THEME [ENERGY.2012.7.1.1] Integration of Variable
Distributed Resources in Distribution Networks
(Deliverable 7.1)
Reliable distribution network
planning model report
Common Deliverable
Reliable distribution network planning model report
Authors
Authors
Organization
Email
Gregorio Muñoz-Delgado Univ. of Castilla-La Mancha (UCLM) gregorio.muñoz.delgado@gmail.com
Javier Contreras
Univ. of Castilla-La Mancha (UCLM) Javier.Contreras@uclm.es
Miguel Asensio
Univ. of Castilla-La Mancha (UCLM) Miguel.Asensio@uclm.es
Access:
Project Consortium
X
Family Projects within topic
ENERGY.2012.7.1.1
European Commission
Public
Status:
Draft version
X
Submission for Approval
Final Version
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Reliable distribution network planning model report
Executive summary
The WP7 activities referring to this Deliverable have been carried out in order to extend them in the
following deliverables with RES generation expansion (Deliverable 7.2) and demand response,
reserves, hybrid storage and electric vehicles (Deliverable 7.3).
This task models how to expand the distribution network adding new assets (lines and substations)
so that the current and future energy supply for the island customers is served at a minimum cost
and with the quality required. The objective function to minimize is the net present value of the
investment cost to add, reinforce or replace feeders and substations, losses cost, and operation
and maintenance cost. The model considers several levels of load in each node and investment
alternatives for each resource to be added, reinforced or replaced. The nonlinear objective function
is approximated by a piecewise linear function, resulting in a mixed integer linear model that is
solved using standard mathematical programming. The proposed model developed in this task
generates various alternative construction plan candidates. The results indicate the practical
applicability of the proposed distribution network expansion planning method. In addition to the
optimization problem, reliability indices and associated costs are computed for each solution. The
implemented model considers that there are several alternatives for each line expansion asset
available depending on the size of the conductors or the transformer’s capacity. The model is multistage and each stage has several load levels, described by a typical daily load curve occurring in
each node at different times. The load is represented as a constant current so that the planning
model becomes a mixed-integer linear programming problem (MILP). Quadratic losses in lines and
transformers are handed by piecewise linearization so that the problem becomes a mixed-integer
linear one, solved by using GAMS/CPLEX. The utilization of MILP techniques ensures fast and
efficient solutions for large-size problems.
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Table of content
AUTHORS ........................................................................................................................... 2
EXECUTIVE SUMMARY ................................................................................................... 3
TABLE OF CONTENT ....................................................................................................... 4
NOMENCLATURE ............................................................................................................. 6
I.
Sets.............................................................................................................. 6
II.
Indexes ........................................................................................................ 6
III. Parameters .................................................................................................. 7
IV. Variables ...................................................................................................... 8
1 INTRODUCTION ........................................................................................................... 10
1.1 Literature review ........................................................................................ 11
1.2 Objectives .................................................................................................. 12
1.3 Document structure ................................................................................... 12
2 OPTIMIZATION PROBLEM FORMULATION ......................................................... 13
2.1 Objective function ...................................................................................... 13
2.2 Constraints................................................................................................. 14
2.2.1 Integrality constraints ...................................................................... 14
2.2.2 Balance equations ........................................................................... 15
2.2.3 Kirchhoff’s voltage law..................................................................... 15
2.2.4 Voltage limits ................................................................................... 15
2.2.5 Capacity limits for feeders ............................................................... 15
2.2.6 Capacity limits for transformers ....................................................... 16
2.2.7 Unserved energy ............................................................................. 16
2.2.8 Investment constraints .................................................................... 16
2.2.9 Utilization constraints ...................................................................... 16
2.2.10 Investment limit ............................................................................... 17
2.2.11 Radiality constraints ........................................................................ 17
2.3 Linearizations............................................................................................. 17
2.3.1 Energy losses .................................................................................. 18
2.3.2 Kirchhoff’s voltage law..................................................................... 19
3 RELIABILITY CALCULATION ................................................................................... 21
3.1 Reliability indexes ...................................................................................... 21
3.2 Reliability cost ............................................................................................ 24
4 DISTRIBUTION EXPANSION PLANNING ALGORITHM ..................................... 26
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5 CASE STUDY ................................................................................................................ 28
5.1 Data ........................................................................................................... 28
5.2 Results ....................................................................................................... 33
5.2.1 Solution 1 ........................................................................................ 33
5.2.2 Solution 2 ........................................................................................ 36
5.2.3 Solution 3 ........................................................................................ 40
5.2.4 Comparative analysis ...................................................................... 43
6 SUMMARY, CONCLUSIONS, AND FUTURE WORK ........................................... 45
6.1 Summary ................................................................................................... 45
6.2 Conclusions ............................................................................................... 45
6.3 Future work ................................................................................................ 46
REFERENCES .................................................................................................................. 47
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Nomenclature
I. Sets
Set of time stages
{
Set of feeder types where
}
Set of new replacement feeders
Set of new added feeders
Set of existing fixed feeders
Set of existing replaceable feeders
Set of available alternatives for feeders
Set of branches with feeders of type
Set of substation nodes
Set of available alternatives for transformers
{
Set of transformer types where
}
Set of existing transformers
Set of new transformers
Set of load levels
Set of load nodes in each stage
Set of nodes connected with node by a feeder of type
Set of system nodes
Q
Set of customer sector where
Z
Set of interruption type;
{
{
}
}
Set of new added line which has not been installed in iteration
Set of new added line which has been installed in iteration
II. Indexes
Index for time stages
Index for available alternatives for feeders and transformers
Index for nodes
Index for nodes
Index for feeder types
Index for transformer types
Index for load levels
Index for piecewise linear sections used for linearization of energy losses
q
Index for customer sectors
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z
Index for interruption types
Index for iterations of expansion planning algorithm
III. Parameters
Annual interest rate
Number of time stages
Capital recovery rate of feeder type .
Investment cost of installing alternative
of feeder type in branch -
Investment cost of expanding existing substations by adding a new transformer or
building a new substation from scratch
Capital recovery rate of new transformers
Investment cost of adding alternative
of new transformers in substation node
Maintenance cost of alternative
of feeder type installed in branch -
Maintenance cost of alternative
of transformer type
installed in substation node
Duration in hours of each load level
Cost of energy supplied by substations at load level
Resistance of alternative
of feeder type installed in branch -
Resistance of alternative
of transformer type
installed in substation node
Lifetime of feeder type
Unserved energy cost coefficient
Power demand at node i at each load level b of stage t
Impedance of alternative
of feeder type installed in branch -
Lower limit for nodal voltages
Upper limit for nodal voltages
̅
̅
Upper limit for currents flow through alternative
Upper limit for energy supplied by alternative
substation node
of feeder type installed in branch of transformer type
installed in
Budget for investments in stage
Number of piecewise linear sections used for energy losses linearization
Slope of piecewise linear section
of energy losses linearization
Upper limit for current flow corresponding to piecewise linear section
losses linearization
Slope of piecewise linear section
feeder type installed in branch -
of energy
of energy losses linearization in alternative
of
Slope of piecewise linear section of energy losses linearization in alternative
transformer type installed in substation node
of
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Upper limit for current flow corresponding to piecewise linear section
of energy
losses linearization through alternative of feeder type installed in branch Upper limit for energy supplied corresponding to piecewise linear section of energy
losses linearization for alternative
of transformer type
installed in substation
node
Positive constant with a enough high value
Average system availability index for each stage
Percentage of customer sector
Cost of an interruption
at node for each stage
associated with customer sector
Customer interruption cost at stage
Cost associated with customer interruptions duration at stage
Customer interruptions duration index for node at stage
Customer interruptions duration target settled by regulation
Cost associated with customer interruptions frequency at stage
Customer interruptions frequency index for node at stage
Customer interruptions frequency target settled by regulation
Duration of interruption type
Expected energy not supply at stage
Cost associated with expected energy not supply at stage
Number of interruption type
which can affect to the node at stage
Total number of customers in node at stage
Cost of not attending SAIFI or SAIDI at stage
System average interruption duration index at stage
System average interruption frequency index at stage
System average interruption duration index target settled by regulation
System average interruption frequency index target settled by regulation
Average failure rate in sections
Penalty factor settled by regulation for not attending CIF or CID
Penalty factor settled by regulation for not attending SAIFI or SAIDI
Number of differences between each solution
Value of variable
obtained by the expansion planning algorithm in iteration
IV. Variables
Present value of investment and operating total cost
Investment cost in stage
Maintenance cost in stage
Supplied energy cost in stage
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Energy losses cost in stage
Unserved energy cost in stage
Binary variable which represents the investment associated with the installation of
alternative of feeder type installed in branch - at stage
Binary variable which represents the investment associated with the expansion of
existing substations by adding a new transformer or building a new substation from
scratch in substation node at stage
Binary variable which represents investment associated with the installation of
alternative of new transformers in substation node at stage
Binary variable which represents the utilization of alternative
installed in branch - at stage
Binary variable which represents the utilization of alternative
in substation node at stage
of feeder type
of transformer type
Current flow through alternative of feeder type installed in branch - at stage for
load level , measured in node , which is greater than 0 if node is the supplier and
0, otherwise.
Energy supplied by alternative
stage for load level
of transformer type
installed in substation node at
Unserved energy in node at stage for load level
Nodal voltage magnitude in node at stage for load level
Current flow corresponding to piecewise linear section
of energy losses linearization
Binary variable corresponding to piecewise linear section
linearization
of energy losses
Current flow corresponding to piecewise linear section of energy losses linearization
through alternative of feeder type installed in branch - at stage for load level
Energy supplied corresponding to piecewise linear section
of energy losses
linearization by alternative of transformer type
installed in substation node at
stage for load level
Binary variable corresponding to piecewise linear section
of energy losses
linearization associated to alternative of feeder type installed in branch - at stage
for load level
Binary variable corresponding to piecewise linear section
of energy losses
linearization associated to alternative of transformer type
installed in substation
node at stage for load level
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1 Introduction
An energy system usually consists of generation units, transmission networks, distribution
networks, consumption centers and control, protection, and regulation equipment [1]. Distribution
networks are an important part of the total electric system, as they supply energy from the
distribution substations to the end users. Distribution networks are typically three-phased and the
standard voltages operating are 30 kV, 20 kV, 15 kV, and 10 kV. Furthermore, most distribution
networks, even though they are topologically meshed, work in a radial way since it is the cheapest
and simplest method from the viewpoint of planning, design and system protection. These
networks have been designed with wide operating ranges, which allows to be passively operated
resulting in a more economical management.
Distribution substations are fed through one or several medium-voltage networks, although
sometimes they can be directly connected with the high voltage network. Each distribution
substation meets the energy by means of one or several primary feeders. Generally, in a
distribution substation can be found: (i) protection devices, (ii) measurement devices, (iii) voltage
regulators, and (iv) transformers [2].
From a centralized standpoint, distribution companies are responsible for operation and planning.
Distribution companies must satisfy the growing demand with quality and in a secure fashion. For
this, planning models are used to obtain an optimal investment plan of minimum cost meeting the
security and quality imposed requirements. These planning models are based on the capacity
distribution network expansion considering: (i) the replacement and addition of feeders, (ii) the
reinforcement of existing substations and the construction of new substations, and (iii) the
installation of new transformers [3]. In this kind of systems, only substations can supply energy
since distributed generation is not considered.
The majority of the interruptions suffered by customers takes place at the distribution system level,
due to its peculiar characteristics and structure [3], so that the quality and reliability of the power
system are important factors to be considered. Distribution system reliability evaluation consists in
assessing how adequately the different parts are able to perform their intended function. Rigorous
analytical treatment of distribution reliability requires well-defined performance indicators, referred
to as metrics. Unfortunately, the reliability vocabulary has not been used consistently across the
industry, and standard definitions are just beginning to be adopted [4].
The distribution business remuneration in the form of rated revenues collected with authorization of
the regulatory authority is a key aspect of regulation. The remuneration model of a regulated
activity should simultaneously contribute to economic efficiency of the system as a whole and
guarantee the viability of the regulated company, assuring a certain quality of service level. The
unbundling of distribution and retail activities has led to a modification of distribution activity’s
remuneration, where cost-of-service remuneration is giving way to incentive-based remuneration.
Under this remuneration scheme, the scope of regulation is to foster an optimal balance between
network operation and maintenance costs incurred by distribution companies and the quality of
supply provided to customers. Under a traditional cost-of-service regulation, an appropriate quality
of service level was maintained by investing in facilities as necessary, with no financial
compensation for customers in case of poor supply. Under the incentive remuneration scheme,
distribution companies are intended to internalize the resulting costs to the customers derived from
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poor quality.
The objective is to check if all customers will be served in a reliable way and at an acceptable cost,
this being defined either by the regulator on behalf of the customers, or by the customers
themselves through customer damage functions. Once these indexes are defined, a reliability
predictive analysis allows the distribution company to optimize its investment in quality
improvement.
In this work, an algorithm based on [3] and [5] is developed to decide the optimal distribution
network expansion planning considering reliability. First, an optimization model is used to obtain a
pool of solutions with different topologies. The optimization model decides the addition,
replacement or reinforcement of different assets such as feeders, transformers and substations.
Moreover, it identifies the best alternative, location, and installation time for the candidate assets.
Next, the algorithm calculates the reliability indexes and their associated costs of each solution of
the pool. Finally, the decision maker choses the best solution considering the total cost and other
factor such as environmental impacts, social factors, etc.
1.1 Literature review
A great variety of models has been proposed in technical literature. The more relevant works are
summarized as follows:
-
In [6] the distribution network expansion through mixed-integer linear programming was
addressed. In [7] the same problem was solved through mixed-integer quadratic
programming.
-
In [8] the distribution network expansion problem was analyzed by a multistage model
formulated though mixed-integer nonlinear programming.
-
In [9] a distribution system planning model was proposed with a mixed-integer nonlinear
programming approach considering the possibility of reinforcing substations and feeders and
installing DG units.
-
In [5] and [10] a multistage distribution network planning problem was proposed through a
mixed-integer linear programming approach.
-
In [11] the planning problem of primary distribution networks was formulated as a multiobjective mixed-integer nonlinear programming considering the system’s reliability costs in
the contingency events.
-
In [12] the distribution network expansion problem was analyzed through a heuristic
algorithm.
-
In [3] a model to solve the multistage problem of a distribution network planning was
presented. In addition, the reliability of solutions was analyzed.
-
In [13] a distribution system planning model for distribution system immersed in a electricity
market was proposed. Reactive energy was considered in the charge flow.
-
In [14] radiality constraints in distribution systems with DG were analyzed and planning
problem was solved through mixed-integer nonlinear programming for a one-stage model.
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-
In [15] the work of [9] was extended through the implementation of a dynamic planning
model considering growing demand. Moreover, the demand was modeled by load levels.
The problem was solved by a genetic algorithm.
-
In [16] a multi-objective optimization model was presented for a distribution system planning
considering different types of DG.
-
In [17] an integrated methodology is proposed for planning distribution networks in order to
improve system reliability which is considered in the objective function.
-
In [18] a multi-objective problem based on benefit maximization associated to the sizing and
location of DG units in a distribution system. In addition, the system reliability was
characterized through the minimization of interruption cost.
1.2 Objectives
The main objective of this work is the implementation of an algorithm for distribution network
planning in which: (i) several solutions are obtained minimizing the costs of investment,
maintenance, production, losses, and unserved energy through an optimization model and (ii)
reliability is analyzed for this solution to determine which expansion plan is the most convenient.
Another distinctive objective is the analysis of the effects of reliability consideration in the
distribution expansion planning problem. For this, case study results are checked.
1.3 Document structure
In Chapter 1, the addressed topic is introduced and available background in technical literature is
presented. Then, the main objectives and the document structure are detailed.
In Chapter 2, the optimization formulation for the distribution network planning problem is
described. For this, the objective function and mathematical constraints are presented. Finally, a
linearization for the nonlinear expressions is defined. In Chapter 3, reliability computation for the
solutions obtained by the optimization problem is explained, and reliability indexes and costs are
formulated. In Chapter 4, the overall algorithm composed of the optimization problem and the
reliability computation is presented.
In Chapter 5, a case study based on a distribution network, composed of 27 nodes and 39
branches, is examined. Data and results are showed and analyzed. In Chapter 6, a summary of the
work is presented, conclusions are drawn, and future work is proposed. Finally, the references
used are presented.
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2 Optimization problem formulation
In this section, the mathematical formulation for the optimization problem of distribution network
expansion planning is presented [19]. The proposed model is built on the distribution network
expansion planning models described in [3], [5] wherein (i) a multistage planning framework is
adopted, (ii) a discretization of the annual load curve into several load levels is used to characterize
the demand, (iii) radial operation of the distribution network is explicitly imposed, (iv) an
approximate network model is used, (v) the costs of losses are included in the objective function,
and (vi) several investment alternatives exist for each asset.
The optimization problem obtained has been formulated using mixed-integer linear programming
for which efficient off-the-shelf software is available and finite convergence to optimality is
guaranteed.
This chapter begins with the description of the objective function and constraints of the optimization
problem for distribution network expansion planning, where assets such as feeders, transformers,
and substations are managed to meet the growing demand. Finally, nonlinear expressions are
linearized, which allows to characterize it as a mixed-integer linear programming problem.
2.1 Objective function
The objective function (2.1) to be minimized represents the present value of the total cost, which
consists of 5 costs terms related to: (i) investment, (ii) maintenance, (iii) production, (iv) losses, and
(v) unserved energy.
∑
∑[
]
(
)
(2.1)
The investment cost is amortized in annual payments during the lifetime of the installed equipment,
considering that once the component is operated during a time equal to its lifetime, there is a
reinvestment in identical equipment, so infinite annual updated payments are used. The remaining
costs related to operation are updated and these costs are indefinitely kept, taking into account an
infinite series of annual payments [20].
Mathematically, these costs are defined as:
∑
{
∑ ∑
∑
∑
∑
(2.2)
}
∑∑ ∑
(
)
∑ ∑ ∑
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(2.3)
Reliable distribution network planning model report
∑
∑ ∑ ∑
∑
(2.4)
(∑ ∑ ∑
(
)
∑ ∑ ∑
)
∑ ∑
(2.5)
(2.6)
{
where
}
and
In (2.2), the investment cost in each stage is formulated as the sum of terms related to (i) the
replacement and addition of feeders, (ii) the reinforcement of existing substations and the
construction of new substations, and (iii) the installation of new transformers. Expressions (2.3)
model the maintenance costs of feeders and transformers. The production costs associated with
substations are characterized in (2.4). Similar to [3], the costs of energy losses in feeders and
transformers are modeled in (2.5) as quadratic terms. Such nonlinearities can be accurately
approximated by a set of tangent lines. This approximation yields piecewise linear functions, which,
for practical purposes are indistinguishable from the nonlinear models if enough segments are
used. Finally, expressions (2.6) correspond to the penalty cost of unserved energy.
It is worth emphasizing that, for each time stage, a single binary variable per conductor in the
feeder connecting nodes and is used to model the associated investment decision making,
namely
. In contrast, two binary variables,
and
and
, as well as two continuous variables,
, are associated with each feeder in order to model its utilization and current flow,
respectively. Note that
between nodes
otherwise.
and
is greater than 0 and equal to the current flow through the feeder
measured at node
only when the current flows from
to , being 0
2.2 Constraints
At this point, constraints associated with the optimization problem of the distribution network
expansion planning are formulated.
2.2.1 Integrality constraints
Investment decisions in new assets are modeled by the following binary variables:
{
}
{
}
(2.7)
{
}
(2.8)
{
}
(2.9)
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For instance, if variable
is equal to 1 and
is equal to
, the distribution company decides
to invest in the replacement of the existing replaceable feeder in branch - by the candidate
conductor for the replacement in stage .
Utilization decisions are also modeled by binary variables:
{
}
{
}
(2.10)
(2.11)
For instance, if variable
is equal to 1 and
is equal to
, the distribution company decides
to use the new added feeder in branch - with a candidate conductor
in stage .
2.2.2 Balance equations
Constraints (2.12) represent the nodal current balance equations, i.e., Kirchhoff’s current law.
∑ ∑ ∑(
)
∑ ∑
(2.12)
These constraints model that the algebraic sum of all outgoing and incoming currents in the node
must be equal to 0 for each stage and load level .
2.2.3 Kirchhoff’s voltage law
The enforcement of the Kirchhoff’s voltage law for all feeders in use leads to the following
expressions:
[
(
)]
(2.13)
This constraints are impose for all types of lines such as existing fixed feeder, existing replaceable
feeder, new replacement feeder, and new added feeder. Note that constraints (2.13) are nonlinear.
2.2.4 Voltage limits
The nodal voltage modules are limited by an upper and lower limit. Mathematically, these
constraints are formulated as follows:
(2.14)
2.2.5 Capacity limits for feeders
The current flow is restricted by the maximum capacity of feeders. This is formulated as follows:
̅
(2.15)
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Constraints (2.15) establish the maximum current flow that can be transport through feeders in use.
If a feeder is not used the current flow is 0.
2.2.6 Capacity limits for transformers
The current supply by substations depends on the number of transformers, which have a maximum
current value that can be supplied.
̅
(2.16)
Constraints (2.16) set the upper bounds for current that can be supplied by transformers in use. If a
transformer is not used the current supplied is 0.
2.2.7 Unserved energy
The variable associated with the unserved energy,
negative. Demand is established as the upper limit:
, is defined as a continuous and non-
(2.17)
2.2.8 Investment constraints
It is considered that during the whole planning horizon is only possible to invest in one of the
candidate alternatives for each equipment.
∑∑
{
}
(2.18)
∑
(2.19)
∑ ∑
(2.20)
∑
(2.21)
As per (2.18)-(2.20), a maximum of one reinforcement, replacement or addition is allowed for each
system component along the planning horizon. Constraints (2.21) guarantee that new transformers
can only be added in substations that have been previously expanded or built.
2.2.9 Utilization constraints
The candidate assets for the reinforcement, replacement or addition only can be used once the
investment is done. Mathematically, this is formulated as follows:
(2.22)
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{
∑
}
(2.23)
∑ ∑
(2.24)
∑
(2.25)
Constraints (2.22)-(2.24) model the utilization of all feeders while explicitly characterizing the
direction of current flows. The utilization of new transformers is formulated in (2.25).
2.2.10 Investment limit
The total investment cost in each stage has an upper limit that cannot be exceeded. Constraints
(2.26) impose this budgetary limit for investments in each stage.
∑
{
∑ ∑
∑
∑
∑
}
(2.26)
2.2.11 Radiality constraints
Generally, distribution networks are radially operated regardless of their topologies. That is,
distribution networks can be topologically meshed but they are operated in a radial way. This
condition is modeled as follows [21]:
∑∑ ∑
(2.27)
It is worth mentioning that constraints (2.27) impose that nodes must have a single input flow as
long as no distributed generation is considered in the model.
2.3 Linearizations
The optimization model for the distribution network expansion planning presented in the previous
section includes nonlinearities which make hard the obtaining of the optimal solution. Instead of
directly addressing the original problem of mixed-integer nonlinear programming, in this work the
approximation of this problem is proposed using a mixed-integer linear programming for which
effective off-the-shelf branch-and-cut software is available. Note that mixed-integer linear
programming guarantees finite convergence to optimality while providing a measure of the distance
to optimality along the solution process.
The nonlinearities are related to (i) quadratic energy losses in the objective function, and (ii) bilinear
terms involving the products of continuous and binary decision variables in the equations
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associated with Kirchhoff’s voltage law. Both nonlinearities are recast as linear expressions by
using a piecewise linear approximation for energy losses and integer algebra results for the bilinear
terms.
2.3.1 Energy losses
Energy losses are modeled in (2.5) by quadratic expressions, which can be approximated by
piecewise linear functions. Figure 2.1 shows the approximation for a quadratic curve through
linear sections.
Quadratic
curveica
Curva
cuadrát
Piecewise linearization
Linealización
a t ramos
Figure 2.1: Quadratic curve of energy losses and piecewise linearization
A general formulation for the piecewise linearization of the quadratic curve is presented as follows
[22]:
∑
(2.28)
∑
(2.29)
(2.30)
(2.31)
(2.32)
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{
}
(2.33)
where
and
.
This piecewise linearization technique has been used to approximate the energy losses in existing
fixed feeders, existing replaceable feeders, new replacement feeders, new added feeders, existing
transformers, and new transformers. Therefore, expressions (2.5) can be linearly formulated as
follows:
∑
(∑ ∑ ∑ ∑
∑ ∑ ∑ ∑
(
)
)
(2.34)
To model linear sections of all energy losses considered in feeders, the following constraints are
used:
∑
(2.35)
(2.36)
(
)
(2.37)
(2.38)
{
}
(2.39)
Analogously, to model linear sections of all energy losses considered in transformers, the following
constraints are used:
∑
(2.40)
(2.41)
(2.42)
(2.43)
{
}
(2.44)
2.3.2 Kirchhoff’s voltage law
Constraints (2.13) model the Kirchhoff’s voltage law for all feeders in use considering existing fixed
feeders, existing replaceable feeders, new replacement feeders, and new added feeders. These
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constraints are only activated when the corresponding line is used. This is nonlinearly modeled
through binary variables and the expression associated with the Kirchhoff’s voltage law. The linear
formulation of these constraints is presented below:
(
)
(
)
(
)
(2.45)
In constraints (2.45)
(2.13). If
is a positive constant big enough and their influence is similar to constraints
is equal to 0 there is no limitation on the value of the expression in brackets in
constraints (2.13), which must be between
and
. In contrast, if
is equal to 1 the
corresponding constraint behaves just like the corresponding constraints in (2.13).
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3 Reliability calculation
To evaluate directly the value of reliability is a difficult task, usually associated with customer
interruption costs and represented through a damage function for each customer type, which is
obtained through surveys.
The problem of using customer damage functions is that customers’ preferences change with time.
The same happens with the continuity of supply levels and the costs of the distribution companies.
On the other hand, the regulators’ task is to serve their customers ensuring them a reliable service,
and establishing minimum reliability levels based on historical data in specific locations and times.
In general, these are rules of thumb based on the perception of the customers, which are easy to
implement and interpret. The levels that are considered acceptable for the continuity of service
from the regulator’s standpoint should be based on the explicit knowledge of the perception of the
customers’ tolerance levels and the cost of the losses suffered by them. This is a very complex
task for the regulator, given the constraints imposed by its size [3].
The customer-based or regulator-based approaches are incorporated into our model to provide the
decision maker with the available information to calculate the cost to achieve the reliability targets
fixed by regulation, and the cost of losses suffered by the customer for a particular reliability index
set.
As proposed in [3], the calculation of reliability indexes and their associated costs considers: (i)
failure rates of system components, which are known for existing components and (ii) duration of
the interruptions as a function of the repair time, service recovery, switching, or isolation states.
Costs related to reliability are calculated for each loading condition, given previous knowledge of
the network.
3.1 Reliability indexes
The most widely used reliability indexes are averages that weight each customer equally.
Customer-based indexes are popular with regulating authorities since a small residential customer
has just as much importance as a large industrial customer. This work distinguishes three types of
customers [3], [4]: (i) residential (ii) commercial and (iii) industrial, since each customer has
different interruption costs.
According to [3], for the calculation of indexes it is considered that each feeder has a circuit breaker
without a recloser at the output of the substation and that each section between nodes has a switch
that enables the reconfiguration of the system after a fault in order to cover the demand in the most
efficient way [8], [23]. Only sustained interruptions are considered in the indexes definition.
To determine the reliability of the network in terms of quality and continuity of supply, it is
necessary to analyze which failures in the network affect each load node. An illustrative example is
shown to describe how a fault at a particular node influences the other load nodes. A fault at a
node is considered in this example as a fault at the node or at its input feeder.
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Reliable distribution network planning model report
2
1
2
3
1
1
3
SS
2
4
5
4
5
Load nodes
Substation nodes
Circuit breaker
Switch
Figure 3.1: Diagram of the illustrative example network
Figure 3.1 shows the illustrative example network composed of 5 load nodes, 1 substation node,
and 2 circuits consisted of 3 and 2 feeders, respectively.
A representative example results from the analysis of a single fault at load node 3. In this case,
circuit breaker 1 is opened and it has to be manually closed since no recloser has been considered
there. Loads linked to nodes 1, 2 and 3 will, therefore, be interrupted. First, the system is
reconfigured modifying switches and circuit breakers to reduce the non-supplied energy. In this
step, the system circuit breaker 1 is closed and switch 3 is opened to isolate the faulted line so,
during the repair time, all loads are satisfied except load 3. When the fault is fixed, switch 3 is
closed and normal operation is reestablished. As a consequence, loads 1 and 2 have not been met
during a time equal to the reconfiguration time and load 3 has not been met during a time equal to
the repair time. Loads 4 and 5 are not affected by this fault.
If the fault occurs in load 1, loads 1, 2 and 3 will not be met during a time equal to the repair time,
since loads 2 and 3 are downstream of the fault. Loads 4 and 5 are also not affected by this fault.
For simplicity, repair times, as well as reconfiguration times have been considered to be
respectively equal for all nodes. With this consideration, a matrix of number of different
reconfiguration and repair interruptions which affect to each node can be built. This is presented in
Table 3.1.
Table 3.1: Number of different interruptions which can affect at each node
Node
Repair
Reconfiguration
1
1
2
2
2
1
3
2
1
4
1
1
5
2
0
Table 3.1 shows how load 3 is affected by two different repair interruptions due to a fault in load 1
and in itself. Additionally, load 3 can be affected by one reconfiguration interruption due to a fault in
load 2. In the same way, it can be observed how node 1 can be affected by one repair interruption
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Reliable distribution network planning model report
due its own fault and can also be affected by two reconfiguration interruptions due to a fault in
loads 2 and 3. This is done in the same way for the others nodes.
The most commonly used indexes in distribution system reliability are [3], [24]-[25]:

Customer Interruption Frequency (CIF) at each load node and for each stage:
∑

(3.1)
Customer Interruption Duration (CID) at each load node and for each stage:
∑

(3.2)
System Average Interruption Frequency Index (SAIFI) is a measure of how many sustained
interruptions an average customer will experience over each stage. For a fixed number of
customers, the only way to improve SAIFI is to reduce the number of sustained
interruptions experienced by customers.
∑
∑
(3.3)
∑

System Average Interruption Duration Index (SAIDI) is a measure of how many interruption
hours an average customer will experience over each stage. For a fixed number of
customers, SAIDI can be improved by reducing the number of interruptions or by reducing
the duration of these interruptions. Since both of these reflect reliability improvements, a
reduction in SAIDI indicates an improvement in reliability.
∑
∑
(3.4)
∑

Average System Availability Index (ASAI) is the ratio of total customer hours in which
service is available divided by the total customer hours in the time period for which the
index is calculated for each stage. ASAI provides the same information as SAIDI. Higher
ASAI values reflect higher levels of system reliability.
[

]
(3.5)
Expected Energy Not Supply (EENS) for each stage:
∑ ∑
(3.6)
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Reliable distribution network planning model report
3.2 Reliability cost
The cost of non-supplied energy can be considered either from the distribution companies’
viewpoint or from the customers’ viewpoint [3]. From the distribution companies’ standpoint, this
cost corresponds to the energy that is not billed during the fault, and from the customers’ view is
related to each customer’s damage function or interruption cost function, being both functions
difficult to measure. This work assumes that the regulator sets limits for the reliability indexes and
penalizes its non-compliance. Thus, distribution companies expand the system seeking the
minimum cost and observing these indexes and their associated penalties for non-compliance [3].
From this perspective, the EENS cost at each stage is:
∑ ∑
(3.7)
The way to penalize the non-attendance of the reliability indexes depends on the specific
regulation. This work specifies that the distribution companies must remunerate the customer
whose reliability index is violated according to its energy bill during the considered period, and with
a penalty factor µ established by the regulator. Therefore, the CID and CIF costs at each stage are:
∑ [
∑ [
∑
∑
]
(3.8)
]
(3.9)
The highest of the previous values is passed on to the affected customers in proportion to their
individual bills. The economic valuation of not attending the average frequency values or the
duration of faults in a considered stage also depends on the specific regulation. In this work, we
consider the penalty applied by the regulator as a percentage ν of the energy bill of the distribution
company’s sold energy. Thus, the SAIFI or SAIDI cost at each stage is:
∑ ∑
[
]
(3.10)
The cost for not complying with the CID or the CIF, which is directly passed on to the affected
customers, must be discounted from the value of the penalty for not satisfying the SAIFI or SAIDI.
The cost associated to the customers’ reliability can be obtained from cost damage functions. The
customer interruption cost (CIC) due to outages in year t is:
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Reliable distribution network planning model report
∑ ∑∑∑
(3.11)
The CIC represents the cost that the customer experience and provides a means to reduce the
allowed revenue for the distribution company if that is above a maximum established by the
regulator.
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Reliable distribution network planning model report
4 Distribution expansion planning algorithm
The difficulty to incorporate reliability into distribution expansion models resides in the fact that it is
necessary to know the network topology in order to calculate the reliability indices that characterize
the system as well as the failure and repair rates and the placement and response of the protective
devices [8]. However, the final optimal topology is the objective of the expansion plan, where
reliability indices are known after expansion in a progressive process [26]. As in [3], costs related to
reliability for each load condition are calculated, given previous knowledge of the network.
In this work, as implemented in [3], first a solution pool is obtained which consists of several
solutions with different topologies. Then, reliability indices and their associated costs are computed.
Finally, with this information and other factors the decision maker chooses the optimal investment
plan among the proposed pool of solutions. Figure 4.1 shows the distribution planning algorithm
flowchart.
Optimization
model
Pool of
solutions
Reliability
index
Reliability
costs
Comparative
analysis
Decision
making
Figure 4.1: Distribution planning algorithm flowchart
Based on [3], the scope of this work consists of the following steps:
1) Traditional multistage expansion planning problem is formulated (2.1)-(2.27).
2) A pool of solutions with different topologies is obtained running the expansion planning
problem in a loop with as many iterations as the number of solutions is set to obtain. In
order to get different solution with different topologies a set of constraints is added to the
optimization problem [27]:
∑
(
( ∑ ∑
)
)
∑
(
(
∑ ∑
)
)
(4.1)
where:
{
( ∑ ∑
)}
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and
{
( ∑ ∑
)}
Reliable distribution network planning model report
Constraint (4.1) avoids obtaining in the iteration the solutions found in previous iterations,
that is, the topology solution obtained in previous iteration can never be chosen again. The
value of the binary variables associated with the new added feeders are stored in two
different sets, one for binary variables equal to 1 and another for binary variables equal to
0. Then, constraints (4.1) force to obtain a solution with a number of differences
with
respect to the each previous solution. In this case, the differences have to be in the binary
variables associated with the set of new added feeders,
.
3)
4)
5)
6)
Evaluate the reliability indices of each topology associated to each obtained solution.
Calculate the reliability costs associated to each obtained solution.
Comparison of topologies of the different solutions of the pool.
Investment decision is adopted by the planners and decision makers considering the cost
and other factor such as environmental impacts, social factors, etc.
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Reliable distribution network planning model report
5 Case study
In this section, a case study is presented and results obtained with the distribution expansion
planning algorithm are analyzed. This study case is based on a system consisting of 27 nodes and
39 branches [3].
Simulations have been implemented on a Dell PowerEdge R910X64 with four Intel Xeon E7520
processors at 8 GHz and 32 GB of RAM using CPLEX 12 [28] under GAMS 24.0 [29]. Then,
system data and results obtained are presented.
5.1 Data
The distribution network used [3] consists of 24 load nodes, 3 substation nodes, and 39 branches.
The distribution network topology is shown in Figure 5.1.
Load node
Existing substation
Candidate substation
Existing fixed feeder
Existing replaceable feeder
Candidate branch to install new feeder
17
1
2
3
4
19
18
5
6
7
8
25
26
9
10
11
12
22
20
21
23
24
27
13
14
15
Figure 5.1: One-line diagram of the distribution network
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16
Reliable distribution network planning model report
The data associated with the optimization problem are:
-
Base power and base voltage of the system are 1 MVA and 13.8 kV, respectively.
The planning horizon is three years divided into yearly stages.
Three load level are considered, , , and , with durations respectively equal to 1095
h/year, 2920 h/year, and 4745 h/year.
A 10% interest rate is set.
The lifetime of all feeders and transformers are 25 and 15 years, respectively.
The costs of energy supplied by all substations,
, are identical and equal to $50/MVAh,
$40/MVAh, and $27.4/MVAh, for load level - , respectively.
For simplicity, the maintenance for all feeders is equal to $450/year.
The cost of unserved energy,
, is $2000/MVAh.
Upper and lower bounds for voltages at load nodes are equal to 1.05 p.u. and 0.95 p.u.,
respectively.
Voltages at substation nodes are set to 1.05 p.u.
A three-block piecewise linearization is used to approximate energy losses.
Table 5.1 shows data for existing fixed feeders. These data comprise maximum current flow,
impedance, and resistance.
Table 5.1: Data for existing fixed feeders
̅
Branch
01
01
02
03
05
05
12
12
(MVA)
(Ω)
(Ω)
6
6
6
6
6
6
6
6
1
1
1
1
1
1
1
1
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
02
05
03
04
06
25
16
26
Table 5.2 shows data for candidate feeders in branches subject to replacement
Table 5.2: Data for candidate replacement conductors
Branch
01
05
05
12
12
05
06
25
16
26
Alternative 1
Alternative 2
̅
̅
(MVA)
(Ω)
(Ω)
(k$)
(MVA)
(Ω)
(Ω)
(k$)
9.6
9.6
9.6
9.6
9.6
0.7
0.7
0.7
0.7
0.7
0.44
0.44
0.44
0.44
0.44
20
21
18
22
19
12
12
12
12
12
0.5
0.5
0.5
0.5
0.5
0.25
0.25
0.25
0.25
0.25
38
39
36
40
37
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Reliable distribution network planning model report
Table 5.3 shows data for candidate feeders in non-existing branches.
Table 5.3: Data for candidate conductors in non-existing branches
Alternative 1
Branch
01
03
04
05
06
07
07
07
08
09
09
09
10
10
10
11
11
12
13
13
14
14
15
15
17
18
20
21
22
23
23
17
19
08
10
07
08
19
26
12
10
13
25
11
21
22
22
26
24
14
20
15
21
16
23
18
25
25
27
23
24
27
Alternative 2
̅
̅
(MVA)
(Ω)
(Ω)
(k$)
(MVA)
(Ω)
(Ω)
(k$)
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
0.76
108
092
090
092
094
096
112
300
098
100
102
305
104
092
090
106
310
102
108
102
110
110
112
106
090
300
315
300
106
094
300
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
0.44
128
112
110
112
114
116
132
320
118
120
122
325
124
112
110
126
330
122
128
122
130
130
132
126
110
320
335
320
126
114
320
Table 5.4 shows data for existing transformers, candidate transformers to install, and the cost of
expanding or building substations.
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Reliable distribution network planning model report
Table 5.4: Data for transformers
Existing transformer
Node
25
26
27
̅
Alternative 1
(k$)
(MVA)
(Ω)
(k$)
15
15
-
0.13
0.13
-
3
3
-
Alternative 2
̅
̅
(MVA) (Ω)
50
70
70
7.5
7.5
7.5
0.25
0.25
0.25
Table 5.5 shows demand data where three load levels
(k$) (MVA) (Ω)
(k$)
1
1
1
-
375
375
375
12
12
12
0.16
0.16
0.16
(k$)
(k$)
2
2
2
600
600
600
are considered.
Table 5.5: Demand data
Demand (MW)
Node
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Stage 1
1.2
0.0
0.0
1.2
1.2
1.2
0.0
1.2
0.0
0.0
1.2
0.0
1.2
0.0
0.0
1.2
0.0
0.0
1.2
0.0
1.2
0.0
0.0
1.2
0.72
0.00
0.00
0.72
0.72
0.72
0.00
0.72
0.00
0.00
0.72
0.00
0.72
0.00
0.00
0.72
0.00
0.00
0.72
0.00
0.72
0.00
0.00
0.72
Stage 2
0.24
0.00
0.00
0.24
0.24
0.24
0.00
0.24
0.00
0.00
0.24
0.00
0.24
0.00
0.00
0.24
0.00
0.00
0.24
0.00
0.24
0.00
0.00
0.24
1.2
1.2
0.0
1.2
1.2
1.2
1.2
1.2
1.2
0.0
1.2
1.2
1.2
0.0
0.0
1.2
1.2
0.0
1.2
1.2
1.2
0.0
0.0
1.2
0.72
0.72
0.00
0.72
0.72
0.72
0.72
0.72
0.72
0.00
2.40
0.72
2.40
0.00
0.00
0.72
0.72
0.00
2.40
0.72
2.40
0.00
0.00
0.72
Stage 3
0.24
0.24
0.00
0.24
0.24
0.24
0.24
0.24
0.24
0.00
0.48
0.24
0.48
0.00
0.00
0.24
0.24
0.00
0.48
0.24
0.48
0.00
0.00
0.24
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
2.4
2.4
2.4
1.2
2.4
2.4
2.4
1.2
2.4
1.2
1.2
1.2
1.2
1.2
1.2
1.2
0.72
0.72
0.72
0.72
0.72
0.72
0.72
0.72
1.20
3.60
3.60
0.72
3.60
1.20
1.20
0.72
1.20
2.40
2.40
0.72
2.40
0.72
0.72
0.72
0.24
0.24
0.24
0.24
0.24
0.24
0.24
0.24
0.48
1.20
1.20
0.24
1.20
0.48
0.48
0.24
0.48
0.48
0.48
0.24
0.48
0.24
0.24
0.24
The data associated with reliability failure calculation are as follows:
For the sake of simplicity, a failure rate of 0.4 failures per year for all lines, interruption duration
for repairs of 2 h, and for reconfigurations of 0.25 h, are assumed. Table 5.6 presents the
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Reliable distribution network planning model report
number of customers
at every node
.
and its percentage distribution as residential, commercial, and industrial
Table 5.6: Number of customer per node and sector participation in load
Stage 1
Node
Stage 2
Res Com Ind
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
100
000
000
010
050
030
000
050
000
000
050
000
010
000
000
050
000
000
010
000
100
000
000
050
0.50
0.00
0.00
0.00
0.20
0.15
0.00
0.10
0.00
0.00
0.50
0.00
0.00
0.00
0.00
0.30
0.00
0.00
0.00
0.00
0.60
0.00
0.00
0.50
0.20
0.00
0.00
0.20
0.20
0.20
0.00
0.20
0.00
0.00
0.40
0.00
0.15
0.00
0.00
0.20
0.00
0.00
0.00
0.00
0.25
0.00
0.00
0.20
0.30
0.00
0.00
0.80
0.60
0.65
0.00
0.70
0.00
0.00
0.10
0.00
0.85
0.00
0.00
0.50
0.00
0.00
1.00
0.00
0.15
0.00
0.00
0.30
Stage 3
Res Com Ind
100
010
000
010
050
030
010
050
050
000
050
050
010
000
000
050
010
000
010
050
100
000
000
050
0.50
0.00
0.00
0.00
0.20
0.15
0.00
0.10
0.10
0.00
0.50
0.40
0.00
0.00
0.00
0.30
0.00
0.00
0.00
0.00
0.60
0.00
0.00
0.50
0.20
0.10
0.00
0.20
0.20
0.20
0.00
0.20
0.20
0.00
0.40
0.40
0.15
0.00
0.00
0.20
0.25
0.00
0.00
0.00
0.25
0.00
0.00
0.20
0.30
0.90
0.00
0.80
0.60
0.65
1.00
0.70
0.70
0.00
0.10
0.20
0.85
0.00
0.00
0.50
0.75
0.00
1.00
1.00
0.15
0.00
0.00
0.30
Res Com Ind
100
010
050
010
050
030
010
050
100
010
070
050
015
015
015
050
015
020
010
050
100
010
010
050
0.50
0.00
0.20
0.00
0.20
0.15
0.00
0.10
0.10
0.00
0.50
0.40
0.00
1.00
1.00
0.30
0.00
0.80
0.00
0.00
0.60
0.00
1.00
0.50
0.20
0.10
0.20
0.20
0.20
0.20
0.00
0.20
0.20
0.00
0.40
0.40
0.15
0.00
0.00
0.20
0.25
0.20
0.00
0.00
0.25
1.00
0.00
0.20
0.30
0.90
0.60
0.80
0.60
0.65
1.00
0.70
0.70
1.00
0.10
0.20
0.85
0.00
0.00
0.50
0.75
0.00
1.00
1.00
0.15
0.00
0.00
0.30
The values given for the interruption costs per customer sector and interruption durations can be
seen in Table 5.7.
Table 5.7: Customer sector interruption cost ($/kW)
Customer sector
Residential
Commercial
Industrial
Reconfiguration Repair
0.1
3
4
5
32
26
The penalty factor for not meeting the
or
is set at 20, and the ν applied as a penalty to
the annual electricity billing statement of a distribution company is set at 0.01. Reliability indexes
must satisfy the limits imposed by the regulator in order for the distribution company to not be
32/48
Reliable distribution network planning model report
penalized. In this work, following values have been adopted:
h/yr,
interruptions per year, and
h/yr.
The number of different solutions obtained,
interruptions per year,
, by the expansion planning algorithm is 3.
5.2 Results
In this section, results obtained by the expansion planning algorithm for the study case are
presented. Then, a comparative analysis is done. Figure 5.2 shows the symbols used to represent
the topology of the solutions obtained.
Node without demand
Node with demand
Existing substation
Uninstalled substation
Feeder in use
Feeder non use
*
New replacement or installation
R1
Alternative 1 in branch subject to replacement
R2
Alternative 2 in branch subject to replacement
A1
Alternative 1 in prospective branch
A2
Alternative 2 in prospective branch
TR1
Alternative 1 for candidate transformer
TR2
Alternative 2 for candidate transformer
Figure 5.2: List of solution symbols
5.2.1 Solution 1
Figure 5.3 shows the topology and operation of the first solution obtained by the expansion
planning algorithm.
17
1
2
3
4
A2*
18
5
6
1
8
18
A2*
A2*
26
3
6
11
10
A2*
9
A1*
22
A2
A2*
20
11
10
23
20
24
A2
(a)
16
12
A1
22
A2
23
21
24
23
A1*
24
R1*
27
27
15
11
10
A1*
A1
A1
A1*
14
R2
9
20
21
27
13
A1
26
A2
A2
A2
21
8
A2
A2
12
22
4
19
7
R2
25
R2
A2
12
6
TR2*
A2
R2*
9
A2
26
3
A2
5
A1
A2
25
2
R2
18
8
A2
R2
1
A1
19
7
A2*
A2*
17
4
A2
5
A1*
A2*
2
R2*
A1*
7
R2*
25
17
19
13
14
15
16
13
(b)
Figure 5.3: Solution 1: (a) Stage 1, (b) Stage 2, (c) Stage 3
33/48
A1*
14
15
(c)
A1*
16
Reliable distribution network planning model report
Table 5.8 shows the total current injections by all transformers in each substation at each load level
of stage . In Table 5.9 the different costs obtained in each stage and the total costs are
presented. The present value of the total cost is 96203.388 k$.
Table 5.8: Solution 1. Injected powers by substations [MVA]
Stage 1
Node
25
26
7.20
6.00
4.32
3.60
Stage 2
1.44
1.20
12.00
08.40
10.56
08.40
Stage 3
2.88
2.16
22.80
14.40
20.64
12.24
6.48
3.84
Table 5.9: Solution 1. Costs [k$]
Investment
Maintenance
Production
Losses
Unserved energy
Stage 1
Stage 2
Stage 3
1901.501
0550.840
1095.691
013.650
013.636
170.909
02437.137
04653.408
82754.605
0039.728
0096.330
2475.953
0.00
0.00
0.00
Total
3548.032
198.195
89845.150
2612.011
0.00
Number of interruptions due to repair and reconfiguration that affects every load node is presented
in Table 5.10 and Table 5.11, respectively.
Table 5.10: Solution 1. Number of repair interruptions at each node
Node
Stage 1
Stage 2
Stage 3
Node
Stage 1
Stage 2
Stage 3
01
02
03
04
05
06
07
08
09
10
11
12
2
0
0
5
1
2
0
2
0
0
1
0
2
3
0
5
1
2
1
2
1
0
1
1
2
3
4
5
1
2
1
2
1
2
1
1
13
14
15
16
17
18
19
20
21
22
23
24
2
0
0
2
0
0
2
0
3
0
0
2
2
0
0
2
2
0
2
3
3
0
0
2
2
3
3
2
2
1
2
3
3
2
3
2
34/48
Reliable distribution network planning model report
Table 5.11: Solution 1. Number of reconfiguration interruptions at each node
Node
Stage 1
Stage 2
Stage 3
Node
Stage 1
Stage 2
Stage 3
01
02
03
04
05
06
07
08
09
10
11
12
6
0
0
3
7
6
0
2
0
0
0
0
6
5
0
3
7
6
1
2
2
0
0
3
6
5
4
3
7
6
1
4
3
6
1
5
13
14
15
16
17
18
19
20
21
22
23
24
0
0
0
2
0
0
0
0
5
0
0
2
1
0
0
2
0
0
0
0
5
0
0
2
2
1
3
4
0
1
0
1
5
0
3
4
Table 5.12 presents the customer interruption frequency at each node for each stage.
Table 5.12: Solution 1. Customer interruptions frequency index
[interruptions]
Node
Stage 1
Stage 2
Stage 3
Node
Stage 1
Stage 2
Stage 3
01
02
03
04
05
06
07
08
09
10
11
12
3.2
0.0
0.0
3.2
3.2
3.2
0.0
1.6
0.0
0.0
0.4
0.0
3.2
3.2
0.0
3.2
3.2
3.2
0.8
1.6
1.2
0.0
0.4
1.6
3.2
3.2
3.2
3.2
3.2
3.2
0.8
2.4
1.6
3.2
0.8
2.4
13
14
15
16
17
18
19
20
21
22
23
24
0.8
0.0
0.0
1.6
0.0
0.0
0.8
0.0
3.2
0.0
0.0
1.6
1.2
0.0
0.0
1.6
0.8
0.0
0.8
1.2
3.2
0.0
0.0
1.6
1.6
1.6
2.4
2.4
0.8
0.8
0.8
1.6
3.2
0.8
2.4
2.4
Table 5.13 presents the customer interruption duration at each node for each stage.
35/48
Reliable distribution network planning model report
Table 5.13: Solution 1. Customer interruption duration index
[h]
Node
Stage 1
Stage 2
Stage 3
Node
Stage 1
Stage 2
Stage 3
01
02
03
04
05
06
07
08
09
10
11
12
2.2
0.0
0.0
4.3
1.5
2.2
0.0
1.8
0.0
0.0
0.8
0.0
2.2
2.9
0.0
4.3
1.5
2.2
0.9
1.8
1.0
0.0
0.8
1.1
2.2
2.9
3.6
4.3
1.5
2.2
0.9
2.0
1.1
2.2
0.9
1.3
13
14
15
16
17
18
19
20
21
22
23
24
1.6
0.0
0.0
1.8
0.0
0.0
1.6
0.0
2.9
0.0
0.0
1.8
1.7
0.0
0.0
1.8
1.6
0.0
1.6
2.4
2.9
0.0
0.0
1.8
1.8
2.5
2.7
2.0
1.6
0.9
1.6
2.5
2.9
1.6
2.7
2.0
The results for the rest of reliability indexes are shown in Table 5.14.
Table 5.14: Solution 1. Reliability indexes
Reliability
index
Stage 1
Stage 2
Stage 3
02.361
02.031
99.977
13.950
02.110
01.907
99.978
27.080
02.320
02.003
99.977
50.102
The different reliability costs obtained for this solution are presented in Table 5.15.
Table 5.15: Solution 1. Reliability costs
Reliability
cost
Stage 1
Stage 2 Stage 3
000.569
000.000
000.152
024.371
270.330
001.092
000.000
000.152
051.187
548.094
002.015
000.000
000.152
091.030
885.570
Total
0003.676
0000.000
0000.456
0166.588
1703.994
5.2.2 Solution 2
Figure 5.4 shows the topology and operation of the second solution obtained by the expansion
planning algorithm.
36/48
Reliable distribution network planning model report
17
1
2
3
4
A2*
18
5
6
A2*
3
4
6
8
11
10
A2*
12
9
A1*
22
A2
A1
7
8
A2
26
A2
R2
A1
12
9
A2
11
10
12
A1*
A2
A1
22
19
A2
25
11
10
4
TR2*
R2*
A2
6
R2
26
A2
A1*
9
3
A2
5
A2
A2
25
2
R2
18
A2
R2
1
A1
A1
7
A2*
A2*
17
19
A2
5
A2*
26
2
R2*
18
8
A2*
1
A1*
A1*
7
R2*
25
17
19
A1
22
A1*
20
21
23
20
24
21
23
27
20
24
14
15
16
13
14
(a)
24
R1*
A2
A2
13
23
27
27
A2*
21
15
13
16
A1*
(b)
14
15
A1*
16
(c)
Figure 5.4: Solution 2: (a) Stage 1, (b) Stage 2, (c) Stage 3
Table 5.16 shows the total current injections by all transformers in each substation at each load
level of stage . In Table 5.17 the different costs obtained in each stage and total costs are
presented. The present value of the total cost is 96283.703 k$.
Table 5.16: Solution 2. Injected powers by substations [MVA]
Stage 1
Node
25
26
7.20
6.00
4.32
3.60
Stage 2
1.44
1.20
12.00
08.40
10.56
08.40
Stage 3
2.88
2.16
22.80
14.40
20.64
12.24
6.48
3.84
Table 5.17: Solution 2. Costs [k$]
Investment
Maintenance
Production
Losses
Unserved energy
Stage 1
Stage 2
Stage 3
1869.552
0791.207
1106.616
013.650
013.636
170.909
02437.137
04653.408
82754.605
0041.355
0092.376
2339.251
0.00
0.00
0.00
Total
3767.375
198.195
89845.150
2472.982
0.00
Number of interruptions due to repair and reconfiguration that affects every load node is presented
in Table 5.18 and Table 5.19, respectively.
37/48
Reliable distribution network planning model report
Table 5.18: Solution 2. Number of repair interruptions at each node
Node
Stage 1
Stage 2
Stage 3
Node
Stage 1
Stage 2
Stage 3
01
02
03
04
05
06
07
08
09
10
11
12
2
0
0
5
1
2
0
2
0
0
1
0
2
3
0
5
1
2
1
2
1
0
1
1
2
3
4
5
1
2
1
2
1
2
1
1
13
14
15
16
17
18
19
20
21
22
23
24
2
0
0
2
0
0
2
0
3
0
0
2
2
0
0
2
2
0
2
1
3
0
0
2
2
3
3
2
2
1
2
1
3
2
3
2
Table 5.19: Solution 2. Number of reconfiguration interruptions at each node
Node
Stage 1
Stage 2
Stage 3
Node
Stage 1
Stage 2
Stage 3
01
02
03
04
05
06
07
08
09
10
11
12
6
0
0
3
7
6
0
1
0
0
0
0
6
5
0
3
7
6
2
1
0
0
0
2
6
5
4
3
7
6
2
1
0
6
2
3
13
14
15
16
17
18
19
20
21
22
23
24
0
0
0
1
0
0
1
0
5
0
0
1
0
0
0
1
0
0
1
1
5
0
0
1
1
0
1
2
0
1
1
2
5
1
0
2
Table 5.20 presents the customer interruption frequency at each node for each stage.
38/48
Reliable distribution network planning model report
Table 5.20: Solution 2. Customer interruptions frequency index
[interruptions]
Node
Stage 1
Stage 2
Stage 3
Node
Stage 1
Stage 2
Stage 3
01
02
03
04
05
06
07
08
09
10
11
12
3.2
0.0
0.0
3.2
3.2
3.2
0.0
1.2
0.0
0.0
0.4
0.0
3.2
3.2
0.0
3.2
3.2
3.2
1.2
1.2
0.4
0.0
0.4
1.2
3.2
3.2
3.2
3.2
3.2
3.2
1.2
1.2
0.4
3.2
1.2
1.6
13
14
15
16
17
18
19
20
21
22
23
24
0.8
0.0
0.0
1.2
0.0
0.0
1.2
0.0
3.2
0.0
0.0
1.2
0.8
0.0
0.0
1.2
0.8
0.0
1.2
0.8
3.2
0.0
0.0
1.2
1.2
1.2
1.6
1.6
0.8
0.8
1.2
1.2
3.2
1.2
1.2
1.6
Table 5.21 presents the customer interruption duration at each node for each stage.
Table 5.21: Solution 2. Customer interruption duration index
[h]
Node
Stage 1
Stage 2
Stage 3
Node
Stage 1
Stage 2
Stage 3
01
02
03
04
05
06
07
08
09
10
11
12
2.2
0.0
0.0
4.3
1.5
2.2
0.0
1.7
0.0
0.0
0.8
0.0
2.2
2.9
0.0
4.3
1.5
2.2
1.0
1.7
0.8
0.0
0.8
1.0
2.2
2.9
3.6
4.3
1.5
2.2
1.0
1.7
0.8
2.2
1.0
1.1
13
14
15
16
17
18
19
20
21
22
23
24
1.6
0.0
0.0
1.7
0.0
0.0
1.7
0.0
2.9
0.0
0.0
1.7
1.6
0.0
0.0
1.7
1.6
0.0
1.7
0.9
2.9
0.0
0.0
1.7
1.7
2.4
2.5
1.8
1.6
0.9
1.7
1.0
2.9
1.7
2.4
1.8
The results for the rest of reliability indexes are shown in Table 5.22.
Table 5.22: Solution 2. Reliability indexes
Reliability
index
Stage 1
Stage 2
Stage 3
02.251
02.004
99.977
13.826
01.913
01.757
99.980
25.840
01.969
01.838
99.979
48.047
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Reliable distribution network planning model report
The different reliability costs obtained for this solution are presented in Table 5.23.
Table 5.23: Solution 2. Reliability costs
Reliability
Stage 1
cost
000.564
000.000
000.152
024.371
270.089
Stage 2
Stage 3
Total
001.042
000.000
000.152
000.000
521.784
001.931
000.000
000.152
000.000
857.891
0003.537
0000.000
0000.456
0024.371
1649.764
5.2.3 Solution 3
Figure 5.5 shows the topology and operation of the third solution obtained by the expansion
planning algorithm.
17
1
2
3
A2*
18
5
6
1
18
8
A2*
2
6
8
11
10
A2*
20
5
6
9
A2
A1*
A1*
23
19
7
8
A2
R2
A1
A1
25
26
A2
R2
R2
12
22
21
4
A2
A1
26
3
A2
R2*
9
2
R2
A2
A1
25
1
18
A2
A2*
A2*
17
A1
7
R2
A1*
26
4
19
A2
5
A2*
A1*
3
R2*
A1*
7
R2*
25
17
4
19
A2
20
24
11
10
23
A1
22
A1*
21
23
A2*
24
R1*
27
A1*
A2
A2
12
A1*
20
24
11
10
A2
27
A2*
A1
9
A2
A1
22
21
27
12
T R2*
A1*
13
14
16
15
13
14
(a)
13
16
15
(b)
14
15
16
(c)
Figure 5.5: Solution 3: (a) Stage 1, (b) Stage 2, (c) Stage 3
Table 5.23 shows the total current injections by all transformers in each substation at each load
level of stage . In Table 5.24 the different costs obtained in each stage and the total costs are
presented. The present value of the total cost is 96290.823 k$.
Table 5.24: Solution 3. Injected powers by substations [MVA]
Stage 1
Node
25
26
27
7.20
6.00
-
4.32
3.60
-
Stage 2
1.44
1.20
-
12.00
08.40
-
10.56
08.40
-
Stage 3
2.88
2.16
-
14.40
14.40
08.40
12.24
12.24
08.40
3.84
3.84
2.64
Table 5.25: Solution 3. Costs [k$]
Investment
Maintenance
Production
Losses
Unserved energy
Stage 1
Stage 2
Stage 3
1890.484
548.837
1405.394
13.650
13.636
170.909
2437.137
4653.408
82754.605
39.822
99.349
2263.592
0.00
0.00
0.00
Total
3844.715
198.195
89845.15
2402.763
0.00
40/48
Reliable distribution network planning model report
Number of interruptions due to repair and reconfiguration that affects every load node is presented
in Table 5.26 and Table 5.27, respectively.
Table 5.26: Solution 3. Number of repair interruptions at each node
Node
Stage 1
Stage 2
Stage 3
Node
Stage 1
Stage 2
Stage 3
01
02
03
04
05
06
07
08
09
10
11
12
2
0
0
5
1
2
0
2
0
0
1
0
2
3
0
5
1
2
1
2
3
0
1
1
2
3
4
5
1
2
1
2
3
2
1
1
13
14
15
16
17
18
19
20
21
22
23
24
2
0
0
2
0
0
2
0
3
0
0
2
2
0
0
2
2
0
2
1
3
0
0
2
2
2
3
2
2
1
2
1
1
2
3
2
Table 5.27: Solution 3. Number of reconfiguration interruptions at each node
Node
Stage 1
Stage 2
Stage 3
Node
Stage 1
Stage 2
Stage 3
01
02
03
04
05
06
07
08
09
10
11
12
6
0
0
3
7
6
0
2
0
0
0
0
7
6
0
4
8
7
1
2
6
0
0
3
4
3
2
1
5
4
1
4
1
2
1
5
13
14
15
16
17
18
19
20
21
22
23
24
0
0
0
2
0
0
0
0
5
0
0
2
0
0
0
2
0
0
0
1
6
0
0
2
0
2
3
4
0
1
0
1
3
0
3
4
Table 5.28 presents the customer interruption frequency at each node for each stage.
41/48
Reliable distribution network planning model report
Table 5.28: Solution 3. Customer interruptions frequency index
[interruptions]
Node
Stage 1
Stage 2
Stage 3
Node
Stage 1
Stage 2
Stage 3
01
02
03
04
05
06
07
08
09
10
11
12
3.2
0.0
0.0
3.2
3.2
3.2
0.0
1.6
0.0
0.0
0.4
0.0
3.6
3.6
0.0
3.6
3.6
3.6
0.8
1.6
3.6
0.0
0.4
1.6
2.4
2.4
2.4
2.4
2.4
2.4
0.8
2.4
1.6
1.6
0.8
2.4
13
14
15
16
17
18
19
20
21
22
23
24
0.8
0.0
0.0
1.6
0.0
0.0
0.8
0.0
3.2
0.0
0.0
1.6
0.8
0.0
0.0
1.6
0.8
0.0
0.8
0.8
3.6
0.0
0.0
1.6
0.8
1.6
2.4
2.4
0.8
0.8
0.8
0.8
1.6
0.8
2.4
2.4
Table 5.29 presents the customer interruption duration at each node for each stage.
Table 5.29: Solution 3. Customer interruption duration index
[h]
Node
Stage 1
Stage 2
Stage 3
Node
Stage 1
Stage 2
Stage 3
01
02
03
04
05
06
07
08
09
10
11
12
2.2
0.0
0.0
4.3
1.5
2.2
0.0
1.8
0.0
0.0
0.8
0.0
2.3
3.0
0.0
4.4
1.6
2.3
0.9
1.8
3.0
0.0
0.8
1.1
2.0
2.7
3.4
4.1
1.3
2.0
0.9
2.0
2.5
1.8
0.9
1.3
13
14
15
16
17
18
19
20
21
22
23
24
1.6
0.0
0.0
1.8
0.0
0.0
1.6
0.0
2.9
0.0
0.0
1.8
1.6
0.0
0.0
1.8
1.6
0.0
1.6
0.9
3.0
0.0
0.0
1.8
1.6
1.8
2.7
2.0
1.6
0.9
1.6
0.9
1.1
1.6
2.7
2.0
The results for the rest of reliability indexes are shown in Table 5.30.
Table 5.30: Solution 3. Reliability indexes
Reliability
index
Stage 1
Stage 2
Stage 3
02.361
02.031
99.977
13.950
02.423
01.986
99.977
27.700
01.844
01.795
99.980
44.655
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The different reliability costs obtained for this solution are presented in Table 5.31.
Table 5.31: Solution 3. Reliability costs
Reliability
Stage 1
cost
000.569
000.000
000.152
024.371
270.330
Stage 2
Stage 3
Total
001.117
000.000
000.202
051.187
548.881
001.798
000.000
000.051
000.000
851.721
0003.484
0000.000
0000.405
0075.558
1670.932
5.2.4 Comparative analysis
Table 5.32 shows a summary of investment, maintenance, production, losses, and unserved
energy costs of each solution.
Table 5.32: Summary of operating and investment costs
Costs
Solution 1
Solution 2
Solution 3
Investment
Maintenance
Production
Losses
Unserved energy
Total
03548.032
00198.195
89845.150
02612.011
00000.000
96203.388
03767.375
00198.195
89845.150
02472.982
00000.000
96283.703
03844.715
00198.195
89845.150
02402.763
00000.000
96290.823
Table 5.33 shows a summary of costs associated with the reliability.
Table 5.33: Summary of reliability costs
Total Reliability Cost Solution 1 Solution 2 Solution 3
0003.676
0000.000
0000.456
0166.588
1703.994
0003.537
0000.000
0000.456
0024.371
1649.764
0003.484
0000.000
0000.405
0075.558
1670.932
The obtained solutions are compared from the distribution company standpoint, not considering for
the comparison the CIC index. Data presented in Table 5.32 show that cost of losses of Solution 3
is lower than cost of losses Solution 2, the investment cost of Solution 3 is greater than investment
cost of Solution 2 and the difference is higher than the previous one. Thus, the total operating and
investment cost of Solution 2 is lower than the one obtained in Solution 3. Analyzing the data
presented in Table 5.33, it can be observed that
is lower for Solution 2, while EENSC, CIFC
and CIDC are similar. In conclusion, Solution 2 is better than Solution 3 since their associated total
costs are lower.
Solution 1 is compared with the previous better solution, that is, Solution 2. In Table 5.32 it can be
seen that cost of losses Solution 2 are lower than the cost of losses Solution 1, while the
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investment cost of Solution 2 is greater than the investment cost of Solution 1. The total costs
associated to Solution 1 are lower than those obtained for Solution 2. Analyzing reliability costs, as
defined in Table 5.33, it can be observed that
is significantly lower for Solution 2 while
EENSC, CIFC and CIDC are similar. Therefore, it might be assessed that, from the distribution
company standpoint, Solution 2 minimizes the investment plan while complying with reliability
criteria.
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6 Summary, conclusions, and future work
This section begins with a summary of the work performed. Then conclusions obtained are
presented. Finally, some future works are proposed.
6.1 Summary
In this work, a distribution expansion planning algorithm has been presented in a centralized view
point. In this algorithm, an optimization model calculates three expansion solutions, among which is
the optimal one. The reliability indexes and their associated costs have been calculated for the
previous set of solutions. Finally, an investment decision has been adopted through a comparative
analysis.
It has been considered the network expansion through the installation of feeders, transformers, and
substation. This allows the distribution company to obtain the optimal strategy to meet a rise in
demand.
The optimization model has been mathematically formulated as a mathematical programming
problem. The objective function consists of costs associated with investment, maintenance,
supplied energy, energy losses, and non-served energy. The minimization of the objective function
is subject to technical, economic, and balance constraints. The planning horizon is three years
divided into yearly stages.
To obtain a particular number of solutions with the optimization problem, non-repeatability
constraints are used, which require the solutions to have a number of differences in order to obtain
different topologies to analyze their reliability.
Initially, the optimization problem has been described. Furthermore, linearizations used to
formulate the problem as a mixed-integer linear programming problem have been presented. Then,
reliability indexes and their associated costs have been defined. After that, the expansion planning
algorithm, which includes the previous optimization problem and the reliability calculations, has
been depicted. The methodology proposed has been illustrated with a study cases composed of a
distribution system of 27 nodes and 39 branches.
6.2 Conclusions
The model presents not only a single plan for the decision-maker but also a set of diversified plans
with different associated costs, topologies, and features. Besides, the most relevant reliability
indices are computed for each solution selected and their associated costs are calculated.
Regulated distribution utilities should strike for an optimal balance between their investment and
O&M costs on one hand and the quality of supply on the other. The regulator may provide financial
rewards or penalties for the DISCOS according to the established reliability objectives. The
obtained results show that significant differences can occur in the cost incurred by a DISCO when
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Reliable distribution network planning model report
the regulator applies penalties for not meeting the imposed reliability levels. It can be stated that
the optimal solution obtained from the distribution expansion planning algorithm may not be the
most cost-effective including the reliability costs in the decision making process. This effect can be
observed in the proposed case study.
Another conclusion is that the construction of new substations improves the reliability indexes. The
number of circuits increases, leading to a reduction of the number of feeders forming each circuit.
Then, a particular fault in a particular feeder affects to a smaller number of feeders.
6.3 Future work
The analysis of this work allows proposing the next future work:
1) Add to the optimization problem the possibility of installing distributed generation. It can be
conventional generators as thermal units or renewable generators as wind, hydro or solar
technology.
2) Deploy a comprehensive set of scenarios to account for all sources of uncertainty in the
model.
3) Use a particular set of constraints to conserve the distribution networks radiality when
distributed generation is considered.
4) Joint distributed generation and distribution network expansion planning with demand
response, reserves, hybrid storage, and plug-in vehicles.
The first three topics will be analyzed in Deliverable 7.2 “Joint RES generation and distribution
network expansion model for insular networks”. Deliverable 7.3 will complete the model delivered
in Deliverable 7.2 by adding other issues relevant to planning in insular distribution systems such
as demand response, treatment of the reserves, hybrid storage technologies and plug-in vehicles.
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Reliable distribution network planning model report
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