Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
Optimal Operation of Distribution Network
with Distributed Generations to Maximize
Social Welfare
M. Nikkhah Mojdehi, J. Zhang, X. Xia
Department of Electrical, Electronic and Computer Engineering,
University of Pretoria, Pretoria, South Africa.
Abstract: Retail electrical power marketers, also known as retailers, typically set up contracts
with suppliers to secure electricity at fixed prices on the one hand and with end users to meet
their load requirements at agreed rates on the other hand. High penetration of distributed
generation (DG) resource is increasingly observed worldwide. From the viewpoint of a retailer,
this paper analyzes the problem of setting up contracts on both the supplier and end-user sides to
maximize profits while maintaining a minimum operational cost of the distribution network with
distributed generations. Proposed two-level optimization model in this paper, can minimize cost
of distribution system with Distributed Generation and maximize profits of retailers. Numerical
simulations are carried out based on an IEEE 33-bus test distribution network. Test results are
included to show the performance of the proposed method.
Keywords: Retailer profit, electricity market, wholesale market, distributed generation (DG),
social welfare, distribution system, DisCo, distribution system operator (DSO).
1. INTRODUCTION
Nowadays increasing demand of electricity forced power
system operators to operate their systems close to thermal,
mechanical and electrical limits. Several solutions could
be considered to alleviate these operation conditions like
increasing generation, transmission and distribution capacity, decreasing consumption by increasing equipments
efficiency, demand management. These scenarios have
their own advantages and disadvantages. Developments
in distributed generation (DG) technologies, information
and communication technologies, environmental concerns,
liberalization in electricity markets and restructuring encourage using DGs in power system. Several definitions
for distributed generations have been presented in the
literature (Zareipour et al 2004 and IEEE 1547 Working
Group 2003). This paper engages in the following definition: distributed generation is an electric power source
connected directly to the distribution network or on the
costumer side of meter (Ackermann 2007).
The potential development of DG is sustained in the
following factors: increasing power quality requirements,
avoiding or shifting investment in transmission lines
and/or transformers, minimizing ohmic losses, protecting
the environment, and maintaining high energy prices at
retail level (Ackermann et al 2000 and Sakis Meliopoulos
2001).
Traditionally, a distribution company (DisCo) purchases
energy from the wholesale market, at a high voltage
level, and then transfers this energy to final customers.
Nevertheless, the restructuring process of the energy sector
has stimulated the introduction of new agents such as
retailers and products, and the unbundling of traditional
Copyright by the
International Federation of Automatic Control (IFAC)
DisCo into technical and commercial tasks, including the
provision of ancillary services 1 . Retailers purchase power
from suppliers and sell it again to end-user customers.
While there has recently been extensive research on market
modeling at the wholesale level, the retail side and the
reaction with DG has received relatively less attention.
Gabriel et al. (2002) analyzed several strategies for determining forward loads for electrical power retailers based
on empirically fit probability distributions for end-user
loads and spot market prices. Contract design problem a
retailer may encounter, both at the supply and end-user
levels, was addressed by Gabriel et al (2006). It provides
a stochastic programming methodology that allows the
retailer to make informed contractual decisions, particularly in respect to contract prices and quantities. Yusta
et al (2005) defined a model which can be applied to
an energy service provider (which supplies electric energy
for eligible industrial customers) in the Spanish electricity market. This model results in a quadratic non-linear
optimization problem, where the objective function to be
maximized is the economic profit obtained by the supplier,
and it calculates the optimal electricity selling prices to
be applied to the customers. The novel market integration mechanism was presented in Jimenez-Estevez et al
(2007), which demonstrates that the design of a market
interface and the consideration of DG units as equivalent
power producers. The proposed scheme shows an adequate
trade-off between technical accuracy (marginal cost theory
and ohmic losses) and practicability (available network
information and information management). Gordijn and
Akkermans (2007) presented four models, based on sev1
International Energy Agency (2002). Distributed generation in
liberalized electricity market.
12195
Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
eral case studies, including distributed generation energy
services to shift demand in peak hours (Spain electricity
market), small-scale hydro plant as a local power producer
(Norway electricity market), distributed generations in
grid energy balancing (Netherlands electricity market) and
distributed generations in active management of distribution networks (UK electricity market). Mashhour and
Moghaddas-Tafreshi (2010) considered distributed energy
resources in wholesale market to maximize their benefits
using a multiperiod optimization model subject to operational and technical constraints.
As mentioned before, DisCo, retailers, DGs and end users
are participants in the distribution market. Relationships
between these participants including money and power
flow, operating of the whole system in respect to an objective function, suitable objective function which satisfies all
participants and purchasing and selling price of electricity
are still need to be studied.
The goal of this paper is to find the sale price of retailers
in the way that not only increases the profit of retailers
but also develops social welfare. By social welfare we mean
minimizing the cost of putting the distribution system into
operation. This goal is implemented in a distributional
network which in itself contains some distributed generation resources in a way that affects the mechanism of
sale business. The costs will include the production cost
of distributed generation units and the cost of purchasing
from the wholesale market. To solve the above problem,
an algorithm has been proposed which is implemented in
two phases.
The rest of this paper is organized as follows. In Section
2, we describe modeling considerations and Section 3
provides mathematical formulations for the simulation.
Section 4 describes our numerical findings and is followed
by the conclusions section.
2. RETAIL ELECTRIC POWER MARKET MODEL
The model which is proposed in this section, considers two
objectives.
- The first one concerns minimizing the cost of distribution system operation, which is a problem that DisCo
always experiences. It concerns how DisCo should use
available resources to minimize the cost of distribution
system operation. In the proposed model, it is assumed
that available resources for Disco are distributed generations and purchased energy from retailers.
- The second is about maximizing the retailer’s profit. It
must be considered that the retailer’s profit be maximized. Retailers, on the one hand purchase power from
wholesale market and distributed generation resources
with spot prices considering the marginal cost of distributed generation units respectively, and on the other
hand sell power to their final consumers with their
expected sale price.
The combination of these two pre-mentioned points formulates a solution in the way of achieving the expected
results. The proposed method in this paper is based on
a loop. In this loop, at first DisCo determines amount of
available power from resources using economic dispatch
to minimize the cost of distribution system operation.
Then retailers make their own strategy to buy electricity
from available resources and sell it to their customers to
maximize their profits.
Meanwhile the goal of this paper is to find the sale price
of retailers in the way that not only increases the retailer’s
profit but also develops social welfare, minimizing the cost
of distribution system operation. The costs will include
the production cost of distributed generation units and
the cost of purchasing from the wholesale market.
So the profit of any retailer will be defined by:
prof it = income − payment
(1)
In (1), income includes the input of each retailer based
on the existing load of its customers and the tariff on
which the contract is based. As it was mentioned in section
1 it is also assumed that in a distribution network the
retailer contract with classes of the final consumers. These
classes are usually classified in terms of their geographical
positions. Payment will include the payout of each retailer
to the power supplier with whom the named retailer
contracts to provide the load of existing consumers in
its own class. The resources from which the retailer in
the distribution network can provide the power are the
wholesale market and distributed generation resources.
It is obvious that if there is no DG in the distribution
network, the retailer will provide all the required power
from the wholesale market with spot prices. But if there
is any distributed generation units in the distribution
network, then we will face a critical question and that is
“what amount of required power would a retailer provide
from the wholesale market and what amount will be
provided from distributed generation units?” As a matter
of fact the combination of the retailer’s purchase from
the wholesale market and the distributed generation units
would be in a way to increase retailer’s profit. But would
this be advantageous for the final consumer? To answer all
these questions, in the following an algorithm is proposed
to reach the maximum social welfare.
3. MATHEMATICAL FORMULATION
As it was mentioned earlier, the presence of distributed
generation resources in the distribution network will bring
about some ambiguities concerning the payout of the retailer and the purchasing strategy carried out by them.
To remove these ambiguities an algorithm is proposed in
this paper that will be accomplished in two phases. In the
first phase of this proposed algorithm an optimal power
flow will be administered from the distribution system
operator (DSO) point of view, in a way that the cost
of power production in the distribution system must be
minimized. By using this method the production portion
of each distributed generation unit and the power injection
portion from the wholesale market to the distribution network classes will be determined. The optimization model
of this optimal power flow (minimization of operation cost
of the whole distribution system) is as follows:
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min
" n
24
X
X
t=1
c=1
LoadN
c,t
Pn
∗ P ricec ∗ PW,t
N
Load
c,t
c=1
Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
#
m
X
+
CDGi,t (Pi,t ) ,
subject to:
(2)
i=1
Loadc,t =
subject to:
PW,t +
m
X
Pi,t = Ploss,t +
n
X
LoadN
c,t ,
(3)
LoadN
c,t
¸
·
βc (P ricec − P riceN
c )
, (7)
∗ 1+
P riceN
c
Loadc,t = PWc,t +
c=1
i=1
(4)
Pi,t ≥
≤ Pi,t ≤
Pimax ,
(5)
for all t = 1, . . . , 24, i = 1, . . . , m, k = 1, . . . , M , and in
which n denotes the total number of classes; t is the
time from 1 to 24; LoadN
c,t is nominal power demand of
class c in hour t; PW,t is the injected power from the
wholesale market at hour t; P ricec shows the price of
electrical energy sold to class c; Ploss,t is real power loss in
distribution network at hour t; Pi,t is power generation of
DG at hour t; m and M are total number of DG units and
nodes in the distribution network respectively; Vk,t shows
2
voltage of node k at hour t, CDGi,t (Pi,t ) = ai Pi,t
+ bi Pi,t +
ci defines the cost characteristic of DG i at hour t. In (2),
optimization variables are PW,t and Pi,t .
The first term in the optimization model (2) represents
the costs of energy purchasing from wholesale market,
and the second term represents the generation costs for
distributed generation units. Constraints (3), (4) and
(5) represent the load balance of the whole distribution
network constraint, voltage constraint on every node of
the distribution network and generating limits which are
specified as upper and lower limits for the real power
outputs of DG units, respectively. Relationships between
voltage of nodes and injected power to the network,
including wholesale market and DGs, can be found in
Nikkhah Mojdehi et al (2009).
In the beginning of implementing this algorithm in the first
phase, we do not have any access to the price of classes;
therefore we need to assume the initial values to determine
the price of the classes. At the first iteration it is assumed
that the price of the classes equals the spot price of the
wholesale market, so a preliminary value for the generation
of distributed generation units and the injection power
from the wholesale market will be achieved.
Using these initial values, the algorithm will enter in its
second phase. In this phase, the main goal is to maximize
the profit of retailers in a way that the income of selling
electricity to classes be maximized and at the same time
costs related to purchasing electricity from the wholesale
market with spot price and purchasing from scattered
resources with limited prices of these production units are
minimized. By fulfilling this part, the price of electricity
will be measured in each class.
The optimization model in this part is as follows:
" n
24
X
X
max
Loadc,t ∗ P ricec −
t=1
n X
m
X
c=1 i=1
c=1
M CDGi,t ∗ PDGi,c,t −P ricespot,t ∗
PDGi,c,t ,
(8)
i=1
Vkmin ≤ Vk,t ≤ Vkmax ,
Pimin
m
X
n
X
c=1
# (6)
PWc,t ,
n
X
PDGi,c,t ,
(9)
c=1
PW,t ≥
n
X
PWc,t ,
(10)
c=1
for all t = 1, . . . , 24, c = 1, . . . , n, i = 1, . . . , m, and in
which PDGi,c,t denotes the power purchased by a retailer
of class c from the distributed generation unit i (DGi ) in
hour t; M CDGi,t = 2 ∗ ai PDGi,c,t + bi defines the marginal
cost of distributed generation unit i in hour t; P ricespot,t
shows the spot price of the wholesale market in hour t;
PWc,t is the power purchased by a retailer of class c from
the wholesale market in hour t; Loadc,t denotes the real
load of class c in hour t; βc is the load elasticity class c;
P riceN
c shows the nominal sale price of class c; P ricec
represents the sale price of class c.
The first term in the optimization model (6) represents the
income of energy selling of each retailer corresponding to
their classes, the second term is the cost of purchasing energy from distributed generation units and the third term
represents the cost of purchasing energy from the wholesale
market. In (6), optimization variables are P ricec , PDGi,c,t
and PWc,t . Constraints (7) and (8) represent the end user
demand function and the load balance of each class respectively. Constraints (9) and (10) guarantee that summation
of purchased power from each DG and wholesale market
by retailers is less or equal to the dispatched power of each
DG and wholesale market by DSO (Pi,t and PW,t ).
Kirschen (2003) and Boisvert et al (2002) presented one of
the existing and most useful models of the function of price
response of loads of network. This demand function shows
how load changes in respect to variations in electricity
prices. In (7) the load function is in linear form based on
nominal variables. In this equation P riceN
c is the selling
price of the nominal electrical power consumed. It is
assumed that (in the situation of nominal demand LoadN
c,t )
the retailer sells electricity at a price which is 10% higher
than the whole cost, that is, it assigns a profit on income
of 10%.
In this equation it is assumed that the response of each
consumer to the price variation, characterized by the
elasticity β , has to be empirically assigned. The price
elasticity of demand, β, measures the degree of response
of the demand to the changes of price in the market, being
defined as the ratio of the variation of the demand quantity
of some commodity or service, in percent, to the variation
of its price P rice , also in percent. It can be expressed as:
∆Load/Load
β=
.
(11)
∆P rice/P rice
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Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
Fig. 1. Demand function
Fig. 2. Daily load variation
The demand is “elastic” if β is higher than 1, and “inelastic” if β is lower than 1. Some studies has been performed
to characterize the end user response to several price
strategies 2 3 , elasticity values ranging from -0.01 to -0.25
have been chosen. Corresponding to Fig. 1 elasticity value
is determined by a negative value and equals to -0.2.
After implementing the optimization model in phase two,
the sale price in each class will be achieved. Then these
results will be applied to optimal power flow in phase
one as input. After optimal power flow in phase one,
the amount of economical generation from distributed
generation units and amount of injection power from the
wholesale market are determined. Then these economical
power generation and injection from optimal power flow
in phase one are applied to optimization problem of phase
two as inputs in order to calculate the selling prices. Until
converging, the sale price of each class must be iterated.
Using the proposed algorithm, it can be seen that without
load elasticity, retailer’s sale price will increase sharply. On
the other hand, load elasticity regulates sale prices and
limits retailer’s profit.
4. SIMULATION AND NUMERICAL RESULTS
An IEEE 33-bus distribution system (Choi et al 2003),
is used to illustrate the proposed model and solution
algorithm. Maximum and minimum limitation for voltage
is assumed 0.9 and 1 pu, respectively. The system includes
four distributed generation units at nodes 4, 7, 25 and
30, three retailers. DG technologies considered in this
paper are Fuel Cell-CHP (FCC), Gas ICE-CHP (GIC),
Gas ICE-Power only (GIP), Microturbine-CHP (MC) and
Microturbine-Power only (MP). Subsequently there will
be three classes of retailing in the mentioned system.
Table 1 shows the cost characteristics of varieties of
DG unit technologies considered in this work 4 . In this
system the number of each load will be determined by the
number of the node attached to it. Table 2 shows the load
classification based on the classes.
Fig. 3. Spot price of wholesale market during a day
The assumed time interval for simulations is 24 hours a
day. Fig. 2 and Fig. 3 show the load variations and spot
prices of the wholesale market during a day respectively.
From Fig. 4 to Fig. 6, it can be seen that the retailer of
each class tends to supply the demand of its class from
the wholesale market principally. Also it can be concluded
that the retailer’s strategy is approximately constant with
applying different distributed generation technologies and
because of high consumed power in each class, retailers
prefer to use this units in off peak hours rather than peak
hours.
Table 1. Distributed generations data with
varieties of technology
DG Technology
FCC
GIC
GIP
MC
MP
a
0.0001
0.0001
0.0001
0.0001
0.0001
b
0.0315
0.0374
0.0777
0.0421
0.0841
c
1.0749
0.4777
0.3483
0.5553
0.4603
Pmax
400
400
400
400
400
Pmin
0
0
0
0
0
Table 2. Load classifications
2
Joskow P. and Tirole J. (2004). Retail electricity competition.
CSEM working paper., University of California Energy Institute,
Tech. Rep., 130.
3 Available at http://www.cne.es.
4 Available at http://www.cbo.gov.
12198
Retailer’s Class
A
B
C
Loads
2, 3, 4, 5, 6, 19, 20, 21, 22, 23, 24, 25
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18
26, 27, 28, 29, 30, 31, 32, 33
Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
Fig. 4. Purchasing strategy of retailer of class A, with Fuel
Cell-CHP technology
Fig. 7. Total profit of each retailer with several DG
technologies
Fig. 5. Purchasing strategy of retailer of class B, with Gas
ICE-CHP technology
Fig. 8. Price of each retailer with several DG technologies
Also from Fig. 7, it can be found that the retailer of class
A has higher profit because of higher demand of its class
than the others. More loads for the retailer do not lead
to more profit and the retailer’s profit is affected by load
forecasting. More loads are equal to more load forecasting
error which can decrease the retailer’s profit significantly.
Fig. 6. Purchasing strategy of retailer of class C, with
Microturbine-Power only technology
Fig. 7 shows that the retailer’s profit without using distributed generation resources is increased and out of evaluated technologies in this paper, Gas ICE-Power only and
Fuel Cell CHP have the highest and the lowest profits for
retailers respectively.
Simulation results are come from this fact that the retailer’s cost is composed of two parts. The first one is the
cost of purchasing power from the wholesale market which
is actually the purchased power multiplied by the price of
the wholesale market which is constant during an hour and
therefore the retailer’s strategy relevant to this price will
not be complicated. The second part of the retailer’s cost
is the cost of purchasing power from distributed generation
resources which includes purchased power from these units
multiplied by their marginal cost. It is worth to mention
that the purchasing price (i.e. the marginal cost) from
these units depends on its active generation power and
is a linear curve with a positive slope. On the other hand,
increase in power purchased from these units leads to
higher price of these units therefore the cost of purchasing
power from these units is increased. In comparison it
should be mentioned that increase in purchasing power
from the wholesale market leads to a lower purchasing
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Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
power than the power supplied by distributed generation
resources mostly because of constant prices.
An important point from these results is the retailer’s
income which includes the price of each class multiplied by
the load of its class. As illustrated in Fig. 8 the retailer’s
price is constant during a period and dose not respond
to hourly variations in wholesale prices and it results to
uneconomical signal 5 , so retailers look for suitable profit
margin with lower cost and higher income.
5. CONCLUSIONS
From simulation results, can be seen that in this modle,
retailers tried to purchase electricity from DGs. It means
that it can be considered as incentives for DGs owners to
invest.
The following conclusions can be drawn from this paper:
- Retailers can experience the most profit without using
distributed generation resources but their profits associated with distributed generation units are maximized
with Gas ICE-Power only technology.
- In this paper, determination of optimal purchasing strategy and the retailer’s profit maximization is implemented with social welfare maximization.
- Without applying subsidiary budget from governments,
distributed generations cannot be supported by retailers.
- In general terms, the more elastic the behavior of customer demand, the lower the profit obtained by retailers,
as customers have an increased response capacity to
price changes. On the other hand, most of the inelastic
customers will tolerate price increase without reducing
considerable consumption, and thus generate more revenues and higher profits.
Gabriel S. A., Conejo A. J., Plazas M. A., and Balakrishnan S. (2006). Optimal price and quantity determination for retail electric power contracts. IEEE Transaction on Power Systems, 21, pages 180–187.
Yusta J. M., Ramirez-Rosado I. J., Dominguez-Navarro J.
A., and Perez-Vidal J. M. (2005). Optimal electricity
price calculation model for retailers in a deregulated
market. International Journal of Electrical Power and
Energy Systems, 27(5-6), pages 437–447.
Jimenez-Estevez G. A., Palma-Behnake R., Torres-Avila
R., and Vargas L. S. (2007). A competitive market
integration model for distributed generation. IEEE
Trans. Power Systems, 22, pages 2161–2169.
Gordijn J., Akkermans T. (2007). Business models for
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1178–1188.
Mashhour E., Moghaddas-Tafreshi S. M. (2010). Integration of distributed energy resources into low voltage
grid: a market-based multiperiod optimization model.
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Nikkhah Mojdehi M., Kazemi A., and Barati M. (2009).
Network operation and reconfiguration to maximize social welfare with distributed generations. EUROCON,
pages 552–557, St.-Petersburg.
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