Boris J. Pavez-Lazo2007

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A 4-state unit definition for the unit commitment problem

Boris J. Pavez-Lazo a,*, Claudio A. Roa-Sepulveda b

a Department of Electrical Engineering, University of La Frontera, Casilla 54-D, Temuco, Chileb Department of Electrical Engineering, University of Concepcion, Casilla 160-C, Concepcion, Chile

Received 28 August 2004; received in revised form 16 December 2004; accepted 23 March 2006

Abstract

This paper proposes a new state definition for the thermal generator units. Being different from the classic On/Off state definition, amodel that considers four possible states in order to commit a unit is proposed. The main difference in the classic UC is that in the on-state mode, the unit can be committed to a maximum or minimum power according to its technical generation limits, but it can be alsocommitted without giving power to the system in synchronism (banking state). Therefore, the unit commitment problem is formulated asa combinatorial optimisation problem with four variables. A more complete and necessary state definition is considered according to theelectric energy non-regulated markets characteristics and particularly because of the Chilean energy markets characteristics. The newUC problem formulation is then solved using the Simulated Annealing technique and several examples show the effectiveness and robust-ness of the proposed method. 2006 Elsevier Ltd. All rights reserved.

Keywords: Unit commitment; On/Off, 4-state; Simulated Annealing

1. Introduction

In the appropriate planning and operation of a modernElectric Power System (EPS) is absolutely important to sat-isfy the demand according to the new market scenarioswhile maintaining system security. No matter if the EPShas being privatised or not, it is featured by a great diver-sity of generators, turbines and boilers and each new unitadded to a system has got a better performance than theprevious ones. These facts turn the EPS operation into adifficult problem to solve due to its high complexity and

non-linearity thus requiring much more advanced tech-niques to get a better solution. If the old paradigm inpower systems was the full utilisation of electric energy toa minimum cost; nowadays, this paradigm has changedto a full utilisation of electric energy to a minimum costfor the user and a maximum profit for investors.

The UC problem has been defined using models rangingfrom the generalised ones [16, good review] to those con-sidering some of the market scenarios [79]. Although allthe proposed models try to be general, the number of pos-sible states of each unit remains low (mainly on or off),especially for thermal power plants (the handling of hydro-power unit is easier for this case). Nonetheless, electricmarkets designed through POOL criteria (where the powersuppliers are exposed to competition of electric power salesand have to aim at an economic operation to obtain max-imum profits) need a more complete unit definition, for

thermal power plants, in order to model them closer toactual operational procedures.When the power system model has been set, its solution

has mainly been approached through classical techniquesbased on methods such as Lagrangian Relaxation (LR),Linear Programming and Dynamic Programming [10].The most common used classical technique has been theLagrangian Relaxation approach [1115]. Since the electricmarket needs to have auditable solutions, the LR techniqueprovides a sound mathematical background to support thisneed. However, convergence problems could arise using

0142-0615/$ - see front matter 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijepes.2006.03.021

* Corresponding author. Tel.: +56 45 325537; fax: +56 45 325550.E-mail address: [email protected] (B.J. Pavez-Lazo).

www.elsevier.com/locate/ijepes

Electrical Power and Energy Systems 29 (2007) 2127
mailto:[email protected]:[email protected]


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this approach, since the unit state definition produces aninteger-variable problem that is hard to handle via LR.Stochastic techniques such as Montecarlo [16] have beenalso used to solve the UC problem, but the unit state def-

inition induces a dimensionality problem. Even with thebig advances to modify the above techniques to improveconvergence, there is still the problem of the limited num-ber of unit state definitions, which could meet the needsof this new market scenario.

Furthermore, the inclusion of restrictions such as powerflow, up/down minimum times and start-up/shut-downcosts has prompted the need of new solution methods thataccept more refined UC models. It is particularly acceptedthat UC is a combinatorial feature problem where the sizeof the search space depends on the number of time slot inthe study, on the definable state number for the different

generating units and on the number of generating units.The size of the solution set is a restriction to be consideredfor obtaining good results due to the high computing cost.Therefore, the search of new optimisation techniquesbecomes important and it has been centred in the use ofArtificial Intelligence (AI) based methods.

Most of the AI work reported in the literature is relatedto Genetic Algorithm (GA), Hopfield Neural Networks(HNN) and Taboo Search (TS) techniques. The mostreported area is GA [1727] that provides good resultsusing an On/Off unit state model (GA binary coding) andsimplified start/down time. Work needs still to be done toovercome the problem of considering a large number of

states (GA decimal coding) and a proper definition ofstart/down times. HNN [2830] provides a solution thatrelies on a Lyapunov type of energy function for assuringconvergence. However, the use of discrete variable (UnitState) provokes HNN convergence instability. AlthoughTS [31,32] provides a good solution for not being trappedinto local minima, the handling of the taboo list is cumber-some in systems with a large solution universe.

In view of the above characteristics and considering thecurrent requirements of electric markets, this paper pro-poses a 4-state generating unit definition for the UC formu-lation. This is different from the standard On/Off model

normally used, to the knowledge of the authors, in the

literature with the exception of [33,34], which describe thetechnical minimum state.

The 4-state generating unit definition produces a dimen-sionality problem that is tackled through the Simulated

Annealing (SA) technique as the optimisation algorithm.This is a technique that has shown advantages in manyworks with respect to classical optimisation techniques[3537,39].

Finally, in order to validate the model a 24-h unit com-mitment for a 5-unit real power system is obtained andcompared to both the results provided by an analyticalsolution and those from the classical On/Off unit model.To evaluate the robustness and the scalability of the algo-rithm we artificially generated test systems of larger sizefrom the 5-unit example.

2. Unit commitment general formulation

2.1. Objective function

The mathematical model used as an objective functionto obtain the unit commitment of thermal units is

FP; t XHh1

XNn1

Ch

nPh

n CSUh

n CSDn

1

Production cost: This term, (2), represents the n unitcosts in hour h as a function of the generated power.

Ch

nPh

n x2nP

h2

n x1nP

h

n x0n 2

Start-up cost: This cost is dependent of both the timeand the unit state when is needed.

CSUhn

b1n t FC b2n when banking

b1n1 eb3nt FC b2n when cooling

&3

Shut-down cost: The CSD values are generally consid-ered constant.

2.2. System constraints

System power balance:

XN

n1

Ph

n Dh 4

Nomenclature

H period under studyh 1,2, . . ., HN number of units

n 1, 2. . .

, NChnPh

n cost of unit n in hour h

CSUhn

unit n start-up costCSDn unit n shutdown costTSUin overall running time of unit nTSDin overall shutdown time of unit n

SUn minimum up time of unit nSDn minimum down time of unit nx0n, x1n, x2n generation cost function coefficients

b1n

, b2n

, b3n

start cost function coefficientsFC fuel costF overall coststn time unit n has been offRh spinning reserve in hour hDh demand of hour h

22 B.J. Pavez-Lazo, C.A. Roa-Sepulveda / Electrical Power and Energy Systems 29 (2007) 2127


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Demand and spinning reserve: The power supplied byall on-units must at least fulfil the demand plus the spin-ning reserve.

XNn1

Ph

nP D

h Rh 5

2.3. Unit constraints

Minimum up and down times: The total number ofhours for which unit n has been running must be biggeror equal to the minimum unit uptime.Xi

TSUin P SUn 6

Similarly, the total number of hours for which unit n hasbeen down must be bigger or equal to the minimum unitdowntime.

Xi

TSDin P SDn 7

Generator technical limits:

Ph

n;min 6 Ph

n6 P

h

n;max 8

Unit initial status: The initial status at the start of thescheduling period must be taken into account.

3. The 4-state model

3.1. Background

It is known that the UC is the problem with the mostdifficult solution that can be found in power system studiesfor its highly non-linear combinatorial characteristic andimposed restrictions. Basically, the UC searches from thetotal of available units of the generating system, the combi-nation of those units that can satisfy the need for energy atminimum cost. In order to reach this gold without consid-ering the solution method used, some common criteriarelated to the On/Off states definition can be summarisedas follow:

(a) The use of priorities, as a function of cost, where themost economic units are chosen to have a high prior-ity to be selected to satisfy the demand. Sadly, thecost calculation is made at maximum power, whichdiminishes the selection of a cheaper unit.

(b) The use of priorities according to the maximumpower. This is related to the definition of Base Unitsto satisfy the demand. This means that its moreimportant the highest power units in order to remainin an on state during the whole planning period. Thiscriterion is not as economical as required when biggerand more expensive power units are required.

(c) The use of priorities according to the minimum up/down times. The restrictions given by Eqs. (6) and

(7) are an important problem during peak or valley

demand periods due to larger up/down times. Inorder to avoid a lack or excess of generation becauseof units disconnection or connection, it is alwaysimportant to maintain units in an on state those hav-ing the higher up/down times. This makes necessaryto maintain slow-acting units operative. This situa-

tion could worsen systems security.(d) The start cost. Given the thermal systems character-istics, it is always cheaper to start a generator unitwhen the thermal systems are still in an operationaltemperature. During the planning period, it may hap-pen that it is decided to turn off a unit that is going tobe needed again and this point will have a cold startcost. One way of getting economical solutions is toavoid a cold start by starting such unit at an earliertime. This criterion does not allow cheaper unitscheduling.

3.2. 4-State model

In the On/Off model, the UC problem is formulated as acombinatorial optimisation problem with variables 0 and 1representing the On/Off states. Considering the above, the4-state model must be formulated as an optimisation prob-lem using the variables 0, 1, 2 and 3 which represent the off,banking, minimum and maximum states (Table 1).

In the 4-state model, the banking, minimum and maxi-mum statuses consider that the unit is operative. Therefore,in any of these three statuses, an up time is considered.

3.3. 4-State model justification

The strong competition between Independent PowerProducers (IPP) to be committed in Chilean NorthernInterconnected System (SING) electric market hasprompted the need for a more elaborated model of theThermal UC. Therefore, two states have been included:minimum and banking.

The banking state was necessary because there are twoequal Combined Cycle Gas Turbine (CCGT) units withdifferent owners having the same penalty factors. For thesake of clarity, banking means that the unit is not syn-chronised but it has all its thermal properties ready to startproducing power. The banking cost is associated to the costof keeping the thermal unit boiler at a temperature suchthat this can be committed at any moment.

Table 1Combinatorial variables definitions

Model Variables Unit status

On/Off 0 Off (Pg = 0)1 On (Pg = Pmax)

4-State 0 Off (Pg = 0)1 Banking (Pg = 0)2 Minimum (Pg = Pmin)

3 Maximum (Pg = Pmax)

B.J. Pavez-Lazo, C.A. Roa-Sepulveda / Electrical Power and Energy Systems 29 (2007) 2127 23


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In order to show the advantages of the proposed 4-statedefinition, the results for the 5-unit system are shown. Forthe sake of a proper economical decision, an hourly ELDbased on the classical lambda-iteration method is only per-formed for the UC optimal solution for both On/Off and 4-state models.

Table 2 shows the Priority List (PL) from a lesser to abigger generation unit cost calculated according to themaximum power output. According to these results, theUC and the Economic Load Dispatch (ELD) calculation

using the On/Off model are done for a load of 705 MWand 750 MW as shown in Tables 3 and 4 for hour 1 and8, respectively.

Table 3 also shows the analytical solution (optimumsolution which is obtained in a manual manner reviewingthe 1023 possible combinations) of the UC and the ELDby means of the 4-state generator model.

Table 3 shows that the most economical solution (bothUC and ELD) corresponds to the one obtained via the 4-state model. This is achieved because it is possible to con-sider units 1 and 4 at minimum generation level.

Table 4 shows a different situation. At this time, the 4-state model induces to the decision of leaving unit 5 inbanking and unit 1 and 3 in off state. Units 2 and 4 gener-ate 750 MW that exactly matches the demand thus givingan UC cost lower than the UC cost using the On/Offmodel. However, when the ELD solution is considered,the total production cost highlights the On/Off modelbecause it gives more alternatives to fulfil the demand.

Table 5 shows the overall 24-h solution of the UC forboth models. From these results, it can be concluded thatit is always possible to obtain a lower overall generatingcost by using the 4-state model. Even though the On/Offmodel could provide a more economical solution for a par-ticular hour.

4. The Simulated Annealing

The Simulated Annealing (SA) technique was proposedby Kirkpatrick, Gelatt and Vecchi in 1983 [38] as a newtechnique to obtain near-to-optimum solutions to optimi-

sation problems. SA has been tested in several optimisationproblems showing great ability for not being trapped inlocal minima.

The SA strategy starts with a high temperature givinga high probability to accept non-improving movements.The temperature and probability levels diminish as longas the algorithm advances to the optimal solution. In thisway, a diversification procedure in the search algorithm isperformed with care in the system energy.

The main key to obtain good solutions in the usage ofSA is the cooling criterion. Questions such as what shouldthe initial temperature be? And what should the coolingprocedure be? are of paramount importance for the good

use of SA. There are several works in literature to answerthose questions, but in general SA gives acceptable solu-tions when the initial temperature is highly associated toa slow cooling procedure. The most important SA param-eters required to solve any optimisation problem are thefollowing:

The number of iterations at a constant temperature(Mo). A low number of Mo will result in being trappedin local minimum.

Cooling strategy (qo). If the annealing temperature isdecreased too fast, the algorithm will be trapped in local

minimum regardless of the proper T and Mo tuning.

Table 2Priority List, 5-unit system

Unit [$/MW] Pmax [MW]

3 5.2 501 8.87 1002 9.53 3005 12.91 275

4 13.09 450

Table 3Comparison between On/Off and 4-state models

Unit Initialconditions

PL solution On/Offmodel

Initialconditions

Analytic solution4-state model

UC ELD UC ELD

Pi [MW] Pi [MW] Pi [MW] Pi [MW]

3 3 50 50 3 0 01 2 100 100 2 25 892 1 300 297.69 1 300 259.925 3 0 0 3 275 137.384 5 450 257.31 5 112.5 218.69

Total [MW] 900 705 712.5 705

Total cost [$] 9896.3 6927.8 8119.8 6905.0

Hour 1.

Table 4

Comparison between On/Off and 4-state modelsUnit Initial

conditionsPL solution On/Offmodel

Initialconditions

Analytic solution4-state model

UC ELD UC ELD

Pi [MW] Pi [MW] Pi [MW] Pi [MW]

3 7 50 50 3 0 01 7 100 100 1 0 02 8 300 300 8 300 3005 10 0 0 7 0 04 12 450 300 12 450 450

Total [MW] 900 750 750 750

Total cost [$] 9896.3 7521.9 9106.3 9106.3

Hour 8.

Table 5Operating cost ($)

On/Off model 4-State model

Best Worst Average Best Worst Average

216284.8 216419.9 216355.1 197700.2 200495.3 198775.5



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The above two annealing parameters should be setthrough sensitivity analyses. These analyses need to bedone in every optimisation problem.

5. Numerical results

In order to validate the model and the proposedmethod, an application to the Chilean Northern Intercon-nected System (SING), a 5-unit system (Tables 79), is pre-sented. Although this system seems to be small (just 5 unitsfulfil the demand), its importance for the Chilean economyis high (10% GDP and 40% exports) and it feeds the copperindustry that produces just about 30% of the world-widemine copper. This fact makes the electric market in thisarea very competitive, thus resulting in new questions suchthe one address in this paper: the unit commitment withmore states than the classical On/Off unit model.

The results obtained via the proposed method are com-pared to the optimal solution manually found(analytic solu-tion) for a 24-h demand curve (out of 1023 UC combinationsper hour) and the classical On/Off unit model. The analyticalsolution is found when searching for the best UC combina-tion that gives the smallest cost per hour (Chilean GridCode).

For all test systems, the simulation conditions and thecooling sequence for SA that we have used an initial temper-ature value of 100, a number of iterations at constant tem-perature (Mo) is set to 10 and a cooling factor (qo) to 0.90.

The economic load dispatch calculation, which uses theclassical lambda-iteration method, is performed only forfeasible solutions.

The results obtained via the proposed method are com-pared to the classical On/Off unit model (see Table 5). Forthe On/Off and 4-state models it consider 20 times run thealgorithm. Although CPU time is not the main concern atthis stage of the research, the algorithm was coded inMATLAB 5.2 in a Pentium IV 2.4 GHz. The averageCPU time was 24.4 s.

To evaluate the scalability of the algorithm, larger sys-tems were generated from the 5-unit test system. For the10-unit test system, the 5-unit test systems were duplicatedand the demand was multiplied by 2. Similarly, other testsystems are generated. The results obtained are shows in

Table 6.In order to consider the algorithm scalability, we shouldnotice that this model considers four states. Therefore, the

maximum number of possible combinations NC given byEq. (9) for a period of 24 h (H) according to the units num-ber N.

NC NSN 1H

9

For example and considering the 5-unit systems, Eq. (9)gives 1.73 1072 possible combinations for a 24-h period.This value is comparable to On/Off model with 10 units.Over 20 units, it is impossible to graphically show the max-imum number of combinations. Therefore, Fig. 1 showsthe magnitude order of combinations for On/Off and 4-state models.

From Eq. (9) and Fig. 1, it can be deduced that the num-ber of possible solutions grows exponentially. It could bealso assumed that the computing time would grow simi-larly. However, Fig. 2 shows that with a 4-state model,the CPU time increases linearly with respect to the numberof units. This curve allows us to demonstrate that the

Table 6Scalability

Numberof units

Worst cost Best cost Average cost Averagetime (min)

10 397368.1 396162.5 396759.0 0.644920 793818.4 787482.6 789583.1 1.194340 1588402.2 1580246.6 1584061.4 2.566860 2383287.8 2376103.8 2379531.9 3.736280 3177075.9 3170540.0 3174518.9 5.0979

100 3982108.5 3965846.1 3971992.3 6.8668

0 10 20 30 40 50 60 70 80 90 100

0

500

1000

1500

Units

On/Off

4-State

Fig. 1. Magnitude order of NC.

10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

CPU-Time(Minutes)

Units

Fig. 2. CPU-time.



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proposed model has a good searching engine despite thehuge search space.

6. Conclusions

The characteristic of the thermal units is that they can

undergo gradual changes of temperature, which is trans-lated in periods of time of several hours necessary to bringthe unit on-line or off-line. In this sense, it is possible todefine more operational states.

The practice for the Northern Chilean Power System(SING), an unit with spinning reserve operates at a levelbelow its maximum power output and in some occasionsthe fuel specific consumption slightly increases with inter-mediate loading due to not being operating in its optimalefficiency point. The difference of fuel specific consumptionwith respect to the maximum power output is obtained bySystems Operators to calculate unit real operational costs.The definition proposed in this paper allows a thermal unit

to be committed at minimum power output when the ther-mal systems are at operating temperature.

On the other hand, the definition of the banking stateincorporates a real action that power systems dispatchersmake when they need that a particular generating maintainits synchronism without producing power for economicalreasons. This type of actions is not considered in theOn/Off model.

Finally, the use of the SA as an optimisation algorithmpermits, without any implementation difficulty, to solve theUC in its complete formulation giving excellent results andhaving low simulation times. With the use of SA, lineal

approximations of the UC formulation are not required.

Acknowledgements

Boris J. Pavez-Lazo gratefully acknowledges the finan-cial support given by the Graduate School of Universityof Concepcion and by the Universidad de La Fronteravia the research project DIDUFRO-INI-110305.

Appendix A. Characteristics and parameters of the 5-unit

system (Tables 79)

The positive sign in column in initial conditions ofTable7 indicates that the unit is operative, whereas the negativesign indicates otherwise.

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b1 b2 b3 a b x2 x1 x0

1 5.4 55 20.588 0.20 8.5 25.988 0.0465 3.333 88.70

2 3.2 65 20.594 0.20 10.9 2 3.794 0.0138 4.437 286.003 1.7 74 10.650 0.18 10.8 12.350 0.0292 3.220 26.004 12.0 76 18.639 0 .18 1 2.2 30.639 0 .0135 5 .706 5 89.055 4.8 101 34.749 0.09 8.2 39.549 0.0346 2.104 355.03

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10 859 22 87011 870 23 80012 875 24 742



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Boris J. Pavez-Lazo2007