Solving the Unit Commitment Problem in Power

Generation by Primal and Dual Methods�

D� Dentcheva�� R� Gollmer�� A� M�oller��

W� R�omisch� and R� Schultz�

Abstract� The unit commitment problem in power plant operation planning is addressed�

For a real power system comprising coal� and gas��red thermal and pumped�storage hydro

plants a large�scale mixed integer optimization model for unit commitment is developed�

Then primal and dual approaches to solving the optimization problem are presented and

results of test runs are reported�

� Introduction

The unit commitment problem in electricity production deals with the fuel cost opti�mal scheduling of on�o� decisions and output levels for generating units in a powersystem over a certain time horizon� The problem typically involves technologicaland economic constraints� Depending on the shares of nuclear� conventional ther�mal� hydro and pumped�storage hydro power in the underlying generation systemfairly di�erent cost functions and side conditions arise in mathematical models forunit commitment� In the present paper we consider a power system comprising coal�and gas��red thermal units and pumped�storage hydro plants� This re�ects the en�ergy situation in the eastern part of Germany� Our work grew out of a cooperationwith the power company VEAG Vereinigte Energiewerke AG Berlin�In our unit commitment model� the objective function is given by start�up andoperation costs of the thermal units� Pumped storage plants do not cause directfuel costs� Their operation� nevertheless� has an impact on the total fuel costs inthe system� Constraints of the unit commitment model comprise output bounds forthe units of the generation system� load coverage over the whole time horizon aswell as provision for a spinning reserve� minimum down times for thermal units andwater balances in the pumped�storage plants� Typical optimization horizons varyfrom several days up to several months�From the extensive literature in unit commitment we mention here ��� �� ��� ����� � ���� and ���� and refer to the comprehensive literature synopsis ���� Thepapers ��� ���� ��� re�ect recent developments of e�cient algorithms based onmodern mathematical techniques�

�This research is supported by a grant of the German Federal Ministry of Education� Science�Research and Technology �BMBF��

�Humboldt�Universit�at Berlin� Institut f�ur Mathematik� ��� Berlin�Konrad�Zuse�Zentrum f�ur Informationstechnik Berlin� Heilbronner Strae ��� ����� Berlin

�

� Modelling

Our mathematical model for the unit commitment problem is a mixed�integer op�timization problem with linear constraints� In the following sections we will tacklethis problem with both primal and dual solution methods� which will lead to di�er�ent model speci�cations and variants� Here� we describe features of the model thatare common to both situations�The optimization horizon is partitioned into �hourly� half�hourly or shorter� timeintervals� whose total number is denoted by T � By I and J we denote the numberof thermal and pumped�storage hydro units in the system� respectively� Then thefollowing variables occuruti � f�� �g � on�o� decisions for the thermal unit i � f�� � � � � Ig

and time interval t � f�� � � � � Tg�pti � output level for the thermal unit i during interval t�stj� w

tj � level for generation and pumping� respectively� for

the pumped�storage plant j � f�� � � � � Jg duringinterval t�

ltj � water level� in terms of energy� in the upper reservoirof plant j � f�� � � � � Jg at the end of interval t�

The objective function reads

F �u�p� ��TXt��

IXi��

FCi�pti�u

ti� �

TXt��

IXi��

SCi�ui�t��� � ���

Here� FCi denotes the fuel cost function for the operation of the i�th thermal unit�With respect to pt

i the function is monotonically increasing and often assumed tobe convex �linear� piecewise linear� quadratic�� Non�convex setups for fuel costs arenot considered in the present paper �for their discussion we refer to ����� Start�up

costs SCi�ui�t�� � SCi�uti� � � � �u

tsii � of the i�th thermal unit are determined by

the preceding down�time t� tsi and will be further speci�ed later on�The constraints of the unit commitment model are formulated as linear equationsand inequalities� The same feasible set can be described by nonlinear expressions�too� Here� we prefer the linear variant in order to apply methodology from mixed�integer linear programming if also the objective is given by linear terms�All variables mentioned above have �nite lower and upper bounds which re�ect thebounded output of all units in the generation system�

pminit ut

i � pti � pmax

it uti� i � �� � � � � I� t � �� � � � � T � � �

� � stj � smaxjt � j � �� � � � � J� t � �� � � � � T � ��

� � wtj � wmax

jt � j � �� � � � � J� t � �� � � � � T � ���

The constants pminit � pmax

it � smaxjt � wmax

jt denote the minimal and maximal outputs�respectively�Load coverage for each time interval t of the optimization horizon leads to theequations

IXi��

pti �

JXj��

�stj �wtj� � Dt� t � �� � � � � T� � ���

where Dt denotes the load to be covered in time interval t� Sudden load increasesor unforeseen conditions �e�g� outage of a unit� have to be compensated on�line�Therefore� for each time interval t� some spinning reserve Rt in the termal units isrequired which leads to the constraints

IXi��

�pmaxit ut

i � pti� � Rt� t � �� � � � � T� � ���

During the whole optimization horizon water balances in the pumped�storage plantshave to be maintained� It is typical of the power system of VEAG that no additionalin� or out�ows arise in the upper reservoirs of the pumped�storage plants such thatthese operate with a constant amount of water� The possible workload for turbinesand pumps is restricted by the water levels and capacities of the upper reservoirs�This is modelled by the inequalities

� � ltj � lmaxj � j � �� � � � � J� t � �� � � � � T � ���

where ltj is connected with the variables stj and wtj by the reservoir constraints

ltj � lt��j � stj � �tjwtj� t � �� � � � � T�

l�j � linij � lTj � lendj � j � �� � � � � J� ���

Here linij � lmaxj and lendj denote the initial� maximal and terminal water levels in the

upper reservoir� and �j is the e�ciency of the j�th pumped�storage plant�Finally� there are minimal down times �i for the thermal units that are mainlydetermined by technological reasons and serve to avoid erosion of the unit by toofrequent changes of thermal stress�

ut��i � ut

i � �� uli� l � t � �� � � � �minft � �i � �� Tg� i � �� � � � � I � ���

t � � � � � � T � �i � ��

� Primal Methods

Our primal approach to the unit commitment problem relies on solving the under�lying large�scale mixed�integer optimization problem by adapted branch�and�boundtechniques� possibly enriched by cutting planes derived from the convex hull of fea�sible points� In order to apply the corresponding methodology and implementationswe formulate the basic model from Section as a mixed�integer linear program�More precisely� we assume that� for each thermal unit i � f�� � � � � Ig� the fuel costsare a�ne�linear functions of the generated power� and the start�up costs are givenby

SCi�ui�t�� � Aimaxfuti � ut��

i � �g� t � � � � � � T ����

where Ai is some positive constant� This piecewise linear convex function can beexpressed in linear terms by introducing an additional variable and adding two linearinequalities to the constraints�The setups for fuel as well as start�up costs are the simplest possible one could thinkof� On the other hand� the real�life data material from VEAG� by means of which we

have validated our models� indicates that both model simpli�cations are tolerable�Staying within the framework of a linear model it is possible to re�ne the linearfuel costs by a piecewise linear setup and to take into account that startup costsvary with the preceding down�time of the unit� The tradeo� for both extensions isa growing number of variables and constraints in the programs to be solved�It is a typical feature of the generation system of VEAG that there are thermalunits which are absolutely identical� Therefore� obviously� a model reformulation ispossible� treating each group of identical blocks by only one status variable ut

i andone output variable pt

i� The status variable now is not a Boolean one but with thenumber of identical blocks being N we would have ut

i � f�� � � � � Ng� indicating thenumber of blocks in on�state� The generalized lower and upper bounds � � � for theoutput pt

i of the group of units remain valid and so does the formulation of start�upcosts�In our test runs we have tried both models� The �rst model corresponds to thepure Boolean setting� the second one to the modi�cation described in the previousparagraph� Though the reduction of the number of integer variables is only bya factor of ��� we observe a considerable reduction of the branch�and�bound treeand the computing times� From this point of view the latter model is superior�On the other hand� the development towards a branch�and�cut algorithm on thebasis of valid inequalities developed in ��� relies on a Boolean structure in theform of a knapsack problem and thus could be carried on for the �rst model only�Further investigation is necessary to decide whether the possible improvement ofbounds by the cuts or the reduction of the number of integer variables by the modelreformulation are preferable�The test runs proved the successful applicability of the primal approach to real�lifemodels of the type outlined above� Our program was developed on the basis ofmodules from the CPLEX Callable Library ��� The details of the runs are given inTable �� We used three parks of generating units� re�ecting three di�erent statesof development� Park � consists of � thermal units and � pumped�storage hydroplants� park has two additional coal��red units and a modi�ed pumped�storageunit� Park comprises another three additional coal��red units and one additionalpumped�storage unit� Computations were done for three selected weeks �� days�with ��hour�intervals �T � �� � on a HP�Apollo ���� ��

variant park � park park week model time quality time quality time quality

holiday � ��� ����� � ��� ���� � ���� ��� �week �� ���� � ��� ���� � �� � ����� �low � ���� ���� � ��� � ���� � ���� �� �� �load ��� ��� � � ���� ���� � ���� �� � �peak � ���� ��� �� ����� � ���� ����� �load ���� ��� ��� ����� � ��� �� �� �

�� No feasible solution exists�

Table �� CPU�time in minutes �HP�Apollo ������ and upper bound

for the deviation of the objective value from the optimum

�

The �gure shows the load� the thermal generation and the use of pumped�storageunits for park in the peak load week�

loadthermal generation

hydro generationhydro pumping

Fig� �� schedule for park in the peak load week

� Dual approach

The dual approach to the unit commitment problem has been widely studied inthe last �� years �see ����� In ��� the authors have addressed unit commitmentwith Lagrangian relaxation for the �rst time� The main idea is to incorporate theloosely coupling constraints linking operation of di�erent units into the objectivefunction by use of Lagrange multipliers� Then the problem decomposes into smallersubproblems� The solution of the relaxed problem provides a lower bound on theoptimal solution of the original problem� The value of the lower bound is a functionof the Lagrangian multipliers� This approach has found a wide application for largesystems� for two reasons� It works fast due to the decomposition of the dual probleminto essentially smaller subproblems� On the other hand� it has been proved in � ��that the duality gap� which occurs by the presence of integrality� becomes small fora large number of units�From now on we assume that the fuel costs are a piecewise linear function of thegenerated power and the start�up costs are given by �����

�

Let us associate the multipliers �� � � IRT with the demand and reserve�constraintrespectively and consider the Lagrange function�

L�p�u� s�w����� �TXt��

IXi��

FCi�pti�u

ti� �

TXt��

IXi��

SCi�ui�t�� �����

�TXt��

h�t�Dt �

IXi��

pti �

JXj��

�stj �wtj���

�t�Rt �

IXi��

�utip

maxi � pt

i�i�

which has to be minimized subject to the constraints � � � � � ��� and � ��� � � ����We obtain the problem�

P ����� � minL�p�u� s�w�����subject to � � � � � ��� and � ��� � � ����

��� �

Denoting the marginal function of P ����� by d����� the dual problem reads�

maxfd����� � ��� � IRT �� � �g� ����

By the separability structure of L�p�u� s�w����� and the constraints with respectto the units� we can decompose the problem P ����� into problems Pi����� and�Pj����� as follows�

Pi����� � minui

TPt��

�minpt

i

fFCi�pti�u

ti�� �tpt

ig� SCi�uti�� �tut

ipmaxi

subject to � � � and � ���

�����

for the coal��red and gas�burning units �i � �� � � � � I� and

�Pj����� � min�sj �wj�

TPt��

��t � �t��wtj � stj�

subject to � ��� � ���� � ��� and � ���

�����

for the pumped hydro storage plants �j � �� � � � � J�� Denoting the marginal functionsof the problems above by di����� and �dj����� respectively� we obtain for d������

d����� �IX

i��

di����� �JX

j��

�dj����� �TXt��

��tDt � �tRt �����

Useful properties of d are its separability structure� concavity and the explicitformulas for computing subgradients� Setting d� �� max

�����d����� and f� ��

min�u�p� F �u�p� it is known by the weak duality theorem that d� � f�� The rela�tive duality gap �f� � d���d� converges to zero as I ��� � ��In order to solve the dual problem ���� e�ciently� a fast non�smooth optimizationmethod� a good initial guess for ������ e�cient algorithms for solving the subprob�lems ����� and ����� and a proper heuristics for computing a primal feasible solutionare needed� We gave our preference to the bundle method described in �����

�

The implemented algorithm works as follows�

� Initial guess for the Lagrange multipliers �� �� Based on a priority list theon�o� decisions are taken to satisfy the demand and reserve constraint foreach time interval� Then the relative production costs of the on�line units areused to initialize �� � is set to be zero�

� Iterative procedure of the bundle method

� Computation of a value and of subgradients of d����� by solving I � Jsubproblems of dimension T � The �rst I problems are solved as fol�lows� The minimization with respect to pt

i is done explicitly or by one�dimensional optimization� The minimization with respect to ui is equiv�alent to the search for a shortest path in the state transition graph of theunit under consideration� and it is carried out by dynamic programming�Nodes passed during the minimal down time are not included in thestate transition graph �cf ����� The next J hydro�storage subproblemsare solved by the algorithm developed in ����

� Bundle�method iteration�

� Determination of a primal feasible solution� The algorithm works in two stepsto satisfy the �possibly violated� reserve constraints� First� we try to satisfy theconstraints by using the pumped�storage hydro plants in those time intervals�where the largest values of Dt � Rt occur� If the reserve�constraints are stillviolated� we modify the schedules of the thermal units by the procedure in���� The idea consists in �nding a period t for which the reserve constraintis violated most and then computing the smallest amount of increase ��t tosatisfy the reserve constraint in this period� This procedure is carried outrecursively until the constraints are satis�ed for all periods�

� Economic dispatch� In a last step we improve the feasible solution found in theprevious step� We solve the primal problem keeping the integer variables �xed�The latter linear optimization problem� referred to as economic dispatch� issolved using the CPLEX Callable Library ���

The encouraging results of the test runs proved the e�ciency of the dual approach�Computations based on the same data �model �� as in the previous section providedthe numerical results reported in Table �

NOA Optimality park � park park tolerance� ���� time gap time gap time gap

holiday week ��� ���� � ���� ��� � ��� ��� �low load week ���� ���� � ��� �� � ��� ���� �

peak load week ���� ��� �� ���� � ��� ��� ��� No primal solution exists�

Table � CPU�time in minutes �HP�Apollo ������ and

upper bound of the relative duality gap

�

LoadThermal generation

PHSP: pumpingPHSP: generation

Fig� � Schedule for park in the holiday week�

Acknowledgement� We wish to thank P� Reeh� G� Schwarzbach and J� Thomas �VEAG

Vereinigte Energiewerke AG� for the outstanding collaboration which made the present

paper possible�

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�