The paper discusses an Integer Linear Programming (ILP) formulation that describes the problem of planning the use of domestic distributed generators, under individual as well as fleet constraints. The planning problem comprises the assignment of time intervals during which the local generator must produce or not. In [1] this ILP is shown to be NP-complete in the strong sense. Heuristic methods have been developed to find solutions in reasonable time.\ud
\ud
In this work a different technique is used to overcome the complexity problems. We use column generation to search the possible decision vectors in a faster way. The ILP is slightly adjusted to facilitate the column generation technique to search in a clever way through the set of possible solutions.\ud
\ud
To measure the results, the column generation technique is compared to an earlier developed heuristic method. Both the quality of the objective function and the speed of the methods are compared

Bakker, V.

Bosman, M.G.C.

Hurink, J.L.

Molderink, A.

Smit, G.J.M.

The paper discusses an Integer Linear Programming (ILP) formulation that describes the problem of planning the use of domestic distributed generators, under individual as well as fleet constraints. The planning problem comprises the assignment of time intervals during which the local generator must produce or not. In [1] this ILP is shown to be NP-complete in the strong sense. Heuristic methods have been developed to find solutions in reasonable time. In this work a different technique is used to overcome the complexity problems. We use column generation to search the possible decision vectors in a faster way. The ILP is slightly adjusted to facilitate the column generation technique to search in a clever way through the set of possible solutions. To measure the results, the column generation technique is compared to an earlier developed heuristic method. Both the quality of the objective function and the speed of the methods are compared

Bakker, Vincent

Molderink, Albert

Hurink, Johann L.

Smit, Gerardus Johannes Maria

NARCIS 

Applying Column Generation to the Discrete Fleet Planning Problem

University of Twente Research Information

1Applying Column Generation to the Discrete FleetPlanning ProblemM.G.C. Bosman, V. Bakker, A. Molderink, J.L. Hurink, G.J.M. SmitDepartment of Electrical Engineering, Mathematics and Computer ScienceUniversity of TwenteP.O. Box 217, 7500 AE, Enschede, The Netherlandsm.g.c.bosman@utwente.nlAbstract—The paper discusses an Integer Linear Programming(ILP) formulation that describes the problem of planning the useof domestic distributed generators, under individual as well asfleet constraints. The planning problem comprises the assignmentof time intervals during which the local generator must produceor not. In [1] this ILP is shown to be NP-complete in the strongsense. Heuristic methods have been developed to find solutionsin reasonable time.In this work a different technique is used to overcome thecomplexity problems. We use column generation to search thepossible decision vectors in a faster way. The ILP is slightlyadjusted to facilitate the column generation technique to searchin a clever way through the set of possible solutions.To measure the results, the column generation technique iscompared to an earlier developed heuristic method. Both thequality of the objective function and the speed of the methodsare compared.Keywords: column generation, schedulingI. INTRODUCTIONDuring the last years the electricity supply chain (theproduction, consumption, distribution, storage and load man-agement with respect to electricity), has changed. For example,the availability of renewable energy sources, the developmentof distributed electricity generators and the demand for moreenergy efficient appliances are increasing [4]. A specific typeof upcoming distributed electricity generators is microCHP(Combined Heat and Power on a domestic scale). A microCHPproduces both heat and electricity for household usage; theelectricity can also be delivered back to the electricity grid.This electricity production, while being connected to thegrid, gives possibilities to increase stability in the grid, toreplace a conventional power plant, and more. To effect thesepossibilities the individual microCHPs need to be controlled,as standalone devices, but also in cooperation with othermicroCHPs or generators.An important part in this control can be contained byonline and offline planning problems. In [1] we presenteddifferent kinds of scheduling models that are derived from themicroCHP problem description. In this paper we focus on theplanning problem that combines the planning of productionruns of the microCHPs of single households equipped with aheat buffer into the larger concept of a so-called fleet. ThisDiscrete Fleet Planning Problem (DFPP) is shown to be NP-complete in the strong sense. In practice this means that it isvery costly to find an optimal solution for instances containing10 houses and more. To overcome these complexity problems,in [2] a heuristic is proposed, that iteratively tries to find abetter planning, by solving independent dynamic programmingformulations for single house problems, and steering theseindividual DPs by changing artificial steering parameters inthe objective function.A. Column generationIn this paper we propose an alternative method that triesto find a good solution in reasonable computation time. Thismethod makes use of a technique called column generation.The idea of column generation is to divide the given problemin two parts: a main problem and a sub problem. In themain problem the original problem is optimized for a muchsmaller part of the solution space, indicated by a set of solutionvectors, the so-called ‘columns’. Based on the solution for thismain problem, a new column is added to this solution space,as long as it improves the objective value in the main problem.In the sub problem this new column is found, by optimizingthe added value of the new column, based on the solution ofthe main problem.Column generation is widely applied in practice. An exam-ple of the use of this technique is given in [3], where it isapplied to solve the Cutting Stock Problem. It is also used tosolve the Vehicle Routing Problem, as in [5].The paper is organised as follows. In Section II the ILPformulation of the DFPP and an earlier developed heuristic aregiven and commented. Also the idea of adopting column gen-eration to this ILP is explained. Section III shows a comparisonbetween the column generation technique and the heuristic.The paper ends with conclusions and recommendations forfuture work in Section IV.II. PROBLEM AND COLUMN GENERATION APPROACHIn this section our column generation approach is presented.Section II-A gives the complete formulation of the originalproblem and introduces a heuristic to solve the problem. InSection II-B a simplified formulation is used to solve theproblem via column generation.2A. ILP model and heuristic approachThe DFPP is given by Equations (1)-(19). The formulationuses time intervals j = 1, . . . , NT and houses i = 1, . . . , N .Note that the constraints are applied to all houses, next to thegiven values for j.maxNT∑j=1N∑i=1π̂jeij (1)s.t. startij ≥ xij − xij−1 j = 2−MR, . . . , NT (2)startij ≤ xij j = 2−MR, . . . , NT (3)startij ≤ 1− xij−1 j = 2−MR, . . . , NT (4)stopij ≥ xij−1 − xij j = 2−MR, . . . , NT (5)stopij ≤ xij−1 j = 2−MR, . . . , NT (6)stopij ≤ 1− xij j = 2−MR, . . . , NT (7)gij = Gimaxxij −Niup−1∑k=0Ĝik+1startij−k+Nidown−1∑k=0Ǧik+1stopij−k j = 1, . . . , NT , (8)eij = αigij j = 1, . . . , NT (9)xij ≥j−1∑k=j−MRi+1startik j = 1, . . . , NT (10)xij ≤ 1−j−1∑k=j−MOi+1stopik j = 1, . . . , NT (11)hli1 = BLi (12)hlij = hlij−1 + gij−1 −Hij−1 −Ki j = 2, . . . , NT + 1(13)0 ≤ hlij ≤ BCi j = 1, . . . , NT + 1 (14)N∑i=1eij ≤ Pupperj j = 1, . . . , NT (15)N∑i=1eij ≥ P lowerj j = 1, . . . , NT (16)xij ∈ {0, 1} j = 1, . . . , NT (17)startij ∈ {0, 1} j = 1, . . . , NT (18)stopij ∈ {0, 1} j = 1, . . . , NT (19)The formulation uses binary decision variables xij (17) torepresent the on/off state of the microCHP in interval j inhouse i. startij (18) and stopij (19) are used to describe theintervals in which the microCHPs are started and stopped inEquations (2)-(7). Now, Equation (8) gives the heat generationof the microCHP, which can be derived from the variables xij ,startij and stopij . The electricity generation is given in (9).Equations (10) and (11) represent technical constraints of themicroCHP; heat buffer constraints are given in (12)-(14). Thetotal electricity output is limited by (15) and (16). Finally, theobjective is to maximize the profit that can be made on anelectricity market with prices π̂j (1).If we let Pupperj ≡ P lowerj ≡ Pj for all intervals j,the DFPP becomes a feasibility problem: can we find anassignment of xij’s, such that the desired pattern Pj is acquired.The profit in this case is fixed, if a solution can be found. Ifwe now allow some deviations from the pattern Pj , Equations(15) and (16) change in:N∑i=1eij − exj ≤ Pj j = 1, . . . , NT (20)N∑i=1eij + slj ≥ Pj , j = 1, . . . , NT (21)where exj and slj are excess and slack variables to measurethe deviation from the desired pattern Pj . As objective we canminimize the deviations from the pattern:minNT∑j=1(slj + exj). (22)In the following, we focus on this variant of the DFPP: theproblem is to minimize the deviation from a fixed productionpattern P .1) Heuristic: A heuristic to solve this variant of the DFPPis described in [2]. It uses a dynamic programming (DP)formulation to solve the planning problem for single houses.The phases in this DP are represented by the intervals and thestates are formed by using similar constraints as in Equations(10)-(14) in the ILP formulation. The objective of the DP is tomaximize the profit of the single house, when electricity pricesπ̂j are used. Now N independent DPs are solved and givealtogether a certain deviation from the total electricity patternP . The idea of the heuristic is to adjust the prices π̂j locallyper house i to find a better global deviation in total electricityproduction. A more detailed description of the adjustments ofthese prices can be found in [2].B. Column generation approachThe limitations of the DP formulation are mainly deter-mined by the sizes of the state spaces of allowed state tuplesfor individual generators. All possible production patterns canbe deducted from these state spaces, including the ones thatare not likely to be chosen. It would be an improvement todecrease the size of the state space by discarding the produc-tion patterns that are not likely to be chosen. However, it isnot an easy task to discard these unlikely production patterns,since it is difficult to determine on beforehand whether apattern will not be chosen. Therefore, the approach to discardpatterns probably leads to a way of almost randomly deletingpatterns, which can impact the optimality of the solution. Moreelaborate approaches can be used, but they need informationon the original state space, which we want to avoid to improvethe speed of the solution technique.On the other hand, once we start with a strongly reducedstate space we can use the information from this small setof patterns to add new patterns to improve the solution. If3the eventual state space that we create is still much smallerthan the original state space, we might have a faster solutionapproach. This is the idea of the column generation technique:starting from a small set of production patterns, we addnew patterns until we are satisfied with the global patternassignment.1) Solving the DFPP: The column generation techniqueconsists if two phases. In the first phase the DFPP is solved;in the second phase a new pattern is added to the current setof feasible patterns.The column generation technique starts with a relativelysmall set of feasible patterns PS. This set grows as the heuris-tic evolves; for now, it suffices to know that the size of the setdoes not change while the heuristic is in the phase of solvingthe DFPP (i.e. assigning patterns to generators to minimize thedeviation from the global predefined production plan P ). Letthe size of this set for a certain visit to this phase be given byNP , such that p1, . . . , pNP are the available patterns to choosefrom. For now we assume that these patterns are feasiblefor all generators (houses). The problem then is to assignexactly one pattern to each generator, such that the globalplan P is approximated as close as possible. To calculatethe deviation to this plan we minimize the mismatch ratherthan the squared mismatch, since we want to avoid quadraticobjective functions. We then get the following formulation:minNT∑j=1(slj + exj) (23)s.t.N∑i=1NP∑p=1ppjyi,p + slj ≥ Pj j = 1, . . . , NT (24)N∑i=1NP∑p=1ppjyi,p − exj ≤ Pj j = 1, . . . , NT (25)NP∑p=1yi,p = 1 i = 1, . . . , N (26)slj , exj ≥ 0 j = 1, . . . , NT (27)yi,p ∈ {0, 1}, (28)where yi,p is a binary decision variable indicating whetherpattern pp is chosen for generator i (in this case yi,p = 1) ornot (yi,p = 0). In Equations (24) and (25) slack and excessvariables slj and exj are introduced to allow an exact matchto the predefined production pattern P . The sum of slackand excess variables is minimized in Equation (23). Finally,Equation (26) requires that exactly one pattern is chosen foreach generator.2) Creating new patterns: The second phase of the columngeneration technique consists of creating new patterns that canbe added to the current pattern set. These new patterns shouldcontribute to the existing set in the sense that they must givepossibilities to decrease the objective value in the first phase(i.e. the sum of slack and excess). A new pattern pg is onlyadded if it improves the existing solution of the DFPP. Tofullfill this requirement, the new pattern pg is only added ifNT∑j=1λjpgj > 1, (29)where λj are the shadow prices associated with Equations (24)and (25).A new pattern should be a feasible pattern. This can beguaranteed by calculating the heat level of the correspondinggenerators following the created pattern and checking whetherthe buffer limits are violated. In the formulation of the secondphase, this is simplified by using two parameter sets, specify-ing in each interval j the minimum number of intervals wherethe generator should have run (MinOnj) and the maximumnumber of intervals where the generator could have run(MaxOnj) up to and including the current interval. Note thatthese parameters MinOnj and MaxOnj are also assumed tofullfill the technical constraints in Equations (10) and (11).Combined with the objective of maximizingNT∑j=1λjpgj , thesecond phase can be formulated as:maxNT∑j=1λjpgj (30)s.t.j∑k=1pgk ≤MaxOnj j = 1, . . . , NT (31)j∑k=1pgk ≥MinOnj j = 1, . . . , NT (32)pgj ∈ {0, 1} j = 1, . . . , NT , (33)where pgj is the binary decision variable to produce in intervalj in pattern g. The objective is maximized, meaning that,if requirement (29) can be met, an optimal solution to theformulation gives an improvement in the first phase.3) The column generation algorithm: Two ILPs are pre-sented for the two phases. ILP1 is the formulation of thefirst phase and ILP2 the formulation of the second phase. Thecomplete solution method is given in algorithm 1.Algorithm 1 Column generationinit PS := {p1, . . . , pNP }solve ILP1solve ILP2whileNT∑j=1λjpgj > 1 doPS ← PS ∪ pgsolve ILP1solve ILP2end whileIf the starting patterns in PS are chosen feasible, allpossible solutions in the eventual set are feasible (in thisversion we assume all houses to have equal characteristics).Due to the feasibility constraints (31) and (32) for the newlycreated pattern, the final solution in the first phase needs not4to be optimal. This is due to the choice for the initial set(and added columns), which could imply that not all necessarycolumns are created in later iterations.Note that we simplified the problem formulation by lookingat the number of generators that are on instead of the totalelectricity production.III. RESULTSTo compare the column generation technique to the iterativeheuristic using DPs we created two sets of problem instances.The column generation approach is modeled using AIMMSmodeling software and solved by CPLEX 11.1. The heuristicapproach is implemented in C++.Both sets of instances make a planning for 48 intervals,meaning that each interval represents half an hour in a one-day-planning. In these instances, we assume that all houseshave similar characteristics, such as heat demand. Due tothe use of a heat buffer we have scheduling freedom in theassignment of runs of the microCHP. In all instances, thiscomes down to the fact that the microCHP in each houseshould run for minimally 25 intervals and maximally 28intervals. The sets of instances differ in the total amount ofhouses that is planned. In the first set, we consider a fleet of50 houses, while in the second set we look at 100 houses. Aninstance is completed by defining the production pattern P .This pattern is a constant pattern (P1 = . . . = PNT ) wherethe values of Pj are set to a constant number of houses (from 0to the number of houses in the fleet). So we have 51 instancesin the first set and 101 instances in the second set.The initial set PS in the column generation approach ischosen to consist of the two vectors MinOn and MaxOn.0 10 20 30 40 5005001,0001,500pattern (# microCHPs on in each interval)mismatchlower boundcolumn generationheuristic methodFig. 1. Mismatch from flat patterns (maximal AIMMS runtime 1 minute periteration)The solutions for both approaches are given in Figures 1 and2, as well as a theoretical lower bound on the deviation fromthe production pattern. In the figures the deviation (mismatch)is plotted against the production pattern. In Figure 3 thecomputation times are given of both techniques in both sets0 20 40 60 80 10001,0002,0003,000pattern (# microCHPs on in each interval)mismatchlower boundcolumn generationheuristic methodFig. 2. Mismatch from flat patterns (maximal AIMMS runtime 1 minute periteration)of instances. The maximum runtime is set to 60 seconds perproblem that is solved in the column generation approach.0 20 40 60 80 1000100200300pattern (# microCHPs on in each interval)computationtime(s)column generation (50 houses)heuristic method (50 houses)column generation (100 houses)heuristic method (100 houses)Fig. 3. Computation timeSince both techniques are applied to achieve good approxi-mations of the optimal solution, they both have to perform wellregarding objective value and speed. In the first two figuresthe column generation outperforms the heuristic approach inthe instances where the lower bound is (close to) 0. Note thatthese are the instances that we want to solve in practice. Inthe instances with a low preferred production pattern P thecolumn generation approach does not find a feasible new pat-tern in the first iteration. Therefore it performs bad comparedto the heuristic. In these instances the heuristic comes closeto the lower bound. The opposite can be said about highervalues of P , where the column generation technique comesclose to the lower bound and the heuristic cannot find largeimprovements.Regarding computational time the column generation tech-5nique is considerably faster than the heuristic. However, theinstances for which the lower bound is close to 0 are moredifficult to solve (but still faster than the heuristic).IV. CONCLUDING REMARKS AND RECOMMENDATIONSIn this paper we have shown a first approach in applyingthe column generation technique to a variant of the DiscreteFleet Planning Problem (DFPP), in which the production ofmicroCHPs of a fleet of houses should be matched to apredefined total production pattern P . First results indicate thatthe solutions improve the results that are obtained by applyingan earlier developed heuristic, as well in objective value as incomputational time.In future work, this concept of column generation can beapplied to the more general DFPP, in which electricity pricesplay a role as well. It is also important to allow more generalhouse configurations, to which not every production patternmight be applicable.V. ACKNOWLEDGEMENTSThis research is conducted within the SFEER project(07937) supported by STW, Essent and GasTerra.REFERENCES[1] M. G. C. Bosman, V. Bakker, A. Molderink, J. L. Hurink, G. J. M Smit,On the microCHP scheduling problem, Proceedings of the Third GlobalConference on Power Control and Optimization (PCO), February 2010,p. 8[2] M. G. C. Bosman, V. Bakker, A. Molderink, J. L. Hurink, G. J. M.Smit, Production planning in a Virtual Power Plant, Proceedings ofthe 20th Annual Workshop on Circuits, Systems and Signal Processing(ProRISC), November 2009, pp. 207-212[3] P. C. Gilmore, R. E. Gomory, A linear programming approach to thecutting-stock problem, Operations Research, Vol. 9, Nr. 6, November1961, pp. 849-859[4] J. Scott, P. Vaessen, F. Verheij, Reflections on smart grids for the future,KEMA consultancy document for the Dutch Ministry of EconomicAffairs, March 2008[5] R. A. Skitt, R. R. Levary, Vehicle routing via column generation,European Journal of Operational Research, Nr. 21, 1985, pp. 65-76

English

https://ris.utwente.nl/ws/files/5459468/paper.pdf

Applying Column Generation to the Discrete Fleet Planning Problem

Abstract

Similar works

Full text

Available Versions

NARCIS

University of Twente Research Information