Reliability constrained multiperiod generation ... - Wiley Online Library

INTERNATIONAL TRANSACTIONS ON ELECTRICAL ENERGY SYSTEMS Int. Trans. Elect. Energ. Syst. 2013; 23:961–974 Published online 12 March 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/etep.1632

Reliability constrained multi-period generation expansion planning of electrical energy resources using MILP Jamshid Aghaei1*,†, Alireza Roosta1, Mohammad Amin Akbari1, Abdorreza Rabiee2{ and M. Gitizadeh1 1

Department of Electrical and Electronics Engineering, Shiraz University of Technology, Shiraz, Iran 2 Department of Electrical Engineering, Islamic Azad University, Damavand Branch, Tehran, Iran

SUMMARY The main goal of Generation Expansion Planning (GEP) is to minimize total costs associated with new power generating units’ installation subject to technical and economical constraints. This paper addresses the GEP with probabilistic reliability criteria. The Loss-of-Load Probability reliability index is explicitly augmented as a new constraint which takes into account the reserve requirements. The outage cost is represented by the Expected Energy Not Served index. Due to nonlinear nature of these reliability indices, the GEP optimization problem with reliability criteria is very complicated to solve. Accordingly, the focus of this work is to deal with the reliability constrained multi-period GEP problem as a Mixed Integer Linear Programming (MILP). The results in the case study indicate the effect of reliability considerations on decision-making process. The simulation results also show the superiority of the proposed MILP-based method in comparison with the well-known metaheuristic algorithms and Dynamic Programming approach in the viewpoints of the accuracy and computational speed. Copyright © 2012 John Wiley & Sons, Ltd. key words:

Generation Expansion Planning (GEP); Reliability Metrics; Expected Energy Not Served (EENS); Loss-of-Load Probability (LOLP); Mixed Integer Linear Programming (MILP)

1. INTRODUCTION Generation Expansion Planning (GEP) problem is an important problem for the decision planners in the power utilities. GEP in fact defines when, where and which new generating unit should be commissioned online in the long term of planning horizon [1,2]. The main goal of GEP is to minimize the total investment, operating and maintenance (O&M) and interruption costs associated with the addition of new power generating units in the planning horizon subject to constraints such as, forecasted demand, acceptable level of reliability, fuel mix and environmental criterions. It is known that GEP problem is a large-scale Mixed Integer Nonlinear Programming (MINLP) problem which its nonlinearity originated from the Probabilistic Production Costing simulation and a setting of nonlinear constraints [3–6] related to nonlinear nature of reliability metrics. Such complicated problem in its nature can only be entirely solved by complete enumeration [6–8]. Therefore, to get the optimal solution, each possible combination of applicable options along a planning period must be examined, which leads to a computational explosion in the GEP problem [9]. During the past decades, mathematical methodologies have been successfully developed and applied to solve the GEP problem. Linear programming, nonlinear programming, mix-integer programming techniques with the linear approximation of an objective function, constraints and certain simplifications are used in [10–14]. References [13] and [14] applied MILP model for the power systems’ GEP. However, *Correspondence to: Jamshid Aghaei, Electronic and Electrical Department, Shiraz University of Technology, Shiraz, Iran. † E-mail: [email protected]; [email protected] { Current address: Department of Electrical Engineering, Faculty of Technology and Engineering, Shahrekord University, Shahrekord, Iran. Copyright © 2012 John Wiley & Sons, Ltd.

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they did not consider outage costs in their models. Moreover, in these studies, deterministic reliability constraints are considered. A well-known algorithm is presented to solve GEP problem based on Dynamic Programming (DP) approach [6,7]. Nevertheless, the so-called ‘the curse of dimensionality’ problem troubles the DP-based approaches in the GEP problems [9,12]. To get a local optimum in the DP method, softwares, e.g. WASP [1] and EGEAS [3], use a heuristic tunnel-based technique where users firstly specify the structure of states and then modify tunnels. As mentioned above, there are some difficulties regarding MINLP problems. For instance, such MINLP problems may not converge in a finite number of iterations due to the nonlinearity of constraints which results in the high computational burden, too. [15]. However, the MINLP problem can be solved using any commercial optimization package [6]. Several modern heuristic techniques have been developed in the last decade to solve the GEP problems, e.g. Evolutionary Programming (EP) [16,17], Simulated Annealing (SA) [18], Tabu Search (TS) [19], Expert Systems (ES) [8], Genetic algorithm (GA) [20]and Differential Evolution (DE) technique [21]. There are some structural problems in the conventional GA; Reference [22] presented an Improved Genetic Algorithm (IGA) to dominate these problems to solve the GEP problem. In this work, execution time and effectiveness of the IGA are compared with the conventional GA, DP and tunnel-based DP algorithm employed in WASP. Authors of [23] suggested Particle Swarm Optimization (PSO) technique and its variant such as Hybrid PSO, Stretched PSO and Composite PSO to the least-cost GEP problem. Different metaheuristic algorithms are implemented to deal with the GEP problem solution in [24]. The main objective of this work is to develop a new methodology based on the metaheuristic approaches to solve the least-cost solution minimizing investments, operation and interruption costs in the multi-stage GEP problem. This work proposed Intelligent Initial Population Generation to reduce the computational time. The results are compared with conventional DP in terms of speed and efficiency. Metaheuristic algorithms have the potential of finding proper feasible solutions, but not guaranteeing to find global optimal solution and accordingly require a large solution space. Therefore, comprehensive numerical computation is often required particularly when the load flow procedure is essential [25]. Reliability is one of the important factors in power system planning for future system capacity expansion [26]. Basically, the evaluation of the power system reliability can be done by using two main approaches: deterministic [11,12] and probabilistic criterion [27,28]. The most common deterministic criterion is the reserve margin that can be set equal to the largest generating unit capacity in the system, or to some fraction of the peak demand. An important drawback of the deterministic criterions is that they do not account for the stochastic nature of unit behavior. For example, it does not consider the failure rate and size of different type of generation units. Deterministic analysis using just reserve margin calculation could lead to over-investment or insufficient system reliability in the GEP. Usually, measurement of reliability or adequacy to make sure that total generation system capacity is sufficient to meet the system demand requirements, is provided by two commonly used indices: (i) Loss-of-Load Probability (LOLP), (ii) Expected Energy Not Served (EENS) [29]. Also, these indices set the reserve requirements and are sensitive to unit size, type, number and Forced Outage Rate (FOR). The contributions of this work with respect to the previous researches in the area can be summarized as follows: • Presenting a hybrid deterministic/probabilistic approach to evaluate the reliability metrics for the reliability constraints of the multi-period GEP problem. An upper bound on probabilities of lossof-load and EENS due to single and double unit contingencies is used to define reliability criteria. The main advantage of this method is that the reliability indices have been formulated as a linear function of the GEP integer and continues variables. • Proposing a 0–1 MILP approach that allows rigorous modeling of (i) non-convex and nondifferentiable outage cost, and (ii) reliability constraints. It overcomes the modeling limitations of the DP approaches and is computationally efficient. The remainder of this paper is organized as follows: In Section 2, the GEP model is introduced in the form of a Mixed Integer Non-Linear Programming (MINLP) problem. In section 3, the procedure of formulation and linearization of the reliability metrics is proposed. Section 4 presents two case studies to demonstrate the effectiveness of the proposed scheme. Some relevant conclusions are drawn in the section 5. Copyright © 2012 John Wiley & Sons, Ltd.

Int. Trans. Elect. Energ. Syst. 2013; 23:961–974 DOI: 10.1002/etep

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RELIABILITY CONSTRAINED MULTI-PERIOD GENERATION

2. PROBLEM FORMULATION Solving a least-cost single objective GEP problem is mathematically equivalent to finding a set of the best decision vectors along the planning period that minimize the total cost subject to technical and economical constraints. 1.2. Objective function The objective function is to minimize the sum of the discounted (present value) investment costs associated with the capacity of newly added generating units (ICt), the operational and maintenance (O&M) costs (MCt) and the outage costs (OCt) for all generating units, minus the salvage costs (SCt) related to the capacity of newly added generation units along the life time of plants. To yield a net present value, all future costs are discounted to the present time with a discount factor d (rate of return) over the time of investment for a unit. The GEP problem is represented by the following formulation [23,24,29,30]: X minCt ¼ ½ICt þ MCt þ OCt SCt (1) t2T

where ICt ¼ ð1 þ dÞ2t MCt ¼

1 X

( ð1 þ d Þ

ð2tþs′ þ0:5Þ

s′ ¼0

N X

Int unt

n¼1 t N X X t¼1

!

unt pn fn þ EESn mn

OCt ¼

ð1 þ d Þ

NX X

) ðpn fn þ EESn mn Þ

(3)

n¼1

" ð2tþs′ þ0:5Þ

s′ ¼0

þ

n¼1

EESn ¼ 8760 pfn pn ; N ¼ 1 X

(2)

J X

Nj

(4)

j¼1 t X N X

g EENSnt þ

t¼1 n¼1

SCt ¼ ð1 þ dÞ2ðTtþ1Þ

NX X

# g EENSnt

(5)

n¼1 N X

dn Int unt

(6)

n¼1

For the newly added generating units, Int is the investment cost [$] of unit n in time period t (a time period = 2 years), unt is the decision variable of unit n in time period t. That is, unt equal to 1 if generating unit n is selected in time period t and 0 otherwise. T is the length of the planning horizon in time period, and set j refers to the type of fuel used, e.g. oil, coal, Liquefied Natural Gas (LNG) and nuclear. N is the total number of newly added generating units, and N j is the maximum construction number of unit type j. fn, mn and pn are the fixed [$/MW], variable [$/MWh] O&M costs and capacity of the generating unit n, respectively. NX is the total number of existing units at the beginning of the planning. pfn is the utilization factor of plant type n. EES is the expected energy served [$/MWh], and set s′ is used to calculate maintenance costs at the middle of each year. EENS is the expected energy not served [MWh], and its cost is g [$/MWh]. dn is the salvage value of the newly added generating unit n. 2.2. Constraints Upper construction limit: This physical constraint reflects the maximum number of yearly construction capability of various types of units to be committed in the expansion plan. Nj X

umt ≤Nj 8t ¼ 1; 2; K T; j ¼ 1; 2; K J

(7)

m¼1

Power demand: The power demand constraint ensures that the sum of the selected units’ capacity among the existing units is equal to peak power demand plus reserve in time period t. Copyright © 2012 John Wiley & Sons, Ltd.


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X X t2t

! þ

an unt pn

n2N

X

an pn ¼ Dt þ Rt

(8)

n2NX

where, an is the availability of unit n [%], Dt is the peak power demand [MW] and Rt is positive variable indicating the reserve value in the time period t [MW]. It is noted that D is a function of price in the view point of generation system or generation planning, since it is a function of generation. At the same time, the generation is a function of price. Therefore, it is called power demand. On the other hand, from the viewpoint of a planner, D is demand, and from a consumer standpoint, it is consumption. Besides, note that in this paper, the D is considered to be known for planner. Indeed, it is assumed that the demand/load forecasts are individually determined using long-term load forecast package prior to solve the GEP problem. Fuel mix ratio: there are different types of units in the GEP. Consequently, decision makers restrict the share of each kind of unit types by following constraint: XX XX umt pm = unt pn ≤Mjmax 8t 2 T; j ¼ 1; 2; . . . ; J (9) Mjmin ≤ t2t m2Nj

Mmin j

t2t n2N

Mmax j

where, and are the minimum and maximum fuel mix ratios of the selected unit type j (i.e. Oil, Coal, LNG and Nuclear). Reserve margin constraint: The capacity of the existing and newly introduced generating units must satisfy the minimum and maximum reserve margins, R min ≤Rt =Dt ≤R max 8t 2 T

(10)

Here, Rmin and Rmax represent the minimum and maximum reserve margins, respectively, which are assumed in percentage. LOLP constraint: The existing units together with the selected units must satisfy the reliability metric criterion, i.e. LOLP. lolpt ≤e8t 2 T

(11)

Where, lolpt is the actual level of LOLP in year t. Also, e is the reliability criterion expressed in LOLP.

3. RELIABILITY METRICS FORMULATION This paper proposes the probabilistic approach to evaluate the reliability metrics (LOLP and EENS) for the multi-period GEP model considering only single and double unit contingencies. To express these reliability metrics in terms of GEP variables, suppose that state vector X (which is composed of binary values, i.e. 0 or 1) is the vector of all generating units in planning horizon T as follows: X ¼ ½un0 ; unt ¼ ½un0 ; un1 ; K; unT

(12)

un0 ¼ 18n 2 NX

(13)

Where, un0 is the state vector of existing units at beginning of the planning period. All of the existing units in each time period along the planning horizon should be scheduled on, so this state vector is equal to 1 for existing unit n (n = 1, 2,.., NX). Thus, the length of vector X will be equal to NX + (T N). The probabilities of random single and double unit contingencies in terms of the GEP variables can be defined by: Y ð1 ukt Ukt Þ8t 2 T; n 2 Nt ; Nt ¼ NX þ t N (14) r1nt ¼ unt Unt k2Nt ; k6¼n

r2nkt ¼ unt ukt Unt Ukt

Y

ð1 ult Ult Þ8t 2 T ; n; k 2 Nt ; k > n

(15)

l2Nt ;l6¼n;k

Where, r1nt is the probability of single contingency in which unit n in time period t is selected but unavailable, r2nkt is the probability of double contingency in which units n and k are selected in time period t but unavailable. Unt is the unavailability (FOR) of unit n in time period t. Nt is the total number of generating units (existing and candidate units) from time period 0 up to time period t. Note that each Copyright © 2012 John Wiley & Sons, Ltd.



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contingency necessarily does not cause loss-of-load, so a new set of binary variables are introduced which should satisfy the following linear inequality constraint. ! ! X X X X 1 Dt pk pi ≤snt ≤1 þ Dt pk pi 8t 2 T; n 2 Nt (16) k2Nt ;k6¼n

=

k2Nt ;k6¼n

i2Nt

=

i2Nt

In (16), s1ntis equal to 1 if the forced outage of unit n in time period t causes loss-of-load and 0 otherwise. In other words, if loss-of-load occurs, (i.e. Dt > Σpk) the lower bound of (16) should be between zero and 1, while the upper bound is greater than 1. Sinces1nt is a binary variable, it should be equal to 1. When there is no loss-of-load, (i.e. Dt < Σpk) the lower bound of (16) should be less than zero, whereas the upper bound is between zero and 1. Therefore, the binary variables1nt should be equal to zero. Similarly, for higher order contingencies, the binary variables can be defined. For instance, for double unit contingencies, the binary variables2nkt should satisfy the following inequalities: ! ! X X X X 1 pl pl Dt i 2 Nt pi ≤snkt ≤1 þ Dt i 2 Nt pi 8t 2 T; n; k l2Nt ; l6¼n;k

2 Nt ; k > n

=

l2Nt ; l6¼n;k

=

(17)

(17), s2nktis

equal to 1 if the forced outage of unit n and k in time period t cause loss-of-load and 0 In otherwise. Ultimately, the reliability metrics LOLP and EENS can be expressed for each time period in terms of the GEP variables as follows: X X X s2nkt r2nkt þ K s1nt r1nt þ (18) LOLPt ¼ n2Nt n2Nt k 2 N t 0 1 k>n BX C X X B C 1 1 2 2 B snkt rnkt ðpn þ pk Rt Þ þ KC snt rnt ðpn Rt Þ þ EENSt ¼ 8760 B (19) C @n2Nt A n2Nt k 2 N t k>n 12 Let,lolp12 t be the true value of LOLPt and eenst be the true value of EENSt due to single and double generating unit contingencies: X X X s2nkt r2nkt s1nt r1nt þ (20) lolp12 t ¼ n2Nt n2Nt k 2 N t k>n 0 1

eens12 t

BX C X X B C 1 1 2 2 B snkt rnkt ðpn þ pk Rt ÞC ¼ 8760 B snt rnt ðpn Rt Þ þ C @n2Nt A n2Nt k 2 N t k>n

(21)

Because of nonlinearity and combinatorial nature of such metrics, they are complicated task from the viewpoint of computational aspects. A solution to overcome this difficulty is to consider only the most probable contingencies, e.g. single and double unit contingencies, and replace the higher order of product term of the contingency probabilities by their upper limits. For instance, by extension of the relation (14) and ignoring higher order terms (greater than second order), the upper limit for the probability of a single contingency can be written as follows: r1nt ≤unt Unt

(22)

By similar procedure in Equation (15), the upper limit on the probability of a double contingency is as follows: r2nkt ≤unt ukt Unt Ukt

(23)

By using (22) and (23) in (20) and (21), respectively, the upper limits of reduced reliability metrics are: Copyright © 2012 John Wiley & Sons, Ltd.


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lolpUL12 ¼ t

X

s1nt unt Unt þ

n2Nt

0 eensUL12 t

X

X

n2Nt

k 2 Nt k > n

s2nkt unt ukt Unt Ukt

(24) 1

BX C X X B C 1 2 B snkt unt ukt Unt Ukt ðpn þ pk Rt ÞC ¼ 8760 B snt unt Unt ðpn Rt Þ þ C @n2Nt A n2Nt k 2 N t k>n

(25)

If the unavailability of the generating units is small, these upper limits reliability metrics reach to a 12 proper approximation of the true value of lolp12 t and eenst metrics. 3.1. Sorting out nonlinearities The above reliability indices are the main source of nonlinearity in the proposed framework for GEP problem. This section provides a linear formulation for the upper bound of reliability metrics [31]. . Consider ’ is the product of n binary variables i.e. ’ = П xi (i = 1, 3.1.1. Linear expression of lolpUL12 t 2. . . m). Therefore, it can be converted to a set of m + 2 linear inequalities using following formulations: 8 ’≥0 > > < ’≤xi ; i ¼ 1; 2; . . . ; m m (26) X > > : ’≥ xi m þ 1 i¼1

Above relations ensure that the product of n binary variables (’) will be equal to 1 if and only if all these binary variables (xi) equal to 1 and zero otherwise. Let reformulate Equation (24) as follows: X X X ynkk Unt Ukt ¼ xnt Unt þ (27) lolpUL12 t n2Nt n2Nt k 2 N t k>n where,

While, for xn, 8 t 2 T, n 2 Nt:

xnt ¼ unt snt 8n 2 Nt ynkt ¼ unt ukt snkt 8t 2 T; ðn; kÞ 2 Nt ; n > k 8 < 0≤xnt ≤snt x ≤u : nt nt xnt ≤snt þ unt 1

(28)

(29)

And, also for ynkt, 8 t 2 T, (n, k) 2 Nt, n > k: 8 0≤ynt ≤snt > > < ynt ≤unt y ≤u > > : nt kt ynt ≤snt þ unt þ ukt 2

(30)

From (27)–(30), one can see that above linear relations are equivalent to the nonlinear reduced met, i.e. (24). Thus, the inequality (31) is always active and set the reserve requirement by ric lolpUL12 t imposing the upper bound to be lower than or equal to the pre-specified reliability criterion e UL12 lolp12 ⩽e t ⩽lolpt

(31)

3.1.2. Linear expression of eensUL12 . Let the continuous variable ’ is the product of binary variable x t and continues variable y that ymin ≤ y ≤ ymax. It can be easily converted to linear inequalities including the binary and continues variables (x, y) as follows: Copyright © 2012 John Wiley & Sons, Ltd.



967

xy min ≤f≤xy max

(32)

y y max ð1 xÞ≤f≤y þ y max ð1 xÞ

(33)

Inequality (32) forces ’ to zero value if the binary variable x is equal to zero, while inequality (33) ensures that when the binary variable x is equal to one, ’ is equal to y. Thus, from the (32) and (33) and according to sets (xi, ynkt) from (29) and (30), the reduced metric can be written as the following linear relations: eensUL12 t 0 1 BX C X X B C B C z ¼ 8760 z þ eensUL12 nt nkt C t B @n2Nt A n2Nt k 2 N t k>n where, 8 t 2 T, n 2 Nt

8 < 0≤znt ≤xnt Un pn P znt ≤Un ðpn Rt Þ þ ð1 xnt ÞUn Pi2Nt pi : z ≥U ðp R Þ ð1 x ÞU n n t nt n nt i2Nt pi

where, 8 t 2 T, (n, k) 2 Nt, k > n 8 < 0≤znkt ≤ynkt Un Uk ðpn þ pk Þ P znkt ≤Un Uk ðpn þ pk Rt Þ þ ð1 ynkt ÞUn Uk Pi2Nt pi : z ≥U U ðp þ p R Þ ð1 y ÞU U n k n k t nkt n k nkt i2Nt pi

(34)

(35)

(36)

in (25). The above linear relations (34) to (36) are equivalent to the nonlinear reduced metric eensUL12 t The other reliability metric, i.e. Expected Load Not Served (ELNS) that refers to average load shedding under loss-of-load (MW) is similar to EENS, and mathematically can be given by: ELNS ¼ EENS=8760:

(37)

4. SIMULATION RESULTS In this section, two case studies (classroom test case and real test case) are considered to study the performance of the proposed MILP-based optimization problem of reliability constrained GEP model. At first, expansion planning of a three-generator system over a six-year horizon (in three 2-year intervals) is studied in detail. This case is simple enough so that the correctness of the results can be readily verified, yet it conveys many interesting features of the proposed MILP formulation of the GEP. Thus, this case examines the effects of reliability aspects, i.e. deterministic and probabilistic approaches, for the pre-selected set of contingencies (single and double contingencies). After that the system with 15 existing generating units and five types of candidate unit (fuel used) [9,22] is scheduled over different time horizons (6-year, 14-year and 24-year). This case highlights the superiority of the proposed scheme with respect to the other methods in the area. Relevant simulations of the case studies are performed in GAMS software package [32] using large-scale commercial CPLEX solver [33] using PC with a 2-GHz Pentium(R) Dual-Core CPU, 1-GB RAM. 4.1. Small-scale test case The generating units’ data of this simple case is taken from [9,22]. For explicit analysis, one 200-MW Oil-unit and one 50-MW LNG-unit with total capacity of 250 MW are assumed at the beginning of the planning horizon. The technical and economical data of the existing units can be found in Table I. The forecasted peak load demand in each time interval (2 years) is shown in Table II. The GEP horizon is considered six years (three 2-year time intervals). Three types of candidate generating units (Oil, LNG Combined Cycle and Coal Bituminous) are considered that their data is shown in Table III. The EENS cost is assumed to be 0.05 [$/KWh]. Copyright © 2012 John Wiley & Sons, Ltd.


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For the sake of simplicity, the maximum capacity limit, i.e. Equation (10), of the candidate power plants is assumed inactive. In this example, four cases are considered, (i, ii) the conventional approach with (i) and without (ii) a deterministic reserve criterion, and (iii, iv) the probabilistic approach with two different LOLP metric criterions over three time period planning. Table IV summarizes generation expansion plans, and Table V shows and compares the results of GEP in each case through the planning periods. Case I: GEP without reserve margin (i.e. Rt = 0). As shown in Table IV, due to no reserve assumption, the least number of the generating units are selected to meet load demand. It can be seen that the peak

Table I. Technical and economical data of the existing plants for the small-scale test case. Unit type Capacity (MW) FOR (%) Fixed O&M cost ($/KW-Month) Maintenance cost ($/KWh) Number of units

Oil

LNG G/T

200 7.0 2.25 0.024 1

50 3.0 4.52 0.043 1

Table II. Forecasted peak load demand for the small-scale test case. Time period Year peak demand (MW)

0

1

2

3

2005 200

2007 750

2009 1250

2011 1900

Table III. Technical and economical data of candidate plants for the small-scale test case. candidate type Capacity (MW) FOR (%) Capital cost ($/MW) Fixed cost ($/KW-Mon) Maintenance cost ($/KWh) Construction limit Life time (years)

Oil

LNG C/C

Coal (Bit.)

200 7.0 812.5 2.2 0.021 1 25

450 10.0 500 0.9 0.035 1 20

500 9.5 1062.5 2.75 0.014 1 25

Table IV. The results of the proposed GEP in the four cases of the small-scale test case. time period

Case I: No reserve Case II: Deterministic reserve Case III: e = 10% Case IV: e = 3%

Copyright © 2012 John Wiley & Sons, Ltd.

Unit type

1

2

3

Oil LNG C/C Coal Bit Oil LNG C/C Coal Bit. Oil LNG C/C Coal Bit. Oil LNG C/C Coal Bit.

0 0 1 0 1 1 1 1 1 1 1 1

0 0 1 1 0 1 0 0 1 1 1 0

1 1 0 1 0 1 1 1 0 0 1 1



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Table V. Comparison of deterministic and probabilistic approach of the proposed GEP in the four cases of the small-scale test case.

Rt (MW) (%) lolpUL12 t lolp12 t (%) LOLPt (%) elnsUL12 (MW) t elns12 t (MW) ELNSt (MW) Outage cost (M$) Total cost($) 109 Execution time (s)

Time period

Case I

Case II

Case III

Case IV

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

0 0 0 18.34 25.99 37.44 18.36 26.12 38.16 62.85 109.2 161.7 63.00 110.5 169.5 27.59 48.39 74.24 1.915 0.195

450 650 700 10.3 4.08 3.54 10.39 4.789 5.390 11.18 7.690 9.710 17.96 10.82 11.91 7.866 4.739 5.216 2.745 0.044

650 650 650 2.3 5.5 9.7 1.88 4.08 6.08 2.18 4.78 8.00 3.515 10.18 9.054 2.933 7.695 12.06 3.915 10.82 21.03 1.714 4.739 9.211 2.703 0.864

650 800 1100 2.3 2.9 0.0 1.88 2.04 0.00 2.18 3.05 1.80 3.515 3.054 0.000 2.930 2.761 0.000 3.915 5.460 3.790 1.714 2.391 1.660 3.059 0.505

load is equal to cumulative installed capacity in each time period. Thus, any number of unit contingencies causes loss-of-load. Therefore, this case is the most unreliable case with actual values of LOLP1, LOLP2 and LOLP3 equal to 18.36%, 26.12% and 38.16%, respectively. Also, in this case, the actual values of ELNS and outage cost are significantly higher than the other cases. Case II: GEP with deterministic reserve criterion. In this case, the pre-specified reserves are considered for each time interval as R1 = 450 MW, R2 = 650 MW and R3 = 700 MW. As shown in Table V, in order to satisfy the reserve requirements criterion, one more LNG C/C unit in the first time period and one more Oil unit in the second period should be installed in addition to the units in the case I. From the row before the last one in Table V, it is observed that scheduling more units to provide reserve criterion causes the total cost to be increased about 43.34% in comparison with case I. This case relatively mitigates outage, and accordingly, the values of LOLP and ELNS are decreased with respect to case I, as shown in Table V. in Equation (31) is imposed with Case III: GEP with e = 10%. In this case, the upper bound of lolpUL12 t e = 10%. As expected, this case provides better outage mitigation with respect to the cases I and II; since, the levels of LOLP and ELNS are improved significantly. Note that, because of imposing same reliability criteria in each time period, the reliability levels (LOLP or ELNS) have been enhanced by increasing the unit number. As seen, the total number of selected units in this case is equal to that of the case II with slightly greater spinning reserves and the lower relative total cost about 1.54%. Consequently, it can be inferred that the GEP with probabilistic reserve criterion could select the combinations of candidate units with minimum total cost and desired reliability level. Case IV: the GEP with e = 3%. As expected, as shown in the first row of Table V, the reserve will increase since the reliability levels are raised. Therefore, more capacity of scheduled units is required to meet the reserve constraint. Thus, the total cost has increased about 59.73%, 11.43% and 13.17% in Copyright © 2012 John Wiley & Sons, Ltd.


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comparison with the cases I to III, respectively. The value of lolpUL12 is equal to zero, indicating that no 3 loss-of-load occurs mainly for great amount of spinning reserve in this period (R = 1100 MW). Any single and double contingency of generating units in this case is substituted by reserve units; therefore, the system no longer incurs loss-of-load. However, as seen from the results of fourth row of Table V, considering the higher order of contingencies will result in loss-of-load, e.g. LOLP3 = 1.80 for the case IV. From the fourth row of Table V, using the proposed linear model of reliability metrics provides relatively good approximation of true value oflolp12 t . Note that by considering reliable units (lower FOR), the difference between upper bound and actual values of reliability metrics can be significantly decreased. Also, it is possible to reach to a desirable level of reliability by incorporating lower reserve and consequently lower total cost in the case of using highly reliable units. The ninth row of Table V explicitly shows that total cost is in contrast with the outage cost of the eighth row. Besides that, the total cost is correlated with the levels of the reserve and reliability. In other words, the lower reserves result in the lower cumulative cost, but with the greater outage cost. 4.2. Real test case The proposed GEP is also studied on a real test system, and the results are compared with those of the other methods in the area. The test system includes 15 existing generating units with the total capacity of 5450 MW, and 18 candidate units in five types (fuel-unit). The characteristics of the existing units as well as candidate units are shown in the Table VI and the Table VII, respectively. To better illustrate the performance of the proposed model, three cases are considered named as Cases V to VII related to short-term (6-year), long-term (14-year) and very long-term (24-year) planning horizon as in many researches in this area [22–24,29,30]. The planning horizons of 6, 14 and 24 years are divided into 3, 7 and 12 time periods, respectively. There are several parameters to be pre-specified, which are related to the GEP problem. In this paper, the lower and upper limits of reserve margins are assumed to 20% and 40%, respectively. Also, the discount factor and LOLP threshold (e) are considered as 8.5% and 1%, respectively. The EENS cost is 0.05 $/KWh. The salvage value for the candidate units is only in the range of 10–20%. The lower and upper bounds of fuel mix, Equation (10), are 0% and 30% for Oil-fired plant, 0% and 40% for LNG-fired plant, 20% and 60% for Coal-fired plant and 30% and 60% for Nuclear. The utilization factors are considered 50% for the oil and LNG CC-fired plants, 70% for the Coal-fired plant and 90% for the nuclear-units [22–24,29,30]. Based on the above assumptions, the proposed MILP framework has been performed for the mentioned three cases. Table VIII shows the optimal number of newly introduced generating units as well as cumulative capacity and peak load to show the amount of reserve requirements in each time period

Table VI. Technical and economic data of existing units for the real test case [22,30]. Name (Fuel Type)

No. of Units

Unit Capacity (MW)

FOR (%)

Opr. Cost ($/Kwh)

Fixed O&M Cost ($/Kw-Mon)

Oil#1(Heavy Oil) Oil#2(Heavy Oil) Oil#3(Heavy Oil) LNG G/T#1(LNG) LNG C/C#1(LNG) LNG C/C#2(LNG) LNG C/C#3(LNG) Coal#1(Anthracite) Coal#2 (Bituminous) Coal#3 (Bituminous) Nuclear#1(PWR) Nuclear#2(PWR)

1 1 1 3 1 1 1 2 1

200 200 150 50 400 400 450 250 500

7.0 6.8 6.0 3.0 10.0 10.0 11.0 15.0 9.0

0.024 0.027 0.030 0.043 0.038 0.340 0.035 0.023 0.019

2.25 2.25 2.25 4.52 1.63 2.00 6.65 6.65 2.81

1

500

8.5

0.015

2.81

1 1

1,000 1,000

9.0 8.8

0.005 0.005

4.94 4.63



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Table VII. Technical and economic data of candidate units for the real test case [22,30]. Caandidate Typr

Construction Upper limit

Capacity (MW)

FOR (%)

Opr. Cost ($/Kwh)

Fixed O&M Cost

Capital Cost ($/Kw)

Life Time (Years)

Oil LNG C/C Coal (Bit.) Nuc.(PWR) Nuc. (PHWR)

5 4 3 3 3

200 450 500 1,000 700

7.0 10.0 9.5 9.0 7.0

0.021 0.035 0.014 0.004 0.003

2.20 0.90 2.75 4.60 5.50

812.5 500.0 1062.5 1625.0 1750.0

25 20 25 25 25

for cases V and VI. It is noted that all the constraints of the problem are satisfied by the obtained results. Also, Table IX compares the costs of the best solution obtained by MILP method with DP and several metaheuristic methods, e.g. EP, SA, TS, ES, GA, ACO, PSO, HA and DE [23,24]. It is noted that there are no results for 14-year horizon for the ACO method. It can be seen that the proposed MILP method has the least total cost in each cases, showing its ability to overcome a probable local optimal trap in a practical long-term GEP. Moreover, from the computational burden (execution time) viewpoint, it can be seen from Figure 1 that the proposed MILP method significantly dominates the metaheuristic methods. Also, the execution time of MILP-based method is much lower than that of DP. For example, in case VI, it takes more than 40.6 h for DP to obtain the optimum solution while it takes only 15.4 min by MILP to get the optimum solution. Consequently, the superiority of the proposed MILPbased method can be logically concluded with respect to the metaheuristic and DP search algorithms based on the results of Table IX and Figure 1.

5. CONCLUSIONS This paper presents a mixed integer linear programming (MILP) model for optimal planning of energy system’s generation capacity with hybrid deterministic/probabilistic reliability criteria. The reliability metrics (LOLP and EENS) are mathematically represented in the form of binary and continuous GEP variables considering random single and double contingencies. The advantage of this formulation is its compatibility with powerful MILP tools. The main advantages of the proposed framework are (i) linear formulations of the GEP optimization problem using linearized reliability metrics; (ii) Considering reliability issues in the GEP model which provides a flexible framework for the planners of power systems to simultaneously cope with both economical and reliability concerns. In view of that, simulation results demonstrate how the GEP problem is affected by power system probabilistic reliability and deterministic reserve criteria. Also, the real test case results show the performance and

Table VIII. Optimal number and cumulative capacity of newly introduced plants in case V and case VI for the real test case. Planning year candidate type Oil LNG C/C Coal .Bit Nuc (PWR) Nuc (PHWR) Peak load (MW) Cumulative capacity (MW) 1

1998 1

(5) 5 (3) 3 (2) 1 (1) 1 (0) 0 7000 (9800) 9300

2000

2002

2004

2006

2008

2010

(0) 5 (2) 0 (1) 2 (1) 1 (0) 0 9000 (12200) 12300

(1) 4 (2) 0 (0) 0 (0) 0 (0) 0 10000 (13300) 13100

5 1 1 1 0 12000 16050

0 1 0 0 0 13000 16500

0 1 1 1 0 14000 18450

0 1 0 0 0 15000 18900

The figures within parenthesis denote the results in case V



972 200 150 100 50

P

A

M IL

H

TS

O

6-year

SA

CO

ES

PS

A

E

EP

G

A

0 D

Execution Time(min)

J. AGHAEI ET AL.

14-year

Figure 1. Execution time of the proposed MILP method against the other methods in the real test case study.

Table IX. Summary of the best results obtained by each solution method in the three cases of the real test case. Total cost 1010 ($) Method1

Case 1

Case 2

Case 3

GA DE EP ES ACO PSO SA TS HA DP MILP

1.2009 1.2009 1.2009 1.2009 1.2009 1.2009 1.2009 1.2009 1.2009 1.2009 1.181

2.1834 2.1834 2.1887 2.1840 2.1859 2.1924 2.1858 2.1797 2.1797 2.152

2.9356 2.9262 2.9262 2.9301 2.9279 2.9262 2.9825 2.9206 Unknown 2.917

1

Metaheuristic and DP methods are taken from [23,24].

effectiveness (i.e. finding an optimal solution in a lower CPU time) of the proposed MILP model with respect to the DP and metaheuristic methods. The research work is under way in order to incorporate reliability metrics in the multiobjective framework and also to simulate GEP uncertainties using the stochastic formulations. 6. LIST OF SYMBOLS AND ABBREVIATIONS t n d int unt fn mn pn j N NX pfn EENS g dn

time period in years type of generating units discount rate, (%) investment cost of unit n in time period t, ($) 1 if generating unit n is selected in time period t and 0 otherwise fixed cost of the generating unit n, ($/MW) variable O&M cost of the generating unit n, ($/MWh) capacity of the generating unit n, (MW) type of fuel set of all generating units, existing as well as candidate set of existing units at the beginning of the planning utilization factor of plant type n, (%) expected energy not served, (MWh) cost of expected energy not served, ($/MWh) salvage value of the newly added generating unit n




an Rt Dt Mmin j Mmax j Rmin Rmax LOLP e X T un0 r1nt r2nkt Unt Nt s1nt s2nkt lolp12 t eens12 t IC SC OC MC

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availability of unit n, (%) reserve value in the time period t, (MW) peak power demand, (MW) minimum fuel mix ratio of the selected unit type j, (%) maximum fuel mix ratio of the selected unit type j, (%) minimum reserve margin which is assumed in percentage, (%) maximum reserve margin which is assumed in percentage, (%) loss-of-load probability the reliability criterion expressed in LOLP, (%) vector of all generating units in planning horizon, 0 or 1 planning horizon in years state vector of existing units at beginning of the planning period, 0 or 1 Probability of single contingency in which unit n in time period t Probability of double contingency in which units n and k are selected in time period t but unavailable the unavailability (forced outage rate) of unit n in time period t the total number of generating units (existing and candidate units) from time period 0 up to time period t 1 if the forced outage of unit n in time period t causes loss-of-load and 0 otherwise 1 if the forced outage of unit n and k in time period t cause loss-of-load and 0 otherwise Real value of LOLPt due to single and double generating unit contingencies, (%) Real value of EENSt due to single and double generating unit contingencies, (MW) investment cost of newly added generating units salvage cost of newly added generating units outage cost of existing and of newly added generating units maintenance cost of existing and of newly added generating units

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