Generation Expansion Planning by MILP considering mid-term scheduling decisions Grigoris A. Bakirtzis, Pandelis N. Biskas, Vasilis Chatziathanasiou Department of Electrical Engineering, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece (
[email protected]) Corresponding author: Pandelis Biskas: Power Systems Laboratory, Division of Electrical Energy, Department of Electrical Engineering, Aristotle University of Thessaloniki, AUTh Campus, 54124, Thessaloniki, Greece, E-mail:
[email protected], Tel.: +30 2310 994352, Fax: +30 2310 996302 Abstract This paper presents a mixed-integer linear programming model for the solution of the centralized Generation Expansion Planning (GEP) problem. The GEP objective is the minimization of the total present value of investment, operating and unserved energy costs net the remaining value of the new units at the end of the planning horizon. Environmental considerations are modeled through the incorporation of the cost of purchasing emission allowances in the units’ operating costs and the inclusion of annual renewable quota constraints and penalties. A monthly time-step is employed, allowing mid-term scheduling decisions, such as unit maintenance scheduling and reservoir management, to be taken along with investment decisions within the framework of a single long-term optimization problem. The proposed model is evaluated using a real (Greek) power system. Sensitivity analysis is performed for the illustration of the effect of demand, fuel prices and CO2 prices uncertainties on the planning decisions. Keywords: Generation Expansion Planning, Mixed-Integer Linear Programming, Value of Lost Load, optimal planning, optimal unit maintenance schedules, mid-term scheduling decisions
1.
Nomenclature
Indices and Sets
i (I )
Index (set) of units
h (I hyd )
Index (set) of hydro units (hydroplants), I hyd ⊆ I
1
m (M )
Index (set) of yearly sub-periods (months)
j (J )
Index (set) of load levels
y (Y )
Index (set) of years within the planning horizon
s (S )
Index (set) of power stations
I old
Set of units commissioned or with firm commissioning plans at the beginning of the planning horizon, I old ⊆ I
I r ,old
Set of units commissioned at the beginning of the planning horizon, eligible for refurbishment, I r ,old ⊆ I
I new
Set of new units, candidates to be selected for addition during the planning horizon,
I new ⊆ I +
= I RES ∪ I hyd I RES
I RES
Set of Renewable Energy Sources (RES) units, I RES ⊆ I .
d h (Dh )
Index (set) of hydroplants/reservoirs immediately upstream of hydroplant/reservoir h ∈ I hyd
Parameters
Y
Last year of the planning horizon
y RES
First year during which the RES quota target is effective
Cost
Present value of total production cost, in €
DF y
Discount factor for year y, in p.u.
DR
Nominal discount rate, in p.u.
IR f
Inflation rate for the supply cost of fuel f, in p.u.
IRCO 2
Inflation rate for the CO2 emissions price, in p.u.
IROM (i )
Inflation rate for the O&M cost (both variable and fixed part) of unit i, in p.u.
Tilife
Expected lifetime of unit i, in years
Ti r ,life
Expected lifetime of the existing unit i after refurbishment, in years
2
Tiini
Age of unit i in the beginning of the planning horizon, in years. If negative, its absolute value represents the year of planned commissioning of the unit (with firm decision at a prior stage).
Ticon
Construction time of unit i, in years.
f (i )
Fuel type of unit i, e.g. ‘coal’, ‘gas’, ‘oil’
IHRi
Incremental heat rate of unit i, in GJ/MWh
eri
CO2 emissions rate of unit i, in T/MWh
cf
Price of fuel f, in €/GJ
cCO2
CO2 emissions price at the first year of the planning horizon, in €/T
cOM (i )
Variable part of the operation and maintenance (Ο&Μ) cost of unit i at the first year of the planning horizon, in €/MWh
cFOM (i )
Fixed part of the Ο&Μ cost of unit i at the first year of the planning horizon, in €/MW-year
h y ,m, j
Duration of load level j of sub-period m of year y, in hours
Ly ,m, j
Amount of load level j of sub-period m of year y, in MW
ctiy
Marginal cost of unit i, including the variable part of the O&M cost and the CO2 emissions cost, during year y, considering inflation, in €/MWh
ctri y
Marginal cost of existing unit i after refurbishment (in case of existing unit), including the variable part of the O&M cost and the CO2 emissions cost, during year y, considering inflation, in €/MWh
Pi max
Capacity of unit i, in MW
Pi r ,max
Capacity of existing unit i after refurbishment, in MW
VLLy
Value of lost load during year y, in €/MWh
y PEN RES
Penalty for not meeting RES quota target during year y, in €/MWh
y ,m
EFORi
Equivalent Forced Outage Rate of unit i, during sub-period m of year y, in p.u.
3
EFORRiy ,m Equivalent Forced Outage Rate of unit i after refurbishment, during sub-period m of year y, in p.u. y
Maximum energy output of hydro unit i during year y, in MWh
Ei
y ,m
Maximum energy output of hydro unit i during sub-period m of year y, in MWh
ξ
Maximum RES penetration, in p.u.
λy
Minimum RES annual production requirement during year y ≥ y RES , in p.u. of annual
Ei
demand. ED y
Energy demand during year y, in MWh
Ciinv
Specific investment cost of unit i, in €/MW
Cir ,inv
Specific investment cost for the refurbishment of existing unit i, in €/MW
y rmin
System minimum reserve margin for year y, in p.u.
y Bmax
Maximum budget for investment in generation expansion for year y, in €
I hy ,m
forecasted net water inflows in hydroplant/reservoir h in sub-period m of year y, in m3/s
qhy ,m
total water discharge of hydroplant/reservoir h in sub-period m of year y, in m3/s
qahy ,m
water flow pumped upstream pumped storage plant h in sub-period m of year y, in m3/s
Vhini
initial volume of water stored in hydroplant/reservoir h at the beginning of the planning horizon, in Hm3
Vhfin
final volume of water stored in hydroplant/reservoir h at the end of the planning horizon, in Hm3
Vhmax
maximum volume of water stored in hydroplant/reservoir h, in Hm3
K
conversion factor, equal to 0.0036 [Hm3s/m3h]
M
large positive value
Variables Continuous variables
4
Pi y ,m, j
Power output of unit i at load level j during sub-period m of year y, in MW
CTi y ,m, j
Hourly operating cost of unit i, including variable O&M cost and CO2 emissions cost, during year y, considering inflation, in €
LNS y ,m, j Load not served during year y, sub-period m, at load level j, in MW y RESdef
Deficit from RES quota target during year y, in MWh
sphy ,m
spillage of hydroplant/reservoir h during sub-period m of year y, in m3/s
vhy ,m
volume of water stored in hydroplant/reservoir h at the end of sub-period m of year y, in Hm3
Binary variables Βinary variable representing the start-up decision (commissioning) of the new unit i in
wiy
year y Βinary variable representing the shut-down decision (decommissioning) of the new unit i
ziy
in year y Βinary variable representing the status of the new unit i in year y (equal to 1 if unit is
uiy
commissioned)
umiy ,m
Βinary variable representing the maintenance status of unit i in sub-period m of year y (equal to 1 when unit is on maintenance)
xiy
Binary variable representing the refurbishment decision of the existing unit i in year y
viy
Βinary variable representing the refurbishment status of the existing unit i in year y (equal to 1 if unit is refurbished)
2.
Introduction Investment decisions are ubiquitous and have always posed great interest to economists, analysts
and researchers, considering optimal investment strategies for the optimal sitting, timing, sizing and
5
type of an investment. Investments on energy production capacity are characterized by: 1) Partial or complete irreversibility: the prospective investor cannot disinvest should the market conditions change adversely. The initial cost of investment is sunk; it cannot be recovered in total. 2) High risks, concerning a) the ongoing uncertainty of the economic environment in which the decisions are made, and b) possible regulatory decisions that may affect the expected income/amortization of the unit. The classical problem that has been employed by researchers for managing the above risks is the Generation Expansion Planning (GEP) problem. Its solution determines the capacity addition schedule (siting, timing, sizing and technology of new plant additions) that satisfies forecasted load demand within given reliability criteria over a planning horizon of typically 10 to 30 years. In its extended formulation, GEP is a large-scale highly-constrained mixed-integer non-linear programming (MINLP) problem, the global optimum of which can be reached only by complete enumeration. Thus, the determination of the proven optimal solution would require the investigation of every possible combination of candidate options over the planning horizon. The enormous calculation overhead of such an approach has forced researchers to employ simplifications of the analytical model and solve it using several sophisticated optimization methods and meta-heuristics during the past decades. Two modeling approaches have been developed for the solution of GEP: a) the micro-approach, and b) the macro-approach [1]. The micro-approach employs analytical and sophisticated methods of operational research and meta-heuristics to cope with complex non-linear transmission constraints and reliability criteria. In micro-approach, the GEP data are set at high level of detail; nevertheless, global optimality cannot be guaranteed. The micro-approach however deals with a problem that is part of a greater macro-economic framework, which considers the planning of the whole economy/energy system based on multi-sector scenarios. Macro-economic approaches reduce the modeling complexities, i.e. ignore the complex features and constraints within the energy sector, usually resulting in linear programming (LP) models. The results of macro-approach models are always macroscopic and rough [1]-[3]. The literature on the solution of the GEP problem using micro-approach modeling is extensive. Two classes of problems have been considered: a) the centralized approach, and b) the decentralized
6
approach. In the centralized approach, the GEP problem is solved centrally: a)
in monopoly situations, by a state-owned or private monopoly-utility to determine the least-cost expansion planning, and
b)
in a deregulated market, by governing or regulating authorities, in order to formulate market designs and policies that lead to the long-term targets of a country (giving motives to certain technologies), concerning the minimization of the overall cost of supplying electricity to the end consumers, the penetration of Renewable Energy Sources, the CO2 emission control, as well as the interactions between emission trading and renewable support schemes, namely the impact of a variety of emission caps and RES support schemes both individually and combined. Several methods have been developed for the solution of the centralized problem, such as
stochastic dynamic programming [4], non-linear programming (NLP) [5], mixed-integer linear programming (MILP) [6], multi-objective programming [7], evolutionary programming (e.g. GAs) [8]-[15], and other heuristics and mathematical approaches [16]-[18]. The formulation of the problem objective and constraints varies in each implementation, incorporating emissions costs and other environmental constraints (NOx, SOx), transmission constraints, reliability criteria, demand-side management programs, reserve margins, location and financial constraints. An overview of the basic features of these approaches is presented in Table 1. In [4] a stochastic dynamic programming model is presented for the solution of the centralized GEP problem, considering uncertainties in demand and fuel prices. The uncertain variables are modeled by Markov chains. In [5] a Non-Linear Programming approach is presented, with a detailed operation model for pumped/storage hydro units and a realistic model of capital costs for hydro plants. The NLP solver “MINOS” is employed for the solution of the relevant problem. In [6] an integrated analysis of GEP and financial planning is provided. All financial constraints are incorporated to the GEP problem, which is formulated as a MIP model. Scenario-based (sensitivity) analysis is employed for the evaluation of several financial planning options. In [7] a multi-objective (multi-criteria) linear programming method is developed incorporating transmission constraints with a DC network
7
representation. The four objectives comprise (a) the investment, operational and transmission costs, (b) the environmental impact, (c) the imported fuel, and (d) the energy price risks. In [8] Evolutionary Programming is employed to handle the nonlinearity introduced by the reliability criterion (Loss of Load Probability, LOLP). A mixed-integer bilinear multi-objective Evolutionary Programming model is presented in [9], with combined investment decisions for generation and transmission expansion. Scenario-based analysis is used to evaluate the effect of fuel prices on the different objectives. In [10] several metaheuristic techniques are applied and compared for the solution of the GEP problem. In [11]-[15] Genetic Algorithms are employed, for the solution of either the single-objective or the multiobjective GEP. Ref. [13] is the only approach handling an AC network representation. GA-based approaches can handle the non-linearity introduced by the LOLP reliability criterion and claim to have the ability to locate the global optimum of the GEP problem within a reasonable computation time. However, they have some structural problems when applied to large-sized problems, such as premature convergence to a local optimum. Finally, several heuristics and mathematical programming approaches have been presented in the literature for the solution of the GEP problem, such as a simple probabilistic peak-shaving method in [16], a heuristic algorithm handling transmission constraints in [17] and a multi-objective optimization model with Monte-Carlo simulation for demand and unit availability uncertainty in [18]. Integrated software packages have also been developed for the solution of the centralized GEP, e.g. the well-known WASP-IV (Wien Automatic System Planning) [19], developed by the International Atomic Energy Agency (IAEA), in which dynamic programming is employed to determine a least cost generation capacity addition plan under specified LOLP target. In the decentralized approach, the strategic GEP problem decision making of a producer within a deregulated (usually oligopolistic) market framework is considered. These models are more complex, since the market operation, and the behavior of rival producers must be modeled appropriately. Game theory is employed for the formulation of nearly all such models, which are solved using stochastic dynamic programming [20]-[21], Lagrangian relaxation, Benders decomposition [22], evolutionary programming [23]-[24], iterative search methods [25], system dynamics [26], heuristic methods [27], and by Mathematical Program with Equilibrium Constraints (MPEC) solvers [28]-[29]. Each of these
8
methods has its pros and cons; an analytical review and comparison of most of the above methods can be found in [29]. The common feature of all the above-mentioned works is that a yearly or two-year time-step is considered during the planning horizon. Only in [21]-[22] a seasonal step (four seasons per year) is used. These time-steps are not able to incorporate mid-term scheduling aspects in the long-term planning framework. However, the incorporation of mid-term scheduling decisions, such as the maintenance scheduling of existing and new units, plays a significant role in the provided solution, and may alter significantly the results (e.g. reliability indices), especially during peak-load periods. In this paper the centralized GEP problem is formulated and solved as a mixed-integer linear programming (MILP) problem. The system reliability criterion is implicitly modeled by embedding in the problem objective function the cost of the Load Not Served (LNS), valued at the Value of Lost Load (VLL).
Environmental aspects are modeled through (a) the incorporation of the cost of
purchasing emission allowances in the thermal unit operating cost, similarly with the short-term optimization modeling presented in [30]-[31], and (b) the inclusion of annual renewable quota constraints and penalties. In addition, a maximum RES penetration constraint is used to model operational reliability requirements. A monthly time-step is employed, allowing the consideration of unit maintenance schedules in the optimization problem. It is assumed that each unit undergoes a onemonth outage for maintenance during every year. The GEP algorithm optimizes the maintenance schedules of both existing and new units, along with the optimization of the new capacity additions. The refurbishment of existing units is efficiently modeled. At the end of the useful life of a generating unit a refurbishment decision may be taken. A refurbishment cost is involved, which potentially (a) decreases the total variable cost of the unit, (b) increases the unit capacity, (c) reduces the unit FOR and (d) extends the expected lifetime of the unit. In this way, the conversion of Open-Cycle Gas Turbines to Combined-Cycle Gas Turbines, the conversion of coal/lignite units to low-carbon coal units with carbon capture and storage systems, or other environmental improvements are modeled efficiently. The proposed model is evaluated using a real (Greek) power system. Scenario-based (sensitivity) analysis is performed for the illustration of the effect of demand, fuel prices and CO2 prices
9
uncertainties on the planning decisions. As shown on Table 1, there are three methods to account for stochastic parameters, as follows: a) scenario-based (sensitivity) analysis, which is used by the most approaches reported in the literature ([6], [9], [21], [23], [27], [29], and also in this paper), b) Monte-Carlo simulation, which is employed in [18] and [26], and c) Stochastic programming approaches, as those presented in [1] and [20], using the theory included in [32]-[33]. The extension of the present model to a Monte-Carlo simulation is straightforward; it solely requires an iterative algorithm, in each iteration of which the proposed model is solved. Considering the already published works in the literature, the main contributions of this paper are: 1) The modeling of the GEP problem as a mixed-integer linear programming (MILP) problem, by borrowing binary variable modeling techniques already used in short-term generation scheduling [34]. 2) A relatively small (monthly) time-step is used, allowing mid-term scheduling decisions, such as unit maintenance scheduling and reservoir management, to be taken along with investment decisions within the framework of a single long-term optimization problem. 3) The efficient modeling of the refurbishment of existing units within the MILP framework.
3.
Model Description and Mathematical Formulation
3.1 Load model description The system load for each sub-period (month) of each year of the planning horizon is represented using a step-wise load duration curve, as shown in Fig. 1. Demand blocks (approximating the load duration curve) represent the demand variations across the hours of each month. The number of steps considered is tailored to represent accurately the sharpness of the load duration curve during the peak hours of each month. 3.2 Problem formulation The GEP problem objective function to be minimized is formulated as follows:
10
y max DF y ⋅ h y ,m, j ⋅ ∑ CTi y ,m, j + VLLy ⋅ LNS y ,m, j + ∑∑ DF y ⋅ cFOM ⋅ uiy (i ) ⋅ Pi i y i y ,m, j
+∑
∑
y i∈I new
Y
∑
DF
DF y ⋅ Pimax ⋅ Ciinv ⋅ wiy + ∑ y
y ⋅ PEN RES
y ⋅ RESdef
}
∑ DF y ⋅ Pir ,max ⋅ Cir ,inv ⋅ xiy +
y i∈I old
y = y RES
DF Y ⋅
{
∑
Min Cost =
1 y ⋅ v − DF ⋅ ∑ ∑ Pi r ,max ⋅ Cir ,inv ⋅ xiy − r ,life − i − Tilife Ti i∈I old y
(1)
Y
1
i
∑ ∑ Pimax ⋅ Ciinv ⋅ wiy − T life ⋅ uiy
i∈I new y
The objective (1) aims at the minimization of the discounted total cost over the planning horizon comprising (a) the operating cost including the cost of power interruptions valued at the value of lost load, VLL (fist summation), (b) the fixed operation and maintenance cost (second summation), (c) the cost of investments in new capacity (third summation), (d) the investment cost for the refurbishment of existing units (fourth summation), (e) the penalty for not meeting the minimum renewable annual energy production requirement (fifth summation), (f) the remaining value of the new installations and of the refurbishment of existing units at the end of the planning horizon, assuming straight line (linear) depreciation of the equipment (last two summations with negative sign). All cash flows in (1) are discounted to the beginning of the planning horizon (base year is assumed to be year “0”, where year “1” is the first year of the planning horizon). The associated formulas regarding the time value of money are:
DF y =
1
(2)
(1 + DR ) y
(
ctiy = 1 + IR f (i )
(
)
y
(
⋅ IHRi ⋅ c f (i ) + 1 + IRCO2
y cFOM 1 + IROM (i ) (i ) =
)
y
)
y
(
⋅ eri ⋅ cCO2 + 1 + IROM (i )
)
y
⋅ cFOM (i )
⋅ cOM (i )
(3)
(4)
The problem constraints are as follows: 1) Energy balance constraint:
y ,m , j h y ,m, j ⋅ ∑ Pi + LNS y ,m, j = h y ,m, j ⋅ Ly ,m, j , y ∈Y , m ∈ M , j ∈ J i
(5)
The term h y ,m, j is included in both sides of equation (5) in order to acquire meaningful shadow
11
prices from the problem solution, in €/MWh. 2) Unit technical constrains: A simple unit FOR modelling is incorporated by derating the unit maximum output by the unit FOR, as follows:
( ) ( (1 − EFORR ) ⋅ P ⋅ v
)
Pi y ,m, j ≤ 1 − EFORiy ,m ⋅ Pi max ⋅ uiy − viy + y ,m i
(
r ,max
i
)
Pi y ,m, j ≤ 1 − EFORiy ,m ⋅ Pi max ⋅ uiy ,
∀i ∈ I old , y ∈Y , m ∈ M , j ∈ J
y i ,
∀i ∈ I new , y ∈Y , m ∈ M , j ∈ J
(6)
(7)
A high FOR (e.g. 80%) is used for RES to account for resource (e.g. wind) unavailability in addition to technical unavailability. In (6) the original maximum power output, or the maximum power output after refurbishment is selected accordingly. 3) Unit operating cost definition: The term CTi y included in the objective function is defined by the following constraints:
CTi y ,m, j ≥ ctiy ⋅ Pi y ,m, j − M ⋅ viy ,
(
)
CTi y ,m, j ≥ − M ⋅ uiy − viy + ctri y ⋅ Pi y ,m, j ,
∀i ∈ I old , y ∈Y , m ∈ M , j ∈ J
(8)
∀i ∈ I old , y ∈Y , m ∈ M , j ∈ J
(9)
∀i ∈ I new , y ∈Y , m ∈ M , j ∈ J
j CTi y ,m,= ctiy ⋅ Pi y ,m, j ,
(10)
where M is a large positive number. Constraints (8)-(9) denote that when the refurbishment decision is taken, the unit variable cost changes (decreases) from ctiy to ctri y . 4) Annual hydro unit energy limitation: The uncertain nature of water inflows is simulated through a fixed annual energy availability per hydro station. This figure is based on historical statistical data for old units, and projections considering the hydro installed capacity for new (prospective) entrants.
∑ h y,m, j ⋅ Pi y,m, j ≤ Eiy ,
∀i ∈ I hyd , ∀y ∈Y
(11)
m, j
Alternatively, instead of the above annual energy availability constraint, the hydro units’ reservoir inter-temporal constraints and the hydraulic coupling among cascaded hydroplants can be simulated as follows:
12
,m vhy= vhy ,m −1 + K ⋅ I iy ,m − qiy ,m − spiy ,m + qaiy ,m + ∀h ∈ I hyd , ∀y ∈Y, ∀m ∈ M K ⋅ ∑ q yh,m + sp yh,m − qa yh,m d d d h h
(12)
d ∈D
∀ h ∈ I hyd , ∀y ∈Y, ∀m ∈ M
(13)
= vh0 Vhini
∀ h ∈ I hyd
(14)
= vhY Vhfin
∀ h ∈ I hyd
(15)
0 ≤ vhy ,m ≤ Vhmax
The forecasted net water inflows are assumed to be known. Constraints (12) represent the monthly reservoir water balance, taking into account the operation (including pumping) of the upstream hydroplants of a given hydroplant, while constraints (13) impose limits on the reservoir stored volume. Constraints (14)-(15) define the initial and final (target) reservoir volumes. When the detailed hydraulic constraints (12)-(15) are included in the model the simple simplified "energy-only" hydro constraints (11) must be omitted to avoid conflicts. It should be noted that constraints (12)-(15) provide an equivalent model with the “energyonly” model of constraints (11), thus, similar results shall be attained, provided that the hydraulic data considered in the first case are converted to the energy quantities included in constraints (11). In fact, it can be proved that the results of both models will be identical, provided that the limit constraints (13) are not activated throughout the planning horizon. 5) Maximum RES penetration constraint: The RES production should remain lower than or equal to a certain percentage of the coincident load, due to system operational reliability reasons:
∑
i∈I
Pi y ,m, j ≤ ξ ⋅ Ly ,m, j , ∀y ∈Y , m ∈ M , j ∈ J
(16)
RES
6) Minimum annual RES energy contribution requirement:
The RES quota target (imposed by
environmental treaties) must be satisfied for all years following the year the target becomes effective:
13
∑ ( h y ,m, j ⋅ ∑ i∈I RES
m, j
+
y ≥ λ y ⋅ ED y , ∀y ≥ y RES Pi y ,m, j ) + RESdef
(17)
where ED y is given by:
= ED y
∑ (h y,m, j ⋅ Ly,m, j ), ∀y ∈Y
(18)
m, j
7) Minimum reserve margin requirement: For load level j, the commissioned unit capacity should be greater than the annual peak load plus a pre-defined reserve margin: y Ly ,m, j } , { ) ⋅ max ∑ ( Pimax ⋅ uiy ) ≥ (1 + rmin m, j
∀y ∈Y
(19)
i
8) Annual budget constraint: The annual budget that could be utilized for generation expansion in new units or for the refurbishment of existing units is limited: y , ∑ ( Pimax ⋅ Ciinv ⋅ wiy ) + ∑ ( Pir ,max ⋅ Cir ,inv ⋅ xiy ) ≤ Bmax
i∈I new
∀y ∈Y
(20)
i∈I old
9) Maintenance constraints: As stated above, it is assumed that each unit has a one-month outage for maintenance during every year. Additionally, a constraint is added in order to force units on the same station to have maintenance works on different sub-periods of the same year.
∑ umiy,m=
1, ∀i ∈ I , ∀y ≥ Y
(21)
m
Pi y ,m, j ≤ Pi max ⋅ (1 − umiy ,m ), ∀i ∈ I , y ∈Y , m ∈ M , j ∈ J
(22)
umiy ,m + ∑ umiy′ ,m ≤ 1, ∀i, i′ ∈ s, ∀s ∈ S , y ∈Y , m ∈ M
(23)
i′≠i
10) Binary constraints: y wiy − z= uiy − uiy −1 , i
wiy + ziy ≤ 1, y
∑
τ= y −Ti +1
∀i ∈ I new , ∀y ∈Y
(24)
∀i ∈ I new , ∀y ∈Y
τ w= uiy , i
(25)
∀i ∈ I new , ∀y ∈Y
(26)
life
y
∑
τ = y −Y +1
zτi ≤ 1 − uiy ,
∀i ∈ I new , ∀y ∈Y
(27)
14
= uiy 0 for y ≤ Ticon , ∀i ∈ I new
(28)
uiy
1 for − Tiini ≤ y ≤ Tilife − Tiini , ∀i ∈ I old & i ∉ I r ,old 0 otherwise
(29)
ziy
1 for= y Tilife − Tiini , ∀i ∈ I old & i ∉ I r ,old 0 otherwise
(30)
1 for y = −Tiini , ∀i ∈ I old wiy = 0 otherwise
(31)
y xiy − z= viy − viy −1 , i
(32)
xiy + ziy ≤ 1,
∀i ∈ I r ,old , ∀y ∈Y
∀i ∈ I r ,old , ∀y ∈Y
(33)
life ini ini 1 for − Ti ≤ y ≤ Ti − Ti , ∀i ∈ I r ,old r ,life 0 for y > Ti
uiy
= uiy viy ,
∀i ∈ I r ,old , y > Tilife
y
(34)
(
)
xτi= viy ,
∀i ∈ I r ,old , ∀y ∈Y
y x= 0 for y ≤ Tilife − Tiini i
∑
(29)
(
)
and y > Tilife − Tiini + 1 , ∀i ∈ I r ,old
(35)
(36)
τ= y −(Tir ,life −Tilife ) +1 y
∑
τ = y −Y +1
zτi ≤ 1 − viy ,
∀i ∈ I r ,old , ∀y ∈Y
(37)
Concerning the new units, relationships (24)-(25) define their commissioning status [28]. Specifically, (26) restricts the life-time of a new unit to Tilife years and (27) ensures that once decommissioned a new unit will not be commissioned again. New units cannot be commissioned prior to their construction time (28). Concerning the old units not eligible for refurbishment, their commissioning binary variables are known and can be considered as input data to the model, as expressed by (29)-(31). Concerning the old units that are eligible for refurbishment, relationships (31)-(35) define the commissioning status of the existing units under refurbishment. Relationship (35) implies that the refurbishment decision can only be taken at the first year after the normal lifetime of an existing unit.
15
(
Also, (36) restricts the life-time extension of a refurbished unit to Ti r ,life − Tilife
)
years and (37)
ensures that once decommissioned a refurbished unit will not be commissioned again.
4.
Test Results
4.1 Cases Description The proposed model has been evaluated on the Greek power system, currently comprising 38 thermal units and 15 hydro plants. The total installed thermal and hydro capacity from existing units and units with firm commissioning plans is equal to 12,103 MW and 3,034 MW, respectively. Existing wind park capacity amounts to 1,200 MW; an equivalent wind park has been regarded here for simplicity. The EFOR of this equivalent wind park has been taken equal to 76%, in order to simulate a utilization factor of 24%. Table 2 summarizes existing unit data. The analytical technoeconomic data of the existing thermal units are given in [28]. The marginal cost presented in column 6 of Table 2 includes the fuel cost and the variable part of the O&M cost (without the CO2 emissions cost in (3)). The system peak load and the annual energy demand during 2010 were 9,793 MW and 52.32 TWh, respectively. Table 3 presents the data of the candidate new entrants. The investment cost, as well as the variable and fixed operation and maintenance (O&M) costs for the candidate new units have been taken from the Annual Energy Outlook 2010 of the U.S. Energy Information Administration [35], considering an exchange rate for €/$ equal to 1.3. Each new hydro unit is assumed to inject 350 GWh per year. The EFOR of the candidate wind parks is linearly increasing with the number of new wind parks (from 76% to 85%), to account for the fact that the new parks will be installed in areas with progressively less production potential (as compared to the already installed ones). The fixed O&M costs shown in column 6 of Table 3 also apply to the existing units (Table 2). The load duration curve has been formed using load data of the Greek Power System of year 2010 [36]. Two load scenarios are considered: (1) a realistic load growth and (2) a high load growth case, in accordance with the scenario of high economic growth [37] elaborated by a working group of the
16
Greek Ministry of Energy. The load data of year 2010 have been taken as base, and the load duration curve has been formed for each month of this year. The durations considered for the step-wise representation of the load duration curve of Fig. 1, are: 1 hour (to capture the monthly peak load), 6 hours, 23 hours, 120 hours, 150 hours, 200 hours, 150 hours and the last step comprises the remaining hours of each month. A Base Case scenario is formulated, in which the problem parameters take the following values: DR =3%, IR f =1% for all fuels (lignite, coal, gas, oil), IRCO 2 =0.5%, cCO2 =15 €/T, VLLy =4,000 y y €/MWh ∀y ∈Y , Bmax =2 billion € ∀y ∈Y , y RES =10, PEN RES =1,000 €/MWh ∀y ∈Y , ξ =0.4, y =0.15 ∀y ∈Y . Year 2011 is the first year of the λ y =0.2 ∀y ≥ y RES , Tilife ≥ 20 ∀i ∈ I new , and rmin
20-year planning horizon. Alternative cases are produced by changing the value of several parameters of the Base Case, i.e. the system load, the CO2 prices, the inflation rate of fuel prices and O&M costs, and the available annual budget for new investments. A summary of the cases’ configuration is presented in Table 4. It should be noted that both wind parks and hydro units are considered in the minimum RES quota target (9), whereas only wind parks are included in the maximum RES penetration constraint (8). 4.2 Numerical Results Table 5 presents the commissioning results of all cases. To facilitate the discussion, the commissioning order of all conventional units has been changed (relative to the MILP solution), so that it coincides with the unit number within each technology. This change was possible because the technical and economic characteristics of new entrant units are identical within each conventional technology (Table 3). Owing to the progressively higher EFOR of the new wind parks discussed before, the commissioning order of the wind parks was not changed and is reported as provided by the MILP solution: the commissioning order of the wind parks is dictated by the EFOR of each park (wind parks with lower EFOR are commissioned earlier). Commissioning of units during the end of the planning horizon (years 18, 19, 20) is presented in grayscale to indicate that no firm commissioning plans should be made for these units, owing to the end-of-horizon effect: the net commissioning cost
17
of these units is very low owing to their high remaining value and the low discounting factor (1). That is, based on the results of Table 5 commissioning plans for only the 17 first years of the planning horizon should be made. The results tabulated in Table 5 indicate that: a) The best new entrants in the Greek power system are the wind parks and the hydro units, given the technology state-of-the-art and the RES quota. However, the effect of increased wind parks on the system reserve requirements has not been considered in this analysis. b) The RES quota requirement (9) plays a significant role in commissioning of more wind parks and hydro units. When the RES quota requirement (9) is relaxed (case 2), less wind parks and hydro units are installed in favor of steam units. Inversely, when the RES quota requirement (9) is increased (case 3), all candidate hydro units and almost all candidate wind parks are commissioned within the 17-year planning horizon. The generation mix throughout the planning period in case 3 is shown in Fig. 2. c) The above observation is more evident in the economic growth cases (4 and 5); when the RES quota requirement (9) is relaxed, all candidate steam units are commissioned within the planning horizon. d) With CO2 price at 15 €/T, the increased load in case of high economic growth (case 4) is covered by steam units and wind parks. When the CO2 price increases, the CCGTs replace the steam units in the future generation mix. This phenomenon is more intense when the CO2 price is 25 €/T (case 7); in this case OCGTs are also commissioned. e) When the inflation rate is increased to 4%, the wind parks and steam units are commissioned earlier and in larger numbers than in cases 3 (30% RES target) and 4 (high Economic Growth), in order to displace expensive generation from existing CCGTs. f) When the budget decreases (case 11), the commissioning of wind parks takes place more gradually, and two CCGTs are constructed instead of one steam unit (compare with case 4). Only when the budget increases (case 12) more wind parks enter commercial operation from the fourth year. Fig. 3 demonstrates how the RES quota target (9) is met in the Base Case. It is evident that the
18
algorithm selects to invest in so many wind parks as required, in order to marginally satisfy the RES target from the 10th year onwards. The effect of the maintenance constraints on the unit schedules is illustrated in Table 6 and Fig. 4. Table 6 presents the results on maintenance schedules for some lignite units. As shown, units on the same station have maintenance works on different months of the same year. Fig. 4 presents the capacity on maintenance for each month of the 2nd year in contradiction with the system load in case 4. As shown, the unit maintenance periods are selected mostly when the system load is low. The mirror effect on the curves in Fig. 4 demonstrates the optimality of the computed unit maintenance schedules. Fig. 5 presents the objective function constituent elements for all cases considered. The present value (PV) of the operating cost constitutes the largest part of the total objective, since it refers to all (both old and new) units, whereas the PV of the investment cost minus the remaining value refers only to the new units. The same cases are executed considering the potential refurbishment of two lignite units (AMYNTAIO 1 & 2), scheduled for de-commissioning on the 8th year of the planning horizon. The refurbishment investment cost is 100.000 €/MW, and extends the units’ lifetime by eight years, decreasing their total variable cost by 5 €/MWh. It is assumed that the units’ maximum production capability and EFOR do not change after refurbishment. The above twelve cases are executed again, and the attained results are quite similar with the ones reported above. In all cases, both units are refurbishmed at the 9th year of the planning horizon. Their scheduling affects mainly the commitment decision of new steam and gas units, which is deferred later on the planning horizon or cancelled (deferred to a year beyond the planning horizon), as shown in the last columns of Table 5.
19
4.3 Interactions between emissions costs and RES support schemes The interaction of emission trading and renewable support mechanism has been an area of recent discussion and debate [38]. For example, a larger share of RES, promoted by a renewable support scheme, reduces the demand for emission allowances and thus reduces carbon prices. Our GEP model can be used to contribute in this discussion, similarly to [39]. Cases no. 2 and no. 5 present the results of deactivating the RES quota obligation in the normal and the high economic growth scenario, respectively. In cases no. 1 and 4, the respective cases are solved enforcing both the RES quota obligation and the emission prices. In order to examine the interactions between the RES minimum annual production target and the emission prices, two new cases are created and solved (named “1a” and “4a”, respectively), by deactivating (zeroizing) the imposed emission cost (equal to 0 €/T) with respect to cases no. 1 and 4. It should be noted that the remarks (b), (c) and (d) of the previous Section are still valid. Additionally, the results of cases 1a and 4a, as compared to cases 1, 2, 4 and 5, lead to the following remarks: a)
When the emission cost is zeroized, the commitment of steam units dominates over all candidate units, as shown in the relevant columns of Table 5 (as compared to commitment decisions of steam units in cases 1, 2, 4 and 5), due to their significantly decreased operating cost.
b)
Generalizing the remark (b) stated in the previous Section, when the RES quota requirement (9) is relaxed (cases 2 and 5) or when the emission cost is zero (cases 1a and 4a), less wind parks and hydro units are installed and/or they are installed at a later stage (Fig. 6), in favor of steam units (Fig. 7).
c)
The main losers in case of zero emission cost are the gas-fired units, which reduce significantly their operation in favor of steam units, as shown in Fig. 8. Actually, in cases 1a and 4a the gas units reach zero or close to zero annual production in year 8 and year 12 (only for case 1a), since all existing steam units are on-line and most candidate steam units are already in operation, as shown in Fig. 7.
d)
As expected, the electricity prices are significantly lower in cases 1a and 4a (Fig. 9), due to the dominant operation of the existing and new steam units.
20
A RES support scheme does not provide an emission price signal that would influence longer-term investment decisions based on a carbon-emissions basis or provide the same level of certainty of emissions reductions, particularly if demand increases. Renewable portfolio standards primarily address the contribution of renewable generation in the resource mix, without focusing on the mix of coal, gas, and other technologies, except to the extent that the contribution of renewable affects the generation dispatch. Our model can be extended by replacing the constant carbon price by a carbon price quota curve, which models the effect of increased demand of emission allowances on the carbon price. The modeling of a carbon price quota curve would require the introduction of additional binary variables in the MILP model [40]. 4.4 Computational, convergence and feasibility issues
4.4.1 Computational issues All tests were performed on a 3.0 GHz Intel I7 processor with 8 GB RAM, running 32-bit Windows. The models were implemented in GAMS 23.2 using the CPLEX 12.0 solver [41]. The size of the model without refurbishment is 306,742 equations and 592,659 variables (of which 24,844 are binary), and the density is 0.001%, whereas with refurbishment it is 537,272 constraints and 815,539 variables (of which 24,898 are binary), and the density equals 0.0005%.
4.4.2 Convergence In theory, a MILP solver can always locate the global optimum of a MILP problem, by solving a large, yet finite number of (convex) LP problems. In practice, MILP solvers, based on branch-andbound and branch-and-cut algorithms can find a near-optimal solution to the MILP problem, using finite computational resources. The quality of the near-optimal solution is controlled by a convergence tolerance or “optimality gap”, which provides the distance of the best feasible solution found so far, from an upper bound of the optimum (of a maximization problem). The upper bound of the optimal solution is computed as the optimum of a relaxed MILP problem, in which the binary constraints on
21
some binary variables are relaxed. The convergence tolerance in all test cases was set to 0.25%. According to GAMS Model and Solver Statistics output, the solver CPU time varies between 7 and 18 minutes. The convergence path (optimality gap against the model execution time) in three representative cases is presented in Fig. 10 for the model without the refurbishment costs and constraints and in Fig. 11 for the model including the refurbishment costs and constraints. As shown, the additional continuous and (mainly) binary variables, as well as the additional constraints related with the refurbishment of the existing units, affect the convergence path of the MIP solver, and have an adverse effect on the model execution time (till the convergence point).
4.4.3 Feasibility of the provided solution Infeasibilities in LP problems may arise whenever the input data are inconsistent and do not define a feasible solution set. In LP problems infeasibility can be easily captured by the appropriate use of artificial variables. With the use of artificial variables there always exists a feasible solution of the extended (with the use of artificial variables) LP problem. The existence of nonzero (basic) artificial variables in the extended LP problem solution implies infeasibility of the original LP problem. With the appropriate use of artificial variables in the GEP problem, infeasibilities are easily captured and reported.
5.
Conclusions – Further research A MILP-based solution to the centralized Generation Expansion Planning problem has been
presented in this paper. Test results on a medium sized real power system (the power system of Greece) demonstrate that it is feasible to solve the centralized GEP simultaneously with the mid-term maintenance scheduling as a single MILP problem with reasonable computational effort. Scenariobased (sensitivity) analysis around a base case scenario was performed by forming several alternative scenarios based on different choices of the problem stochastic parameters, such as load growth, fuel
22
prices etc. The extension of the presented model to a multi-stage stochastic MILP that correctly captures the problem uncertainties will be another topic of the authors’ future work. Further, another direction of future work will be the MILP-based solution of the decentralized strategic GEP problem. Finally, the combined Generation and Transmission expansion planning problem, correctly addressing both the generation and the transmission expansion planning problems capturing their interrelations, is a topic for further research.
6.
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Cyber., 39 (2009) 1086-1096. [10] S. Kannan, S.M.R. Slochanal, N.P. Padhy, Application and comparison of metaheuristic techniques to generation expansion planning problem, IEEE Trans. Power Syst., 20 (2005) 466475. [11] J.B. Park, Y.M. Park, J.R. Won, K.Y. Lee, An improved genetic algorithm for generation expansion planning, IEEE Trans. Power Syst., 15 (2000) 916-922. [12] S. Kannan, S. Baskar, J.D. McCalley, P. Murugan, Application of NSGA-II algorithm to generation expansion planning, IEEE Trans. Power Syst., 24 (2009) 454-461. [13] P. Murugan, S. Kannan, S. Baskar, Application of NSGA-II algorithm to single-objective transmission constrained generation expansion planning, IEEE Trans. Power Syst., 24 (2009) 1790-1797. [14] H.T. Firmo, L.F.L. Legey, Generation expansion planning: an iterative genetic algorithm approach, IEEE Trans. Power Syst., 17 (2002) 901-906. [15] J. Sirikum, A. Techanitisawad, V. Kachitvichyanukul, A new efficient GA-Benders’ decomposition method: for power generation expansion planning with emission controls, IEEE Trans. Power Syst., 22 (2004) 1092-1100. [16] A.S. Malik, B.J. Cory, P.D.C. Wijayatunga, Applications of probabilistic peak-shaving technique in generation planning, IEEE Trans. Power Syst., 14 (1999) 1543-1548. [17] M.S. Sepasian, H. Seifi, A.A. Foroud, A.R. Hatami, A multi-year security constrained hybrid generation-transmission expansion planning algorithm including fuel supply costs, IEEE Trans. Power Syst., 24 (2009) 1609-1618. [18] H. Tekiner, D.W. Coit, F.A. Felder, Multi-period multi-objective electricity generation expansion planning problem with Monte-Carlo simulation, Electric Power Systems Research, 80 (2010) 1394-1405. [19] International Atomic Energy Agency (IAEA), WIEN Automatic System Planning (WASP) Package, A Computer Code for Power Generation System Expansion Planning, Version WASPIV User’s Manual. Vienna, Austria, IAEA, 2001. [20] A. Botterud, M.D. Ilic, I. Wangensteen, Optimal investments in power generation under
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centralized and decentralized decision making, IEEE Trans. Power Syst., 20 (2005) 254-263. [21] T. Barforoushi, M.P. Moghaddam, M.H. Javidi, M.K. Sheikh-El-Eslami, Evaluation of regulatory impacts on dynamic behaviour of investments in electricity markets: a new hybrid DP/GAME framework, IEEE Trans. Power Syst., 25 (2010) 1978-1986. [22] J.H. Roh, M. Shahidehpour, Y. Fu, Security-constrained resource planning in electricity markets, IEEE Trans. Power Syst., 22 (2007) 812-820. [23] J. Wang, M. Shahidehpour, Z. Li, A. Botterud, Strategic generation capacity expansion planning with incomplete information, IEEE Trans. Power Syst., 24 (2009) 1002-1010. [24] A.S. Chuang, F. Wu, P. Varaiya, A same-theoretic model for generation expansion planning: problem formulation and numerical comparisons, IEEE Trans. Power Syst., 16 (2001) 885-891. [25] A.J.C. Pereira, J.T. Saraiva, A decision support tool for generation expansion planning in competitive markets using system dynamics models, IEEE Bucharest Power Tech Conf., June 28th-July 2nd 2009. [26] A.J.C. Pereira, J.T. Saraiva, A decision support system for generation expansion planning in competitive electricity markets, Electric Power Systems Research, 80 (2010) 778-787. [27] V. Nanduri, T.K. Das, and P. Rocha, Generation capacity expansion in energy markets using a two-level game-theoretic model, IEEE Trans. Power Syst., 24 (2009) 1165–1172. [28] F.H. Murphy, Y. Smeers, Generation capacity expansion in imperfectly competitive restructured electricity markets, Oper. Res., 53 (2005) 646-661. [29] S.J. Kazempour, A.J. Conejo, C. Ruiz, Strategic generation investment using a complementarity approach, IEEE Trans. Power Syst., 26 (2011) 940-948. [30] J.P.S. Catalao, S.J.P.S. Mariano, V.M.F. Mendes, L.A.F.M. Ferreira, A practical approach for profit-based unit commitment with emission limitations, Electric Power Systems Research, 32 (2010) 218-224. [31] J. Xie, J. Zhong, Z. Li, D. Gan, Environmental-economic unit commitment using mixed-integer linear programming, European Transactions on Electrical Power, 21 (2010), 772-786. [32] A.J. Conejo, M. Carrion, J.M. Morales, Decision Making under Uncertainty in Electricity Markets", Springer publisher, 2010.
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[33] P. Kall, S.W. Wallace, Stochastic Programming, John Wiley and Sons, 1994. [34] C.K. Simoglou, P.N. Biskas, and A.G. Bakirtzis, Optimal self-scheduling of a thermal producer in short-term electricity markets by MILP, IEEE Trans. Power Syst., 25 (2010) 1965–1977. [35] U.S. Energy Information Administration, Assumptions to the Annual Energy Outlook 2010, available: http://www.eia.doe.gov/oiaf /aeo/assumption/pdf/electricity_tbls.pdf. [36] The
Hellenic
Transmission
System
Operator,
available:
http://www.desmie.gr
and
http://emarketinfo.desmie.gr/. [37] Energy Ministry of Greece, “Energy Scenarios Analysis for the penetration of RES in the Greek power system”, available: http://www.ypeka.gr/LinkClick.aspx?fileticket=oXNSJpc0O3Y%3D &tabid=285&language=el-GR (in Greek). [38] P.R. González, The interaction between emissions trading and renewable electricity support schemes. An overview of the literature, Mitigation and Adaptation Strategies for Global Change, 12 (2007) 1363-1390. [39] L. Bird, C. Chapman, J. Logan, J. Sumner, W. Short, Evaluating renewable portfolio standards and carbon cap scenarios in the U.S. electric sector, Energy Policy, 39 (2011) 2573-2585. [40] S. de la Torre, J. M. Arroyo, A. J. Conejo, J. Contreras, Price maker self-scheduling in a poolbased electricity market: a mixed-integer LP approach, IEEE Trans. Power Syst., 17 (2002) 1037-1042. [41] General Algebraic Modeling System, available: www.gams.com.
Grigorios A. Bakirtzis received the Dipl. Electr. Eng. Degree from the Department of Electrical Engineering, Aristotle University of Thessaloniki, Greece, in 2004. He is currently a Ph.D. candidate with the same university. Since May 2006 he is with Siemens Hellas, Automation & Drives department. His research interests are in power system planning and economics. Pandelis Biskas received his Dipl. Eng. degree from the Department of Electrical Engineering, Aristotle University, Thessaloniki, in 1999, and his Ph.D, in 2003 from the same university. He also performed his Post Doc research from March 2004 till August 2005 in the same university. From March 2005 till July 2009 he was a power system specialist at the Hellenic Transmission System
26
Operator (HTSO), Market Operation Department. Currently, he is a Lecturer at the Aristotle University of Thessaloniki, in the Department of Electrical and Computer Engineering. His research interests are in power system operation & control, in electricity market operational and regulatory issues, and in transmission pricing. Vasilis Chatziathanasiou received the Diploma in Electrical Engineering in 1978 and the Ph.D in Nuclear Technology in 1989 both from the University of Thessaloniki, Greece. In 1978 he joined the Electrical Engineering Department of the University of Thessaloniki where he is currently Assistant Professor. His main field of interest is analysis of coupled magneto-thermal fields in power systems, underground cables and electrical machines operational parameters, application of intelligent agents in the power industry, heat transfer in production and transmission of electrical energy, thermography and thermal behavior of integrated inductors.
27
TABLE CAPTIONS Table 1. Relevant features of works reported in the literature and the model proposed in this paper Table 2. Existing unit data Table 3. Candidate new units’ data Table 4. Cases configuration Table 5. Unit commissioning schedules for all study cases Table 6. Unit maintenance schedules – Base Case (year 1)
FIGURE CAPTIONS Fig. 1. Piecewise approximation of monthly load duration curve Fig. 2. Generation mix in case 3 Fig. 3. RES target achievement in Base Case Fig. 4. Capacity on maintenance and system load in case 4 (year 2) Fig. 5. GEP Objective constituent elements (all cases) Fig. 6. Wind and hydro units production under different emissions cost and RES target configurations Fig. 7. Steam units production under different emissions cost and RES target configurations Fig. 8. Gas units production under different emissions cost and RES target configurations Fig. 9. Yearly average electricity prices under different emissions cost and RES target configurations Fig. 10. Optimality gap convergence path for representative cases – without refurbishment Fig. 11. Optimality gap convergence path for representative cases – with refurbishment
28
Fig. 1. Piecewise approximation of monthly load duration curve
Fig. 2. Generation mix in case 3
29
Fig. 3. RES target achievement in Base Case
Fig. 4. Capacity on maintenance and system load in case 4 (year 2)
30
Fig. 5. GEP Objective constituent elements (all cases)
Fig. 6. Wind and hydro units production under different emissions cost and RES target configurations
31
Fig. 7. Steam units production under different emissions cost and RES target configurations
Fig. 8. Gas units production under different emissions cost and RES target configurations
32
Fig. 9. Yearly average electricity prices under different emissions cost and RES target configurations
Fig. 10. Optimality gap convergence path for representative cases – without refurbishment
33
Fig. 11. Optimality gap convergence path for representative cases – with refurbishment
34
Ref.
Model/approach
Transmission constraints No No No Yes (DCLF) No Yes (DCLF) No No No Yes (ACLF) No No No Yes (DCLF) No
Stochasticity
Uncertainties modeled
Centralized Centralized Centralized Centralized Centralized Centralized Centralized Centralized Centralized Centralized Centralized Centralized Centralized Centralized Centralized
Transmission planning No No No No No Yes No No No No No No No Yes No
Yes No scenario-based No No scenario-based No No No No No No No No Monte-Carlo
Centralized
No
No
scenario-based
Demand, fuel prices Financial planning options Fuel prices Demand, unit availability Demand, Budget, CO2 price, RES target, fuel price
Decentralized
No
No
Yes
Decentralized
No
No
Decentralized
No
Decentralized Decentralized Decentralized
Point of view
[4] Stochastic dynamic programming [5] NLP [6] MILP [7] Multi-objective linear programming [8] Evolutionary programming (EP) [9] Multi-objective EP [10] Metaheuristic techniques [11] GA [12] Multi-objective GA [13] GA [14] Iterative GA [15] GA – Benders’ decomposition [16] Probabilistic peak-shaving [17] Heuristic [18] Multi-objective optimization This MILP paper Stochastic dynamic programming [20] (discrete Markov chain) Dynamic Programming & game [21] theory Lagrangian relaxation and Benders [22] decomposition [23] GA (bi-level optimization problem) [24] Game theory (Cournot) [25] System dynamics
Maintenance Refurbishment Implementation schedules of existing units time step No No Yearly No No Yearly No No Yearly No No Yearly No No Two-year No No Single-period No No Two-year No No Yearly No No Two-year No No Two-year No No Yearly No No Yearly No No Yearly No No Yearly No No Five-year Yes
Yes
Monthly
Demand
No
No
Yearly
scenario-based
Demand, market price, regulatory interventions,
No
No
Yes (DCLF)
No
-
No
No
No No No
Yes (DCLF) No No
scenario-based No No
Regulatory policies Price volatility, units’ reliability, demand, investment & operation costs Demand, line outages Rival offering, rival investment
No No No
No No No
Seasonal (four seasons / year) Seasonal (four seasons / year) Single-period Single-period Yearly
No
No
Yearly
No No No
No No No
Yearly Single-period Single-period
[26]
GA
Decentralized
No
No
Monte-Carlo
[27] [28] [29]
Heuristic (supply function) MPEC (Cournot) MPEC (supply function)
Decentralized Decentralized Decentralized
No No No
Yes (DCLF) No Yes (DCLF)
scenario-based No scenario-based
Table 1. Relevant features of works reported in the literature and the model proposed in this paper 35
Table 2. Existing unit data
36
Table 3. Candidate new units’ data
37
Table 4. Cases configuration
38
Table 5. Unit commissioning schedules for all study cases
39
Table 6. Unit maintenance schedules – Base Case (year 1)
40
