WP EN2014-19

A Clustered Unit Commitment ProblemFormulation for Integration inInvestment Planning Models

K. Poncelet, A. van Stiphout, E. Delarue, W. D’haeseleer,G. Deconinck

TME Working Paper - Energy and EnvironmentLast update: October 2014

An electronic version of the paper may be downloaded from the TME website:http://www.mech.kuleuven.be/tme/research/

A Clustered Unit Commitment Problem Formulation forIntegration in Investment Planning Models

Kris Poncelet*,#, Arne van Stiphout+, Erik Delarue*,

William D’haeseleer*, Dirk van Hertem+

#Ir. Kris Poncelet, Leuven, Belgium, kris.poncelet@mech.kuleuven.be

*University of Leuven (KU Leuven) Energy Institute TME Branch, EnergyVille

+University of Leuven (KU Leuven) Energy Institute ELECTA Branch, EnergyVille

October 16, 2014

Abstract

Unit commitment (UC) models determine the optimal scheduling of a given set of powerplants to meet the electricity demand, taking account of the operational constraints of thepower system. Recently, most formulations of the UC problem are based on mixed-integerlinear programming (MILP). Solving large-scale unit commitment problems is computation-ally demanding and therefore restricted to time periods of one day to one week. This workingpaper presents a unit commitment model in which power plants are grouped into clustersdepending on the technology and the year of investment. This way, binary commitmentvariables can be replaced by integer commitment variables, reducing computing times con-siderably. Therefore, the UC problem formulation presented here is suitable for integration inenergy/power system planning models. Depending on the computational resources available,the integer commitment variables can be replaced by linear variables, thereby reducing com-puting times drastically, albeit at the expense of reduced accuracy. In addition to applicationin planning models, this model is useful when data on individual power plants is not available.

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Nomenclature

Abbreviations

FCR frequency containment reservesFRR frequency restoration reservesaFRR automatic frequency restoration reservesmFRR manual frequency restoration reservesRR replacement reservesGHG greenhouse gasesLP linear programmingMILP mixed integer linear programmingUC unit commitmentVRES variable renewable energy sources

Sets

I (index i) set of technologiesR (index r) set of intermittent renewable technologiesP (index p) set of pumped storagesV (index v) set of vintagesT (index t) set of time periods

Parameters

∆t time step used in the model [h]ηmargi,v marginal efficiency of plants of technology i and vintage v [φ]ηmaxi,v rated efficiency of plants of technology i and vintage v [φ]ηmini,v efficiency of plants of technology i and vintage v generating at the minimal

operating point [φ]ηPSp Round-trip efficiency of pumped storage p [φ]AFi,v, availability factor of technology i and vintage v [φ]Dt electricity demand in period t [MW ]

EPSp maximum energy content of the reservoir of pumped storage p [MWh]

EPSp minimum energy content of the reservoir of pumped storage p [MWh]

EFi emission factor of fuel consumed by plants of technology i [ton COeq2 /MWhp]

FITr feed-in tariff for renewable technology r [EUR/MWh]FSmFRRi,v binary parameter indicating whether the technology is capable of starting up

/shutting down sufficiently fast to provide non-spinning mFRR (1) or not (0) [φ]FSRRi,v binary parameter indicating whether the technology is capable of starting up

/shutting down sufficiently fast to provide non-spinning RR (1) or not (0) [φ]MCMDi,v minimum continuous maintenance duration of technology i and vintage v [∆t]MDTi,v minimum down-time of power plants of technology i and vintage v [∆t]MUTi,v minimum up-time of power plants of technology i and vintage v [∆t]Ni,v number of power plants of technology i and vintage v [φ]Pi,v maximum power output of a single plant of technology i and vintage v[MW ]Pi,v minimum power output of a single plant of technology i and vintage v[MW ]

PCO2 price of emission permits [EUR/ton COeq2 ]

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P Fi fuel price of fuel consumed by plants of technology i [EUR/MWhp]

P PSp rated power output of pumped storage plant p [MW ]

QFCR,up/downt demand for upward/downward FCR in period t [MW ]

QaFRR,up/downt demand for upward/downward aFRR in period t [MW ]

QmFRR,up/downt demand for upward/downward mFRR in period t [MW ]

QRR,up/downt demand for upward/downward RR in period t [MW ]

RDi,v maximum downwards ramp rate of plants of technology i and vintage v withintime step ∆t [∆MW/MWnom]

RDRFCRi,v maximum downwards ramp rate to provide FCR [∆MW/MWnom]

RDRaFRRi,v maximum downwards ramp rate to provide aFRR [∆MW/MWnom]

RDRmFRRi,v maximum downwards ramp rate to provide mFRR [∆MW/MWnom]

RDRRRi,v maximum downwards ramp rate to provide RR [∆MW/MWnom]

RUi,v maximum upwards ramp rate of plants of technology i and vintage v withintime step ∆t [∆MW/MWnom]

RURFCRi,v maximum upwards ramp rate to provide FCR [∆MW/MWnom]

RURaFRRi,v maximum upwards ramp rate to provide aFRR [∆MW/MWnom]

RURmFRRi,v maximum upwards ramp rate to provide mFRR [∆MW/MWnom]

RURRRi,v maximum upwards ramp rate to provide RR [∆MW/MWnom]

SUCi,v start-up costs of plants of technology i and vintage v [EUR/MW ]SDCi,v shut-down costs of plants of technology i and vintage v [EUR/MW ]V oLL value of lost load [EUR/MWh]V OMi,v variable operations and maintenance cost [EUR/MWh]V RESr,t profile of generation of renewable technology r (if no curtailment) [MW]

Variables

cci,v,t CO2 tax costs of plants of technology i and vintage v in period t [EUR]ccurtr,t cost relared to curtailment of renewable energy [EUR]cfi,v,t fuel cost of plants of technology i and vintage v in period t [EUR]cllt cost related to load shedding in period t [EUR]csui,v,t start-up costs of plants of technology i and vintage v in period t [EUR]csdi,v,t shut-down costs of plants of technology i and vintage v in period t [EUR]curtr,t curtailed energy from renewable technology r in period t [MW ]cvomi,v,t variable operations and maintenance costs of plants of technology i and

vintage v in period t [EUR]epsp,t energy level of pumped storage p in period t [MWh]geni,v,t combined power output of units of technology i and vintage v in period t

[MW ]genpsp,t power generation of pumped storage p during period t [MW ]llt load shedding in period t [MW ]

nend,mainti,v,t number of units ending a maintenance period in period t [φ]nmainti,v,t number of units in maintenance in period t [φ]noni,v,t number of committed units of technology i and vintage v in period t [φ]nsdi,v,t number of units shutting down in period t [φ]

nsd,mainti,v,t number of units shutting down to go into maintenance in period t [φ]

nsd,mFRRi,v,t number of units contracted to provide non spinning downward mFRR [φ]

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nsd,regi,v,t number of units shutting down in period t, exclusive units shuttingdown to go into maintenance [φ]

nsd,RRi,v,t number of units contracted to provide non spinning downward RR [φ]

nstart,mainti,v,t number of units going into maintenance from being offline in period t [φ]nsui,v,t number of units starting up in period t [φ]

nsu,mFRRi,v,t number of units contracted to provide non spinning upward mFRR [φ]

nsu,RRi,v,t number of units contracted to provide non spinning upward RR [φ]pumppsp,t power consumption of pumped storage p during period t [MW ]

rFCR,up/downi,v,t provision of upward/downward FCR from [MW ]

raFRR,up/downi,v,t provision of upward/downward aFRR [MW ]

rmFRR,s,up/downi,v,t provision of upward/downward spinning mFRR [MW ]

rmFRR,ns,up/downi,v,t provision of upward/downward non-spinning mFRR [MW ]

rRR,s,up/downi,v,t provision of upward/downward spinning RR [MW ]

rRR,ns,up/downi,v,t provision of upward/downward non-spinning RR [MW ]rampdowni,v,t ramp down of units of technology i and vintage v in period t [MW ]rampupi,v,t ramp up of units of technology i and vintage v in period t [MW ]sui,v,t capacity starting up between period t and t+1 [MW ]sdi,v,t capacity shutting down between period t and t+1 [MW ]tc total operational cost [EUR]

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1 Mathematical formulation

UnitsThe time resolution of the model is denoted by the time step ∆t, expressed in hours. All generationand load data is expressed in [MW], giving the average generation or load during the time step∆t. Costs are expressed in [EUR].

1.1 Costs

Objective functionThe objective function of the UC model is to minimize the total operational cost (1). Operationalcosts consist of fuel costs, costs related to GHG emissions, VOM costs, start-up costs, shut-downcosts, costs due to load shedding and curtailment costs1.

min(∑

i

∑v

∑t

(cfi,v,t + cci,v,t + cvomi,v,t + csui,v,t + csdi,v,t

)+∑t

(cllt) +∑r

∑t

(ccurtr,t))

(1)

Generation costsThe different cost components in the objective function are further specified. Fuel usage andcorresponding GHG emissions are dependent on the efficiency of the plant. Generally, this efficiencyis not constant over the range of power output, but follows a non-linear relationship. As discussed in[1], a piecewise linear approximation of this relationship can be implemented. In case of a concaveproduction curve, this requires additional integer variables. In this model, the approximation isrestricted to a single linear function assuming a convex production function:

cfi,v,t = P Fi ·

(noni,v,t · Pi,vηmini,v

+geni,v,t − (noni,v,t · Pi,v)

ηmargi,v

)·∆t (2)

Where ηmargi,v is derived from the linear interpolation between the minimal and the nominal oper-ating point:

ηmargi,v =∆cfi,v

∆geni,v=

(Pi,v

ηmaxi,v− Pi,v

ηmini,v

)(Pi,v − Pi,v)

(3)

A similar equation is used to describe the GHG emissions and corresponding costs:

cci,v,t = PCO2 · EFi ·

(noni,v,t · Pi,vηmini,v

+geni,v,t − (noni,v,t · Pi,v)

ηmargi,v

)·∆t (4)

In contrast, the VOM costs are assumed to be directly proportional to the power output:

cvomi,v,t = geni,v,t · V OMi,v ·∆t (5)

1Although there is no system cost related to curtailment of renewable energy, a curtailment cost can be used tomodel the impact of e.g. feed-in tarrifs.

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Start-up costs are assumed to be direcly proportional to the number of start-ups and the size ofthe unit. No distinction is made between hot, warm and cold starts. Similarly, shut-down costsare proportional to the number of shut-downs.

csui,v,t = nsui,v,t · Pi,v · SUCi,v (6)

csdi,v,t = nsdi,v,t · Pi,v · SDCi,v (7)

Lost-load costs are related to the amount of load shedding and the value of lost load:

cllt = llt · V oLL ·∆t (8)

Finally, the curtailment costs are proportional to the amount of energy curtailed and the feed-intariff:

ccurtr,t = curtr,t ·∆t · FITr (9)

1.2 System Constraints

Market clearingThe market clearing equation ensures that demand and supply are balanced in each period:∑

i,v

(geni,v,t) +∑r

(V RESr,t − curtr,t) +∑p

(genpsp,t − pumppsp,t) + llt = Dt ∀t (10)

As the potential generation from intermittent renewables (VRES) is driven by wheather conditions,the profile of maximal feed-in of VRES (V RESr,t) is exogenously determined depending on theinstalled capacities of onshore and offshore wind turbines, solar PV panels and run-of-river hydroplants. Curtailment of renewables is possible via the variable curtr,t. Curtailment is limited to themaximal potential VRES generation in every timestep:

0 ≤ curtr,t ≤ V RESr,t ∀t (11)

Similarly, the amount of lost load is limited by the electricity demand.

0 ≤ llt ≤ Dt ∀t (12)

ReservesTo deal with unforeseen events, demands for different categories of reserves can be imposed. In thismodel description, we follow the terminology of ENTSO-E2 with respect to the different reservecategories, being frequency containment reserves (FCR), frequency restoration reserves (FRR) andreplacement reserves (RR). Freaquency containment reserves are activated automatically and haveto be fully available within 30 seconds. On the other hand, activation of FRR can either beautomatic (aFRR) or manual (mFRR), where aFRR have to ba fully available within 30 secondsand mFRR within 15 minutes. Finally, RR are used to replace the FRR, such that new imbalancescan be restored by these FRR. The requirements for RR are less stringent, with full activationtimes up to one hour. A detailed description of the different reserve categories is presented in the

2European Network of Transmission System Operators for Electricity.

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network code on load-frequency control and reserves [2] For each reserve category, a demand forupward and downward reserves is imposed. In this model, the provision of reserves is restrictedto dispatchable power plants. Due to the required speed of activiation, FRR and aFRR can beexclusively provided by online units (spinning reserves). On the other hand, mFRR and RR canbe provided by spinning units as well as fast-starting units.∑

i,v

rFCR,up/downi,v,t ≥ Q

FCR,up/downt ∀t (13)

∑i,v

raFRR,up/downi,v,t ≥ Q

aFRR,up/downt ∀t (14)

∑i,v

(rmFRR,s,up/downi,v,t + r

mFRR,ns,up/downi,v,t ) ≥ Q

mFRR,up/downt ∀t (15)

∑i,v

(rRR,s,up/downi,v,t + r

RR,ns,up/downi,v,t ) ≥ Q

RR,up/downt ∀t (16)

1.3 Technological Constraints

Conventional power plants are subject to technical constraints. Modeling these constraints in UCmodels is typically done at power plant level. Certain constraints (e.g. minimum up and downtimes, provision of spinning reserves) require the use of binary variables to track the commitmentstatus of each plant in each period. In other words, a minimum of one binary commitmentvariable per plant and per time step is needed. To avoid this large amount of integers variables,this model groups plants into clusters depending on the technology and the vintage (investmentyear), similar as in [3]. Consider as an example a power system consisting of 3 groups of plantsof respectively 10, 20 and 40 units. The total number of potential states per time step equals(210× 220× 240) = 1.181E + 21 in case of using binary commitment variables. Grouping of similarplants reduces the number of combinations to (11 × 21 × 41) = 9471. It must be noted that thisreduction of states decreases as the number of groups is increased. Furthermore, by linearizing theinteger commitment variables, the model presented can be used as an addition to a linear capacityexpansion model. Due to the fact that technical constraints are treated at a group level, ratherthan at power plant level, the formulation deviates from typical MILP UC problem formulations.

Logic conditionsBefore dealing with the technological constraints, some logical conditions need to be implemented.First, the amount of online units in each group is limited to the number of plants belonging tothis group (Ni,v). As the model presented here is a unit commiment model, the number of plantsbelonging to each group is an input parameter. In case this model is integrated in an expansionplanning model, the number of plaints belonging to each group relates to the investment decision(variable), and is therefore not known a priori. Some planning models invest in technologies ratherthen single units. In this case, the number of plants belonging to each group can be found bydividing the capacity by a pre-defined unit size (dependent on the technology).

noni,v,t ≤ Ni,v ∀t (17)

The following equation establishes a link between the number of shutdowns, start-ups and onlineunits:

noni,v,t+1 = noni,v,t + nsui,v,t − nsdi,v,t ∀t (18)

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Furthermore, the amount of units that can start up and shut down in each period is restricted tothe remaining offline and online units respectively:

nsui,v,t ≤ Ni,v − noni,v,t ∀t (19)

nsdi,v,t ≤ noni,v,t ∀t (20)

Generation limitsEach plant is restricted to operate within an operating range delimited by a maximum and mini-mum operating point. As a consequence, the combined generation of a group of power plants alsofollows these restrictions:

geni,v,t ≤ noni,v · Pi,v ∀t (21)

geni,v,t ≥ noni,v · Pi,v ∀t (22)

Minimum up and down times Each plant has a minimum up and down time. This is respre-sented in equations (23)-(24):

nsui,v,t ≤ Ni,v − noni,v,t −∑

z=1:MDTi,v−1

nsdi,v,t−z ∀t (23)

nsdi,v,t ≤ noni,v,t −∑

z=1:MUTi,v−1

nsui,v,t−z ∀t (24)

The above equations are more stringent than the logic equations (19)-(20), therefore ommittingthe need to incorporate these equations.

Ramping limitsWhen considering a single power plant, its power output can either change due to ramping upor down if the plant is online and remains online or due to starting up or shutting down if thecommitment status of the plant changes. In contrast, if a group of power plants are considered,the power output of this group can change due to ramping of some units and simultaneous changesin the commitment status of some other units (equation (25)).

geni,v,t+1 = geni,v,t + rampupi,v,t − rampdowni,v,t + sui,v,t − sdi,v,t (25)

The ramping variables in equation (25) are restricted. First, to avoid excessive thermal stress,power plants can change their output at a limited rate (i.e. ramping rate) (equations (26)-(27)).These equations presume that only power plants which are online at period t (noni,v,t), and do notshut down between period t and t+1 (nsdi,v,t), can contribute to the ramp from period t to period

t+1 (rampup/downi,v,t )3. Furthermore, if spinning reserves are contracted for this period, these limit

the actual ramp that can take place by the group of generators as there should be a sufficientmargin to provide the contracted reserves if called upon.

rampupi,v,t + rFCR,upi,v,t + raFRR,upi,v,t + rmFRR,s,upi,v,t + rRR,s,upi,v,t ≤ (noni,v,t − nsdi,v,t) ·RUi,v · Pi,v ∀t (26)

3Following a similar, but slightly different reasoning, one could argue that only units which remain online inperiod t+1 subtracted by the units that have come online between period t and t+1 could contribute to the ramp,i.e. (non

i,v,t+1 − nsui,v,t). From equation (18), this proves to be equivalent.

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rampdowni,v,t +rFCR,downi,v,t +raFRR,downi,v,t +rmFRR,s,downi,v,t +rRR,s,downi,v,t ≤ (noni,v,t−nsdi,v,t)·RDi,v ·Pi,v ∀t (27)

Second, the ramp is restricted by the operating range of the group of power plants:

rampupi,v,t + rFCR,upi,v,t + raFRR,upi,v,t + rmFRR,s,upi,v,t + rRR,s,upi,v,t ≤((noni,v,t − nsdi,v,t) · Pi,v

)−(geni,v,t − sdi,v,t

)∀t (28)

rampdowni,v,t + rFCR,downi,v,t + raFRR,downi,v,t + rmFRR,s,downi,v,t + rRR,s,downi,v,t ≤(geni,v,t − sdi,v,t

)−((noni,v,t − nsdi,v,t) · Pi,v

)∀t(29)

Above equations state that the margin to provide ramping and spinning reserves is restricted bythe generation level of units that remain online on the one hand, and the maximum/minimumstable generation level of these units on the other hand.

Furthermore, from equation (25), the change in cluster power output can also change due to thestart-up or shut-down of additional units. The number of start-ups and shutdowns have beenlimited by equations (23) and (24) respectively. In addition, some additional constraints areneeded to limit the actual power shut down/started up. First of all, units have to operate abovethe stable generating level at all times (equations (30)-(31)). Second, equations (32)-(33) enforcethe logic condition that units cannot exceed the rated capacity. Finally, plants are assumed to startup within/shut down from a range specificied by the minimum operating point and the rampinglimits (equations (34)-(35)).

sui,v,t ≥ nsui,v,t · Pi,v ∀t (30)

sdi,v,t ≥ nsdi,v,t · Pi,v ∀t (31)

sui,v,t ≤ nsui,v,t · Pi,v ∀t (32)

sdi,v,t ≤ nsdi,v,t · Pi,v ∀t (33)

sui,v,t ≤ nsui,v,t · (Pi,v +RUi,v · Pi,v) ∀t (34)

sdi,v,t ≤ nsdi,v,t · (Pi,v +RDi,v · Pi,v) ∀t (35)

Reserve requirementsTo some extent, the provision of spinning reserves is restricted by equations (26)-(29). However,to include the stringent temporal requirements for the provision of spinning reserve products,additional constraints are needed. Following the conventions of the ENTSO-E, the contractedprimary and secondary reserves are expected to be fully available within 30 seconds when calledupon. Therefore, the time frame is significantly shorter than the time periods of 15 minutes or 1hour typically used in UC models. Therefore, the maximum ramps are adapted to correspond tothe considered time frame:

rFCR,up/downi,v,t ≤ (noni,v,t − nsdi,v,t) · Pi,v ·RURFCR

i,v /RDRFCRi,v ∀t (36)

Similar equations hold for the provision of FRR and RR reserves. However, one should note thatthe margin for providing FRR reserves is reduced by the contracted FCR and so on.

raFRR,up/downi,v,t + r

FCR,up/downi,v,t ≤ (noni,v,t − nsdi,v,t) · Pi,v ·RURaFRR

i,v /RDRaFRRi,v ∀t (37)

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rmFRR,s,up/downi,v,t +r

aFRR,up/downi,v,t +r

FCR,up/downi,v,t ≤ (noni,v,t−nsdi,v,t)·Pi,v ·RURmFRR

i,v /RDRmFRRi,v ∀t (38)

rRR,s,up/down+rmFRR,s,up/downi,v,t +r

aFRR,up/downi,v,t +r

FCR,up/downi,v,t ≤ (noni,v,t−nsdi,v,t)·Pi,v·RURRR

i,v /RDRRRi,v ∀t

(39)Moreover, the non spinning upward/downward mFRR and RR are constrained by the number ofunits contracted to start up/shut down and the range in which they can start-up or shut downwithin the required time (equations (40)-(45)).

FSmFRRi,v ·nsu,mFRRi,v,t ·Pi,v ≤ rmFRR,ns,upi,v,t ≤ nsu,mFRRi,v,t · (Pi,v +RURmFRRi,v ·Pi,v) ·FSmFRRi,v ∀t (40)

FSRRi,v · nsu,RRi,v,t · Pi,v ≤ rRR,ns,upi,v,t ≤ nsu,RRi,v,t · (Pi,v +RURRR

i,v · Pi,v) · FSRRi,v ∀t (41)

FSmFRRi,v ·nsd,mFRRi,v,t ·Pi,v ≤ rmFRR,ns,downi,v,t ≤ nsd,mFRRi,v,t ·(Pi,v+RURmFRRi,v ·Pi,v)·FSmFRRi,v ∀t (42)

FSRRi,v · nsd,RRi,v,t · Pi,v ≤ rRR,ns,downi,v,t ≤ nsd,RRi,v,t · (Pi,v +RURRR

i,v · Pi,v) · FSRRi,v ∀t (43)

rmFRR/RR,ns,upi,v,t ≤ n

su,mFRR/RRi,v,t · Pi,v ∀t (44)

rmFRR/RR,ns,downi,v,t ≤ n

sd,mFRR/RRi,v,t · Pi,v ∀t (45)

Here, FSmFRR/RRi,v is a binary parameter such that only units which can start up/shut down

sufficiently fast (FSmFRR/RRi,v = 1) can participate in the provision of non-spinning reserves.

Finally, equations (23)-(24) need to be adapted to take into account that units contracted forproviding non-spinning reserves need to be available to start up/shut down when called upon:

nsui,v,t + nsu,mFRRi,v,t + nsu,RRi,v,t ≤ Ni,v − noni,v,t −∑

z=1:MDTi,v−1

nsdi,v,t−z ∀t (46)

nsdi,v,t + nsd,mFRRi,v,t + nsd,RRi,v,t ≤ noni,v,t −∑

z=1:MUTi,v−1

nsui,v,t−z ∀t (47)

Maintenance and outagesDifferent possibilities exist to model periods of maintenance and (un)forced outages. Here, fourdistinct approaches are presented. The first method to take into account the limited availabilityof a power plant is to derate the nameplate capacity:

geni,v,t ≤ AFi,v · noni,v,t · Pi,v ∀t (48)

A second approach limits the yearly generated output. This leads to an optimal scheduling of themaintenance periods. ∑

t

(geni,v,t ·∆t) ≤ AFi,v ·∑t

(Pi,v ·∆t) (49)

However, both approaches do not take into account the discrete character of maintenance periods,i.e., during a continuous maintenance period, a power plant is either available or not. To accountfor these discrete effects, a third approach can be used (equations (50)-(53)). Equation (50) imposesthe minimum level of maintenance. Furthermore, equations (51)-(52) ensure the continuity of themaintenance period. Here, a distinction is made between units going into maintenance after havingbeen online for a period (nstart,mainti,v,t ) and units which shut down to go into maintenance (nsd,mainti,v,t ).The variable tracking the number of units shutting down is therefore split up into two variables

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(equation (53)). Finally, equation (54) amends equation (46) to take into account that some arein maintenance. ∑

t

(nmainti,v,t ·∆t) ≥ (1− AFi,v) ·∑t

(Ni,v ·∆t) (50)

nmainti,v,t+1 = nmainti,v,t + nstart,mainti,v,t + nsd,mainti,v,t − nend,mainti,v,t ∀t (51)

nmainti,v,t ≥MCMDi,v∑

z=1

(nstart,mainti,v,t−z + nsd,mainti,v,t−z ) ∀t (52)

nsdi,v,t = nsd,mainti,v,t + nsd,regi,v,t ∀t (53)

nsui,v,t + nsu,mFRRi,v,t + nsu,RRi,v,t ≤ Ni,v − nmainti,v,t − noni,v,t −MDTi,v−1∑

z=1

nsd,regi,v,t−z ∀t (54)

Pumped storagesPumped storages are restricted by a seperate set of operational constraints. A first equation linksthe energy content of the pumped storage in different periods:

epsp,t = epsp,t−1 −( genpsp,t√

ηPSp

+ pumppsp,t ·√ηPSp)·∆t ∀t (55)

In this equation, the energy content of pumped storage reservoir is depleted when turbining, whilethe storage is filled during pumping. It is assumed here that the efficiency of the turbine andthe pump are equal, and together make up the round trip efficiency of the pumped storage, ηPSp .Furthermmore, the electrical capacity of the pump and turbine are assumed to be identical andequal to the rated capacity of the pumped storage plant:

genpsp,t ≤ P PSp ∀t (56)

pumppsp,t ≤ P PSp ∀t (57)

Finally, the storage basin can store a maximal amount of potential energy, and can be depletedup to a minimum level.

EPSp ≤ epsp,t ≤ EPS

p ∀t (58)

References

[1] Kenneth Van den Bergh, Kenneth Bruninx, Erik Delarue, William D’haeseleer, A Mixed-Integer Linear Formulation of the Unit Commitment Problem. KU Leuven, TME BranchWorking Paper 2014-07, April 2014, http://www.mech.kuleuven.be/en/tme/research/

energy_environment/Energy_and_environment

[2] ENTSO-E, Supporting Document for the Network Codeon Load-Frequency Control and Reserves. June, 2013.https://www.entsoe.eu/fileadmin/user upload/ library/resources/LCFR/130628-NC LFCR-Supporting Document-Issue1.pdf

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[3] Bryan Palmintier, Impact of Unit Commitment Constraints on Generation Expansion Plan-ning with Renewables. Power and Energy Society General Meeting, 2011 IEEE, http:

//ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6038963&tag=1

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