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Abstract—A Bilevel Stochastic Programming Problem (BSPP)
model of the decision-making of an energy hub manager is
presented. Hub managers seek ways to maximize their profit by
selling electricity and heat. They have to make decisions about: i)
the level of involvement in forward contracts, electricity pool
markets and natural gas networks and ii) the electricity and heat
offering prices to the clients. These decisions are made under
uncertainty of pool prices, demands as well as the prices offered
by rival hub managers. On the other hand, the clients try to
minimize the total cost of energy procurement. This two-agent
relationship is presented as a BSPP in which the hub manager is
placed in the upper level and the clients in the lower one. The
bilevel scheme is converted to its equivalent single-level scheme
using the Karush–Kuhn–Tucker (KKT) optimality conditions
although there are two bilinear products related to electricity
and heat. The heat bilinear product is replaced by a heat price-
quota curve and the electricity bilinear product is linearized
using the strong duality theorem. In addition, Conditional Value
at Risk (CVaR) is used to reduce the unfavorable effects of the
uncertainties. The effectiveness of the proposed model is
evaluated in various simulations of a realistic case study.
Index Terms —Bilevel stochastic programming, energy hub,
hub manager, electricity pool, forward contract, Conditional
Value at Risk.
N
OMENCLATURE
Indices
Scenario index
f Forward contract index
t
Time index
k
Forward contract block
A. Najafi, H. Falaghi and M. Ramezani are with the Faculty of Electrical
and Computer Engineering, University of Birjand, 97175 Birjand, Iran (e-
mail: arsalan.najafi@birjand.ac.ir;falaghi@birjand.ac.ir;
mramezani@birjand.ac.ir)
J. Contreras is with E.T.S. de Ingenieros Industriales, University of
Castilla-La Mancha, 13071 Ciudad Real, Spain (e-mail:
Javier.Contreras@uclm.es)
This work was supported in part by the Ministry of Science of Spain, under
Project ENE2015-63879-R, and the Junta de Comunidades de Castilla-
La Mancha, under Project POII-2014-012-P and Grant PRE2014/8064.
j
Heat price quota curve block
c
Client index
Rival scenario index
r
Rival index
Variables
(,)
P
Ct
Pool market purchase cost
(,)
P
Pt
Purchased or sold energy in the pool
(,)
f
Pft
Purchased energy from forward contracts
()
F
Ct
Forward contract cost
(, )
CHP
e
Pt
Electric energy produced by the CHP unit
(, )
Boil
h
Pt
Heat energy produced by the boiler
(, )
gas
CHP
Pt
Gas entering the CHP unit
(, )
gas
Boil
Pt
Gas entering the boiler
(, )
CHP
h
Pt
Heat generated by the CHP unit
(, )
s
hs
cj
Offering heat price in each step
()
s
h
c
Heat price offered by the hub manager
(, )vcj
Binary variable associated with offering heat
price
(,)
f
vfk
Binary variable associated with forward
contract
(, )
CHP
vt
Binary variable associated with CHP
ON/OFF state
()
r
e
Electricity price offered by the rivals
(, )
r
h
cr
Heat price offered by rivals
()
s
e
c
Electricity price offered by the hub manager
(, ,)
s
e
Pc t
Electric energy sold by the hub manager
(,, )
s
h
Pct
Heat energy sold by the hub manager
(, )
m
xc
Supported electricity by the hub manager
(, , )
r
x
cr
Supported electricity by each rival
A Stochastic Bilevel Model for the Energy Hub
Manager Problem
Arsalan Najafi, Hamid Falaghi, Member, IEEE, Javier Contreras, Fellow, IEEE, Maryam Ramezani,
Member, IEEE
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()
m
zc
Supported heat by the manager under study
(, )
r
zcr
Supported heat in percent supported by each
rival manager
(,, , )
h
Hct j
Supported heat in each step of heat price-
quota curve
(, )
h
Scj
Percent of supporting heat demand in heat
price-quota curve
(, ,)Rc t
Revenue of selling electricity by the manager
under study
(, )
e
cr
Dual variable associated with electricity
()
h
c
Dual variable associated with heat
(, , )
r
ucr
Binary variable used to linearize the
complementary slackness electricity equation
of the rivals
(, )
r
kcr
Binary variable used to linearize the
complementary slackness heat equation of
the rivals
(, )
m
uc
Binary variable used to linearize
complementary slackness electricity equation
of the manager under study
()
m
kc
Binary variable used to linearize the
complementary slackness heat equation of
the manager under study
CVaR
Conditional value at risk
Value at risk
()
Auxiliary variable for risk
Parameters
B
oil
Conversion efficiencies from gas to heat
through the boiler
CHP
h
Conversion efficiencies from gas to heat
through the CHP unit
CHP
e
Conversion efficiencies from gas to
electricity through the CHP unit
Gas dispatch factor
ˆ
()
e
Dc
Total expected electricity demands
ˆ
()
h
Dc
Total expected heat demands
(,)
p
t
Electricity pool price
(,)
f
f
t
Forward contract price
(, ,)
h
Dc t
Heat demands
(, ,)
e
Dc t
Electrical demands
g
as
Gas price
()
Scenario probability
()
Rival scenarios probability
Confidence level
Risk coefficient
,maxf
P
Upper bound of forward contracts
,minf
P
Lower bound of forward contracts
r
h max,
Maximum heat selling offer price by the hub
manager
r
h min,
Minimum heat selling offer price by the hub
manager
(, )
s
hs
cj
Maximum price of each block in heat price
quota curve
,max
B
oil
h
P
Maximum generation of heat energy by the
boiler
12
,
M
M
Sufficiently large numbers
Sets
F
Set of candidate contracts that can be signed
Number of scenarios
T
Number of time periods
J
Number of heat price-quota curve blocks
Number of rival scenarios
NC
Number of clients
R
Number of rival hub managers
K Number of forward contract blocks
I.
INTRODUCTION
An energy hub is a new concept used in multi-carrier energy
systems. The energy hub is a simple model that can receive,
send, convert and store different types of energy. These
actions are done by various components such as a Combined
Heat and Power (CHP) unit, heat and electrical storage,
transformers, boilers and electronic devices. Linking multi-
carrier energies is the main issue of an energy hub concept [1,
2]. An energy hub aims at feeding the loads via multi-energy
inputs and outputs. Various types of energy at the input port
of the energy hub provide the decision maker with more
flexibility to satisfy the various energy loads. Hence, an
energy hub provides the possibility of profiting from a number
of prospective advantages over conventional decoupled
energy supplies, adding more flexibility in load supplying or
peak shaving [3]. In addition, energy hubs are not restricted to
any system size. This enables the integration of an arbitrary
number of energy carriers and products, allowing for high
flexibility in system modeling [4]. In the past, only electric
energy was important and retailers were intermediaries
between producers and potential clients ]5[ . Currently, hub
managers can play the same role, due to the emergence of
multi-carrier energy systems or natural gas markets. Hence,
maximizing the energy management profit is the main purpose
of hub managers acting as retailers in restructured power
systems. For a medium-term time horizon, retailers face
uncertain pool prices and client demands. On the other hand,
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clients may choose a rival retailer to purchase electricity in a
fully competitive environment. By extending the concept of
the retailers’ problem to allow for more energy carriers,
energy hub managers have a similar role except for the
different types of energy involved. Hub managers deal with
more types of energy such as electricity, heat, wood energy,
etc. They may also participate in another market, i.e., natural
gas market. Therefore, hub managers have more difficulties in
making decisions to procure and sell energy as well as how to
make price offers for different types of energy. Thus, the
medium-term decision making of a hub manager is about the
optimal involvement in electricity and other markets, as well
as the optimal selling prices to clients, in order to maximize
the expected profit for a specific risk level of profit variability.
Decisions in restructured power systems have so far been
limited to maximize the profit or to procure energy for
consumers [6]. For instance, [7] has presented a general
decision making framework for retailers and [8] has
considered a single client providing a mixed-integer nonlinear
decision-making procedure. Previously, numerous papers
focused on various types of energies either in traditional or
restructured power systems. This is known as the energy hub
concept [9,10]. Few aspects of energy hubs have been
investigated in several papers as follows. In [11], the planning
of energy hubs in a region with natural gas and electric
energies has been presented in order to determine the optimal
number and size of the required components of the hub.
Similarly, [12] has investigated the expansion planning of an
energy hub. A model has been proposed in [13] to determine
the best components to consider reliability and economic
behavior of an energy hub where the maximum loss of load
probability and adequacy indices have been studied under
single contingency conditions. Some works have studied how
to model the operational features in their research studies.
Namely, ]14[ has studied an energy hub in a smart home
considering a CHP unit and an electric vehicle. The main
objective has been to minimize consumers’ cost by controlling
the usage of energy carriers. In relation to smart homes,
similar papers can be found in ]15[ and ]16[ . The impact of
small-scale energy storage has been investigated in [17].
Reference ]18[ has developed a model to consider the
dynamic variations of the thermal loads in energy hubs using
Markov chains and Monte Carlo simulation. A goal
programming method has been proposed in [19] to optimize
the power flow between interconnected power systems.
Another formulation has been presented in [20] in order to
model an energy hub using Mixed Integer Linear
Programming (MILP). The proposed formulation has taken
into account storage losses and operational limits. In [21], a
model has been presented for the energy hub power flow. This
model has been obtained from a set of nonlinear equations
showing the hub connections. Reference [22] has developed a
framework for the placement and control of residential energy
systems using MILP considering electric energy and natural
gas carriers. Economic dispatch considering uncertainty of
wind turbines has been studied in [23]. In [24] energy
management of hub inputs has been conducted aiming to
minimize the total procured energy cost for a short-term time
horizon using MILP. In addition, several papers have
investigated other energy hub problems related to reliability
and electric vehicles [25, 26].
In this paper, a model using bilevel stochastic programming
to model an energy hub is proposed to consider both the hub
manager’s profit and the consumers’ cost. This concept has
been previously used in the retailer problem [27] and is
extended here to model an energy hub containing more energy
carriers. The proposed bilevel model takes into account the
reaction of consumers to heat and electricity selling prices.
Finally, the BSPP is converted to an equivalent single-level
stochastic programming.
The main contributions of this paper are as follows:
A bilevel stochastic programming model of an energy hub is
defined, where the maximization of the profit of the hub
manager and the minimization of the cost to the clients are
the objectives of the upper and lower levels, respectively.
A linear model is obtained to consider the bilinear terms
from selling electricity and heat.
The reaction of clients to heat and electricity selling prices in
a fully competitive market is obtained.
Risk aversion in the BSPP of the hub manager is considered
to decrease the unfavorable effects of risk in the decision
making process.
II
. BILEVEL MODELING FRAMEWORK
The decision-making problems of the hub manager and the
clients can be combined into a single bilevel stochastic
programming problem. The BSPP is used to define a decision-
making problem involving two optimization levels. In this
case, the hub manager is at the upper level and the clients are
at the lower one. The modeled hub manager tries to maximize
their profit by selling heat and electric energy to the clients
whilst the clients try to minimize their costs by procuring
electric and heat energy from the hub manager and also from
its rivals. The complexity of the decision making at the upper
level is due to uncertainty in pool market prices and clients’
demands. The hub manager procures the energy from two
input carriers: electricity at uncertain prices and natural gas at
a fixed price. These carriers have to offer heat and electricity
prices to the clients to maximize their profit. Lower prices
result in lower profits and higher prices result in a lesser
willingness of the clients to deal with the manager and a
greater willingness of the clients to deal with rival hub
managers.
Fig. 1 depicts the upper and lower levels and the ways of
procuring energy. The hub under study has a CHP unit and a
boiler, which are self-production units fed by natural gas. The
hub manager procures electricity in three ways: electricity
pool market, forward contracts and a CHP unit. Heat is also
obtained from a boiler and a CHP unit. Clients have access to
the selling prices of heat and electricity and, consequently,
decide to procure the energy in order to minimize their cost.
To create this model, some assumptions are made as
follows:
Clients cannot purchase energy from the electricity
pool and only can procure their required energies by
the managers.
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The electricity prices offered by the rivals are
independent from the uncertainty in electricity pool
prices.
The electricity prices and heat prices offered by the
hub manager are similar to the retailers’ market rules
and have fixed prices and are independent from the
uncertainty in electricity pool prices but dependent
on each client.
The hub manager problem rules are assumed to be like
a retailer one. Hence, the clients procure their
required energies by fixed tariffs by extending
retailer market rules.
III. P
ROBLEM MODELING
A. Electricity pool market
The two ways of procuring electric energy by the hub
manager are forward contracts and the electricity pool market.
Retailers or hub managers may sell energy in the electricity
pool in order to increase their profit. The cost or revenue of
the energy traded in the pool is described as follows [28]:
(1)
(,) (,) (,)
PpP
Ct tPt
where
(,)
P
Ct
,
(,)
P
Pt
and
(,)
p
t
are the total cost or
revenue of trading, the energy traded and electricity pool price
in scenario
and period t, respectively.
(,)
P
Pt
may be
either positive or negative to represent the purchase or sale of
energy, respectively.
Signing forward contracts is a conventional way to procure
part of the clients’ need for electric energy. In forward
contracts, electricity is generated by an external agent and
purchased by the hub managers. Forward contracts have fixed
prices at the beginning of the decision making time horizon.
The method presented in [5] is used to model the forward
contract as given in Fig. 2 and the following equation:
(2)
1
() (,,) (,,),
K
Fff
fFtk
C t ftkP ftk t
where
(,, )and (,, )
ff
Pftk ftk
are the power and price of
block k at time t and contract f, respectively.
()
F
Ct
is the
total cost of contracts at time t and K is the number of blocks.
The amount of purchased power from contract f (
(,)
f
Pft) is
obtained as follows:
(3)
1
(,) (,,) (,)
K
ff
f
k
Pft Pftkvfk
(4)
(,) 0,1
f
vfk
B. CHP unit and boiler
Assume an energy hub as the one in Fig. 1.
(, )
CHP
e
P
t
is
the electric energy produced by the CHP unit, which is
defined as follows:
(5)
(, ) (, )
CHP CHP gas
eeCHP
Pt Pt
where
(, )
gas
CHP
Pt
and
CHP
e
are the gas entering the CHP unit
and conversion efficiencies from gas to electricity through the
CHP unit, respectively. The boiler is also fed by natural gas
and generates heat. The relation between the input and the
output of the boiler is described as follows:
(6)
(, ) (, )
Boil Boil gas
hBoil
Pt Pt
where
(, )
Boil
h
P
t
,
(, )
gas
Boil
P
t
and
B
oil
are the heat produced
by the boiler, the gas entering the boiler and the conversion
coefficient from gas to heat through the boiler, respectively.
The amount of heat produced by the CHP unit,
(, )
CHP
h
Pt
),
is calculated as follows:
(7)
(, ) (, )
CHP CHP gas
hhCHP
Pt Pt
where
CHP
h
is the heat conversion coefficient through the
CHP unit.
In addition, the dependency on the electrical and thermal
outputs of the CHP unit is modeled by defining a feasible
operation region bounded with coordinates
(,),
AA
AH P
(,),
BB
B
HP (,)
CC
CH P and ( , )
DD
DH P
, indicating the heat
and electricity outputs at each point as presented in [28,32].
For instance,
(,)
AA
AH P
are the heat and electricity outputs
of the CHP unit in coordinate
A.
C. Energy hub modeling
Energy hub is a concept describing a multi-carrier energy
system including electric energy, gas, heat, etc., that can be
F
f
P
1
2
1j
F
fj
P
)(MWP
F
f
F
f 1
F
f 2
F
f 3
3
1j
F
fj
P
Fig. 2: Forward contract blocks.
Fig. 1: Bilevel scheme of the problem.
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converted, stored, and transmitted. An energy hub relates the
input and output energies in a predefined area. Here, there are
two inputs (gas and electric energy) and two outputs (heat and
electric energy). The electric energy is transmitted to the
output in two ways: directly from the input to the output or by
a CHP unit. Heat output can also be produced through the
boiler and the CHP unit (see Fig. 3).
The coupling matrix is defined as follows:
(8)
1
0(1)
out
in
in
out CHP
e
ee
out CHP Boil
g
as
hh
C
P
P
P
P
P
P
where
is the dispatch factor to specify the share of natural
gas,
gas
P
, entering the CHP unit or the boiler.
D. Offering heat price
The offering heat price, as a control variable, is very
important to increase profit in the upper level. Higher prices
decrease the clients’ willingness to buy from the upper level
and increase their tendency toward rival managers. Hence, the
heat selling offer price by the hub manager is a stepwise
function between
r
h min,
and
r
h max,
as shown in Fig. 4.
The minimum and maximum prices proposed by the rivals
are the upper and lower bounds of the prices on the horizontal
axis. By getting closer to
,max
,
r
h
the heat provided will
decrease and vice versa.
(9)
(, 1)(, ) (, ) (, )(, )
sss
hs hs hs
cj vcj cj cjvcj
(10)
1
(, ) 1
J
j
vcj
where (, )
s
hs
cj
and (, )
s
hs
cj
are the offered price to client
c and the maximum price of block j, respectively. Eq. (9)
declares that
(, )
s
hs
cj
is positioned between minimum and
maximum bounds of the blocks. Eq. (10) guaranties that only
one block is selected.
(11)
1
() (, )(, )
J
ss
hhs
j
ccjvcj
(12)
(, ) 0,1vcj
Each block shows a specific step of offering heat price. The
offered prices represent the selected blocks. The manager can
offer only one price. Therefore, only one block should be
selected among all the blocks. In (12)
()
s
h
c
is the price
offered through all of the blocks and it is equal to the price of
the selected block.
E. Uncertainty characterization
Three uncertainty sources are taken into account: electricity
market prices, electricity demands and the prices offered to
supply electricity by the rival managers. Heat demand
uncertainty is neglected in the simulation for the sake of
simplicity.
Due to the lack of information about the future, there is
uncertainty in pool prices. Moreover, clients’ demands and the
electricity prices offered by the rivals are independent from
the upper-level decisions. The uncertainty of the upper level is
related to scenario
, which includes pool prices and
electricity and heat demands. Note that the summation of the
probabilities over all scenarios has to be equal to 1. On the
other hand, the prices offered by the rivals are a function of
, which is described as follows:
,1 ,
scenario : ( ),..., ( )
rr
eeNC
where scenario
is the set of rivals’ scenarios,
,1
()
r
e
is a
random variable showing the price of electricity offered by
rival r to client 1 in scenario
which is unknown to the hub
manager under study. NC is the total number of clients. The
summation of probabilities of all rivals’ scenarios has to be
equal to 1:
(13)
1
() 1
where
and ( )
represent the total number of rival
scenarios and the probability of rival scenario
,
respectively. Since we assume that gas price fluctuations are
very low, gas and heat prices are considered to be fixed.
IV. PROBLEM FORMULATION
A. Bilevel formulation
Upper level
in
e
P
gas
P
out
e
P
out
h
P
1
e
chp
h
chp
B
oil
Fig. 3: A sample energy hub with two carriers in the input and output [32]
100
0
Percent of supplying heat enrgy (%)
($ / )
s
h
M
Wh
,min
r
h
,max
r
h
,1
r
h
,2
r
h
Fi
g
. 4: Heat-
p
rice
q
uota curve.
