1

Chance-Constrained Day-Ahead Hourly Scheduling in Distribution System Operation
Yi Gu1 , Student Member, IEEE, Huaiguang Jiang2, Member, IEEE, Jun Jason Zhang1 , Senior Member,
IEEE, Yingchen Zhang2 , Senior Member, IEEE, Eduard Muljadi2, Fellow, IEEE, and Francisco J.
Solis3 , Senior Member
1

arXiv:1711.10687v1 [] 29 Nov 2017

Dept. of Electrical and Computer Engineering, University of Denver, Denver, CO, 80210
2
National Renewable Energy Laboratory, Golden, CO, 80401
2
School of Mathematical and Natural Sciences, Arizona State University, Glendale, AZ, 85306

Abstract—This paper aims to propose a two-step approach for
day-ahead hourly scheduling in a distribution system operation,
which contains two operation costs, the operation cost at substation level and feeder level. In the first step, the objective is to
minimize the electric power purchase from the day-ahead market
with the stochastic optimization. The historical data of dayahead hourly electric power consumption is used to provide the
forecast results with the forecasting error, which is presented by
a chance constraint and formulated into a deterministic form by
Gaussian mixture model (GMM). In the second step, the objective
is to minimize the system loss. Considering the nonconvexity
of the three-phase balanced AC optimal power flow problem
in distribution systems, the second-order cone program (SOCP)
is used to relax the problem. Then, a distributed optimization
approach is built based on the alternating direction method of
multiplier (ADMM). The results shows that the validity and
effectiveness method.

Index terms— Renewable energy integration, second-order
cone program, Gaussian mixture model, optimal power flow,
stochastic optimization, alternating direction method of multiplier, Gaussian mixture model
N OMENCLATURE
C
f1
f2
β
cDA , cRT , cP V
cs
GDA
GRT
GR
λt
GDL
t
Pij
zij
Gerr
Vi
n
ǫ

Total cost of the two steps.
Operation cost at the substation level.
System loss at feeder levels.
The system loss weight.
The unit price of the day-ahead market, realtime market and renewable generation.
The price to resell the redundant power
generated to the electric market.
The electric power obtained from the dayahead market.
The RT power consuming.
The renewable generation.
An occurrence probability helps to decide if
the systems needs to buy the electric power
at time t.
The demand load at period t.
The loss of branch lines from bus i to j.
The impedance from bus i to j.
Forecasting error model of the renewable
energy.
The voltage at bus i.
The amount of the clusters in the model,
n = 1, · · · , N .
The weight of each cluster in GMM.

x
Ui , Ci
pi , qi
Vi , Ii
si
Ω
Gf

x = [x1 , x2 , · · · , xq ]T is the q-dimensional
data vector.
The parent node and the children nodes at
node i, i ∈ NB .
Active and reactive power at bus i.
The voltage and the current.
Injection power at bus i.
Complex impedance.
The forecasted renewable generation.
I. BACKGROUND

AND

M OTIVATION

In general power system operation, the day-ahead hourly
scheduling [1]supplies the customers playing a more active
role. They can be provided more benefits, such as lower
system operation cost, advanced system reliability, and lower
volatility in hourly [2]. The high penetration of the renewable
energy [3] can lead to a lower net load power consumption
from the bulk power system, however, it brings increasingly
stochastic deviations in net load profiles [4]. Compared with
the transmission systems, the distribution systems often work
in an unbalanced polyphase state because of the asynchrony,
asymmetry and the diversity of the load, which brings bigger
challenge for the distribution system operation. In this paper,
a stochastic optimization based two-step approach for dayahead scheduling is proposed to minimize the distribution
system operation cost, which consists of the cost of net load
consumption and the system loss.
At the substation level, the high penetration of the renewable
energy can reduce the electric power [5] purchasing from the
day-ahead market for the distribution systems [6]. In [7]–[9],
the stochastic programming optimization (SPO) can provide
many potential benefits to the transmission systems. With
the renewable energies, this paper focuses on minimizing
the electric power purchasing from the day-ahead market at
the substation level in the first step. The historical data of
the day-ahead electric power purchasing is used to generate
the forecast results and the forecasting errors, which can be
formulated as a chance constraint for the SPO in distribution
systems. According to the distribution of the forecasting errors,
the chance constraint can be formulated into a deterministic
form by Gaussian Mixture Model (GMM) [10], and the global
minimum of the convex problem can be determined.
In [2],the variability of the renewable energy [11] is managed by hourly demand response in day-ahead scheduling,
without considering the stochastic net load deviation in an

system, which equals to the operation cost at feeder level. β
is used to limit the system loss weight in the optimization.The
consymption of the load cost model at the substation level is
simulated as:

hour, which dramatically impacts the operation cost of system
loss. However, the single line model is used compute the
system loss, which ignores the three-phase balanced configuration in distribution systems. In the second step, this
paper focuses on minimizing the cost of the system loss at
the feeder level with the three-phase balanced model. The
characteristic of the nonconvexity is a critical problem for the
three-phase balanced AC optimal power flow. The heuristic
methods are always applied to solve the problem, but which
are hardly avoided falling into the local minimums [12]. Based
on the alternating direction methods of multipliers (ADMM), a
distributed method is provided to solve the AC optimal power
flow problem.
This paper is organized as follows: the proposed approach
is described in Section II. In Section III, the operation cost
in substation level is analyzed. In Section IV, the operation
cost in feeder level is analyzed. In Section V, the numerical
results are presented in IEEE standard distribution systems.
The conclusion is Section VI.
II. A RCHITECTURE

S UMMARY
A PPROACH

AND

OF THE

f1 =

t=1

AT THE

where the time intervals is defined as t = 1, 2, , N T .
cDA , cRT and cP V describe the unit price in the day-ahead,
real-time market and solar generation, respectively. cs is the
established price to sell the redundant power generated by
distribution systems to the electric market, GDA and GRT
are the electric power amount purchased from the day-ahead
and real time market, GP V is the amount of the solar power
generation in the distribution systems. λt is an occurrence
probability which decides if the distribution systems needs
to buy the electric from real-time market at time t. GDL
is
t
V
the demand load at time t. cDA GDA
+ cP V GP
is the baset
t
case generation cost consists of the time-dependent power
consumption by day-ahead scheduling and the renewable
RT
presents the deviation
energy generation cost. λt · cRT
t Gt
power purchased from real-time market for compensation. And
V
− GDL
+ GP
(1 − λt ) · cs (GDA
t ) means the redundant energy
t
t
can be resold to the market in a lower price.
Several months of hourly power forecasting data in dayahead scheduling and actual net load power consumption
are used to provide the forecast results [4], [17], [18].
The available hourly forecasting power in day-ahead market
GDA (N T + 1) can be now described as following (Gerr is
the day-ahead forecasting error) [19]:

P ROPOSED

GDA (N T + 1) = GDA (N T ) + Gerr (N T ) · GDA (N T ) (3)
Subject to:

S UBSTATION L EVEL

GDA + λGRT + GP V = GDL

(4a)

cs < cDA < cRT
t

(4b)

GRT,min ≤ GRT ≤ GRT,max
GDA,min ≤ GDA ≤ GDA,max

(4c)
(4d)

GP V,min ≤ GP V ≤ GP V,max

(4e)

λt =

The total cost is calculated as (1), the sum of the operation
cost at the substation level and feeder level:
C = min(f1 + βf2 )

(2)

o

V
+ (1 − λt ) · cs (GDA
+ GP
− GDL
t
t
t )

In Fig. 1, the proposed method consists two steps, the
stochastic optimization of the electric power purchased from
the day-ahead market [13] at the substation level and the
system loss optimization for the three-phase balanced AC
optimal power flow [14], [15] at the feeder level [16].
Fig. 1 shows the first step at the left, the historical data
of the electric power purchased from the day-ahead market is
employed to generate the forecast results and the distribution
of the forecast errors. Then, combined with the day-ahead
hourly scheduling model, an objective formulation is simulated
with an chance constraint. A GMM based approach is used
to convert the chance constraint into a deterministic problem.
Finally, the day-ahead hourly optimal operation cost can be
obtained at the first step.
In the rest side of Fig. 1, a three-phase balanced AC optimal
power flow is simulated to compute the distribution system at
the feeder level. After the day-ahead hourly purchased electric
power is determined, the three-phase balanced distribution
system model is built to minimum the system loss. Considering
the nonconvexity of the AC optimal power flow, an inequality
constraint is built based on SOCP to relax the problem into a
convex problem. After that, the objective function of system
loss can be derived with ADMM. Finally, the three-phase
balanced AC optimal power flow is used to minimize the
system loss successfully.
III. O PERATION C OST

NT n
X
V
RT RT
(cDA GDA
+ cP V GP
t
t ) + (λt · ct Gt )

(

V
1 GDA
+ GP
< GDL
t
t
t
V
0 GDA
+ GP
≥ GDL
t
t
t

P r(f (Gerr ) ≤ 0) > α

(1)

(4f)

(4g)

where (4g) indicates that day-ahead forecasting error [20]
should be fulfilled with the probability α, and the chanceconstraint [21] can be converted in a deterministic formulation
with the GMM.

where C is the total cost, f1 is the purchased electric power
from the day-ahead market, which equals to the operation cost
at the substation level, f2 is the system loss of the distribution
2

Day-ahead Hourly
Scheduling Model

Forecast
Results and
Errors

Second-order Cone Programming
Based Relaxation

Gaussian
Mixed Model

Objective Function of Threephase Unbalanced AC Optimal
Power Flow

Objective Function
with Chance
Constraint

Objection Function
Derivation with ADMM and
Computation

Objective Function with
Deterministic Constraint

Final Results

Results of Day-ahead Hourly
Scheduling
Substation Level
Stochastic Optimization at Hourly Level

Convex Relaxation

Three-phase Unbalanced
Distribution System Model

Alternating Direction Method of Multipliers

System Data
Collection

Historical
Purchased
Power Data

Feeder Level
AC Optimal Power Flow at Minutes Level

Fig. 1. The flowchart of proposed approach.

2) M-Step: In M-step, the parameters are recalculated and
estimated to maximize the quantity of the expectation in (7).

A. GMM with Expectation Maximization.
Comparing with the regular GMM, a definition of expectation maximization EM based GMM is described. It is
used to model the forecasting error model of the renewable
generation. We have known that GMM is a particular form of
the finite mixture model. For (5), more than one components
is calculated as the sum with different weights (ǫ):
p(x|Θ) = ǫp(x|θ)

θ(t+1) = arg max R(θ|θt )

(7)

The EM based GMM can decide the amount of clusters
based on minimum description length (MDL) [24], [25]. The
MDL criterion is frequently used on the field of selection
in [26], because of the improved method is less sensitive to
the initialization.

(5)

ǫ in (5) is calculated in [22] and has been described to be
non-negative, that the sum equals to 1. Each component in
the GMM model is a normal distribution and obeys to θn =
(µ, Σ), which indicates the means vector and the covariance.
According to [23], the parameters of the mixture model
is computed with expectation maximization (EM), the
expectation-step (called E-step) and maximization-step (called
M-step) are described as below.
1) E-Step: The algorithm is ended when the function in (6)
reaches the convergence.

IV. O PERATION C OST

AT THE

F EEDER L EVEL

When the substation level cost is determined, the objective
function of the three-phase balanced AC optimal power flow
is to minimum the system loss. As we know, the influence
of the three-phase distribution system is small enough, the
second order cone programming (SOCP) is used to calculate
the system loss here [27]. which can be defined as follows [28],
[29]:
X
F =
Pij ,
(8)
E

R(θ|θt ) = Ex,θt [logL(θ; x)]

where Pij is a branch loss from bus i to j, and Pij = |Iij |2 rij ,
Iij is the complex current from bus i to j, and zij is the
complex impedance zij = rij + ixij . E is the branch set of
the distribution system, which can be represented with the set
of buses and branches: G = [V, E].

(6)

Where L(θ; x) = p(x|θ). x is a set of the observed data
from the given statistic model and the θ is unknown parameters
along with the likelihood function in (6).
3

TABLE I
T IME CONSUMING OF DIFFERENT ALGORITHMS BASED ON THREE

350

INDIVIDUAL SYSTEMS

300

IEEE 13-bus
1.78 s
189.43 s
7.20 s

IEEE 34-bus
5.23 s
221.34 s
13.16 s

IEEE 123-bus
14.66 s
459.21 s
25.58 s

Total Cost ($)

Method
Proposed Method
Simulated annealing
Interior-Point

Comparison with Different Alpha

Alpha=95%
Alpha=90%
Alpha=80%

250
200
150
100
50

Based on the branch flow model, the SOCP relaxation
inequalities can be represented as follows

0
0

(9)

Vi,min ≤ Vi ≤ Vi,max ,

(10a)

Iij ≤ Iij,max .

(10b)

min f (x) + g(z)

(11a)

s.t. x ∈ Kx , z ∈ Kz
Ax + Bz = c

(11b)

x,z

B. The performance of the proposed approach
It is assumed that the renewable energy resources are located
in bus 7, 23, 29, 35, 47, 49, 65, 76, 83, and 99 [30]. The result
of the three-phase balanced AC optimal power flow is shown in
Fit. 2(b). The errors of the primal residual and the dual residual
are less than 0.5 × 10−3 after 5 iterations. After 30 iterations
(less than 0.2 second), the curves of primal residual and the
dual residual are coinciding and stable, which demonstrates
the high speed and effectiveness of the convergency of the
proposed approach.
In Fig. 3, the comparison is made with different α in (4g)
during 24 hours. The results shows that the total operation cost
of the system is higher with a lower α, which indicates that
a higher accuracy of error forecasting model can help to save
more money.

V. T HE N UMERICAL R ESULTS
250
251

30

28

26

46

108

45

27

43

44

23
21
19
20

38

36

35

18 135

37

9

10
7

149

1

8
12

150

152

13

5

52

53

54
55

94

6

95

93
195

610

56

85

79
77 78

76
80

84

76
88

90

92

16

74
72

17

4

75
73

57

96

34
15

3

71
70
69

67

160

60
58

59

61
2

99

98

197
97

62

39

68

14

11

451
450

100

101

66

40

22

114

104
103

102

63

41

113

107

106

64

65

105
42

24

112

110

300

151

109

47

25 48

350
111

51
50

49

25

As shown in Fig. 2(a), the IEEE 123-bus distribution
system contains 118 basic branches, 85 unbalanced loads, 4
capacitors, and 11 three phase switches (6 initially closed
and 5 initially opened), which is the topology taken in the
preliminary results. The simulation platform is based on a
computer server with a Xeon processor and 32 GB ram. The
programming language are Python and Matlab.

Then, a dual problem is build based on the objective
function (8) and (11a) with ADMM and solved in parallel.

29

20

A. The test bench

The standard optimization problem of ADMM is defined as
follows:

31

15

Fig. 3. The total cost with different Alpha.

where Sij indicate the complex power flow Sij = Pij + iQij ,
lij := |Iij |2 , and vi := |Vi |2 . The basic physical constraints
can be defined as follows:

32

10

Time(Hours)

|Sij |2
≤ lij ,
vi

33

5

91

81
87

89

86

82

83

C. Performance comparison

(a)

As in Table I, the different algorithms are used on different
individual power systems (IEEE trans 13, -34 and -123 Bus
). The proposed approach obtains a shortest time consuming
on all of the three power systems, which demonstrates our
method can work more efficiently than others.

-3
2.5 x 10

Primal Residual
Dual Residual

2

Error

1.5

1

VI. C ONCLUSION

0.5

0

1

5

10

15

20

25

30

35

Iteration Number

(b)
Fig. 2. (a) The IEEE 123-bus distribution system. (b) The residual error
curves of the three-phase balanced AC optimal power flow.

4

In this paper, a stochastic optimization based approach
is proposed for the chance-constrained day-ahead hourly
scheduling problem in distribution system operation. The
operation cost is divided into two parts, the operation cost
at substation level and feeder level. In the operation cost at
substation level, the proposed approach minimizes the electric
power purchase from the day-ahead market with a stochastic
optimization. In the operation cost at feeder level, the system

loss is presented with the three-phase balanced AC optimal
power flow [31]. In our work, the detailed flowchart and the
description of the stochastic optimization with the forecasting
errors is improved, which helps to describe our approach
detailedly. And the detailed description of the derivation from
the chance constraint into the deterministic form with GMM
is provided. The SOCP relaxation is used to solve the problem
with three-phase balanced distribution system.
In the future, we will focus on improving an advanced
approach to develop the distribution system optimization based
on an three-phased unbalanced system [32]–[35].

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5

