Elsevier Editorial System(tm) for International Journal of Electrical Power and Energy Systems Manuscript Draft Manuscript Number: IJEPES-D-13-00427R1 Title: Computationally Efficient Composite Transmission Expansion Planning: A Pareto Optimal Approach for Techno-Economic Solution Article Type: Research Paper Keywords: transmission expansion planning, generation planning, genetic algorithm, congestion management, social welfare, linear matrices Corresponding Author: Dr. Rajiv Shekhar, Ph. D. Corresponding Author's Institution: Indian Institute of Technology Kanpur First Author: Neeraj Gupta, Ph. D. Order of Authors: Neeraj Gupta, Ph. D.; Rajiv Shekhar, Ph. D.; Prem K Kalra, Ph. D. Abstract: This paper presents an integrated approach for composite transmission expansion planning incorporating: (i) computationally efficient linear matrices, (ii) a novel Demand/Energy Not Served (DNS/ENS) and Generation Not Served (GNS) calculation approach, to circumvent the time intensive iterative procedures. A self-tuning mechanism based on stochastic Roulette wheel (RW) simulation procedure supports the reduction of network congestion. It establishes a trade-off between technical and economic criteria using the theory of marginal value (marginal reduction in interruption cost and marginal increment in the investment) for the incremental updating method. A hybrid of deterministic (N-1) and probabilistic (critical N-2) contingency scenarios have been simulated for security of the system. Results show that existing lines and generators capacity are necessary to update for economic operation for minimizing interruption cost and to achieve optimal investment. Modified 5-bus 24-bus and 118-bus IEEE systems are taken to show the generalization of Methodology.

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Abstract— This paper presents an integrated approach for composite transmission expansion planning

incorporating: (i) computationally efficient linear matrices, (ii) a novel Demand/Energy Not Served

(DNS/ENS) and Generation Not Served (GNS) calculation approach, to circumvent the time intensive

iterative procedures. A self-tuning mechanism based on stochastic Roulette wheel (RW) simulation

procedure supports the reduction of network congestion. It establishes a trade-off between technical and

economic criteria using the theory of marginal value (marginal reduction in interruption cost and

marginal increment in the investment) for the incremental updating method. A hybrid of deterministic

(N-1) and probabilistic (critical N-2) contingency scenarios have been simulated for security of the

system. Results show that existing lines and generators capacity are necessary to update for economic

operation for minimizing interruption cost and to achieve optimal investment. Modified 5-bus 24-bus and

118-bus IEEE systems are taken to show the generalization of Methodology.

Index Terms— transmission expansion planning, generation planning, genetic algorithm, congestion

management, social welfare, linear matrices.

1. Introduction

During the last few decades, rapid changes in the electricity industry around the globe necessitate a robust

and optimal transmission infrastructure to supply electricity. The existing literature provides a wide variety of

TEP methodologies in the complex deregulated environment [1], [2], where, very few technical papers have

discussed composite TEP (generation and transmission planning are carrying simultaneously) [3], [4]. The

methodologies developed for TEP can be classified according to different domains such as (i) modeling, (ii)

optimization method, (iii) reliability, (iv) congestion management, (v) AC power planning, (vi) competition and

electricity market, (vii) uncertainty analysis, (viii) distributed generation (integration of wind farms and other

renewable generators), (ix) environmental impact (x) Coordinated TEP and composite TEP, and (xi) security

constrained TEP. A comprehensive review is presented in [1]-[10]. Some researchers have used above described

domains separately in TEP [1]-[9], however, rarely integrated -even at some extent- on a single platform [9]. In

this regard, a number of technical papers and reports have discussed the transmission system planning issue as a

set of optimization problems [1] – [8], [10]-[14], where variables are discrete, for example, capacity of

generators and lines, location of lines etc. [10]-[13]. Over the last two decades, numerous articles and books

1 Prof. Rajiv Shekhar (Corresponding Author) is with the department of Materials Science & Engineering, Indian Institute of Technology Kanpur, India.

Computationally Efficient Composite Transmission

Expansion Planning: A Pareto Optimal Approach

for Techno-Economic Solution

Neeraj Gupta, 1Rajiv Shekhar, Prem K. Kalra

*ManuscriptClick here to view linked References

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have been written on the development of search techniques for optimizing TEP, focusing on both traditional

(linear, quadratic programming, mix-integer, heuristic etc.) and non-traditional optimization techniques (GA,

swarm, meta-heuristic etc.) [1] – [13]. Using these techniques, TEP has, over the years, evolved from cost-based

to value-based approaches [9] – [14]. In the value-based approach, Min-cut-max-flow (MCMF) algorithm and

load flow based curtailment strategy have been used to calculate expected demand/ energy not served

(EDNS/EENS) [1], [3], [8], [12], as reliability measures. The iterative computational requirement of

EDNS/EENS forced planners to find a novel, simplified non-iterative approach [1], [12], [14]. An analytical

review of TEP models and reliability measures are given in [1], [15]. The TEP methodology given in [15]

demonstrates a new non-iterative EDNS/EENS calculation approach, which might be useful in the long term

TEP procedure with MCS based probabilistic contingency analysis. Here minimization of the sum of

investment, operational and interruption costs were carried out to determine optimal TEP [1] – [15]. Most of the

research papers in this regard have used deterministic N-1 contingency based TEP methodology, while very few

publications have reported work based on N-2 and MCS based contingency approach [1], [12], [15].

Developing countries such as India are going through a rapid change of industrial development process,

resulting in a large demand. India plans to install 74 GW generation capacity by 2017 [16]. Clearly, increase in

generation capacity has to be complemented with increased transmission capacity. Moreover, electric power

systems are getting more and more complex due to bottlenecks in transmission networks, primarily because of

uncertain demand growth and increased heterogeneity of power generation processes [1], [14]. The Central

Electricity Regulatory Commission (CERC) in India has estimated that 10.3% to 12.9 % power deficit is due to

unreliable transmission network [16], [17], which, in turn, may also increase the risk of blackouts Thus, a

robust and optimal transmission infrastructure to supply electricity reliably is of prime importance under smart

operation of the power industry. This, in turn, requires analysis of more severe contingency scenarios other

than N-1 contingency scenarios, especially in a country such as India where demand far outstrips supply [1],

[12], [14], [18]. Along with, it should attract the investors for a huge investment by giving them profit

maximizing signal [4], where loss should be minimum due to contingency and congestion in the network. This

E-mail: vidtan@iitk.ac.in and phone: (+91) (512) 259 7016.

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motivates us to propose a techno-economic computational efficient planning methodology (incorporating

demand and probabilistic outage of lines), which incorporates reliability evaluation modeling [7], [12], [13],

[19], congestion management criterion [12], uncertainty analysis, security of the systems, value assessment [12],

and market based approach [3], [16].

The TEP approach followed by previous investigations has several shortcomings [1]-[22]. First, the capacity

of all possible new alternative lines and generators were specified a priori (only locations have been selected

during the optimization process). Second, the calculation procedure of ENS (DNS) is iterative, thus expensive

to implement with probabilistic contingency analysis. Third, transmission service provider‘s and generator‘s

benefits are not incorporated simultaneous with customer‘s profit in the concept of social welfare, for example,

non-zero ENS also implies unutilized generation capacity or generation not served (GNS) and wheeling loss

(WL), the cost of which must be accounted for in TEP. Fourth, the computational efficiency of the algorithm

should be better to analyze sever probabilistic contingency scenarios along with N-1 contingency scenarios.

Fifth, the Techno-Economic planning criteria are rarely available for the composite planning of generators and

transmission network, which results excessive investment. Six, generally during the planning of the power

systems, reliability level is decided ―a priori‖ based on experts knowledge. Thereafter, simulation is carried out

to achieve least cost solution satisfying the decided reliability level. Selection of reliability level does not have

any analytical procedure, which may leads to sub-optimal solution. In the proposed methodology, well known

and globally accepted ―Marginal cost‖ based approach is demonstrated to trade-off reliability (least interruption

cost) and economics (least investment) [16].

Unfortunately, the proposed TEP in [12] is computationally-intensive. For a modified IEEE-5 bus power

system, it takes seven days (calculation is processed on Computer E-series (VPCEC15FG), 2.13 GHz, 4 GB

ram) to give an optimal solution for a case study incorporating 450 GA iterations with 30 population size,

twelve demand scenarios, and 1000 MCS contingency scenarios. A similar exercise for a 24-bus IEEE power

system takes more than 14 days. To make it computationally efficient, linear sensitivity factors have been

incorporated: (1) GPF (generation participation factor) to replace the iterative ELD calculation [23], (2) PTDF

(power transfer distribution factor) matrix to replace multiple DC-load flow calculations [24], (3) LODF (Line

outage distribution factor) and GLODF (generalized line outage distribution factor) matrix for transmission

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lines contingency analysis [25], and (4) BBIM (bus-branch incidence matrix) [16], [26] for the calculation of

ENS and GNS. The overall scheme is implemented on the modified IEEE-5 bus, IEEE-24 bus and IEEE-118

bus test power systems to show the generalization at large networks. Here, equation (8) is used (constrained by

equation (7)) to establish a better trade-off between economics and reliability [16]. The proposed methodology

finds an optimal solution by using the marginal cost (investment) theory [16].

2. Methodology

The proposed methodology has been implemented with the following assumptions:

a. Forecasted peak hourly load curve at all buses is defined to incorporate 8760 seasonal scenarios [16].

b. Single-stage planning of 10 years is demonstrated [12].

c. 8.5% compound load growth is selected for target year [12].

d. Probabilistic N-2 contingencies are incorporated along with N-1 contingency scenarios.

e. Location of generators and possible candidate alternative lines are pre-specified based on topological

conditions, expert knowledge and resource availability [16].

f. Old generators and transmission lines are free to update with new capacities, which are to be calculated by

the planning procedure.

g. Demand is assumed completely elastic and variation of 20% is permitted from mean value over the year

[16].

2.1 Objective Function

The objective function (J) includes sum of operational cost, interruption cost (cost of expected ENS, GNS,

WL, and outage cost of generation), and investment for setting up the new lines and generation capacities.

TN

T1,q q T,rl,s s q 1

s g

(t) * (t) *

(t) *

8760 EENS EGNS

EWLt 1

Interruption cost w.r.t customer, generators, transmission owner

C *TL C2C * EGO (t)

J

C EENS(t) C EGNS(t)

C EWL(t)

nN 8760

p T,p p pp 1 t 1

g 8760

G,s G,s G.s s ss 1 t 1

Investment and operating cost in setting up transmission lines

Investment and operating

*OCF *TL * F

C1 C2 *OCF EPG EG

cost in setting up generation capacity

(1)

Subscript T stands for transmission line and G stand for generator, where t is the time at which quantities are

measured. EENS, EGNS and EWL represents Expected Energy Not Served (MWh), Expected Generation Not

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Served (MW) capacity, and Expected Wheeling Loss (MWh), respectively. CEDNS, CEGNS, CEWL, Crl,s are the costs

of EDNS, EGNS, EWL, and revenue loss (rl) of sth generator due to outage respectively, in units of k$/MWh.

Capital investment cost of qth transmission line is CT1,q, and CT2 is annual operating and maintenance cost of line

(k$/MW/km). Length of the qth transmission line (km) TLq. N is the number of transmission lines, where

subscript ‗T‘ belongs to total (updated existing + new) and ‗n‘ belongs to new transmission lines. EGOs(t) is the

expected outage of generator at bus s at time instant t due to force outage rate (FOR). C1 is capital cost of the

particular facility represented by subscripts. First subscript represents type of facility (‗T‘ or ‗G‘), where second

subscript stands for index of facility, i.e., C1T,q and C1G,s are respectively for transmission line (in k$/km) and

generator (in k$/MW). C2 is operating and maintenance cost of the particular facility represented by subscripts,

i.e., C2T,p and C2G,s are respectively for line (in k$/MWh/km) and generator (in k$/MWh, including fuel cost).

EPGs and EGs are proposed expected generation capacity of generator at bus-s calculated from simulation and

pre-specified capacity of generators at bus-s, respectively. Fp represents capacity of pth transmission line (MW)

and OCFp=(1-FORp)/FORp is operating cost factor of qth transmission line computed by forced outage rate

(FOR), based on climatic and geographical conditions. Minimization of fuel cost is reflected in ELD thus does

not included in J.

2.2 Constraints

TEP problem is dealt as constrained optimization problem, a list of equality and inequality constraints are:

0s sG D (2)

* fPTDF G T (3)

f (max) f f (min)T >T T (4)

minG G (5)

RWR N (6)

0 1con,Np . (7)

MEC MI (8)

0 20baseD D * . (9)

1ib (10)

Equation (2) represents supply-demand constraint, where, equation (3) is used for load flow calculation.

Equation (4) constrained the flow in transmission lines and equation (5) is used for operating generators above

minimum limit. Here, probabilistic RW simulation is used to mitigate congestion in the network (as discussed

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in section 2.8), where number of total rotations of RW in one complete cycle (RRW) is defined by equation (6),

where N is the number of transmission lines – existing and proposed both - having a congestion probability

greater than 0.1, as shown by equation (7). Equation (8) provides techno-economic solution and establishes a

trade-off between reliability (interruption cost) and economics (investment) of the system. Instead of using

conventional economic load dispatch (ELD) iteratively for all demand scenarios, GPF matrix is used to

calculate optimal operating points of the generators, where change in all demand scenarios from base demand

follows equation (9). Equation (10) is used to remove infeasible chromosome formation in GA population,

where more than one bit in ith chromosome (bi) should be 1.

2.3 GPF Calculation and Application

To eliminate the iterative computations of economic load dispatch (ELD), GPF model is used. In this

strategy, first the base optimal generation (Gsbase

) of all generators is calculated by ELD for based demand

scenario. Thereafter, other optimal operating points are calculated for the reasonably small changes in loads

(multiple demand scenarios) using GPF by 1/ / 1/s s jj g

GPF c c

. Here, subscript s stand for bus, cs is the

cost coefficient associated with the quadratic term of the generator‘s cost function at bus s [23], [16] and j

belongs to all generators in the power system. The change in sth generator‘s power ( sG ) subject to change in

system demand ( D ) is given by *s sG GPF D . Steps for calculating ELD of generators for multiple demand

scenarios are:

1. GPF for all generators is calculated.

2. Out of all demand scenarios generated from load curves, first scenario is chosen as the base demand. For

this base demand ELD is run to get basesG for all generators.

3. The base demand is subtracted from all remaining demand scenarios to obtain the change in demand ( D ).

4. Change in optimal power generation is calculated for remaining demand scenarios D .

5. Economic generation at all generator buses are defined without iteration by bases s sG G G .

From the simulation we observed that for 8760 demand scenarios, above describe algorithm reduces

computational time by approximately 67%, compare to ELD. Incorporation on PTDF matrix instead of DC-load

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flow procedures for multiple demand scenarios reduced 78% computational time. Many publications have

shown the importance of the PTDF matrix [16], [24].

2.4 GLODF Matrix Calculation and Application

The PTDF matrix approach is much faster than the DC- load flow approach; however, it is still unwieldy for

the outage study of large networks. Power system in Fig. 1 shows that for the outage of two transmission lines,

20 matrix elements of PTDF matrix need to be stored in the memory. In contrast, GLODF matrix requires only

10 matrix elements to be stored, and is therefore computationally efficient for large power systems. The details

of GLODF matrix have been given by Guo et al [25].

Fig. 1. Example of a 5-bus power system

Step by step calculation of power flow in transmission lines is given below:

1. Given base BBIM (considering all lines on-line), where rows represent buses and columns represents

transmission lines. It shows the relationship of buses with the nature of transmission lines (incoming or

outgoing). BBIM element has value 1 and -1, if lines are incoming to bus-s and outgoing from bus-s,

respectively. For unconnected line to bus-s BBIM element has value 0.

2. Consider multiple outages of transmission lines according to probabilistic contingency.

3. Diagonal reactance matrix for online transmission lines (XM) is formed, which has reactance of online

transmission lines on the diagonal.

4. Diagonal reactance matrix for contingent transmission lines (X0) is formed, which has reactance of

contingent (off-line) lines on the diagonal.

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5. Segregate the base case BBIM in two BBIMs ( and ), where BBIM has number of columns associated

with contingent transmission lines and has remaining columns associated with on-line transmission lines.

6. GLODF matrix is calculated as, here, E is identity matrix (dimension equal to the number of contingent

lines)

1

1 1 1 10

T TMGLODF X B E X B (11)

7. Power flow in remaining (online) lines after outage of multiple transmission lines is calculated as:

, , ,

post pre pref M f M f OT T GLODF T (12)

Here, ,pref MT and ,

postf MT are the pre- and post- contingency power flow in on-line transmission lines,

respectively. ,pref OT is the pre-contingency power flow in contingent transmission lines. The above descried

approach is very efficient to calculate change in lines flow for 8760 demand scenarios.

2.5 Linear calculation approach for ENS, GNS and WL

Method presented in [9] to calculate ENS and GNS are explored here and derived as:

, , , , , , , ,s f s i c s i f s j c s ji cm j cn

Diff T T T T

(13)

Here, cm is constrained in-coming lines and cn is constrained out-going lines at bus-s Diffs is the sum of

overflow in in-coming and out-going transmission lines and can have one of the three values given below:

: 0

: 0

0 :

s s

s s s

GNS if Diff

Diff ENS if Diff

Otherwise

(14)

We construct the vector Tover

to represent overflow in transmission lines, where the kth element is defined as

, , , , , , , ,

, , , ,

:

0 :

f k s c k s f k s c k soverk

f k s c k s

T T if T TT

if T T

(15)

Tf,k,s = amount of power flow in kth transmission line (either incoming or outgoing transmission lines to any

bus ‗s‘). Sum of the elements of vector Tover

represents WL for a particular demand scenario (this provides

monetary loss to transmission owner, which he realized due to the constraint in the network. He might make

some profit from this, if wheel on the network). For all buses, equation (13) can be represented in matrix form

[25] as

[Diff] = [BBIM][Tover

] (16)

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Diff is a matrix for multiple demand scenarios. After contingency of transmission lines BBIM is replaced with

Using the above formulation to access reliability of power systems, ENS, GNS and WL calculation algorithm

for multiple demand scenarios is given in the steps outlined below. Here it is assumed that BBIM is given for

the network in normal operating scenario, and post contingency power flow in lines have been calculated for all

demand scenarios, using above models.

1. For the given multiple contingencies, is calculated.

2. Power flow in transmission lines are calculated by GLODF matrix if there is no generator contingency,

otherwise GPF and reformed PTDF matrices are used.)

3. Tover

is calculated using equation (15).

4. The multiplication of the Tover

with gives a matrix Diff shown by equation (16).

5. Apply equation (14) to form matrices GNSs and ENS

s.

6. Sum of columns of GNSs and ENS

s matrices represents the system GNS and ENS, respectively for each

demand scenario.

From the simulation results shown in [16], it is noted that the matrix model for calculating ENS and GNS does

not alter the output of the simulation presented in [12]; however it is computationally more efficient and save

approximately more than 85% comparative to linear optimization based load curtailment strategy (LCS). Since

for each load scenario LCS has to run multiple times for computing ENS, however, proposed algorithm

calculates ENS in a single step.

2.6 Contingency Analysis

Generally, N-1 contingency analysis is carried out under composite TEP methodologies [1], [21], [27].

Although, N-1 secured system is economic (less investment) but not reliable for more than single outage of

lines/generators, which are very frequent in highly developing countries like Indian and China. In these

countries demand growth is very uncertain with heterogeneity in generation process, which changes operating

scenarios frequently. This forces more stress on the transmission network and results in more than single outage

of lines frequently [17], [18], [27]. N-1 contingency secured power system resist for all single outage (off-

peak/peak time), but unable to maintain reliability under more than single contingency. Thus, in a highly

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deregulated power system, to maintain grid code efficiently against vulnerable operating scenarios, a highly

reliable network is of prime importance. Our proposed methodology is comparatively fast and thus N-1

contingencies are incorporated with critical (highly probable) N-2 contingencies. It must be pointed out

that higher contingency scenarios (N-3, N-4, Monte Carlo simulation) can be incorporated very easily in the

proposed methodology Process of finding a set of contingency scenarios (Z) for simulation is as follows:

1. Outage probability of all transmission lines is known.

2. All transmission lines are the elements of set X for N-1 contingency analysis.

3. Make a set of all combinations of two transmission lines Y_2.

4. Calculate outage probability of all elements in set Y_2, where each element is the combination of two

lines.

5. Rank the elements of set Y_2, in higher to lower outage probability sequence.

6. Calculate average outage probability of the set Y_2.

7. Select all elements of set Y_2, which are above than average outage probability to form a set Y.

8. All lines are removed from the set X, which are also included in the set Y. For example in Fig. 1, set for

single outage analysis of lines X = TL 1, TL 2, TL 3, TL 4, TL5, TL 6, TL 7. If we chose a set Y =

(TL 1, TL 4), (TL 1, TL 6) for severe N-2 contingency analysis. Then, new set Xn = TL 2, TL 3, TL5,

TL 7. Here three lines (TL 1, TL4 and TL 6) are removed from the set X.

9. Final contingency set Z is the union of set Y and Xn, Thus in upper example, Z = (TL 1, TL 4), (TL 1,

TL5), TL 2, TL 3, TL5, TL 7.

A similar procedure is used to generate severe probabilistic N-2 contingency scenarios for generators outage

analysis along with N-1 contingencies. Reliability of the reduced network (after each contingency scenario) is

checked for all demand scenarios by calculating EENS, EGNS and EWL. During the contingency analysis,

condition of islanding is taken care off, since system islanding may be the cause of ill-conditioned PTDF,

GLODF matrices. In this case, to simulate methodology properly, at least on transmission line is placed from

islanded bus to nearby grid connected bus. Thereafter, capacity of that line is defined by proposed methodology.

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2.7 Detection of Islanding after Contingency of Lines

In power systems operation islanding is always undesirable. Since Islanded system may suffer from deficit

generation or surplus of economic generators. This may lead to social loss by increasing electricity price.

Therefore, in this paper, Islanding is checked for each contingency scenario, and if islanding is realized by the

network an extra line is permitted to remove islanding. Isolation of demand or/and generator buses or a part of

power system from the power grid (islanding) is detected by an efficient method described by Sun et al. [26].

The incidence matrix (A) is used to compute islanding, where ai,j element will be ‗1‘ if line is connected

between ‗i‘ and ‗j‘ nodes, otherwise ‗0‘. In the matrix A, the sum of elements in a row gives the number of

connected transmission lines to the associated bus, which is called degree of bus (BD). A bus is ―islanded‖ from

the grid if it has zero BD. However, the problem is to identify islanding in the system when all buses have non-

zero BD. For that we use either the upper or lower triangular matrix (AU or A

L) of A. Now consider A

U, where the

last row is removed (reduced AU matrix). Here, we calculate the sum of ‗1‘s in each row (IL). A non-zero sum

signifies that the associated bus is connected to the grid. According to Fig. 1, for the outage of three lines (TL

4, 5 and 6), the resulting A and AU matrices are given below.

1 2 3 4 5

1 0 1 1 0 0 2

2 1 0 1 0 0 2

3 1 1 0 0 0 2

4 0 0 0 0 1 1

5 0 0 0 1 0 1

Dbus B

A

The number of islanded regions can be defined as the addition of one to the number of zeros in IL vector.

Here, one zero in IL suggests the presence of two ―islands‖ in the power system the detailed analysis is shown

in [26]. Here, bus-1, bus-2 and bus-3 are in one island, while bus-4 and bus-5 are in other island.

2.8 Techno-Economic Solution Mechanism (A Pareto Approach) for Capacity Selection

Composite TEP design problem involves simultaneous solution of multiple conflicting objective functions [28],

[29]. The most common objectives which arise in power systems planning are to minimize the overall

investment (economics) and simultaneous maximization of the reliability of system (continuity of supply).

Tradeoff between two quantities are explained by Fig. 2, where a number of plausible solutions (points ‗A‘,

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‗B‘, ‗C‘ and ‗D‘) of a hypothetical composite TEP design are shown. Solution point ‗A‘ shows that the system

is economical (minimum investment), however less reliable (large interruption cost), on the other hand, solution

point ‗B‘ provides highly reliable system (least interruption cost) with high investment cost (uneconomical).

Both solution points are useful particular planner criterion according to developed and developing countries

(developed countries require more economic solution). Solution point ‗C‘ cannot be the choice of planner –not a

good solution (higher investment with less reliability). In multi-objective planning strategy these multiple

solutions form a pareto-optimal front (shown by long dashed line in Fig. 2), where any point may be the

solution. To achieve this strategy two level (bi-level) optimization procedure [27] is carried out. In first level,

GA generates a chromosome where each bit shows the location of candidate lines. In second level marginal

value of the investment is checked with respect to the marginal value of the interruption cost (measure of

reliability), and if both are equal pareto-optimal solution point is selected for the generated chromosomes. This

is achieved stochastically using roulette wheel simulation and congestion management strategy as described in

next section.

Fig. 2. A typical two conflicting objective problem in Power system planning (solution points A, B and D are

pareto-optimal optimal solution)

2.9 Roulette Wheel Simulation for congestion management

It can be noticed from the previous studies that the TEP for the given generation capacities is divided into two

sections. The first one concerns with the location selection of transmission lines, while the second is associated

with finding the appropriate capacities of selected transmission lines. Given generation plan cannot be optimal

or at this non-optimal plan new transmission lines for plan cannot be optimal. If we want to optimize overall

power system, generation planning should be simultaneously calculated along with TEP. Thus, third section of

the composite TEP belongs to optimal capacity of the generators.

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First issue is resolved with the help of optimization procedures, assuming the capacity of the lines a priori.

But now, a more sophisticated approach to resolve both the issues have been introduced, namely, RW selection

based on stochastic approach. This procedure is prominently and effectively applied in genetic algorithm (GA)

optimization procedure [12] and adopted for composite TEP to manage the congestion stochastically. It

improves the planning efficiency and optimality and can be used with any optimization formulation (linear or

non-linear or meta-heuristic) and generalized for any size of the network. After calculating power flow in all

lines multiple times (equal to the number of operating scenarios) all congested transmission lines -- where

power flow is greater than their respective transmission capacities -- have value 1, while the un-congested lines

are assigned a value 0. Congestion probability of each transmission line is then computed. Now, an imaginary

RW [12], [16] is constructed where each transmission line in the network is represented by a separate segment

on the surface of RW. The area of each segment is proportional to the congestion probability of associated

congested line (RW is rotated N times). At the end of each rotation, the segment at which the pointer stops, and

the corresponding transmission line, is noted. After rotation of N times, capacity of all lines is updated

according to

o

N N N NF F m F (17)

FN0 is base capacity of the N

th transmission line, mN is number of times out of N that the roulette pointer stops

at the segment corresponding to the Nth transmission line, and

NF represents value by which the transmission

capacity is increased in the Nth transmission line. The updated capacity of the network has strong correlation

with the congestion probability, thus efficiently and optimally reduces congestion after each iteration of

network capacity assignment. It is important to ensure that the transmission capacities are not over-specified

because it would result in superfluous investment. Consequently, the process of capacity updating of a

transmission line is terminated, once the probability of congestion goes below 0.1. In every step, after the RW is

rotated N times, the marginal expected cost 1 /i iEC EC FMEC and marginal investment

1 /i iinv invT T FMI are calculated, where , , 1N i N iF F F . EC

i and EC

i-1 are expected cost in i

th iteration

and expected cost in (i-1)th iteration, respectively. FN is updated network capacity of the network in one

complete simulation of RW selection and dFN is change in the capacity of the network. i

invT and 1i

invT are

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transmission investment in ith iteration and transmission investment in (i-1)

th iteration, respectively. Final

outcome of TEP is in fact subjected to the economies of scale i.e., if long-run marginal social cost is below or

equal to the long-run investment, TEP is appropriate. Thus, when MEC equals MI, an optimal solution has been

found where the total cost is minimized. This method provides a tuning mechanism for better Techno-economic

planning, where pareto optimal solution has been found.

2.10 Genetic Algorithm (GA)

GA is very well known optimization search method to solve complex power system planning problem, where

most of the design variables are discrete (binary). In TEP the states of transmission lines are represented as

binary variable, where 1 and 0 represent the active and inactive lines respectively. In GA every chromosome has

multiple bits, where each bit represents a particular transmission line and generator. The length of each

chromosome is equal to the number of bits which is the sum of number of new alternative transmission lines:

the candidates to be planned. Initial population is generated randomly keeping constraints like all bits in the

chromosome should not be 0. In this paper generated population has thirty chromosomes. From the second

iteration of GA, the population generation follows the law of natural selection which has sequential process

such as selection, reproduction, crossover and mutation as described in [16]. If the generated chromosomes are

similar in new population, then mutation with high mutation probability is carried out until all chromosomes get

different. This is required to avoid the multiplicity of the similar simulation. As a result, the new chromosomes

are available to compute associated fitness value to achieve optimal solution. The process of creating new

population is continued until one of the stopping criteria is satisfied, i.e., maximum number of iterations,

solution is achieved, does not change the value of objective function in consecutive 50 iterations. Two level (bi-

level) optimization procedures [27], [28] are used to find the global optimal solution. In level-I, GA is used to

generate candidate lines, where, capacity of the respective selected lines and generators are optimized in level-II

as shown in Fig. 3.

3. Proposed TEP algorithm

Proposed composite TEP methodology is described below and corresponding flow chart is shown in Fig. 3:

1. Initial population of chromosomes having the length equal to the set of proposed lines is generated. Each

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chromosome satisfies criteria defined by equation (10).

2. Demand matrix (D) for 8760 demand scenarios are generated for target year and accordingly economic

power generation matrix G is calculated using GPF model. Both matrices will have 8760 columns and

number of demand and generation buses, respectively.

3. One chromosome from GA population is taken and base capacity (Fj0) of 25 MW is updated initially to all j

candidate lines and total investment in calculated.

4. Base PTDF, BBIM, GPF matrices, power flow in lines are calculated for all 8760 demand scenarios using

ELD (conventional model).

5. From the multiple contingency cases, one case is taken and thereafter lines and generators are removed

accordingly from the network.

6. If system is in islanding mode, connect one line from islanded bus to the nearby bus, connected to network

(to secure uninterrupted power supply).

7. Power flow (Tf) in all online transmission lines are calculated using PTDF, LODF and GPF matrices for

8760 demand scenarios and stored for further calculation, under three contingency cases

a) For no outage of generators (only lines outage), power flow in all online lines are calculated by GLODF.

Procedure is described above.

b) For outage of slack generator (replace the slack generator with the next higher capacity generator) and/ or

others generators (no outage of lines), calculate G for remaining generators using GPF and new PTDF

matrix. Calculated Tf in the network.

c) For the case where generator and line outage occurs are considered, step (b) is followed.

8. ENS, GNS and WL is calculated using equation (14)-(16) and values are stored.

9. Go to step 5 until last contingency scenario is taken.

10. EENS, EGNS and EWL and corresponding expected cost (EC) is calculated.

11. Marginal expected cost (MEC) and marginal investment (MI) are calculated, and If MEC is equal to MI,

then go to step 18.

12. The probability of congestion for each active line is calculated using stored data in step 7.

13. Imaginary RW is constructed. Active lines with pcon,j ≤ 0.1 are not considered to update

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14. Capacities of the transmission lines are updated according to equation (17)

15. Impedance of the transmission lines are updated according to updated capacity

16. Expected generation is calculated from the data stored in step 7 for all 8760 demand scenarios and

contingency cases, and required optimal generation is calculated by 1.6* 3* ( )Ri i iG ERG std ERG ,

incorporating 16% reserve margin. Here, ERGi is difference of expected and existing capacity of ith

generator and std is standard deviation of generation distribution to incorporate maximum uncertainty.

17. Investment in the updated lines and generators capacity is calculated and go to step 5.

18. Compute associated objective function by equation (1).

19. Go to step 3 until whole population is analyzed.

Fig.3 Flow chart of the proposed TEP methodology Fig.4. IEEE 24-bus power system

20. Genetic algorithm sequential process like selection, reproduction, crossover and mutation is applied to

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generate new population of best candidate lines.

21. When stopping criteria met, GA is terminated and at this point optimum solution is found.

4. Case Study

The proposed methodology is first compared with the procedure described in [12]. For that both methodologies

are applied on the same network (modified 5-bus IEEE test power system, given in [12]), here, we have

observed that except computational time there is no change in the optimal solution. Methodology presented in

[12], has limitation to apply on large power systems for multiple contingency scenarios. Though, the presented

methodology is free from computational inefficiency, thus applied on 24-bus IEEE power system (Fig.4) and

118-bus IEEE Power system. We have also observed that without optimizing the capacity of generators,

transmission investment is high enough to give uneconomic solution (lead to over investment). Therefore, in

this paper, capacity of the proposed generators has been simulated for multiple transmission network plans. To

show the effect of generation planning simultaneously with TEP, simulations were carried out on the same

power system (modified 5-bus IEEE test power system), with similar operating scenarios as shown in [12]. In

modified 5-bus power system, pre-specified capacity of generation was 250 MW, 150 MW, 200 MW and 250

MW, respectively at bus 1, 2, 6 and 7, however calculated optimal value (using the proposed methodology) is

184 MW, 211 MW, 187 MW and 240 MW respectively. It shows that the specified generation capacities at

buses 1 and 2 vary significantly from the pre-specified generation capacities. On the other hand investment in

transmission network is minimized further. To show comparison, simulation is carried out first to set up new

lines where generation capacity is not optimized (denoted by WOG in Fig. 5). Thereafter, optimization of the

generation capacity has been achieved simultaneously with TEP (denoted by WG in Fig. 5). It is evident from

the results (shown in Fig. 5) that the generation planning along with TEP decreases the overall updated

transmission and generation capacities by more than 30 %, and thus the total investment. This strategy gives

appropriate signal to the generator‘s investors that under outage of lines they will be online always to play

efficiently in market, because capacity is calculated using step 16 (given in section 3). This planning strategy

increases the reliability of power network with decreasing investment.

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Fig. 5. Comparison of TEP, without generation planning (WOG) and with generation planning (WG)

To show the applicability of the proposed methodology on a large power system, a 24 bus-IEEE power

system, is used. It has 10 generators (with 1800 MW total capacity) and 35 existing transmission lines,

capacities of which are given in Fig. 4. Here it is noted that only existing generators are planned to update with

new capacity for the total system forecasted demand of 4259 MW. Total number of new alternative lines (NTL)

is 82, shown in [16]. Due to its computational efficiency, the proposed TEP methodology is able to handle a

relatively higher number of critical N-2 contingency cases along with N-1 contingencies. Updated and new

transmission capacities are given in Table-I. Associate proposed economic generation capacity is shown in

Table-II. In the methodology to realize discrete values of the transmission lines step increment (NF ) is taken

as 25 MW. Here, generator‘s optimal capacity is calculated based on ELD, thus, approximated to the nearby

standard value after simulation. Here, we can observe that 12 new lines have to be set-up with capacity update

of 18 existing lines comprising total capacity of 3650 MW. Here to follow economies of scale of the

transmission lines 50 MW lines are replaced by 75 MW capacities, where the total investment is $77.67m.

Required number of parallel transmissions lines are also shown in Table-I. These are constrained by the ROW

constraints; all existing transmission routes except TL 12, 20, 29, 31, 32 and 34 (can have maximum three

parallel lines) are constrained for only two parallel line, and new routes can have maximum three parallel lines.

Table-I

New lines and updated capacity of existing lines

Capacity # of proposed

parallel lines

Existin

g

Updated Total

TL 1 75 75 150 1*75

TL 3 75 50 125 1*75

TL 12 75 150 225 2*75

TL 14 100 100 200 1*100

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TL 16 75 50 125 1*75

TL 17 75 75 150 1*75

TL 19 100 50 150 1*75

TL 20 75 150 225 2*75

TL 21 50 50 100 1*75

TL 23 100 100 200 1*100

TL 24 75 100 175 1*100

TL 28 50 100 150 1*100

TL 29 75 150 250 2*75

TL 31 100 200 300 2*100

TL 32 75 175 250 2*100

TL 33 100 75 175 1*75

TL 34 75 125 200 2*75

NTL (1-3) 0 100 100 1*100

NTL (1-8) 0 75 75 1*75

NTL (2-4) 0 150 150 2*75

NTL (3-9) 0 125 125 2*75

NTL (6-7) 0 125 125 2*75

NTL (7-8) 0 150 150 2*75

NTL (7-9) 0 100 100 1*100

NTL (13-16) 0 225 225 1*100+2*75

NTL (14-15) 0 225 225 1*100+2*75

NTL (15-19) 0 125 125 2*75

NTL (15-21) 0 100 100 1*100

Total updated network capacity 3650

TL (Existing Transmission Line), NTL (New Transmission Line)

Table-II

Required Expected Economically Optimal Capacity of the Generators in MW

Bus 1 2 7 13 15 16 18 21 22 23

Existing 150 150 200 250 100 250 250 200 150 350

Updated 400 350 400 550 350 450 375 425 400 450

Total 550 500 600 800 450 700 625 625 550 800

EMP ($) 7 8 7 7 7 6.5 7 7.2 7.8 7.3

EMP = Expected Marginal Price

The remaining expected cost in the system, comprising EENS (74 MWh), EGNS (81 MW), and EWL (88

MW) costs etc. is $37.33m. This condition is most economic condition which is calculated by step-10 of the

methodology. Total investment in the generators is $1767.62m. Methodology based on GA is generally

expensive but due to its advantageous features, it is accepted and here simulation took 11 hours approximately

on core i3 processor with 4 GB RAM in MATLAB environment (using MatPower [30]). Solution gives

approximately flatter than expected marginal price profile at buses (Table-II). Since methodology follows

congestion management and resulting system is efficiently secured for unseen contingencies without loss of

economic market.

20

The proposed methodology is also tested on an IEEE 118-bus power system, given in [16]. This system has

54 existing generators, 91 loads and 186 existing transmission lines. FOR, lines length, impedances and existing

capacity of each transmission line is given in [16]. Total existing capacity of the transmission network is 29,600

MW with 10,000 mile circuit length to meet the load of 4,827 MW and 8,090 MW generation. Generation has

enough excess capacity to meet the 8.5% compound load growth. Here, 10 new generators (NG) are proposed to

be planned at the buses (31, 46, 87, 90, 102, 103, 107, 110, and 118). Capital cost to update old and new

generators per MW capacity, outage cost and O&M cost is also given for each generator (existing and new) in

[16]. A total of 167 candidate alternative lines are available [16], from which 53 new lines, totalling 7875 MW,

are selected after simulation. To achieve optimal solution at pareto-optimal front, 24 existing transmission lines

((from bus-to bus) 8-5, 8-9, 9-10, 30-17, 26-25, 34-37, 55-56, 63-59, 60-61, 64-65, 68-116, 77-78, 80-97, 88-89,

94-100, 101-102, 110-111, 105-108, 104-105, 105-106, 89-90, 63-64, 24-70, 22-23) are required, also, to update

with 2900 MW capacity. Due to ROW constraints limit on lines capacity are 300 MW, however, most of the old

lines have reached the prescribed maximum limit. Simulation took 128 hours to give optimal results with the

same processor mentioned above. In the final solution, the remaining EENS, EGNS and EWL are 300 MW, 305

MW and 310 MW, respectively. The value of objective function is $770m with investment in transmission and

generation system is $230m and $400m, respectively. The total cost including cost of EENS, EGNS and EWL is

$140m. Proposed generation capacity of new 10 generators are 173, 100, 173, 258, 368, 173, 173, 182, 47, 173

MW respectively calculated for the security of system duringN-1 and N-2 critical contingencies. Some old

generators (thirteen) are required to update along with new generators to secure economic operation of the

power system during the contingencies, with 300, 275, 100, 104, 100, 165, 190, 197, 275, 355, 95, 76, 155 MW

capacity at buses 10, 26, 25, 49, 46, 55, 65, 66, 80, 89, 100, 113, 116, respectively. Here updating of existing

generators ensures the operational economics. Further noticed that the update of old transmission lines and

generation capacities along with new lines and generators achieve optimally economic solution with higher

reliability level. In this scheme marginal price of all generator buses are approximately flat due to use of

economic load dispatch strategy.

5. Conclusion

A new probabilistic composite TEP methodology which does not require a priori specification of generation

21

and transmission lines capacities has been presented. Proposed methodology gives perfect alignment between

generation planning and TEP. To speed-up the TEP methodology four linear matrices have been introduced in

this paper, i.e., GLOF, PTDF, GPF and BBIM to remove DC-load flow calculations for multiple demands and

changing network topology due to contingencies. Proposed methodology is validated with the results presented

in [12] on modified IEEE 5 bus network. It shows that updating the existing generators and lines are necessary

for economic operation of the power system. Also it reduces the involved investment in the network. Since a

matrix model of calculating ENS, GNS and WL are proposed which reduced computation time drastically.

Results of the proposed TEP methodology have also been shown on an IEEE 24-bus and IEEE 118-bus power

systems. Results shown conclude the generalization and practical importance of the methodology for the large

power systems planning. GA based method secure the nonlinear optimization for discrete variables and security

checked is done for N-1 and probable N-2 contingency cases. Case study has been shown that probabilistic

congestion based model flatter the price profile of all generator buses. Thus, an economic market can be

established. Here, we can observe several lower capacity transmission lines. In real situation, these lines may be

uneconomical due to difficulties in acquiring ROW(s) to construct new transmission lines, and the economies of

scale. Avoiding of low size transmission lines can be achieved by pruning the "smaller capacity" transmission

lines, which is described in detail in a forthcoming paper. To make the proposed methodology more realistic,

important aspect of power system e.g. congestion cost, re-dispatch cost [15], [16] and AC-load flow analysis

[22] needs to be included with multi-stage [10] composite TEP, which is the scope of further papers.

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