig. 1. Voltage constraint tightening due to the uncertainty margins.

Feasible area for |Ibr,i|

Uncertainty margin

Ii,max

Ii,max −ΩI

|Ibr|

0

ΩI

ΩI

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

Ωlower

V,Ωupper

V,ΩIbr based on the obtained solution. If the

maximum changes in the tightenings between two subsequent

iterations are below certain thresholds ηΩ

V, ηΩ

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 Ωlower

V,Ωupper

V,ΩIbr . 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 1−quantile 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 1−quantile

Vk,1-%

bus,j,t .

The upper 1−quantile Vk,1-%

bus,j,t and the lower quantile

Vk,%

bus,j,t of the voltage magnitudes, as well as the 1−quantile

of the current magnitudes are evaluated based on the empirical

distribution. The empirical uncertainty margins are then given

by

Ωupper

V j,t =|Vk,1-

bus,j,t|−|Vk,0

bus,j,t|,(21a)

Ωlower

V j,t =|Vk,0

bus,j,t|−|Vk,

bus,j,t|,(21b)

Ωupper

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

Ωupper

Vj,t ,

Ωlower

Vj,t ,

ΩIbr,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

Ωlower

VΩupper

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 = 1∠0◦

m= 1,Ωm−1

ibr = Ωm−1

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 Ωm

V i,Ωm

Ibr

and check tightenings

max|Ωm

V i −Ωm−1

V i | ≤ ηΩ

V

&

max|Ωm

ibr −Ωm−1

ibr | ≤ ηΩ

I

Stop

Yes

No

Yes

No

Multi - period BFS-OPF Uncertainty tightenings

VP F

bus

Vk

bus

Ωm

V i

Ωm

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 2−3and

3−4will 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 2−3, 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 2−3,

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= 10−4.

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 ηΩ

V=ηΩ

I= 10−3, 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

·10−2

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|Ωm

V i −Ωm−1

V i |

max|Ωm

ibr −Ωm−1

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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