non-convex power plant modelling in energy optimisation

European Journal of Operational Research 171 (2006) 1113–1126

www.elsevier.com/locate/ejor

Non-convex power plant modelling in energy optimisation

Simo Makkonen a,*, Risto Lahdelma b

a Process Vision Ltd, Melkonkatu 18, FIN-00210 Helsinki, Finlandb University of Turku, Department of Information Technology, Lemminkaisenkatu 14 A, FIN-20520 Turku, Finland

Available online 10 March 2005

Abstract

The European electricity market has been deregulated recently. This means that energy companies must optimisepower generation considering the rapidly fluctuating price on the spot market. Optimisation has also become more dif-ficult. New production technologies, such as gas turbines (GT), combined heat and power generation (CHP), and com-bined steam and gas cycles (CSG) require non-convex models. Risk analysis through stochastic simulation requiressolving a large number of models rapidly. These factors have created a need for more versatile and efficient deci-sion-support tools for energy companies.

We formulate the decision-problem of a power company as a large mixed integer programming (MIP) model. Tomake the model manageable we compose the model hierarchically from modular components. To speed up the optimi-sation procedure, we decompose the problem into hourly sub-problems, and develop a customised Branch-and-Boundalgorithm for solving the sub-problems efficiently. We demonstrate the use of the model with a real-life application.� 2005 Elsevier B.V. All rights reserved.

Keywords: Energy management; Deregulation; Energy market; Optimisation; Mixed integer programming

1. Introduction

The recent deregulation of the European elec-tricity market has involved the unbundling of thepower generation, transmission, trade and retailbusinesses, and in some cases splitting very largeutilities into smaller competing companies. Sepa-

0377-2217/$ - see front matter � 2005 Elsevier B.V. All rights reservdoi:10.1016/j.ejor.2005.01.020

* Corresponding author. Tel.: +358 9 25320 300; fax: +358 925320 360.

E-mail addresses: [email protected] (S.Makkonen), [email protected] (R. Lahdelma).

ration of monopolistic network operations fromthe other (competitive) businesses is particularlyimportant, because this prevents unfair competi-tion through cross-subsidisation. These changeshave created a highly volatile power market, wherecompetition is intense. Consequently, several newinstruments for trading and risk hedging have ap-peared on the market. In particular, the traditionalphysical long-term contracts have been replacedby contract portfolios consisting of a large numberof short-term contracts. This means that the pro-duction and trade must be optimised accurately

ed.

mailto:[email protected]

mailto:[email protected]

1114 S. Makkonen, R. Lahdelma / European Journal of Operational Research 171 (2006) 1113–1126

enough to meet the spot market variations, and itis also necessary to repeat the optimisation daily,hourly or ever more frequently in order to reactto the rapidly changing situations.

The growing emphasis on environmental as-pects is also changing the face of the energy indus-try. For example the European energy industrymust consider the already launched green certifi-cates and be prepared to manage emissions tradingby year 2005. In some cases emissions trading willinfluence energy production and production plan-ning dramatically. There is a pressure to move intomore advanced and environmentally friendlypower generation technologies, and also to switchto less harmful primary energy sources.

A similar deregulation development is takingplace on the gas market. Gas is an increasinglypopular alternative fuel at power plants due toits environmental friendliness in relation to otherfossil fuels. Gas is also, for some end-users, a di-rect substitute for electricity. Thus, variations inthe availability and price of gas affect both the pro-duction of, and demand for, electricity.

Fig. 1 illustrates the new business environmentin energy planning. Due to the recent changes, en-ergy companies must boost their production andretail businesses to survive under the stronger

PowPlan

Productionplan

Electricity

Emissio

Variable

Variable CO2 a

electricity price

Technicalconstraints

Fig. 1. The business environm

competition on the volatile power market. Toadapt to the new business environment, energycompanies need financial and human resourcesand new kinds of decision support tools. Electric-ity traders need tools for quantifying the risksand determining the use of different risk hedginginstruments. The retailers must manage customerportfolios and dynamic pricing of contracts onthe market. The producers must optimise theiroperations subject to volatile prices for electricityand fuel (especially gas), and they must managethe new challenges associated with CO2 allowancesand emission trading. This means that on a dereg-ulated market, optimisation of the power genera-tion should consider both the spot and forwardprice of electricity and the CO2 allowances giventhe physical, technical and power reserve con-straints of power generation today.

The new decision-support tools must be moreversatile, accurate, and efficient than the presentgeneration. Versatility is needed in that it mustbe easy to maintain the model and keep the modelparameters up to date. Accurate modelling ofnew production technologies, such as gas tur-bines (GT), combined heat and power genera-tion (CHP), and combined steam and gas cycles(CSG) may require non-convex models, which

erts

Fuelmixture

Heat / Steam

n

Variablefuel price

Variable load

llowance price

ent in energy planning.

S. Makkonen, R. Lahdelma / European Journal of Operational Research 171 (2006) 1113–1126 1115

makes the optimisation problems much more diffi-cult than before. Optimisation must be faster thanbefore for two reasons. Firstly, rapid re-optimisa-tion is required when the situation on the marketchanges. Secondly, advanced computations, suchas risk analysis through stochastic simulation, re-quires the solving of a large number of models rap-idly (Makkonen and Lahdelma, 1998; Spinney andWatkins, 1996).

In this paper, we focus on the production andtrade planning problem of a power company withplants that require non-convex modelling. We for-mulate the problem as a large mixed integer pro-gramming (MIP) model. To make the modelmanageable we compose the model hierarchicallyfrom modular components. We present a genericmethod of encoding different kinds of non-convexpower plants in the model. To speed up the optimi-sation procedure, we decompose the problem intohourly sub-problems and develop a customisedBranch-and-Bound algorithm for solving the sub-problems efficiently. We demonstrate the use ofthe model with a real-life application.

The non-convex energy optimisation modelintroduced in this paper is an extension of theElectricity and Heat Trade Optimisation (EHTO)system first introduced by Lahdelma and Makko-nen (1996). EHTO is embedded in the commercialGENERIS energy information and decision sup-port system, where the overall energy model iscomposed and configured hierarchically fromgraphical model components. The basic function-ality of EHTO is to optimise the combined heatand power acquisition for a given time horizonbased on forecasts for sales volumes and spotmarket prices. The embedded Power Simplex opti-misation code is used to solve the models(Lahdelma and Hakonen, 2003). EHTO has beenused for analysing the risks of electricity trade onthe liberated market through stochastic simulation(Makkonen and Lahdelma, 1998), for analysinghow various old and new types of electricity con-tracts can be used in cooperation (Makkonenand Lahdelma, 2001), and for multicriteria deci-sion support on the deregulated energy market(Makkonen et al., 2003). GENERIS has beendeveloped by Process Vision Ltd, and it is cur-rently in use at more than 50 utilities in seven

European countries. The non-convex optimisationmodel introduced in this paper has been installedat one medium size power company in Finland.

This paper is organized as follows. In Section 2,we present the GENERIS energy information sys-tem. In Section 3, we present how convex powerplant models are treated in EHTO. In Section 4,we extend the EHTO model to handle non-convexpower plant models and show how the model issolved by using an efficient tailored Branch-and-Bound algorithm. In Section 5, we demonstratethe use of the model in a real-life case and analysethe system performance.

2. The GENERIS energy information system

2.1. Overall system architecture

GENERIS is a commercial energy informationand decision support system for different kinds ofparticipants in the deregulated electricity market:grid operators, electricity traders, retailers, net-work operators and producers. The system in-cludes a configurable set of tools for the differentparticipants. In the case of a producer or trader,the application set contains an on-line measure-ment data management system, a trade supportapplication, a contract and portfolio managementapplication, a forecasting model for energy de-mand, monitoring and reporting applications,business-to-business and in-house communicationmodules (EDIEL, ETSO ESS, etc.) and the EHTOoptimisation application.

A difficult practical problem in a real-life deci-sion support system is to guarantee the integrityand availability of reliable data. In GENERIS,this problem is solved through an overall energydata model that connects the different tools to-gether and provides centralised data validationand management services. This means that the dif-ferent applications, such as the EHTO Optimiser,can concentrate on their specific tasks.

2.2. The EHTO optimiser

The EHTO Optimiser is suitable for a widerange of different optimisation and planning tasks


starting from operative short-term optimisationin real-time and extending to strategic long-termplanning and risk analysis models based on sce-nario techniques and stochastic simulation. TheEHTO Optimiser can be used for planning theoptimal energy acquisitions and sales for a givenplanning horizon from 1 hour up to several years.

The basic functionality of EHTO is to optimisethe combined heat and power acquisition for agiven time horizon based on forecasts for salesvolumes and spot market prices. The modelprovides the optimal production and trading planfor electricity and heat. The objective function is tomaximize the difference between sales income andenergy procurement costs. EHTO can handle bothfixed sales based on heat and electricity load fore-casts and open sales to the market based on priceforecasts. The procurement costs include purchasecosts and production costs. The production costsmay depend on fuel prices, electricity spot marketprice, outdoor temperature, and many otherparameters. Several generation technologiesincluding industrial power plants, co-generation

Fig. 2. Graphical configuration of the GENERIS EHTO energy manare light, and CHP components are two-coloured. The model contain(rectangles), contract nodes (circles) and demand nodes (triangles).

plants and combined district heat productionplants are supported. The power plants can alsohave multi-fuel options, including fuels whoseprice varies hourly.

The overall energy optimisation model is con-figured graphically from predefined model compo-nents (Fig. 2). The system can contain three kindsof supply and demand components: power compo-nents, heat components and CHP components.According to its type, each supply or demand com-ponent is connected through a contract to a powerbalance component, a heat balance component, orboth. The supply side components include differentkinds of short and long-term purchase contracts,and various production facilities, such as hydropower plants, condensing power plants, districtheating stations, and different types of CHPplants. The demand side components include var-ious kinds of open and fixed sales contracts withdifferent types of customers or customer groups.There is also a model component for the spot mar-ket, which acts simultaneously as a supply anddemand component.

agement model. Power components are dark, heat componentss five production plants (squares), a heat and a power balance


3. The energy management model

3.1. The overall energy management model

The EHTO system generates the overall energymanagement model automatically based on thegraphical configuration (Fig. 2) and compo-nent-specific settings. The model is composed ofsubsystems u 2 U, which correspond to the vari-able supply and demand side components. Themodel is specified for a sequence of hours t 2 T

that form the planning horizon. The resultingmodel is formulated as

minXu2U

CuðxuÞ

s:t: Hxt ¼P t

Qt

� �; t 2 T ;

xu 2 Xu; u 2 U :

ð1Þ

Here, x is the vector of all decision variables, xu isthe vector of variables related to component u, andxt is the vector of variables related to hour t. Theobjective function is formulated as minimisationthe sum of subsystem-specific net cost functionsCu(xu). In this representation possible demand-side(sales) components will contribute to the objectivefunction with a negative cost. The supply and de-mand components are connected to commonpower and heat balances through hourly balanceconstraints t 2 T. The balance constraints are rep-resented by a time-invariant technology and trans-mission matrix H and forecasts Pt,Qt for the netdemand of power and heat. The subsystem-specificconstraints are represented by the sets Xu. Both theconstraints Xu and objective function terms Cu(xu)may contain dynamic dependencies. Examples ofdynamic constraints are the storage constraintsfor a hydro power and the energy package con-straints in long-term power contracts. Examplesof dynamic cost function terms are the start-upand shut-down costs for thermal power plants.

3.2. Decomposition into hourly models

Using various decomposition and co-ordinationtechniques, the overall energy management modelcan be decomposed into hourly models

minXu2U

CtuðxtuÞ

s:t: Hxt ¼P t

Qt

� �;

xtu 2 X tu; u 2 U :

ð2Þ

Here, the subsystem-specific objective functionsterms have been decomposed into hourly termsCt

uðxtuÞ and the subsystem-specific constraints havebeen decomposed into hourly constraints repre-sented by the sets X t

u.The applicable decomposition techniques de-

pend on what kind of dynamic dependencies arepresent in the model. In the simplest case withno dynamic dependencies, each hourly model issolved once in sequence. When dynamic dependen-cies are present, the decomposition algorithm mayrequire solving the hourly models repetitively withdifferent parameters. Suitable techniques includee.g. dynamic programming, Lagrangian decompo-sition, Dantzig-Wolfe decomposition, Benders�decomposition, and various heuristic techniques.The interested reader could refer, for example toBaldick (1995), Dantzig (1963), Guan and Luh(1992), Guan et al. (1995), Lahdelma and Makko-nen (1996, 2001), Lahdelma and Ruuth (1989,1993), Lautala et al. (1992), and Wang et al.(1995).

The hourly model (2) consists of submodels minCt

uðxtuÞ subject to xtu 2 X tu that are combined to-

gether by linear constraints. The standard EHTOoptimiser assumes that the hourly submodels areconvex for each component. Convexity of a sub-model means that X t

u is a convex set and CtuðxtuÞ

is a convex function of xtu 2 X tu. The convexity

assumption is used in EHTO to speed up the opti-misation of the hourly models dramatically.Firstly, because the submodels are convex, thenthe hourly model (2) is also convex. This meansthat it is possible to use efficient convex optimisa-tion techniques for solving (2) instead of muchslower non-convex techniques. Secondly, if allthe submodels are linear programming (LP) prob-lems, then (2) is also an LP problem. Solving LPproblems is, in general, much more efficient thansolving general convex problems of roughly thesame size. As we will show next, it is possible torepresent convex power plant and trade contract


models as compact LP models with good accuracy.Thirdly, the resulting LP model has a special struc-ture which can be utilized for solving the problemmuch more efficiently than by using any genericLP algorithms.

3.3. Convex power plant modelling

As illustrated in Fig. 3, the hourly power pro-duction P t

u and heat production Qtu of a CHP plant

must be in the characteristic operation area X tu,

ðP tu;Q

tuÞ 2 X t

u: ð3Þ

Assuming that the area is convex and that theoperating costs Ct

u ¼ CtuðP t

u;QtuÞ are a convex func-

tion of power and heat production, we can repre-sent the characteristic operating area of the CHPplant as a convex combination of extreme charac-teristic points ðctj; ptj; qtjÞ through

Ctu ¼

Xj2Ju

ctjxtj;

P tu ¼

Xj2Ju

ptjxtj;

Qtu ¼

Xj2Ju

qtjxtj;

Xj2Ju

xtj ¼ 1;

xtj P 0; j 2 Ju;

ð4Þ

where Ju is the index set of variables associatedwith plant u. The minimum cost function of theplant is then the lower envelope of the convexpolytope defined by (4). With a sufficiently dense

Q (c3,p3,q3)

P

(c4,p4,q4)

(c5,p5,q5)

(c6,p6,q6)

(c1,p1,q1)

(c2,p2,q2)

Fig. 3. Feasible operating area of a CHP plant.

set of characteristic points, the above formulationapproximates any convex cost function with arbi-trarily good precision. In practice, the characteris-tic points can either be determined empirically(based on test runs) or calculated based on an ana-lytical model. In either case, the number of pointswill be reasonably small.

This formulation allows the shape of the char-acteristic to change hourly, but assumes that thesame number of points jJuj are used for each hour.If a plant needs fewer points at some hours, extrapoints can be effectively disabled by fixing those xtjto zero. It is typical that the number of character-istic points and their (p,q)-coordinates remainfixed from hour to hour, but due variable fuelprices, the c-coordinate varies from hour to hour.In such cases it is most convenient to representthe characteristics of the power plant as (r,p,q)coordinates, where r is the fuel consumption. Thec-coordinate is computed from r based on thehourly fuel price Ct according to

ctj ¼ Ctrtj: ð5Þ

In addition to CHP plants, the energy modelmay contain separate power and heat components.Such components include condensing power plants,hydropower, heat plants, and various purchase andsales contracts for heat and power. All these can bemodelled as special cases of the convex CHP plantmodel (4) with either qtj ¼ 0 (in power components)or ptj ¼ 0 (in heat components).

Substituting convex models (4) for each compo-nent into (2) gives an LP model with a specialstructure. The EHTO optimiser contains an imp-lementation of the specialised Power Simplex

algorithm for solving such models efficiently.According to Lahdelma and Hakonen (2003)Power Simplex solves real-life problems 20–190times faster than the tabular simplex code by Flan-nery et al. (1988) and 20–100 times faster than theefficient sparse simplex code by Lahdelma et al.(1986). The Power Simplex algorithm can onlysolve problems that contain one heat and onepower balance as in (1) and (2). Since heat cannotbe transported over long distances, Power Simplexapplies only for local CHP planning. Problemswith multiple heat balances can be solved withcomparable efficiency using the Extended Power


Simplex algorithm by Rong et al. (2005) and tri-generation problems using the Tri-CommoditySimplex algorithm by Rong and Lahdelma (2005).

4. Non-convex power plant modelling

There are situations where the convexity of apower plant cannot be assumed. If the marginalefficiency of the power plant is an increasing func-tion of p or q, this results in a non-convex charac-teristic. It is also possible that the operating area inthe (p,q) plane is non-convex. These situations arecommon with advanced production techniques,such as in backpressure plants with condensingand auxiliary cooling options, in gas turbines,and in combined gas and steam cycles. Non-con-vexity can also result e.g. from a fuel constraintthat requires a more expensive fuel to be used be-fore a cheaper one. A complex power plant canalso have a number of alternative operating modesthat shift some or all of the characteristic points.This makes the characteristic non-continuous(and thus non-convex).

Fig. 4. Non-convex characterist

We next show how to extend the convex energysystem model to incorporate non-convex powerplant models and how to solve the non-convexmodel efficiently.

4.1. Defining non-convex CHP plants

A well-known technique to handle non-convexmodels is to divide the model into convex submod-els and to encode these submodels as alternativemodel components. Fig. 4 illustrates the character-istic operating area of a CHP plant consisting ofone boiler and a backpressure turbine with op-tional reduction bypass, condensing operationand auxiliary cooling. The characteristic consistsof 9 points. The fuel consumption at each (p,q)point is specified through two consumption rates,rp for electricity and rq for heat as

rj ¼ rppj þ rqqj: ð6Þ

A single combined consumption ratio would ofcourse be sufficient for specifying the fuel con-sumption. Separate ratios for power and heat are

ic of a backpressure plant.

Table 1Allocation of characteristic points to convex sub-areas

Area P1 P2 P3 P4 P5 P6 P7 P8 P9

A1 1 1 1 1 1A2 1 1 1 1 1A3 1 1 1 1


given in order to determine how the productioncosts should be allocated for power and heat afterthe optimisation.

The characteristic area consists of different sub-areas. The line between points 1 and 2 is the back-pressure line where the consumption rates arerather good. If more heat is needed, it is possibleto open a reduction valve and let some of the highpressure steam bypass the turbine. This brings theoperation towards points 8 and 9 with good effi-ciency. More power can be extracted from the tur-bine by moving into the condensing area, towardspoint 3 with a much higher consumption ratio andconsequently lower marginal efficiency. Finally, ifeven more power is needed, it is possible to startthe auxiliary cooling, which means moving to-wards points 4, 5, 6, and 7.

The characteristic surface for the CHP plant isnot convex. This can be detected easily by comput-ing the angles between adjacent triangular facets.Concave angles appear along line segments 1–3and 2–3. Therefore it is necessary to divide the areainto convex sub-areas. The first convex sub-areaA1 is formed by the combined backpressure, con-densing and reduction area (1,8,9,2,3). The borderline of the remaining auxiliary cooling area is non-convex at point 3. Thus it must be split into twosub-areas, A2 = (1, 3, 6, 5, 4) and A3 = (2, 7, 6, 3).A compact way to represent this subdivision is toindicate for each characteristic point to which con-vex sub-areas it belongs as shown in Table 1. Anadvantage with this representation is that itis not necessary to repeat common border pointsbetween two or more sub-areas.

4.2. Notation

Next, we formalize the hourly non-convexpower plant model. To make the notation simpler,we omit the superscript t from all future formulas.We use the following notation:

Index sets

U set of components (cogeneration plants,trade contracts, demand side managementcomponents)

U* set of non-convex plants (i.e., plants withmore than one area)

A characteristic areas of all componentsAu characteristic areas of component uAj characteristic areas that contain point jJ index set of characteristic points in all

componentsJu index set of characteristic points of com-

ponent u

Parameters

cj production cost at characteristic pointj 2 J

cp± penalty for slack or surplus powercq± penalty for slack or surplus heatpj power generation at characteristic point

j 2 J

qj heat generation at characteristic pointj 2 J

Decision variables

xj variable associated to characteristic pointj 2 J

xp± slack or surplus variable for powerxq± slack or surplus variable for heatya zero-one variable determining if area a is

in use (a 2 Au, u 2 U*)

4.3. The non-convex hourly model

The non-convex hourly model is summarised inEqs. (7)–(15).

minXj2J

cjxj þ cp�xp� þ cpþxpþ

þ cq�xq� þ cqþxqþ ð7Þs:t:

Xj2J

pjxj þ xp� � xpþ ¼ P ; ð8ÞXj2J

qjxj þ xq� � xqþ ¼ Q; ð9ÞXj2Ju

xj ¼ 1; u 2 U ; ð10Þ


xj P 0; j 2 J ; ð11Þxp�; xq�; xpþ; xqþ P 0; ð12Þ

xj 6Xa2Aj

ya; j 2 Ju; u 2 U �; ð13Þ

Xa2Au

ya ¼ 1; u 2 U �; ð14Þ

ya 2 f0; 1g; a 2 Au; u 2 U �: ð15ÞThe objective (7) minimises the net acquisition

costs for power and heat. The net acquisition costsinclude production costs for physical plant com-ponents, purchase costs for trade components,income (negative costs) from possible demand-side-management components, and penalties forslack and surplus energy. Eqs. (8) and (9) are thehourly balance constraints for power and heat,correspondingly. Non-negative slack and surplusvariables (12) are included in the balance con-straints to allow the supply to differ from thedemand at given (penalty) cost. For each compo-nent u the net cost and the power and heat produc-tion is expressed as a convex combination of thecost at characteristic points (cj,pj,qj), j 2 Ju. Theconvex combination is encoded for each compo-nent by a set of xj variables, whose sum is one(10) and that are non-negative (11). The xj vari-ables act as interpolation coefficients in the objec-tive function and the balance constraints.

The Eqs. (7)–(12) form the convex Power Sim-plex formulation of the model (Lahdelma andHakonen, 2003). This model is sufficient when allmodel components are convex. However, it allowsrunning non-convex components in the area be-tween the convex sub-areas, which is physicallyimpossible.

To handle non-convex components correctly,the model is augmented with constraints that disal-low plants to operate between two or more areas.This is implemented through zero-one variables yafor each area a 2 Au in each non-convex compo-nent u 2 U*. The ya variable equals one whenthe corresponding area is used and zero otherwise.Constraints (14) force exactly one area per compo-nent into use. Constraints (13) disallow operationin other areas or between areas by forcing xj-vari-ables belonging to disallowed areas to zero. Itshould be observed that each characteristic point

may belong to several areas. For this reason anxj-variable is forced to zero only if all correspond-ing ya variables are zeroes (i.e. their sum is zero).

4.4. Solving hourly non-convex model

The non-convex model (7)–(15) can be solved asa general mixed integer programming (MIP) prob-lem. The most common technique to solve MIPproblems is to apply the Branch-and-Bound (B &B) algorithm. The B & B algorithm is based on abest-first tree search. Each node in the search treeconsists of an LP problem that is formed as a lin-ear relaxation of a corresponding MIP problem.The initial (root) node is the LP relaxation of theoriginal MIP problem and it is formed simply byignoring the non-convex constraints (15). Themain loop of the B & B algorithm manages thesearch tree, keeps track of the so far best MIPsolution and chooses, in turn, the most promisingunexplored leaf node to solve. Solving an LP prob-lem in a node has three possible outcomes:

1. The node is LP-infeasible. This implies thatalso the corresponding non-convex problem isinfeasible.

2. The node is LP-optimal, but MIP-infeasible.This means that some of the non-convex con-straints (15) are not satisfied, i.e., at least some0/1 variable ya has a fractional value.

3. The node is LP-optimal and MIP-feasible. Thismeans that the non-convex constraints (15) are(coincidentally) also satisfied.

Case 1 means a dead-end in the search tree. Noadditional search from this node is needed. In par-ticular, if the root node is LP-infeasible, then theoriginal MIP problem is infeasible.

Case 2 leads into the branch step of the algo-rithm. If the solution of the current node is betterthan the best MIP solution so far, it is necessary tocontinue the search from this node. In generic B &B implementations two child nodes are createdand inserted to the search tree; one with the addi-tional LP constraint ya = 0 and another one withya = 1. Thus, a binary search tree is formed, andthe maximum depth of the tree equals the numberof 0/1 variables.


In case 3, if the new solution is better than thecurrently best MIP solution, it replaces the bestMIP solution. Then the search tree is also pruned,i.e., nodes whose solutions are not better than thenew solution are removed. The solution of a childnode cannot be better than the solution of the par-ent node, because more and more constraints areadded deeper in the tree. Thus, it is not worthlooking for better solutions from nodes wherethe LP solution is not better than the currently bestMIP solution.

After the tree has been fully explored, thebest found MIP solution is the optimum tothe original MIP problem. If no MIP solutionhas been found, the original MIP problem isinfeasible.

To utilize the special structure of the problem,we have implemented a customised B & B algo-rithm. The algorithm differs from the generic B& B algorithm in three aspects:

• The 0/1 variables and associated constraints(13)–(15) are treated implicitly. This meansthat MIP-infeasibility is determined directlybased on the xj variables. MIP-infeasibility

BBOptimise(r) // r

best_node:= NULL; // so

node_heap:= NULL; // le

LP_Optimise(r); // o

if LP_Optimal(r)if MIP_Feasible(r)

return r; // LP

else

node_heap.Insert(r); // in

end if

end if

while not node_heap.Empty() do

node:= node_heap.GetFirst(); // re

u:= InfeasiblePlant(node); // u

for a in u.Areas do // ma

MakeChild(node,u,a); // cr

end for

end while

return best_node; // op

end BBOptimise

is caused by plants that are operatingbetween two or more areas. Therefore theLP relaxation contains only the constraints(7)–(12) and the additional constraints inthe branch step are handled by fixing a sel-ection of the xj variables to zero based on(13).

• The branch step is not implemented based on asingle infeasible 0/1 variable, but based on anentire plant u. Instead of forming two childnodes, multiple child nodes are formed, onefor each convex area a 2 Au.

• In generic implementations of B & B, thesolution of LP models is normally delayed,because solving LP models is computation-ally quite expensive. However, the PowerSimplex algorithm can solve the childnodes extremely efficiently by reusing the oldbasic solution of the parent node. Therefore,we can solve the LP models associated withnew nodes immediately when the nodes arecreated.

The customised B & B algorithm is summarizedin the following pseudo-code.

is root node

far best MIP solution

af nodes of tree to be explored

ptimise LP problem

relaxation solves MIP problem

sert r into heap

move and return first from heap

is an infeasible plant

ke a child for each area

eate & solve child, test case 1,2,3

timal MIP solution or NULL

Node 0Plant 1 is

MIP-infeasible

Node 2

LP-Infeasible

Node k1Plant 3 is

MIP-infeasible...

Node 1Plant 2 is

MIP-infeasible

Node k1+1

LP-infeasible

Node k1+2

MIP-feasible

Node k1+k2Plant 3 is

MIP-infeasible

... Node k1+k2+1

LP-infeasible

Node k1+k2+2

MIP-feasible

Node k1+k2+k3Plant 2 is

MIP-infeasible

...

Branch on plant 1

Branch on plant 2 Branch on plant 3

Branch on plant 3 Branch on plant 2

area 1 enabledarea 2 enabled area k1 enabled

area k2 enabledarea 1 enabledarea 2 enabled

area 1 enabledarea 2 enabled

area k3 enabled

Fig. 5. Sample search tree created by the customised B & B algorithm.


The MakeChild-routine creates a new LP prob-lem by adding the constraints xj = 0 for eachj 2 JunJa. This means that only the variables corre-sponding to area a of plant u are enabled. Make-Child then solves the LP model starting from theold basis of the parent node. Finally, MakeChildtests for the three cases listed above and actsaccordingly. In case 2, if the objective functionvalue is better than the so far best MIP solution,the new node is inserted into the heap ordered byits objective function value. In case 3, if a betterMIP solution is found, the best MIP solution isupdated and the search tree is pruned. Fig. 5 illus-trates the search tree created by the customised B& B algorithm. The maximum depth of the treeequals the number of plants with non-convexcharacteristics.

Table 2Properties of the cogeneration plants

Plant Type Areas Characteristic points

A BP 3 9B CSG 8 24C CSG 14 30Total 25 63

5. Real-life example

In the following, we demonstrate non-convexpower plant modelling and optimisation with areal-life application. The example is based on theenergy management model of the Tampere Elec-tricity Board (TEB), which is the municipal powercompany of the city of Tampere, Finland. TEBproduces both electricity for the national networkand district heat for the city of Tampere. Districtheat is produced at three regional cogenerationplants and two major district heating plants. The

cogeneration plants consist of one backpressure(BP) plant (A) and two combined steam and gascycle (CSG) plants (B, C). Reserve heating plantsalso exist, but these are excluded from the optimi-sation model. TEB has also some hydro powerplants, which are similarly excluded from themodel. The hydro power plants are not adjustableand therefore they do not affect the optimisation.The overall model is illustrated in Fig. 2.

The characteristic operating region of eachcogeneration plant is non-convex. Each non-con-vex characteristic is modelled, as described in theprevious section, as a union of multiple convexoperating regions. Fig. 4 illustrates the non-linearcharacteristic for the backpressure plant A. Thecharacteristic of plant A contains 3 areas that aredefined by 9 characteristic points (p, q, c). Table2 summarizes the properties of the three cogenera-tion plants (A, B, C).

To demonstrate the performance of non-convexpower plant optimisation, we solved different vari-ants of the model using a 730-hour (one month)


planning horizon. The basic variant of the modelincludes all three cogeneration plants (ABC), twodistrict heating plants, and the possibility to tradeelectricity on the spot market. We formed threesmaller models (BC, AC, AB) by shutting downeach of the cogeneration plants in turn. Shuttingdown a plant means that the optimisation systemeliminates the corresponding convexity constraintand variables from the optimisation model. Table3 summarises the dimensions of the four non-con-vex test models. The table shows for each hourlymodel the number of linear constraints (m), thenumber of continuous variables (n), the numberof areas in the plant characteristics, and the num-ber of different combinations of areas from differ-ent plants. The number of areas would equal thenumber of required zero-one variables if the modelwere encoded as a regular MIP model. The num-ber of combinations reflects better the complexityof the model. In the worst case, the Branch-and-Bound algorithm may have to enumerate throughall combinations and solve the corresponding LPproblems at the leaf nodes of the search tree, plusthose at the inner nodes.

To compare the complexity of non-convexpower plant modelling with convex modelling,we also formed convex variants of the four testmodels. This was done simply by merging theareas in each characteristic. The convex test

Table 3Dimensions of the test models

Model m n Areas Combinations

ABC 5 77 25 (3+8+14) 336BC 4 67 22 (0+8+14) 112AC 4 52 17 (3+0+14) 42AB 4 46 11 (3+8+0) 24

Table 4Results on solving the four 730-hour non-convex models and their co

Model Non-convex

Nodes Nodes/hour Iters Iters/node

ABC 33,215 45.5 49,896 1.50BC 2046 2.8 3121 1.53AC 730 1 66 0.09AB 762 1.04 128 0.17

models have the same number of constraints andvariables as their non-convex variants.

We solved the eight test models using theGENERIS EHTO system in a 600 MHz PentiumIII PC under the Windows 2000 operating system.To reduce the effect of random variations in CPUtime measurements, we solved each model fivetimes and computed the average CPU time. Table4 shows the results on the test runs. For eachmodel, we have listed the number of Power Sim-plex iterations and the CPU time in seconds. Forthe non-convex models, we also listed the numberof Branch-and-Bound nodes where an LP problemhas been solved. The number of nodes excludesnodes that have been eliminated in the boundphase without calling Power Simplex to solvethem. The last column shows the ratios betweenthe CPU times for solving the non-convex andconvex models.

The results show that the non-convex modelstake much more time to solve than their convexvariants. The largest model (ABC) takes 69 timesmore CPU time to solve than the correspondingconvex model. With the smaller models the differ-ence decreases, but is still significant. This is con-sistent with the theoretically exponential timecomplexity of the non-convex models with respectto the number of non-convex power plants andareas.

However, the solution times are still very rea-sonable even with the largest non-convex modelwhere finding a solution took 10.5 seconds. Basedon the number of nodes per hour, we observed thatthe Branch-and-Bound prunes the search tree verywell. In the ABC model with 336 potential combi-nations, only 46 nodes need to be solved on aver-age. In the BC model with 112 combinations, onlythree nodes were solved on average. In the AC

nvex variants

Convex Comparison

CPU(s) Iters CPU(s) CPU Ratio

10.5 42 0.152 690.667 30 0.098 6.80.128 66 0.080 1.60.098 3 0.048 2.0


model all hourly problems and in AB, almost allhourly problems were solved directly by their ini-tial LP relaxation!

The solution time depends ultimately on thenumber of simplex iterations that are required.The number of iterations per solved LP model isvery small as indicated by the Iters/node columnin Table 3. A large share of the LP problems issolved immediately by a restored and adjustedold basis without any iteration.

6. Conclusions

Changes on the energy market, increasing envi-ronmental awareness, and new energy productiontechnologies create a need for new kinds of deci-sion-support tools for energy companies. Energycompanies need to be able to solve in real-timemore detailed and realistic models that are basedon reliable on-line information. In this paper, wehave presented a technique to model differentkinds of combined heat and power productionplants more accurately using non-convex optimi-sation models. We have also demonstrated with areal-life application, that non-convex optimisationcan be implemented very efficiently using suitabledecomposition techniques and customised embed-ded optimisation algorithms.

The performance of the modelling and solutiontechnique is applicable to real-life needs. Thesolution time of the optimisation method is linearwith respect to the length of the time horizon butexponential with respect to the number of plantsand areas within plant characteristics. In real-time operative use, the time horizon can be any-thing from a few hours to several weeks. Withcurrent PC-hardware, the presented concept wasproven fast enough for real-time operative usein a real-life case with three non-convex cogenera-tion plants containing a total of 25 areas. Thesolution time for a 730-hour model with threenon-convex power plants was only 10.5 secondson a 600 MHz Pentium III. The system is alsosuitable for long-term planning as a full year-model can be solved in about two minutes. Ifthe model is applied in long-term risk analysis,the annual model needs to be solved multiple

times based on different scenarios. In such appli-cations, the solution time may be several hours,which is still feasible.

The exponential time complexity with respect tothe number of plants and areas sets practical limi-tations on the proposed method. More sophisti-cated search and pruning techniques in theBranch-and-Bound algorithm would probablyallow solving somewhat larger problems with suf-ficient speed. In much larger problems, it may benecessary to apply heuristic and/or near-optimalsolution methods.

Future studies could consider modelling andoverall optimisation of fuel mixtures with con-straints, e.g. annual maximum usage. The tech-niques could also be used for modelling emissionlimits and extended to consider the trade of emis-sion certificates. Additionally, further researchcould explore how the model can be better adaptedfor evaluating different strategies in terms of multi-ple uncertain criteria (Lahdelma et al., 2005; Mak-konen et al., 2003).

Acknowledgements

We thank Ilhan Or, Kevin Doyle, Ender Gur-gen and an anonymous referee for valuable com-ments that have helped us to improve the paper.

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