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LUSYM: a unit commitment model formulated as a mixed-integer linear program

Kenneth Van den Bergh, Kenneth Bruninx, Erik Delarue, William D‘haeseleer

TME WORKING PAPER - Energy and Environment Last update: March 2016

An electronic version of the paper may be downloaded from the TME website:

http://www.mech.kuleuven.be/tme/research/

LUSYM: a unit commitment model formulated as a mixed-integer

linear program

Kenneth Van den Bergh, Kenneth Bruninx, Erik Delarue and William D’haeseleer

University of Leuven (KU Leuven) - Energy Institute

March 2016

Abstract. This paper presents a state-of-the-art deterministic unit commitment (UC) model, for-mulated as a mixed-integer linear program (MIP), and referred to under the name LUSYM (LeuvenUniversity System Modeling). A UC model determines the optimal scheduling of a given set of powerplants to meet the electricity load, taking account of the operational constraints of the electricitysystem. The presented formulation is tight and compact, and includes power plant constraints,flexible load, renewables curtailment, storage units, transmission grid constraints, spinning reservesand must-run constraints. The model is able to solve large-scale electricity systems within reason-able run times. Run times are reduced by means of a tight and compact formulation, efficient datahandling and the use of best-in-class MIP solvers.

Keywords. Unit commitment; mixed-integer linear programming; tight and compact formulation.

Nomenclature

Sets

I (index i) set of power plantsJ (index j) set of storage unitsL (index l) set of transmission linesLac (index lac) set of AC transmission linesLdc (index ldc) set of DC transmission linesLpst(index lpst) set of AC transmission lines with phase shift transformer

M (index m) set of must-run groupsN (index n) set of nodesS (index s) set of reserve zonesT (index t) set of time steps

Parameters

αlpst maximum phase shifter angle at line lpst [◦]∆t time resolution [h]ηcj charging efficiency of storage unit j

ηdj discharging efficiency of storage unit j

Agridl,n incidence matrix of the grid {-1,0,1}

Amustm,i matrix linking power plant i to must run group m {0,1}

Aplantn,i matrix linking power plant i to node n {0,1}

Astorn,j matrix linking storage unit j to node n {0,1}

Arsrs,i matrix linking power plant i to reserve zone s {0,1}

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Arsrs,n matrix linking node n to reserve zone s {0,1}

AVi,t availability of power plant i at time step t {0,1}Dn,t electricity demand at node n at time step t [MW]DCDFlac,ldc DC line distribution factors

F 0lac zero-imbalance flow in AC transmission line lac [MW]

F lac maximum power flow through AC transmission line lac [MW]

F ldc maximum power flow through DC transmission line ldc [MW]LCCn cost of load curtailment at node n [∆t EUR/MWh]

LSn,t maximum storable load consumption at node n at time step t [MW]

LSEn,t maximum storable load energy content at node n at time step t [MWh]LSEn,t minimum storable load energy content at node n at time step t [MWh]

MCi marginal generation cost of power plant i [∆tEUR/MWh]MDTi minimum down time of power plant i [∆t]MUTi minimum up time of power plant i [∆t]MRMm must-run requirement of group of power plants mMRPi must-run requirement of power plant iNCi generation cost at minimum output of power plant i [∆tEUR/h]OPi required planned outages of power plant i [∆t]PCj maximum charging power of storage unit j [MW]PDj maximum discharging power of storage unit j [MW]

PEj maximum energy content of storage unit j [MWh]PEj minimum energy content of storage unit j [MWh]

PSDFlac,lpst phase shifter distribution factorsPTDFlac,n power transfer distribution factorsRAC lol reserve allocation cost for load curtailment [∆tEUR/MW]RACcur reserve allocation cost for renewables curtailment [∆tEUR/MW]RCi ramping cost of power plant i [EUR/MW]RCCn renewables curtailment cost at node n [∆tEUR/MWh]RDi maximum ramp-down rate of power plant i [MW/∆t]RUi maximum ramp-up rate of power plant i [MW/∆t]SECn simultaneous gross export capacity of node n [MW]SICn simultaneous gross import capacity of node n [MW]SDCi shut-down cost of power plant i [EUR]SUCi start-up cost of power plant i [EUR]SDi maximum shut-down rate of power plant i [MW/∆t]SUi maximum start-up rate of power plant i [MW/∆t]SR+

s required upward spinning reserve in reserve zone s [MW]SR−s required downward spinning reserve in reserve zone s [MW]RESn,t available renewable generation at node n at time step t [MW]TCl transmission cost for line l [∆tEUR/MWh]

Variables

αlpst,t phase shifter angle at line lpst at time step t in [◦]costrcn,t renewables curtailment cost at node n at time step t [∆tEUR/h]

costlcn,t load curtailment cost at node n at time step t [∆tEUR/h]

costgeni,t generation cost of power plant i at time step t [∆tEUR/h]

costrampi,t ramping cost of power plant i at time step t [∆tEUR/h]

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costrsrn,t reserve allocation cost for load and renewables curtailment at node n

at time step t [∆tEUR/h]coststarti,t start-up cost of power plant i at time step t [∆tEUR/h]

coststopi,t shut-down cost of power plant i at time step t [∆tEUR/h]

costtransl,t transmission cost of line l at time step t [∆tEUR/h]

fl,t power flow through line l at time step t [MW]flac,t power flow through AC line lac at time step t [MW]fldc,t power flow through DC line ldc at time step t [MW]

gi,t generation of power plant i above minimum output at time step t [MW]geni,t power generation of power plant i at time step t [MW]lcn,t load curtailment at node n at time step t [MW]lsn,t storable load at node n at time step t [MW]lsen,t storable load energy at node n at time step t [MWh]ofi,t forced outage of power plant i at time step t {0,1}opi,t planned outage of power plant i at time step t {0,1}pn,t power injection in the grid at node n at time step t (MW)pcj,t charging power of storage unit j at time step t [MW]pdj,t discharging power of storage unit j at time step t [MW]pej,t energy level of storage unit j at time step t [MWh]r+i,t upward spinning reserve from power plant i at time step t [MW]

r−i,t downward spinning reserve from power plant i at time step t [MW]

rcurn,t reserve from renewables curtailment at node n at time step t [MW]

rloln,t reserve from load curtailment at node n at time step t [MW]

rcn,t renewables curtailment at node n at time step t [MW]vi,t start-up status of power plant i at time step t {0,1}wi,t shut-down status of power plant i at time step t {0,1}zi,t on/off-status of power plant i at time step t {0,1}

1 Introduction

The unit commitment (UC) problem can be defined as the scheduling of electric power generatingunits over a daily to weekly time horizon in order to minimize operational system costs (Hobbset al., 2001). The unit commitment solution must respect the technical and operational limits of theelectricity system, such as power plant constraints and reserve requirements. The unit commitmentmodel gives for each generation unit and each time step the unit commitment (UC) decision, i.e.,the on/off-status, and the economic dispatch (ED) decision, i.e., the power output if on-line. TheUC decision is typically taken hours to days before the actual delivery, since most power plantscannot start-up quickly. The ED decision is typically taken minutes to hours before the actualdelivery, as changing the power output of an online plant requires less time than bringing a powerplant online. The UC decision problem translates into a more complex mathematical formulationthan the ED decision, due to the binary nature of the on/off decision.1

The UC problem can be addressed from a system perspective or a generator perspective. Accordingto the system perspective, the operational cost for the whole system is minimized while guarantee-ing supply-demand balance in the system. One speaks of a Security Constrained Unit Commitment

1Including non-spinning reserves in the economic dispatch decision introduces a binary on/off-decision in the EDproblem.

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(SCUC) if also security and transmission constraints are considered. This system perspective cor-responds to a vertically integrated environment, in which a regulated monopolist schedules thegeneration portfolio at minimum operational cost (i.e., cost-based unit commitment), or to an un-bundled environment with a centralized and controlled Electricity Pool model (i.e., bid-based unitcommitment, e.g., the PJM market). A SCUC can also be used as a proxy for the market outcomeof a liberalized market based on a decentralized model with bilateral trading and possibly PowerExchanges (e.g., the European electricity market), under the assumption of perfect competition.According to the generator perspective, the profit of one generator is maximized given its gener-ation portfolio and an electricity price. This type of UC problem is referred to as Price BasedUnit Commitment (PBUC) or Self-Unit Commitment. The generator perspective corresponds toa deregulated market environment in which generators are responsible for the UC decision of theirgeneration portfolio.

UC models are partial equilibrium models, focusing solely on the electricity sector. Interactionsbetween the electricity sector and other sectors in the economy are neglected (e.g., fossil fuel pricesare imposed as exogenous parameters to UC models, neglecting the relation between fossil fuel-firedelectricity generation and fossil fuel prices). UC models can be distinguished from other electricitygeneration models based on the classification parameters proposed by Ventosa et al. (2005):

• Degree of competition: UC models can correspond to a competitive market or a regulatedvertically integrated monopoly. Strategic behavior of market players (e.g., an oligopolisticmarket) is not represented in UC models.

• Time frame. UC models are short-term operational models with a time frame of days to weeksand a time resolution of minutes to hours. No long-term decisions, such as investments in newgeneration capacity, are usually considered.

• Generation system: The main distinctive characteristic of UC models is the high level of detailwith which the generation units are represented, considering each power plant individuallywith its technical limits.

• Demand flexibility: UC models often use an inelastic demand, which is a fair assumptionfor short-term analyses. However, an increasing amount of academic literature deals withincluding demand flexibility in UC modeling, see for instance De Jonghe et al. (2014) andPatteeuw et al. (2015).

• Uncertainty: The various sources of uncertainty in the electricity sector (e.g., power plantoutages, renewable forecast errors) can be addressed with a deterministic or a stochasticapproach. In a deterministic unit commitment (DUC) model, generation units are scheduledbased on expected values of probabilistic input parameters. Possibly, reserves are scheduledto deal with deviations from the expected values. In a stochastic unit commitment (SUC)model, the full probabilistic distribution of the uncertain parameters is taken into account. ASUC model is computationally more challenging than a DUC model. A detailed overview ofunit commitment under uncertainty is given by Tahanan et al. (2015).

• Transmission constraints: Transmission constraints can be implemented in UC models bymeans of an AC power flow model, a DC power flow model or a trade-based model. However,often grid constraints are neglected in UC models, assuming the considered electricity systemis a copper plate.

The UC problem is a non-convex and non-linear problem. The non-convexity is caused by thebinary nature of the on/off decision and non-linearities occur due to, amongst others, non-linear

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generation cost curves and non-linear transmission constraints. All this makes the UC problema difficult problem to solve. Over the course of the last decades, different mathematical methodshave been used to solve the UC problem. Sheble and Fahd (1994); Sen and Kothari (1998); Yamin(2004); Padhy (2004); Farhat and El-Hawary (2009); Saravanan et al. (2013) give an overview ofthese different methods, of which the most important are Exhaustive Enumeration (i.e., listingall possible combinations of on/off-states and picking the most optimal one), Priority Listing (i.e.,committing generation units in order of increasing marginal generation cost until the electricity loadis met), Dynamic Programming (i.e., optimization-based method that searches for the minimumcost solution by solving simpler subproblems), Lagrangian Relaxation (i.e., the Lagrangian of theoptimization problem is solved), Mixed-Integer Programming (i.e., optimization-based method tosolve a mixed-integer problem by means of the branch-and-bound method), Decommitment Method(i.e., starting with all units online and switching off units) and more recently meta-heuristic methodssuch as Fuzzy Systems, Genetic Algorithms, Artificial Neural Networks, Evolutionary Programming,Tabu Search and Ant Colony Search Algorithms.

A mathematical method that gained importance due to dramatic improvements in solver perfor-mances is mixed-integer programming (MIP) (Hobbs et al., 2001). MIP is an operational researchmethod in which certain variables are restricted to be integers. The advantage of UC MIP is twofold:(1) the MIP solver returns a feasible solution (if feasible solutions exist and can be found by the MIPsolver), and (2) the level of optimality is known (the MIP solver returns the optimality gap betweenthe MIP solution and the lower bound to the UC problem). The disadvantage of UC MIP is longerrun times, compared to faster methods such as Priority Listing. However, due to improvements incommercial solvers and model formulations, UC MIP models are nowadays often used in industryand academia.

This paper presents a deterministic linearized mixed-integer formulation of the security constrainedunit commitment problem (further referred to as MIP UC).2 The formulation is implemented in thelatest release of Gams (Gams, 2016) and solved with the latest release of Cplex MIP solver (IBM,2016) or Gurobi MIP solver (Gurobi Optimization, 2016). Processing of input and output datahappens in Matlab (MathWorks, 2016), using the Matlab-Gams coupling as described by Ferriset al. (2011). The model is able to solve the unit commitment problem for large scale electricitysystems (several hundreds of generation units, about hundred time steps) within several hours.3

Several academic and commercial UC models exist. Examples of academic UC models are theWILMAR model developed by mainly Scandinavian research institutes (Meibom et al., 2003), theELMOD model developed at the Dresden University of Technology (Leuthold et al., 2008), andDispa-SET developed by the EU JRC Institute for Energy and Transport (Hidalgo Gonzalez et al.,2014). Commercial unit commitment models are Plexos (Energy Exemplar, 2015), Promod (VentytxEnergy, 2015), Antares (Doquet et al., 2008) and Bid (Poyry, 2015). All these UC models differslightly, depending on the purpose of the model and the setting in which it is developed. Note thatthis is anything but an exhaustive list of available UC models.

The paper continues as follows. Section 2 describes the MIP formulation of the UC problem indetail. Section 3 discusses the validation of the LUSYM model and section 4 concludes.

2The acronyms MIP and MILP refer both to mixed-integer linear programs. Non-linear mixed-integer programsare referred to with the acronym MINP (or MINLP). This paper deals with linear mixed-integer programs, herereferred to with the acronym MIP.

3Simulations run on an Intel(R) Core(TM) i7-2620M CPU@2.7GHz, 8 GB RAM. The exact run time heavilydepends on the considered instance and stopping tolerance.

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2 Mathematical formulation

The LUSYM model can be split up in two parts; (1) the unit commitment model itself with hourlyor quarter-hourly time resolution and a daily or weekly time horizon (see section 2.1), and (2) ayearly scheduling of power plants outages with daily or weekly time resolution (see section 2.2).The outage scheduling happens ex-ante the unit commitment scheduling.

2.1 Unit commitment

A MIP of the UC problem consists of one objective function, i.e., minimize total operational systemcost, subject to several system constraints. The mixed-integer formulation of the unit commitmentproblem is extensively described in the literature. Arroyo and Conejo (2000) present a mixed-integerlinear formulation, based on three binary variables per generation unit (i.e., on/off-state, start-upstatus and shut-down status). Carrion and Arroyo (2006) present an updated version of the mixed-integer linear formulation, requiring only one binary variable per generation unit (i.e., on/off-state).Both formulations are equivalent, but the former is tighter. Tightness refers to how good thefeasible area of a mixed-integer problem is approximated by the relaxed version of the formulation(the better the feasible area is approximated, the tighter the formulation). An important conceptin this regard is the convex hull. The convex hull is defined as the smallest convex feasible regioncontaining all feasible integer points of a MIP (Wolsey, 1998). As such, the relaxed version of anperfectly tight formulation describes the convex hull of the MIP. Another important characteristic ofa unit commitment formulation is its compactness. Compactness refers to the number of variablesand equations needed to describe the problem (the fewer variables and equations, the compacterthe formulation). A mixed-integer problem solves faster the more compact and tight its formulationis.

Several authors propose a tight and compact formulation of (part of) the MIP UC formulation.Ostrowski et al. (2012) describe a tighter formulation of the generation limits and ramping limits ofpower plants. Morales-Espana et al. (2013a) present a formulation which is simultaneously tighterand compacter, with special focus on the generation limits of power plants. Rajan and Takriti (2005)discuss a tight formulation of the minimum up and down times. Morales-Espana et al. (2013b) fo-cus on a tight and compact formulation of the start-up and shut-down trajectories of power plants.Frangioni et al. (2009) present a tighter formulation of the linear approximation of the productioncost curve. Damcı-Kurt et al. (2013) present a tight description of the two-period ramping con-straint. Yang et al. (2015) show that splitting up the power output of a generation unit in theminimum output and the output exceeding the minimum output allows a tighter and compacterformulation. Zheng et al. (2015) presents two cuts to tighten the unit commitment formulation,based on a natural understanding of the problem and the generalized flow cover inequality.

The MIP UC formulation presented in this paper is based on the papers mentioned above. Thenomenclature can be found the outset of the paper.

Objective functionThe objective function of the UC model is to minimize total operational system cost, consisting ofgeneration costs, start-up costs, shut-down costs, ramping costs, load curtailment costs, renewablescurtailment costs, reserve allocation costs for load and renewable curtailment and transmission

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costs.min

∑i,t

(costgeni,t + coststarti,t + coststopi,t + costramp

i,t

)+

∑n,t

(costlcn,t + costrcn,t + costrsrn,t

)+∑l,t

costtransl,t

(1)

The generation costs include fuel costs, CO2 emission costs and variable operations and maintenancecosts. The generation cost of a power plant is time-depending (due to changing fuel and CO2

emission prices) and output-depending (due to the output-dependent generation efficiency). Thenon-linear cost curve is linearized as follows:4

costgeni,t = NCi zi,t +MCi gi,t ∀i,∀t (2)

The start-up and shut-down cost follow from, respectively:5

coststarti,t = SUCi vi,t ∀i,∀t (3)

coststopi,t = SDCiwi,t ∀i,∀t (4)

The ramping cost follows from:

0 ≤ costrampi,t ≥ RCi (gi,t − gi,t−1) ∀i,∀t (5)

0 ≤ costrampi,t ≥ RCi (gi,t−1 − gi,t) ∀i,∀t (6)

The transmission cost follows from:

0 ≤ costtransl,t ≥ TCl fl,t ∀l,∀t (7)

0 ≤ costtransl,t ≥ −TCl fl,t ∀l,∀t (8)

The load curtailment cost follows from:

costlcn,t = LCCn lcn,t ∀n, ∀t (9)

The renewables curtailment cost follows from:

costrcn,t = RCCn rcn,t ∀n, ∀t (10)

The reserve allocation cost for load and renewables curtailment follows from:

costrsrn,t = RAC lol rloln,t +RACcur rcurn,t ∀n, ∀t (11)

Note that the reserve allocation cost for spinning reserves from conventional units is implicitly takeninto account in the generation cost.

4The cost curve can be approximated with multiple linear intervals, but this increases run times drastically whileaccuracy only increases slightly (Arroyo and Conejo, 2000; Frangioni et al., 2009).

5A more advanced formulation of the start-up cost takes account of the off-line time of the power plant anddistinguishes between hot starts, warm start and cold starts (Arroyo and Conejo, 2000; Morales-Espana et al., 2013b).

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Market clearing constraintThe market clearing constraint imposes demand-supply balance at each node for each time step. Thesupply-demand balance consists of generation from conventional units, (dis)charging from storageunits, generation from renewables, the electricity load and the flexible load (curtailable and storable)and injections in the electricity grid.∑

i

Aplantn,i (zi,t Pi + gi,t) +

∑j

Astorn,j (pdj,t − pcj,t) +RESn,t − rcn,t =

Dn,t − lcn,t + lsn,t + pn,t ∀n,∀t(12)

Renewables curtailmentElectricity generation from renewables (and cogeneration units) is mainly driven by other factorsthan the electricity demand (e.g., weather conditions, subsidies) and is therefore only to a limitedextend dispatchable. Renewable generation can be curtailed in the energy market or scheduled asdownward reserve, but renewables curtailment is limited by the available renewable generation:

0 ≤ rcn,t + rcurn,t ≤ RESn,t ∀n, ∀t (13)

Renewables curtailment is non-negative:

rcn,t , rcurn,t ≥ 0 ∀n,∀t (14)

Flexible loadTwo types of flexible load are considered: curtailable load and storable load. Load curtailment (orload shedding) can be scheduled in the energy market or as upward reserve, and is limited by theload:

0 ≤ lcn,t + rloln,t ≤ Dn,t ∀n, ∀t (15)

Load curtailment is non-negative:

lcn,t , rloln,t ≥ 0 ∀n, ∀t (16)

Storable load is characterized by an energy limit, a power limit and an energy balance equation,respectively:

LSEn,t ≤ lsen,t ≤ LSEn,t ∀k, ∀t (17)

0 ≤ lsn,t ≤ LSn,t ∀n, ∀t (18)

lsen,t = lsen,t−1 + ∆t lsn,t ∀n,∀t (19)

Power plant generation limitsA power plant can only generate power within a certain power range. It is important to highlightthat the power plant output is defined as geni,t = zi,t P i +gi,t. The lower limit for the power outputabove the minimum power output is:

0 ≤ gi,t − r−i,t ∀i,∀t (20)

The upper generation limit for power plants with MUTi ≥ 2 is given by:

gi,t + r+i,t ≤ (P i − P i) zi,t − (P i − SUi) vi,t − (P i − SDi)wi,t+1 ∀i ∈MUTi ≥ 2,∀t (21)

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If MUTi = 1, Eq. (21) is replaced by:

gi,t + r+i,t ≤ (P i − P i) zi,t − (P i − SUi) vi,t −max(SUi − SDi, 0)wi,t+1 ∀i ∈MUTi = 1, ∀t (22)

gi,t + r+i,t ≤ (P i − P i) zi,t − (P i − SDi)wi,t+1 −max(SDi − SUi, 0) vi,t ∀i ∈MUTi = 1,∀t (23)

Eqs. (20)-(23) describe the convex hull of the power plant generation limits (Morales-Espana, 2014).

Additional generation limit constraints can be imposed by considering multiple time steps (Os-trowski et al., 2012), however it is not sure that these constraints result in a speed-up given thatthey make the formulation tighter but less compact. These additional constraints are:

gi,t + r+i,t ≤ (P i − P i) zi,t+Ki +

Ki∑k=1

(SDi − P i + (k − 1)RDi)wi,t+k

−Ki∑k=1

(P i − P i) vi,t+k ∀i,∀t = 1, ...T −K

(24)

with Ki = min{MUTi ; (P i−SDi)/RDi + 1 ; T − t}. Eq. (24) is only tightening Eq. (21) if K ≥ 2.

Forced or planned power plant outages can force power plants to stay offline (see section 2.2).

zi,t ≤ AVi,t (25)

Finally, generation and reserve scheduling variables are non-negative or binary.

geni,t , gi,t , r+i,t , r

−i,t ≥ 0 ∀i,∀t (26)

zi,t , vi,t , wi,t ∈ {0, 1} ∀i,∀t (27)

Power plant ramping limitsThe basic ramping-up and ramping-down constraints are, respectively:

gi,t + r+i,t − gi,t−1 ≤ RUi zi,t + (SUi − P i −RUi) vi,t ∀i,∀t (28)

gi,t−1 − gi,t + r−i,t ≤ RDi zi,t−1 + (SDi − P i −RDi)wi,t ∀i,∀t (29)

Additional ramping constraints can be imposed by considering more time steps (Ostrowski et al.,2012), however it is again not sure that these additional constraints result in a speed-up as theymake the formulation tighter but also less compact. Additional ramping-up constraints are:

gi,t + r+i,t − gi,t−1 ≤ RUi zi,t − (RUi − SDi + P i)wi,t+1 + (SUi − P i −RUi) vi,t

∀t,∀i ∈ RUi > SDi − P i & MUTi ≥ 2(30)

gi,t + r+i,t − gi,t−2 ≤ 2RUi zi,t + (SUi − P i −RUi) vi,t−1 + (SUi − P i − 2RUi) vi,t

∀t,∀i ∈MUTi ≥ 2 & MDTi ≥ 2(31)

Analogously, additional ramping-down constraints are:

gi,t−1 − gi,t + r−i,t ≤ RDi zi,t + (SDi − P i)wi,t −RDi vi,t − (RDi − SUi + P i) vi,t−1

∀t,∀i ∈ RDi > SUi − P i & MUTi ≥ 2(32)

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gi,t−2 − gi,t + r−i,t ≤ 2RDi zi,t + (SDi − P i)wi,t−1 − 2RDi (vi,t−1 + vi,t)+

(SDi − P i +RDi)wi,t ∀t,∀i ∈MUTi ≥ 2 & MDTi ≥ 2(33)

Power plant minimum up and down timesThe minimum down time and up time constraints are given by, respectively:

1− zi,t ≥t∑

t′=t+1−MDTi

wi,t′ ∀i,∀t (34)

zi,t ≥t∑

t′=t+1−MUTi

vi,t′ ∀i,∀t (35)

In addition to the above constraints, the following logic relationship between the different powerplant statuses is needed:

zi,t−1 − zi,t + vi,t − wi,t = 0 ∀i,∀t (36)

Eqs. (34)-(36) describe the convex hull of the minimum up and down time constraints (Rajan andTakriti, 2005).

Must-run constraintsMust-run constraints can be imposed to a subset of power plants:∑

i

Amustm,i (zi,t P i + gi,t) ≥MRMm

∑i

Amustm,i AVi,t P i ∀m ∈MRMm > 0,∀t (37)

A must-run constraint can also be imposed to a single power plant:

zi,t P i + gi,t ≥MRPiAVi,t P i ∀i ∈MRIi > 0, ∀t (38)

Note that the must-run constraint can be overruled by power plant outages (i.e., must-run constraintcan be violated when the power plant is undergoing a forced or planned outage).

Spinning reserve constraintsSpinning reserve constraints can be imposed to reserve zones, consisting of one or multiple nodes.Upward spinning reserves can be delivered by online power plants, load curtailment and curtailedrenewable generation. Downward spinning reserves can be delivered by online power plants and re-newables curtailment. Downward reserves can also be delivered by renewables curtailment. Upwardand downward spinning reserve requirements are given by, respectively:∑

i

Arsrs,i r

+i,t +

∑n

Arsrs,n (rloln,t + rcn,t) ≥ SR+

s ∀s, ∀t (39)

∑i

Arsrs,i r

−i,t +

∑n

Arsrs,n r

curn,t ≥ SR−s ∀s, ∀t (40)

Storage unit constraintsDifferent storage technologies, such as pumped hydro storage and electric batteries, can be im-plemented in the unit commitment model by the same set of equations. The energy balance of astorage unit is given by:

pej,t = pej,t−1 + ∆t pcj,t ηcj −

∆t pdj,t

ηdj∀j,∀t (41)

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The energy level of a storage unit and its charging and discharging power rates are limited:

0 ≤ pcj,t ≤ PCj ∀j,∀t (42)

0 ≤ pdj,t ≤ PDj ∀j,∀t (43)

PEj ≤ pej,t ≤ PEj ∀j,∀t (44)

Grid constraintsA DC power flow representation of the electricity grid is implemented, including phase shiftingtransformers (PST) and high voltage direct current (HVDC) lines. The DC power flow is a linearizeddescription of the electricity grid characteristics, respecting Kirchhoff’s voltage and current laws.Power transfer distribution factors (PTDFs) give the linear relationship between power injectionsin the grid and flows through transmission lines .

flac,t =∑n

PTDFlac,n injn,t +∑lpst

PSDFlac,lpst αlpst,t +∑ldc

DCDFlac,ldc fldc,t + F 0lac ∀l,∀t (45)

∑n

pn,t = 0 ∀t (46)

The line flows and phase shifter angles are constrained as follows:

−αlpst ≤ αlpst,t ≤ αlpst ∀lpst, ∀t (47)

−F lac ≤ flac,t ≤ F lac ∀lac, ∀t (48)

−F ldc ≤ fldc,t ≤ F ldc ∀ldc, ∀t (49)

Also a trade-based grid representation is implemented. In a trade-based grid representation, onlyKirchhoff’s current law is respected. There is also no longer a difference between the representationof AC lines and DC lines. If the trade-based grid representation is used, Eqs. (45)-(49) are replacedby:

pn,t =∑l

Agridl,n fl,t ∀n,∀t (50)

−F l ≤ fl,t ≤ F l ∀lac, ∀t (51)

In a trade based grid model, an additional constraint can be imposed on the gross import or exportin a node (in order to compensate for the lower accuracy of a trade based grid model compared toa DC power flow grid model). ∑

l∈Limpn

|Agridl,n fl,t| ≤ SICn ∀n, ∀t

∑l∈Lexp

n

|Agridl,n fl,t| ≤ SECn ∀n, ∀t

(52)

A detailed discussion of the different electricity grid models and a derivation of the power transferdistribution factors (PTDF), phase shifter distribution factors (PSDF) and direct current distribu-tion factors (DCDF) can be found in Van den Bergh et al. (2014).

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2.2 Power plant outages

The unit commitment scheduling is preceded by a power plant outage scheduling. One distinguishesbetween planned outages (e.g., yearly maintenance) and unplanned or forced outages (e.g., technicalfailure). Planned outages can be scheduled at moments of low residual load (i.e., load minusrenewables generation). The planned outage scheduling problem is formulated as a deterministicmixed-integer linear program, with a time frame of one year and a time step of one day. Forcedoutages are randomly distributed during the year. It is assumed that planned and forced outageslast for one day.

Objective functionThe objective function of the planned outage scheduling is to minimize the maximum generationmargin (i.e., the margin between available generation capacity and load). In other words, outagesare planned during moments with large generation margins. The planned outage scheduling isperformed for each node separately. Since the time step is one day, the daily average load andrenewables generation is used in the power plant outage module.

min M (53)

M ≥∑i

Aplantn,i opi,t P i +RESn,t −Dn,t ∀t (54)

The binary variable opi,t is zero if power plant i undergoes a planned outage at day t, otherwiseopi,t is one.

Total duration of planned outagesThe number of days with a planned outage is imposed as follows:∑

t

1− opi,t = OPi ∀i (55)

The planned outage schedule opi,t and the forced outage schedule ofi,t (which is determined ran-domly) combine into the power plant availability AVi,t as follows:

AVi,t = opi,t ofi,t ∀i,∀t (56)

3 Validation of the model

The LUSYM model is validated in two different ways: (1) by comparing its performance withexisting unit commitment models and (2) by comparing its simulation results with historicallyobserved generation data.

3.1 Benchmarking with existing unit commitment models

The performance of the unit commitment model, in terms of optimality of simulation results and runtimes, is compared with three commercial unit commitment models (the names of the commercialmodels are not disclosed for confidentiality reasons). This benchmarking exercise is repeated everyyear (from 2013 to 2016) for LUSYM in order to track its progress.

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As benchmarking case, a 2030 scenario of the Central Western European electricity sector (i.e.,Belgium, France, Germany, Luxembourg and the Netherlands) is considered (ENTSO-E, 2014).The electricity system contains 552 power plants and 4 pumped storage units. The network modelconsists of 5 nodes and 7 lines, represented by a trade-based network model. One full year issimulated in weekly blocks of 168 hours with an hourly time resolution. No power plant outagesand reserve requirements are imposed. Load can be curtailed at a cost of 10,000 EUR/MWh andrenewables at a cost of 0 EUR/MWh. All simulations are run on an Intel(R) Core(TM) i7-2620MCPU@2.7 GHz with 8 GB RAM. The LUSYM model is solved by Gurobi with a 1% stoppingtolerance.

Fig. 1 gives an overview of the performance of the different models, in terms of optimality (i.e.,minimization of generation cost) and run times. First, it is clear that LUSYM model performssimilarly to the commercial UC-2 (which is also a MIP-based model), both in terms of optimalityand run times. Note that one should compare the run time of UC-2 with the run time of the LUSYM2013 version. LUSYM outperforms the more heuristic-based models UC-1 and UC-3 in terms ofoptimality with about 15%, while the run time is about the same for UC-1 and considerably shorterfor UC-3. Second, LUSYM shows a remarkable speed-up over the course of the years. This speed-upis caused by three factors: (1) improvements in solver performances (the 2013 benchmarking casewas solved by Gurobi 5.5, the 2016 by Gurobi 6.5), (2) a more compact and tight formulation ofthe unit commitment problem, and (3) a more efficient implementation of the code in Matlab andGams.

(a) Total operational system cost.

(b) Run times.

Figure 1: Comparison of the LUSYM model performance, in terms of optimality of simulationresults and run times, with three commercial unit commitment models: UC-1, UC-2 and UC-3 (thenames of the commercial models are not disclosed for confidentiality reasons).

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3.2 Benchmarking with historical generation data

The simulation results of LUSYM are compared with historical generation data for the 2013 Eu-ropean electricity system (ENTSO-E, 2015). The benchmarking case considers the full ENTSO-Earea, consisting of 32 countries: Austria, Belgium, Bosnia-Herzegovina, Bulgaria, Croatia, CzechRepublic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Ireland, Italy, Latvia,Lithuania, Luxembourg, Macedonia, Montenegro, Netherlands, Norway, Poland, Portugal, Roma-nia, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland and the United Kingdom. The genera-tion portfolio contains 2,692 power plants and 23 pumped storage units. The network model consistsof 32 nodes and 56 lines, represented by a trade-based network model. A full year is consideredin daily blocks with an hourly time resolution. No reserve requirements are imposed. Load andrenewables can be curtailed at a cost of 3,000 EUR/MWh. The unit commitment model is solvedby Cplex with a 1% stopping tolerance.

(a) Nuclear generation (TWh/year).(b) Lignite generation (TWh/year).

(c) Coal generation (TWh/year). (d) Gas generation (TWh/year).

(e) Oil generation (TWh/year). (f) Renewable generation(TWh/year).

(g) Net pumped storage consump-tion (TWh/year).

(h) Net import (TWh/year).

Figure 2: The LUSYM model is able to reproduce historical generation data acceptably well (casestudy of the 2013 European electricity system). The black bars refer to historical data, the whitebars refer to simulation results.

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The model is calibrated in order to match simulation results with historically observed generation.Generation from nuclear power plants and lignite-fired power plants is reduced for some countriesby increasing the number of planned outages and decreasing the installed capacity.

Fig. 2 shows the simulation results and the historical observed generation per country for eachgeneration technology. Although it is not possible to match historical generation perfectly due toinherent simplifications of a deterministic unit commitment model, such as the lack of uncertainty,the LUSYM model is able to reproduce historical generation data acceptably well. The model takesabout 17 hours to run on a laptop (Intel(R) Core(TM) i7-2620M CPU@2.7 GHz with 8 GB RAM).

4 Summary

This paper presents LUSYM, a state-of-the-art mixed-integer linear program (MIP) of a determinis-tic security-constrained unit commitment model. The presented formulation is tight and compact,including power plant constraints, renewables curtailment, flexible load, renewables curtailment,storage units, transmission grid constraints and spinning reserve constraints. The model also in-cludes a planned outage scheduling module. The model is implemented in Gams and solved withthe latest release from Cplex or Gurobi MIP solver.

Benchmarking simulations show that the model is competitive with existing commercial unit com-mitment models, in terms of optimality and run times, and that the model is able to reproducehistorically observed generation data. The run time of LUSYM is limited by a tight and compactformulation, an efficient implementation and the use of best-in-class MIP solvers.

Acknowledgements

The authors would like to thank German Morales-Espana for his valuable remarks and comments.Any remaining errors are the sole responsibility of the authors.

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