Probabilistic Dynamic Multi-objective Model for Renewable and

Non-renewable DG Planning

Alireza Soroudi∗,a,b, Raphael Cairea, Nouredine Hadjsaida, Mehdi Ehsanb

aGrenoble Electrical Engineering Laboratory (G2Elab), Saint Martin d’Heres, France.bElectrical Engineering Department of Sharif University of Technology, Tehran, Iran.

Abstract

This paper proposes a probabilistic dynamic model for multi-objective distributed gen-

eration planning which also considers network reinforcement at presence of uncertainties

associated to the load values, generated power of wind turbines and electricity market

price. Monte Carlo simulation is used to deal with the mentioned uncertainties. The

planning process is considered as a two-objective problem. The first objective is the min-

imization of total cost including investment and operating cost of DG units, the cost

paid to purchase energy from main grid and the network reinforcement costs. The sec-

ond objective is defined as the minimization of technical risk, including the probability

of violating the safe operating technical limits. The Pareto optimal set is found using

NSGA-II method and the final solution is selected using a max-min method. The model

is applied on two distribution networks and compared with other models to demonstrate

its effectiveness.

Key words: Stochastic uncertainty modeling, Distributed Generation, wind turbine,

Pareto optimality, Technical risk, Monte Carlo Simulation

1. Introduction

1.1. Motivation

The role of Distributed Generation (DG) units has become much more important with

the deregulation of power industry. These units have been become an interesting option

for Distribution Network Operators (DNOs) to meet the requirements of their customers.

∗Corresponding authorEmail address: alireza.soroudi@gmail.com (Alireza Soroudi)

Journal

Investments in distributed generation enhances the environmental and technical benefits of

DNOs [1]. Thus, there is a clear need to enhance the current DG planning methodologies

to include an appropriate treatment of various DG technologies and uncertainty handling.

This need motivates the work proposed in this paper.

1.2. Literature review

The existing works of the literature dealing with DG planning problem have consid-

ered many of its technical and economical aspects. These aspects are as follows: emission

reduction [2], active loss reduction [3, 4], reducing the cost of curtailed energy [5], in-

creasing the reliability of power supply [6], voltage profile improvement [7, 8], reducing

the risk of overloading the distribution feeders [9], maximizing the DG penetration level

[10], enhancing the social sustainability [11], reducing the construction period [12] and

reducing the cost of energy purchased from power market [13]. The DG planning prob-

lem has a multi-objective structure. The Pareto optimality concept can be used to deal

with the multi objective problems in which the objective functions are incommensurate

(see Appendix for more details). The multi-objective models proposed in the literature

are either static or dynamic. The static models assume that all investments are done at

the beginning of the planning horizon while the dynamic ones consider the timing of the

investment decisions. The importance of dynamic planning is that the value of money

changes with time and if DG units can postpone some network investment. In [5], a static

multi-objective model is proposed which considers the cost of investment and operation,

cost of energy purchased, cost of energy losses and the cost of energy not supplied. This

model does not consider the uncertainties. The static models of the literature which con-

sider the uncertainties are either use fuzzy arithmetic [14] or probabilistic modeling [15]

or a mixed probabilistic-possibilistic method [16]. In [13], a dynamic model is proposed

for integration of DG units in distribution networks but it does not consider the reinforce-

ment of network along with DG investment. The authors have already proposed dynamic

models [2, 9], which considers simultaneous DG and network investment but do not deal

with the uncertainties associated to the DG planning problem. The gap that this paper

tries to fill is considering the uncertainties of input variables in addition to simultaneous

consideration of DG and network investment.

2

1.3. Contributions

This paper proposes a multi-objective model for determining the optimal sizing, locat-

ing and investment timing in DG units and network which is not only dynamic but also

considers the presence of uncertainties associated with different parameters. The Monte

Carlo simulation is used to deal with the effects of uncertainties of stochastic wind power

generation, electricity market price and load values on technical and economic issues of

the planning problem.

1.4. Paper organization

In Section 2, the problem formulation is described. The proposed solution algorithm

is presented in section 3. In section 4, the proposed model is applied on a distribution

system and the simulation results are given and discussed. Finally, the conclusions are

given in section 5.

2. Problem formulation

2.1. Decisions variables

The decision variables are the number of non-renewable DG units and wind turbines,

to be installed in bus i in year t, i.e. ξdgi,t ,ξwi,t, respectively; the binary investment decision

in feeder ℓ in year t, i.e. γℓt , and finally the number of installed HV/MV transformers in

year t, i.e. ψtrt . The constraints and the objective functions are described as follows:

2.2. Uncertainty modeling

In this work three major uncertain parameters including load value, wind turbine

generated power and the energy price from the main grid are considered. The modeling

of these parameters is described as follows:

2.2.1. Load

The daily load variation over the long-term is modeled as a load duration curve with

Nh demand levels where in each demand level, i.e. h, a DLFi,t,h is associated which shows

its average magnitude. The uncertainty of DLFi,t,h is modeled using a normal probability

density function (PDF). Assuming a base load of SDi,base in the first year of the planning

3

horizon and a demand growth rate, i.e. α, the apparent demand in bus i, in year t and

in demand level h, in each Monte Carlo experiment, i.e. e, is calculated as follows:

SD,ei,t,h = SD

i,base ×DLF ei,t,h × (1 + α)t (1)

Each DLF ei,t,h is calculated as follows:

DLF ei,t,h = µD

i,t,h + λD,ei,t,h × σD

i,t,h (2)

where λD,ei,t,h is a random variable generated for demand in bus i, in year t, demand level

h and each Monte Carlo experiment, i.e. e, using a normal PDF with a mean of 0 and a

standard deviation of 1. The forecasted values of demand level factors and their standard

deviations are µDi,t,h and σD

i,t,h respectively. For simplicity, it is assumed that all buses

follow the same load duration curve but it would not affect the generality of the proposed

framework.

2.2.2. Wind turbine generation

The generation schedule of a wind turbine highly depends on the wind speed in the

site. The variation of wind speed, i.e. v, can be modeled using a Weibull PDF and its

characteristic function which relates the wind speed and the output of a wind turbine

[16].

PDF (v) = (k

c)(v

c)k−1exp(−(

v

c)k) (3)

where k is the shape factor and c is the scale factor of the Weibull PDF of wind speed in

the zone under study.

The generated power of the wind turbine is determined using its characteristics as follows:

Pwi,t,h =

0 if v ≤ vcutin or v ≥ vcutout

v−vcutin

vrated−vcutin

Pwi,r if vcutin ≤ v ≤ vrated

Pwi,r else

(4)

where Pwi,r is the rated power of wind turbine installed in bus i, Pw

i,t,h is the generated

power of wind turbine in bus i and demand level h, vcutout is the cut out speed, vcutin is the

cut in speed and vrated is the rated speed of the wind turbine.

4

2.2.3. Electricity market price

The price of electricity purchased from the main grid is determined by market oper-

ation. This value changes during each demand level. The variation of this quantity is

modeled by a peak price, i.e. ρ, multiplied by a factor named Price Level Factor, i.e.

PLFt,h. Price Level Factors are uncertain due to the behaviors of the market players.

The spot price is considered to follow normal PDF [17], as follows:

PLF et,h = µρ

t,h + λρ,et,h × σρt,h (5)

where µρt,h and σρ

t,h are the forecasted value of price level factor and its standard deviation,

respectively. λρ,et,h is a random variable generated for electricity price in year t, in demand

level h and Monte Carlo experiment, i.e. e, using a normal PDF with a mean of 0 and a

standard deviation of 1.

2.3. Constraints

The constraints that should be satisfied in each Monte Carlo experiment are described

in this section, as follows:

2.3.1. Power flow equations

The power flow equations must be satisfied in each Monte Carlo experiment, i.e. e, in

each demand level h and year t, is as follows:

−PD,ei,t,h +

Ndg∑

dg=1

P dgi,t,h +

Nw∑

w=1

Pw,ei,t,h = V e

i,t,h

Nb∑

j=1

Y tijV

ej,t,hcos(δ

ei,t,h − δej,t,h − θtij) (6)

−QD,ei,t,h +

Ndg∑

dg=1

Qdgi,t,h +

Nw∑

w=1

Qw,ei,t,h = V e

i,t,h

Nb∑

j=1

Y tijV

ej,t,hsin(δ

ei,t,h − δej,t,h − θtij)

where PD,ei,t,h, Q

D,ei,t,h, , are the active and reactive power demand, respectively. The P dg

i,t,h,

Qdgi,t,h are the active and reactive power generated by DG units, respectively. The Pw,e

i,t,h,

Qw,ei,t,h are the active and reactive power generated by wind turbine units in Monte Carlo

experiment e, respectively.

P grid,et,h = Vslack,t,h

Nb∑

j=1

Y tijV

ej,t,hcos(δ

ei,t,h − δej,t,h − θtij) (7)

Qgrid,et,h = Vslack,t,h

Nb∑

j=1

Y tijV

ej,t,hsin(δ

ei,t,h − δej,t,h − θtij)

5

where P grid,et,h and Qgrid,e

t,h are the active and reactive power imported from the grid, in year

t and demand level h for Monte Carlo experiment e.

2.3.2. Operating limits of DG units

Each DG should be operated considering its resource limits, i.e.:

P dgi,t,h ≤

t∑

year=1

ξdgi,year × Pdg

lim (8)

Pdg

lim is the operating limit of dg technology in MW.

The power factor of DG unit is kept constant in all demand levels [18] as follows:

cosϕdg = Const. (9)

2.4. Objective functions

The objective functions are formulated and described in this section, as follows:

2.4.1. Technical Risks

The first objective function is the technical risk. In each Monte Carlo iteration, the

following two operating limits are checked: voltage limits and thermal limits of feeders.

If any of these limits is violated it is considered as a technical risk in the network. These

limits are explained as follows: The voltage of each bus in each demand level h and in each

year t and Monte Carlo experiment e, should be kept within the safe operating limits.

Vmin ≤ V ei,t,h ≤ Vmax (10)

Vmin and Vmax are the minimum and maximum permissible limits of voltage, respectively.

To maintain the security of the feeders and substation, the flow of current passing through

them should be kept below their thermal limit. The thermal limits of existing feeders and

substation are Iℓ and Str, respectively where these limits change if any investment is done

in feeder ℓ or substation transformer until year t. This constraint is described as follows:

Ieℓ,t,h ≤ Iℓ + Capℓ ×t

∑

year=1

γℓyear (11)

Sgrid,et,h ≤ Str + Captr ×

t∑

year=1

ψtryear

6

where Capℓ, Captr are Capacity limit of added feeders and transformers, respectively.

The number of experiments in which at least one of the two constraints (10) or (11) is

violated, is recorded as NRt,h. The technical risk in year t, denoted by TRt, is defined

as the average of technical risk over different demand levels. Here, the duration of each

demand level, i.e. τh, is used as a weighting factor as follows:

TRt =

∑Nh

h=1NRt,h

NEt,h× τh

∑Nh

h=1 τh(12)

The objective function to be minimized is proposed here as the weighted average of

maximum yearly technical risk and its average value over the planning horizon as:

OF1 = w1 maxt

(TRt) + w2

T∑

t=1

TRt

T(13)

where w1,2 will be specified by the planner in order to control the importance of sever-

ity and average value of technical risks. By minimizing the OF1, the algorithm tries to

simultaneously improve the overall satisfaction of the technical network constraints, rep-

resented by 1T

∑Tt=1 TRt, and the severity of technical dissatisfaction over the planning

horizon, represented by maxt(TRt).

2.4.2. Total costs

The next objective function, i.e. OF2, to be minimized is the total costs including the

cost of electricity purchased from the main grid, investment and operating costs of the

DG units and also reinforcement costs of distribution network. As it will be described in

section 3.1, in each year t and each demand level h, Monte Carlo simulation is done for

NEt,h experiments and cost of purchasing electricity from the grid in demand level h and

year t, in Monte Carlo experiment e, is determined as follows:

GCet,h = PLF e

t,h × ρ× P grid,et,h × τh (14)

where τh is the duration of demand level h.

The expected value of GCt,h until experience, e, is calculated as follows:

GCe

t,h =

∑em=1GC

mt,h

e(15)

7

The number of required experiments for convergence of GCe

t,h is known as NEt,h then

total grid cost, i.e. TGC, is calculated as follows:

TGC =T∑

t=1

Nh∑

h=1

GCNEt,h

t,h × (1

1 + d)t (16)

where d is the discount rate and T is the planning horizon. The total operating costs of

the DG units, i.e. DGOC, is calculated as:

DGOC =T∑

t=1

Nb∑

i=1

Ndg∑

dg=1

Nh∑

h=1

τh × (OCw × Pwi,t,h +OCdg × P dg

i,t,h)× (1

1 + d)t (17)

where OCdg and OCw are the operating cost of non-renewable and wind DG technologies,

respectively. P dgi,t,h is the active power generated by DG unit in bus i, year t and demand

level h. Ndg is the number of DG technologies considered for planning.

The total investment cost of the DG units, i.e. DGIC, is calculated as:

DGIC =T∑

t=1

Nb∑

i=1

Ndg∑

dg=1

(ξwi,t × ICw + ξdgi,t × ICdg)× (1

1 + d)t (18)

where ICw, ICdg are the investment cost of non-renewable and wind DG technologies,

respectively. The number of buses in the network is denoted by Nb.

The reinforcement cost of the distribution network is the sum of all costs paid for installa-

tion and operation of new feeders and transformers. The total feeder reinforcement cost,

i.e. LC, and substation reinforcement cost, i.e. SC, are calculated as follows:

LC =T∑

t=1

Nℓ∑

ℓ=1

Cℓ × Lℓ × γℓt × (1

1 + d)t (19)

SC =T∑

t=1

Ctr × ψtrt × (

1

1 + d)t

where Cℓ is the cost of feeder reinforcement for feeder ℓ in $/km and Ctris the cost of

transformer investment. Lℓ is the length of newly added feeder in km.

Thus, OF2 is defined as:

OF2 = LC + SC +DGIC +DGOC + TGC (20)

It should be noted that in (20), the mean values of TGC and DGOC are added to invest-

ment cost of DG units and also the network reinforcement costs.

8

3. The proposed method

In this section, first the concept of Monte Carlo Simulation method is described and

then the fundamentals of multi-objective optimization and the solution methods are given,

as follows:

3.1. Monte Carlo simulation

The Monte Carlo Simulation (MCS) is a tool for simulating the behavior of uncertain

parameters which have probabilistic nature. This means there exists a Probability Density

Function (PDF) that describes the behavior of these parameters. The main concept

of MCS method is described as follows: suppose a multi variable function, namely y,

y = f(x1, ..., xn), in which x1 to xn are random variables with their own PDF. The

problem is, knowing the PDFs of all input variables, i.e. x1 to xn, how the PDF of y can

be obtained? The concept of MCS is obtaining the PDF of ye using the PDFs of input

variables xi using the following steps [19]:

Step 1. e = 1, Avg = {}.

Step 2. For each input variable xi, generate a value, i.e. xei , using its PDF.

Step 3. Calculate ye using ye = f(xe1, ..., xen)).

Step 4. Calculate Ye =1e

∑em=1 ym

Step 5. Store Ye in Avg.

Step 6. Check if Ye is converged then go to step (7); else e = e+ 1 then go to step (2).

Step 7. End

At the end of these steps, the PDF of the output function, y, is estimated as a normal

PDF [20] with a mean and standard deviation calculated as follows:

µy = µYe(21)

σy =

√

∑em=1 (ym − µy)2

e

3.2. The two stage solution method

As it is already described in Section 1.1, the DG planning problem has a multi-objective

structure. In multi-objective problems, the decision maker (planner) tries to find Pareto

9

optimal set of solutions (please see Appendix). To solve the problem formulated in section

2, a two-stage solution algorithm is proposed, in which the first stage finds the Pareto

optimal front and in second stage helps the planner to find the best solution, as follows:

3.2.1. Stage I: finding Pareto optimal front using NSGA-II

This work uses the Non-dominated Sorting Genetic Algorithm (NSGA-II) proposed

in [21]. The mechanism of this algorithm is described as the following steps:

Step 1. Randomly create an initial population with the size Np.

Step 2. Using the crossover and mutation operators of GA [22] on the current population

creat a population with the same size. Construct the mixed population with the

size of 2Np.

Step 3. Calculate OF1 and OF2 for each solution using (1) to (20).

Step 4. Find the Pareto optimal front for each solution, i.e. Xn using (27) and (28).

Step 5. Calculated the crowding distance, i.e. CDn, using (22).

CDn =

NO∑

k=1

LDkn

NO

(22)

where LDkn is the local diversity of solution n in Pareto front k. It is defined

as the distance between the given solution and its nearest neighbors in a given

Pareto front, as follows:

LDkn =

|fk(Xn+1)− fk(Xn)|+ |fk(Xn)− fk(Xn−1)|

2MDk

(23)

where Xn+1 and Xn−1 are the two solutions in neighborhood of Xn, when all

of the solutions are sorted, regarding the objective function k. The maximum

distance of solutions in the Pareto front is obtained as follows:

MDk =2NPmaxn=1

(fk(Xn))−2NP

minn=1

(fk(Xn)) (24)

In order to maintain the diversity of solutions in Pareto optimal front, in minimum

and maximum solutions of the given front will be assigned with the biggest local

diversity, calculated as follows:

LDk2NP

= LDk1 = max

n(LDk

n) (25)

10

Step 6. Generate the next population using the solutions of lower fronts up to size Np. If

all solutions are from the same front, choose the solution with the lower CDn.

Step 7. iter=iter+1.

Step 8. If iter < itermax go to Step 2, else go to Step 9.

Step 9. End

3.2.2. Stage II: Final solution selection

The next step after finding the Pareto optimal solutions, is to select the best solution.

In this paper, max-min method [14, 23] is used. For each solution Xn, the value of

fmaxk

−fk(Xn)

fmaxk

−fmink

is calculated which shows the ability of solution Xn in minimizing objective

function fk.NPmaxn=1

{

NO

mink=1

{

fmaxk − fk(Xn)

fmaxk − fmin

k

−µrefk

}}

(26)

where fmaxk and fmin

k are maximum and minimum values of the objective function k,

in solutions of Pareto optimal set. The µrefk is the minimum required satisfaction for

objective function k which is determined by the planner according to its requirements.

This means that the ability of each solution in minimizing every objective function is

checked and then it is compared to the minimum satisfaction requirement of the planner

thus the solution which its minimum success in minimizing all objectives is maximum will

be chosen as the best solution.

4. Simulation results

In this section, the proposed methodology is applied to two distribution networks.

The first one is a 9-bus test system and the second one is a real large scale 574-node

distribution network.

4.1. Assumptions

The forecasted load and price duration curve are depicted in Fig. 4. The dura-

tion of each demand level, τh, is assumed to be 365 hr. The other simulation as-

sumptions are presented in Table 1. The proposed framework enables the planner to

consider various DG technologies. In this paper, without loss of generality, only gas

and wind turbines are considered. The gas turbine (GT) has the investment cost, i.e.

11

ICdg = 500000$/MV A, and operating cost, i.e. OCdg = 50$/MWh [24] and wind tur-

bines (WT) have ICw = 1227000$/MV A, OCw = 45$/MWh with the capacity of 1 MVA.

The capacity of substation transformers, i.e. Captr, that can be added to the substation

is 10 MVA and the cost of each transformer, i.e. Ctr, is 0.2M$. The cost of reinforcement

for each feeder, i.e. Cℓ is 0.15M $/km[24]. The planning horizon is assumed to be 8 years.

4.2. Case-I: 9-bus distribution network

The proposed method is applied on a 132/33 kV 9-bus distribution network which is

shown in Fig.3. The technical data of this network can be found in [2].

4.2.1. Choosing the best solution

• In this case, the best solution is found assuming that µrefk=1,2 = 0, this means that

none of the objective functions is more important than the other one. The best

solution is found using the (26) and the value of objective functions for this solution

is OF1 = 0.11 and OF2 = 6.73 × 107$. The planning scheme for this solution is

given in Table 2.

• In this case, it is assumed that one of the objective functions is more important

for the planner. For example, the planner is looking for a solution in which the

technical risk is more minimized. The best solution is found assuming that µrefk=1 =

0.9, µrefk=2 = 0, this means that the best solution will be chosen from the part of

Pareto front which its solutions have more than 90% satisfaction in minimizing the

OF1. The best solution is found using the (26) and the value of objective functions

for this solution is OF1 = 0.02 and OF2 = 6.90 × 107$. The planning scheme for

this solution is given in Table 3.

4.2.2. Comparing the proposed model with other methods

The proposed dynamic model is coded in MATLAB and solved using the NSGA-II

method, Simulated Annealing (SA) [25] and also Particle Swarm Optimization (PSO) [26]

and static model. The static model is totally similar to the dynamic model except that all

investments are obliged to be done at the beginning of the planning horizon. The number

of Pareto optimal solutions found by the proposed model is 16 where the variation ranges

12

of objective functions are given in Table 4. The Pareto optimal front obtained by each

model is depicted in Fig. 5. The solutions found by the proposed model dominate the

solutions of other methods. This means that for every solution in other fronts there exists

at least one solution in the Pareto optimal front of the proposed model which has the less

objective functions compared to it.

4.3. Case-II: 574-node real distribution network

The second case is a 20-kV, 574-node distribution system, depicted in Fig.6, which

is extracted from a real French urban network. This system has 573 sections with total

length of 52.188km, and 180 load points. This network is fed through one substation.

These data have been extracted from reports of Electricite de France (EDF) [27] and

more details can be found in [28]. The Pareto optimal front is found using the method

described in Section 3.2 and depicted in Fig.7. The objective functions of each solution are

described in Table 6. Using the (26) the best solution is #10 since it has the maximum

of minimum minimizing satisfaction (according to Table 6. The characteristics of this

solution are given in Table 7 and Table 8. In Table 7, the investment costs including DG

and network investments are specified as well as the timing of the investment decision. The

comparison between the selected solution and the solution #1 (no technical risk) shows

that for decreasing teh technical risk from 0.0063 (in solution # 10) to 0 (in solution #

1) the planner should pay 112507469.7004 − 96867855.5489 = 1.563 × 107$ more. This

clearly shows the benefit of the proposed algorithm. In Table 8, the timing of investment

and also the optimal location of DG units are given for both of the wind and gas turbine

technologies.

4.4. Advantages and disadvantages of the proposed method

4.4.1. Advantages:

The advantages of the proposed solution method are as follows:

• It determines the optimal timing, location and DG technology for a given distribu-

tion network.

13

• The uncertainties of renewable DG technologies (like wind turbine), electric loads

and electricity prices are modeled and their impacts have been investigated.

• The Pareto optimal front found by this method dominates the Pareto front found

by other methods.

• The algorithm can help the planner to determine how much should be paid for

reducing the technical risks.

4.4.2. Disadvantages:

One of the major drawbacks of the proposed method is its computation burden which

increases with the size of the network and also the number of uncertain variables. Al-

though it is always interesting to increase the computational performance of the algorithm

but since the planning models are usually off line, this would cause no severe problem.

Another method for handling the uncertainties is scenario based modeling [29]. If the

number of scenarios is too high then scenario reduction method can be applied to reduce

the computational burden while maintaining the accuracy [30, 31].

5. Conclusion

The proposed planning model considers the DG option and network investment si-

multaneously and provides the DNO a set of Pareto optimal solutions. The dynamic

nature of the proposed model and its ability to model the uncertainties associated to the

planning problem makes it suitable to be used in practical applications. The flexibility

of the proposed probabilistic model enables it to consider other sources of uncertainties

and other objective functions. The performance of the proposed model is assessed by

applying it on a distribution network. The numerical results and comparison show that

the proposed method is promising. The future work will be focused on decreasing the

computational burden of the proposed model.

Appendix

The multi-objective problems can be generally described as follows [13]:

14

min F (X) = [f1 (X) , ..., fNO(X)] (27)

Subject to:

{G (X) = 0, H (X) ≤ 0}

X = [x1, · · · , xm]

Where NO is the number of objectives.

Suppose X1 and X2 belong to the solution space. X1 dominates X2 if:

∀k ∈ {1...NO} fk (X1) ≤ fk (X2) (28)

∃k′ ∈ {1...NO} fk′ (X1) < fk′ (X2)

All solutions which are not dominated by any other solution is a member of first Pareto

front or optimal front or non-dominated front. To find the remaining Pareto fronts, all

solutions of first front are discarded and the same process is repeated for the remaining

solutions.

List of Symbols and Abbreviations

Indices

i Bus

h Demand level

e Monte Carlo experiment

V ariables

SD,ei,t,h Apparent load in bus i, year t, demand level h and Monte Carlo experiment e

PDi,base Apparent base load in bus i, year t=1

Pwi,t,h Active power of wind turbine in bus i, year t and demand level h

Sgrid,et,h Apparent power magnitude passing through substation, in year t, demand level h

and Monte Carlo experiment e

15

Iℓ Capacity limit of existing feeder ℓ

Capℓ Capacity limit of potential feeder ℓ

Str Capacity limit of existing transformer

Captr Capacity limit of potential transformer

Cℓ Cost of reinforcement for feeder ℓ in $/km

Ctr Cost of investment in transformer

GCet,h Cost of energy purchased from grid in year t and demand level h and Monte Carlo

experiment e

Ieℓ,t,h Current magnitude passing through feeder ℓ, in year t, demand level h and Monte

Carlo experiment e

CDn Crowding distance of solution n

vcutout Cut out speed of the wind turbine

vcutin Cut in speed of the wind turbine

d Discount rate

τh Duration of demand level h in hours.

DLF ei,t,h Demand level factor in Monte Carlo experiment e, in bus i, year t and demand level

h.

µDi,t,h Forecasted mean value of demand level factor in bus i, year t and demand level h.

µρt,h Forecasted mean value of price level factor in year t and demand level h.

ICdg/w Investment cost of non-renewable/renewable(wind) DG units

γℓt Investment decision in feeder ℓ and year t

ψtrt Investment decision for transformer installation in year t

16

ξdg/wi,t Investment decision for non-renewable/renewable DG installation in bus i in year t

Lℓ Length of feeder ℓ in km

LDkn Local distance of solution n regarding objective function k

MDk Maximum distance between solutions regarding objective function k

NEt,h Number of Monte Carlo experiments in year t, demand level h

Nb Number of buses in the network

Nℓ Number of feeders in the network

Nh Number of demand levels

Np Number of population

NO Number of objective functions

NRt,h Number of Monte Carlo experiments with technical risk in year t, demand level h

OCdg/w Operating cost of non-renewable/renewable(wind) DG units

PLF et,h Price level factor in year t , demand level h and Monte Carlo experiment e

vrated Rated speed of the wind turbine.

Pwi,r Rated power of wind turbine installed in bus i

QDi,base Reactive base load in bus i and year t=1.

σDi,t,h Standard deviation of forecasted value of demand level factor in bus i, year t and

demand level h.

σρt,h Standard deviation of forecasted value of price level factor in year t and demand

level h

TRt Technical risk in year t

DGOC Total operating costs of DG units

17

TGC Total grid cost

DGIC Total investment cost of the DG units

LC Total feeder reinforcement cost

SC Total substation reinforcement cost

Acknowledgment

The authors greatly appreciate the valuable advices and financial supports provided

by the SYREL and GIE-IDEA groups of the Grenoble-INP University, during the study.

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List of Figure Captions:

• Figure 1: The flowchart of the Monte Carlo Simulation

• Figure 2: The flowchart of the proposed algorithm

• Figure 3: The distribution network under study

• Figure 4: The variations of forecasted load and electricity price in each demand

level

• Figure 5: The comparison between the obtained Pareto optimal fronts of different

methods

• Figure 6: The geographical view of a real 574-bus urban network in case-II

• Figure 7: The Pareto optimal front found in case-II

22

Table 1: Data used in the study

Parameter Unit Value

w1 0.55

w2 0.45

ρ $/MWh. 65

α % 3

d % 9

Vmax Pu 1.05

Vmin Pu 0.95

vcutin m/s 3

vcutout m/s 25

vrated m/s 13

c 8.75

k 1.75

σDi,t,h 0.01× µD

i,t,h

σρt,h 0.1× µρ

t,h

NP 30

Maximum iteration 1000

Table 2: Investment plan in best solution of case I

WT GT Feeder Substation

ξwi,t t i ξdgi,t t i γℓt t ℓ ψtrt t

1 2 2 1 2 4 1 3 2 1 7

1 5 4 1 3 6 1 5 6

1 6 6

1 8 8

23

Table 3: Investment plan in best solution of case II

WT GT Feeder Substation

ξwi,t t i ξdgi,t t i γℓt t ℓ ψtrt t

1 1 3 1 1 4 1 1 2 1 3

1 2 2 1 3 6

1 2 3

1 4 7

1 6 9

Table 4: The comparison between the proposed model and other methods

Method # of solutions maxOF1 minOF1 maxOF2($) minOF2($) Run time (s)

Proposed 16 0.8357 0.0109 6.99× 107 6.71× 107 18920

SA 24 0.9274 0.0843 7.06× 107 6.72× 107 12482

PSO 21 0.9683 0.1564 7.03× 107 6.75× 107 16234

Static 22 0.9236 0.3264 7.06× 107 6.80× 107 15992

Table 5: The Planning scheme of solution #11 in scenario II

Year Bus FC SC

t WT GT (105$) (105$)

1 63 5.7639 0

2 371 7.2362 0

5 57 8.6461 0

6 18.8580 2

8 142 574 25.7470 0

24

Table 6: The solutions of Pareto optimal front in case II

Solution OF1 OF2 µ1 µ2 min(µ1, µ2)

1 112507469.7004 0.0000 0.0000 1.0000 0.0000

2 84683648.7369 0.0214 1.0000 0.0000 0.0000

3 107572367.7060 0.0009 0.1774 0.9585 0.1774

4 98451569.6933 0.0013 0.5052 0.9407 0.5052

5 89576596.1460 0.0154 0.8241 0.2797 0.2797

6 90525850.4494 0.0106 0.7900 0.5059 0.5059

7 87554243.1986 0.0185 0.8968 0.1364 0.1364

8 93732213.0991 0.0080 0.6748 0.6280 0.6280

9 97907630.9501 0.0052 0.5247 0.7551 0.5247

10 96867855.5489 0.0063 0.5621 0.7059 0.5621

11 110394545.2631 0.0005 0.0759 0.9754 0.0759

max 112507469.7004 0.0214 1.0000 1.0000 0.5621

min 84683648.7369 0.0000 0.0000 0.0000 0.0000

Table 7: The characteristic of best solution in case II

Year DGIC(M$) SC(M$) LC(M$) TRt

1 4.931 0 3 0.00479

2 4.181 0 9 0.00450

3 1.227 0.2 6 0.00407

4 0 0.2 9 0.00639

5 0.25 0 21 0.00755

6 0 0 9 0.01437

7 0 0 18 0.00407

8 0 0 30 0.00465

25

Table 8: The investment decisions regarding DG units in case II

WT GT

year Bus year Bus

1 180,121,288 1 180,288,360,280,73

2 396,467,574 2 216,574

3 432 5 121

Generate , ,, , , , , e D e

i t h i t hρλ λ

, 0t hNR =

Solve the equations (1) to (20)

Is any of (10) or (11) violated ? Yes

1e e= +No

, , 1t h t hNR NR= +

Monte Carlo Converged?

Yes

No

hh N<1h h= + Yes

t 1; h 1;= =

t<TNo

Yes

End

No

1t t= +Save , ,, e

t h t heNE GC= %

1 ,e =

Figure 1: The flowchart of the Monte Carlo Simulation

26

Randomly Generate an initial set of N solutions

Iteration=1

Calculate OF1 and OF2 for each solution

Calculate the fitness for each solution

Save the best N solutions in memory

Is stopping criteria met ?

Crossover Mutation

Store new solutions

New solutions

Union

�������������������

����

Pareto optimal solutions

Yes

Select the best solution

Stage II

Stage I

Monte Carlo Simulation

Figure 2: The flowchart of the proposed algorithm

Figure 3: The distribution network under study

27

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Hours

Fo

reca

sted

val

ues

of

load

an

d e

lect

rici

ty p

rice

µDi,t,h

µρt,h

Figure 4: The variations of forecasted load and electricity price in each demand level

0 0.2 0.4 0.6 0.8 16.7

6.75

6.8

6.85

6.9

6.95

7

7.05

7.1x 10

7

Technical risk (OF1)

Co

st (

$) (

OF

2)

Proposed dynamic model SA dynamic modelPSO dynamic modelStatic model

Figure 5: The comparison between the obtained Pareto optimal fronts of different methods

28

Figure 6: The geographical view of a real 574-bus urban network in case-II

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

8.5

9

9.5

10

10.5

11

x 107

Technical risk (OF1)

Co

st (

$) (

OF

2)

Figure 7: The Pareto optimal front found in case-II

29