European Journal of Operational Reearch 176 (2007) 1874 1895 O.R. Application CO 2 emiion trading planning in combined heat and power production via multi-period tochatic optimization Aiying Rong *, Rito Lahdelma Univerity of Turku, Department of Information Technology, Lemminkäienkatu 14 A, FIN-20520 Turku, Finland Received 15 December 2004; accepted 4 November 2005 Available online 19 January 2006 www.elevier.com/locate/ejor Abtract The EU emiion trading cheme (ETS) taking effect in 2005 cover CO 2 emiion from pecific large-cale indutrial activitie and combution intallation. A large number of exiting and potential future combined heat and power (CHP) intallation are ubject to ETS and targeted for emiion reduction. CHP production i an important technology for efficient and clean proviion of energy becaue of it uperior carbon efficiency. The proper planning of emiion trading can help it potential into full play, making it become a true winning technology under ETS. Fuel mix or fuel witch will be the reaonable choice for foil fuel baed CHP producer to achieve their emiion target at the lowet poible cot. In thi paper we formulate CO 2 emiion trading planning of a CHP producer a a multi-period tochatic optimization problem and propoe a tochatic imulation and coordination approach for conidering the rik attitude of the producer, penalty for exceive emiion, and the confidence interval for emiion etimate. In tet run with a realitic CHP production model, the propoed olution approach demontrate good trading efficiency in term of profit-to-turnover ratio. Conidering the confidence interval for emiion etimate can help the producer to reduce the tranaction cot in emiion trading. Comparion between fuel witch and fuel mix trategie how that fuel mix can provide good tradeoff between profit-making and emiion reduction. Ó 2005 Elevier B.V. All right reerved. Keyword: Stochatic optimization; CO 2 emiion trading; Combined heat and power (CHP) production; Energy 1. Introduction Mitigation of the environmental impact of energy production and ue ha become an integral part of energy policy planning. Conequently, the requirement for environmentally ound energy production * Correponding author. E-mail addre: aiying@it.utu.fi (A. Rong). 0377-2217/$ - ee front matter Ó 2005 Elevier B.V. All right reerved. doi:10.1016/j.ejor.2005.11.003
A. Rong, R. Lahdelma / European Journal of Operational Reearch 176 (2007) 1874 1895 1875 technologie ha gained much ground in the energy buine. Recently the dicuion ha centered on the climate change. Combined heat and power (CHP) production i a leading technology to repond to the market demand and environmental concern becaue of it high energy efficiency. The EU commiion encourage the ue of more efficient energy technologie, including CHP technology, producing fewer emiion per unit of output. Thu, the EU commiion announce to raie the hare of electricity produced by CHP technology from 9% to 18% during the year 1997 2010 (CEC Commiion of the European Communitie, 1997). An EU-wide emiion trading cheme (ETS) i alo tarting in 2005 to fulfill the EU commitment under the Climate Protection Protocol in Kyoto to cot-efficiently reduce the emiion of greenhoue gae by 8% during the period 2008 2012 compared with the 1990 level (Commiion of the European Communitie, 2000). CHP production mean the imultaneou production of ueful heat and electric power. When team or hot water i produced for an indutrial plant or a reidential area, electricity can be generated a a by-product. Vice vera, urplu heat from an electric power plant can be ued for indutrial purpoe, or for heating pace and water. CHP plant make the maximum ue of the fuel energy content by producing electricity and heat together with minimum watage. The CHP plant can achieve a total efficiency of over 90%, while in conventional condening power plant the efficiencie remain around 40%. Primary energy conumption in CHP production a compared with correponding generation in eparate procee i typically lowered by one third. The decreae in fuel conumption reduce the burden of energy production on the environment. That i, the CO 2 emiion are reduced at the ame rate a the ue of fuel i reduced. Moreover, a wide range of fuel can be ued with modern CHP technology. Multi-fuel CHP plant can ue, for example, olid fuel (coal, peat, and wood reidue), liquid fuel (oil), gaeou fuel (natural ga) and even fuel with a low calorific value and high moiture content (wate, bio-fuel). The ETS tate that indutrial activitie that emit ignificant amount of CO 2 mut have a permit to do o. Such indutrie will be allocated allowance for pecific amount of greenhoue ga emiion for the relevant obligation period baed on national allocation plan of individual member countrie. The individual producer can meet their compliance target by reducing their emiion or by trading allowance within the EU. The producer mut pay a penalty price for exceive emiion and have to make up the deficit by buying the lacking allowance in the beginning of the ubequent obligation period. Ideally, ETS will caue emiion to be reduced where it can be done mot cot-efficiently. The ETS provide both challenge and opportunitie for the foil fuel baed energy ector, including CHP intallation. A large number of exiting and potential future CHP intallation will be ubject to ETS and targeted for emiion reduction. The high energy efficiency and low emiion make CHP production technologie environmentally friendly olution compared with many other production form. The flexibility in fuel choice facilitate fuel witch (change into fuel with lower pecific CO 2 emiion) and fuel mix a reaonable alternative for CHP producer to reduce their emiion. Evaluation of option for complying economically with the emiion target i complicated by many uncertaintie involved in CHP production and emiion trading. In CHP, the heat and power production follow a joint characteritic, which mean that the production planning of both commoditie mut be done in coordination. Under the deregulated electricity market, the power production hould repond to the volatile pot price on the market, while heat mut till be produced to balance the demand. In addition, fuel price and allowance price play an important role in fuel choice. Proper planning of emiion trading can help the potential of CHP production into full play, making CHP technology become a true winning technology under ETS. Generally, emiion trading hould be coordinated with other cloely related operational deciion. Different emiion compliance option can alo be employed in coordination. Under the US Clean Air Act Amendment (CAAA) of 1990, Lee et al. (1994) conidered the coordination of SO 2 emiion trading with energy and pinning reerve tranaction and conumption of take-or-pay fuel. They ditributed adaptively the emiion target for the entire planning horizon into hort-term operational target, which were, in turn,
1876 A. Rong, R. Lahdelma / European Journal of Operational Reearch 176 (2007) 1874 1895 enforced in the aociated unit commitment and dynamic dipatch ubproblem. Manetch (1994) propoed for long-term unit commitment and dipatch a method for integrating production planning with determination of SO 2 compliance option, uch a witching into low ulfur coal and intalling crubber. Thee paper emphaize the production planning of the power ytem under the contraint of emiion control. Until now, mot publihed paper that deal with CO 2 emiion trading in the energy ector are from the viewpoint of policy planning (Kunh et al., 2004; Hauch, 2003; Söderholm and Strömberg, 2003). They do not addre the trading problem itelf. In thi paper, we tudy the CO 2 emiion trading planning problem of an individual CHP producer at the operational level. We formulate the CO 2 emiion trading planning of a CHP producer a a multi-period tochatic optimization problem and propoe a olution approach that optimize CHP plant operation and CO 2 emiion trading in coordination. During each trading period, the future CHP production until the end of the planning horizon i optimized baed on cenario for heat demand, power price and allowance price. Baed on the optimized production plan the CO 2 emiion during the obligation period are etimated to determine how much allowance hould be traded (bought or old). The trading trategie are related to the rik attitude of the deciion maker (DM). The propoed method can be ued to evaluate the relative efficiency of different emiion compliance option uch a fuel witch and fuel mix. Thi paper extend the idea preented in our early work (Rong et al., 2004) in five way. Firtly, we explicitly conider the rik attitude of the DM in the problem formulation. Intead of maximizing the expected profit, we maximize the expected utility of the profit. Secondly, an hourly CHP production planning model replace the previou aggregated model in production planning. Thirdly, we conider the tranaction cot in the emiion trading and propoe a trading trategy that depend on how uncertain the emiion etimate are. Fourthly, we explicitly deal with the penalty for exceive emiion in the olution approach baed on optimality condition. Finally, we etimate the emiion uing a weighted average with allowance price a weight. Our early method did not apply weight in the etimation. Thi paper i organized a follow. In Section 2, we decribe the characteritic of CHP production and the uncertaintie involved in the CHP production and emiion trading planning problem under the deregulated energy market. In Section 3, we formulate the CO 2 emiion trading planning problem for a CHP producer a a multi-period tochatic optimization problem. In Section 4, we preent the olution approach for integrating CHP production planning and emiion trading and propoe the correponding trading trategie. In Section 5, we report the reult on numerical experiment and compare the relative efficiency of the fuel witch and fuel mix trategie. 2. Characteritic of CHP production and uncertaintie in emiion trading planning 2.1. Characteritic of CHP production The primary concern of a CHP producer i to produce heat to atify variable demand. Normally heat production mut meet the demand on an hourly bai. In CHP technology, heat and power production i linked together. The level of heat production determine the range in which the power generation can be adjuted and alo the marginal cot function for power generation. A CHP plant can be repreented by a joint characteritic that define the dependency between production cot and heat and power generation a hown in Fig. 1. Becaue the production cot are principally determined by the fuel conumption, the characteritic can alternatively pecify the dependency between the fuel conumption and heat and power production. The characteritic can be either convex or non-convex. For the convex CHP plant, the characteritic operating region can be repreented a a convex combination (ee, e.g., Bazaraa and Shetty, 1993) of extreme point (c j, p j, q j ), which are the corner point of the triangular facet in Fig. 1. A non-convex characteritic can be
A. Rong, R. Lahdelma / European Journal of Operational Reearch 176 (2007) 1874 1895 1877 Fig. 1. Feaible operating region of a CHP plant. divided into multiple convex ub-region, which are encoded a alternative model component. The ame modeling technique applie alo to other energy acquiition component, uch a eparate heat and power plant, purchae contract, and demand-ide-management component. The intereted reader can refer to Lahdelma and Hakonen (2003) for convex CHP plant modeling and to Makkonen and Lahdelma (2006) for non-convex modeling. On the liberalized power market, the rational producer hould adjut power generation o that the marginal production cot equal the market price. A a reult, the producer hould optimize it heat and power production for each hour againt the mot recent forecat for heat demand and pot market price for power. CHP production planning i further complicated by the need to control the CO 2 emiion. The amount and type of conumed fuel determine the caued CO 2 emiion. Modern multi-fuel CHP plant are able to ue different fuel and witch between fuel rapidly. In the fuel mix mode, everal fuel can be ued imultaneouly within certain limit. The dipatch of fuel i governed by ome rule. Generally the cheapet fuel i burned firt unle there are other pecial requirement. The fuel price and fuel mix affect the hape of CHP plant characteritic. The producer can adjut the production level, chooe between different fuel, and trade emiion allowance to balance it emiion with allowance. Thu, the CHP production planning problem mut be olved in coordination with the emiion trading planning. 2.2. Uncertaintie in integrated CHP production and emiion trading planning In the planning problem we conider three main ource of uncertainty: heat demand, power price and allowance price. Fig. 2 illutrate the uncertaintie and their dependencie in the planning problem. The heat demand depend almot entirely on local condition. Municipal power plant generate mainly ditrict heat. The uncertainty in ditrict heating demand i almot entirely due to local weather condition,
1878 A. Rong, R. Lahdelma / European Journal of Operational Reearch 176 (2007) 1874 1895 Fig. 2. Determination of heat demand, power price and emiion allowance price. i.e., temperature, wind, etc. Indutrial power plant generate proce heat. Thi demand depend on how the indutrial proce i run. The pot price for power i formed on the market a the equilibrium between power upply and demand. On the Nordic power market (Nord Pool, 2004) the mot ignificant factor affecting the pot market price i the inflow to hydropower ytem. Other important factor are eaonal variation, variation in fuel price, producer deciion, conumer behavior, and import & export. The heat demand and power price are omewhat correlated. Cold weather will increae the demand for both heat and power, and conequently alo the power price. However, thi dependency i not very trong, becaue the heat demand i determined locally and power price on the entire market area. The price of emiion allowance will be determined by their upply and demand and the pot deciion of individual trader throughout EU. If all actor on the allowance market have the ame information, the allowance price hould all time reflect their common undertanding about the future price development. Thi mean that the pot price for allowance i the bet poible etimate alo for the future price. Factor
A. Rong, R. Lahdelma / European Journal of Operational Reearch 176 (2007) 1874 1895 1879 that affect the EU-wide allowance market are energy conumption, utilization of the Joint Implementation (JI) and Clean Development Mechanim (CDM), and political deciion about favoring and difavoring ome energy form. The allowance cot will increae the marginal cot of the foil fuel baed power production, which make allowance price and power price omewhat correlated. 3. Multi-period CHP production and emiion trading planning model We aume that the DM are rik avere and their preference tructure in term of the profit i repreented by an increaing concave utility function U(Æ). The planning horizon i divided into period t =1,..., for CO 2 emiion trading. The trading period can be, e.g., week or month. During each trading period, the emiion level can be affected by adjuting the fuel mix and power production level and balanced by emiion trading. Index + 1 refer to the time after the planning horizon when the producer can till try to ell their urplu allowance and mut make up for any deficit. The CHP production at different plant mut be planned at much finer granularity than the emiion trading. For thi reaon each trading period i divided into hour h 2 H t. At the beginning of the planning horizon, the multi-period production and emiion " trading planning model to maximize the expected utility of the profit can be tated a max E U X X z pr ðx h ; c p hþþ X!# z tr f t ; c f t þ z tr f þ1 ; c f þ1 ð1þ t¼1 h2h t t¼1.t. hx; Ei 2XðQÞ; ð2þ hf; Ei 2F. ð3þ Here the function z pr (Æ) i the net profit from hourly CHP production and z tr (Æ) i the net profit from emiion trading during a trading period. x, f and E are variable in the model. The vector x determine the CHP production in each hour, vector f determine allowance trade in each trading period and calar E i the cumulated emiion during the entire planning horizon. c p h, cf t and Q are tochatic parameter. c p h i the hourly power price, c f t i the allowance price in period t and vector Q contain the heat demand for each hour in the planning horizon. The et X(Q) repreent the contraint of the CHP production proce that depend on the heat demand. The et F repreent the contraint of the emiion trading proce. To define the detail of the model, we introduce the following notation. Index et B et of CHP plant and other upply or demand component modeled a CHP plant H, H t et of hour in the planning horizon and in each trading period t =1,...,, correpondingly J, J b et of extreme point of the characteritic operating region in all plant and in plant b 2 B K et of fuel Parameter d ratio of allowance tranaction cot to allowance price g k pecific CO 2 emiion of fuel k 2 K Q, Q h vector of heat demand during the planning horizon and the demand for hour h 2 H c F# penalty price for exceive emiion at the end of the planning horizon (period +1) c F+ emiion allowance purchae price at the end of the planning horizon, c F þ ¼ c f þ þ1 c F emiion allowance ale price at the end of planning horizon, c F ¼ c f þ1 c f t emiion allowance price in period t =1,..., +1 c f t þ emiion allowance purchae price c f t þ ¼ð1þdÞc f t for period t =1,...,, including penalty for after lat period c f þ þ1 ¼ð1þdÞcf þ1 þ cf #
1880 A. Rong, R. Lahdelma / European Journal of Operational Reearch 176 (2007) 1874 1895 c f t c p h c q h c r k;h p j,h q j,h r k,j,h emiion allowance ale price in period t =1,..., +1,c f t ¼ð1 dþc f t power price in hour h 2 H heat price in hour h 2 H price of fuel k 2 K in hour h 2 H characteritic power coordinate j 2 J in hour h 2 H characteritic heat coordinate j 2 J in hour h 2 H conumption of fuel k 2 K at extreme point j 2 J in hour h 2 H Variable f, f t vector of allowance trade during the planning horizon and trade in trading period t =1,..., +1 ft þ ; ft emiion allowance purchae and ale in trading period t =1,..., +1 x, x h vector of deciion variable determining the production during the planning horizon and in hour, h 2 H, x h =[x 1,h,..., x jjj,h ] x j,h contribution of extreme point j 2 J to CHP production in hour h 2 H x p h power production in hour h 2 H E cumulated emiion during the planning horizon E +, E F 0, F t cumulated emiion exceeding and falling below the final allowance level F initial allocation and cumulated emiion allowance level at the end of trading period t =1,..., 3.1. The production model The production model X(Q) determine the hourly production of heat and power a a um of the production at different CHP plant and poible trade uing variou purchae and ale contract. Each plant model conit of a equence of hourly ubmodel that may be linked together by dynamic contraint, uch a tart-up and hut-down contraint, ramp contraint and torage contraint. The production model X(Q) can be ubdivided into hourly model X t (Q t ), which can be olved eparately uing uitable decompoition and coordination technique. The applicable decompoition technique depend on what kind of dynamic contraint are preent. In thi application we aume that no dynamic contraint are preent and that the hourly plant model are convex. Thi mean that we can olve the production model imply by olving the hourly model independently uing a convex olver. The mot efficient olver for hourly convex CHP production planning problem are Power Simplex (PS) (Lahdelma and Hakonen, 2003), and the envelope-baed algorithm ECON & ECOFF (Rong and Lahdelma, in pre). PS ha been implemented a part of the EHTO NEXUS energy optimization ytem (Lahdelma and Makkonen, 1996), which i in commercial ue at everal Finnih energy companie. Dynamic contraint and non-convex CHP model would require more ophiticated olution technique for the production model, but would not affect the emiion trading model or the overall olution approach. Non-convex production planning problem can be olved, e.g., by uing the Branch and Bound (BB) technique. Makkonen and Lahdelma (2006) olved non-convex planning problem by PS-baed BB (PBB) and Rong and Lahdelma (2005c) developed envelope-baed BB (EBB) algorithm for non-convex model. Rong and Lahdelma (2005b) analyzed the rik involved in CHP production expanion planning under the emiion trading cheme uing the production model imilar to that by Rong and Lahdelma (in pre). The model by Rong and Lahdelma (in pre) addree the CHP production under the deregulated power market and the modeling technique i imilar to that by Lahdelma and Hakonen (2003). Here we adopt a model that i imilar to that by Rong and Lahdelma (in pre). The hourly CHP production i modeled a a convex combination of characteritic extreme point for each hour h 2 H:
X x j;h ¼ 1; b 2 B; ð4þ j2j b X p j;h x j;h x p h ¼ 0; ð5þ j2j X q j;h x j;h ¼ Q h ; ð6þ j2j x j;h P 0; j 2 J. ð7þ In thi formulation, the convex combination for each plant i encoded by a et of x j,h variable, indicating the operating level of each plant in term of extreme point of the operating region, whoe um i one (4) and that are non-negative (7). The power balance (5) determine the net amount of power x p h that can be traded on the market at price cp h. The heat balance (6) tate that the demand Q h mut be atified. The emiion during the planning horizon are the um of the hourly emiion from the conumed fuel: E ¼ X h2h X j2j A. Rong, R. Lahdelma / European Journal of Operational Reearch 176 (2007) 1874 1895 1881 X g k r k;j;h!x j;h. ð8þ k2k The hourly profit from the CHP production i the power and heat ale revenue minu the production cot. The production cot are computed a a convex combination of fuel cot at the extreme point. z pr ðx h ; c p h Þ¼cp h xp h þ cq h Q h X! X c r k;h r k;j;h x j;h. ð9þ j2j k2k 3.2. The trading model The trading model F determine the allowance trade in each trading period a follow: F t ¼ F t 1 þ f t ; t ¼ 1;...; ; ð10þ E ¼ F þ f þ1 ; ð11þ f þ t ¼ maxf0; f t g; t ¼ 1;...; þ 1; ð12þ f t ¼ maxf0; f t g; t ¼ 1;...; þ 1. ð13þ E þ ¼ maxf0; E F g; ð14þ E ¼ maxf0; F Eg. ð15þ Contraint (10) determine the cumulated allowance at the end of each trading period. Contraint (11) require that the emiion trading after the lat period balance the emiion. Contraint (12) and (13) determine the amount of allowance bought and old during each trading period. The combination of contraint (12) and (13) diallow the activity of buying and elling allowance to be done imultaneouly. Contraint (14) and (15) determine the emiion exceeding and falling below the allowance level at the end of the planning horizon. The combination of contraint (14) and (15) implie that E + E = 0 and the reult of allowance trading at the end of planning horizon take one of three reult: exactly balance, fall below or exceed the realized emiion. The trading profit during a period i either the revenue from elling or the negated cot of buying allowance. The purchae price after the end of the planning horizon (c F+ ) include the penalty for exceive emiion.
1882 A. Rong, R. Lahdelma / European Journal of Operational Reearch 176 (2007) 1874 1895 z tr ðf t ; c f t Þ¼ cf þ t f þ t þ c f t f t ; t ¼ 1;...; ; ð16þ z tr ðf þ1 ; c f þ1 Þ¼ cf þ þ1 f þ þ1 þ cf þ1 f þ1 ¼ cf þ E þ þ c F E. ð17þ 4. Solution approach We olve the production and trading problem uing cenario analyi. Scenario analyi ha proved to be an effective approach to addre planning problem under uncertainty (Marana et al., 1997; Mulvey and Shetty, 2004). We want to olve the production and trading planning model at each trading period t * 2 {1,..., } during the planning horizon. Thi mean that for period 1;...; t 1 the deciion variable and tochatic parameter of the model (1) (17) are fixed to their already realized value. Thu, optimization conider variation in the variable and tochatic parameter only for period t *,...,. 4.1. Scenario-baed repreentation of the problem The uncertaintie of the operating environment were repreented in the model (1) (17) by tochatic parameter with ome joint probability ditribution. When conidering the production and trading planning problem in the beginning of a trading period t *, we approximate the future uncertaintie by a et of cenario. Each cenario define a et of value for the tochatic parameter at each period in the future. The et of cenario capture both the uncertainty of each tochatic parameter and the dependency information between them. To facilitate the repreentation, we extend the cenario alo for the pat period 1;...; t 1 to coincide with the realized hitory. In the current application the tochatic parameter are time erie (vector) for heat demand (Q), power price (c p ), and allowance price (c f ). The heat demand and power price contain hourly value, but the allowance price contain weekly or monthly value. To repreent the cenario-baed model formed at period t *, we augment the previou notation by inerting both a cenario index and period index t * a the lat two ubcript for the tochatic parameter and variable of the model. For example, Q h! Q h;;t denote the heat demand in hour h in cenario generated at the beginning of trading period t *. Let S denote the number of cenario generated at each period. In the cenario repreentation the objective function to maximize the expected utility become max 1 S þ X t¼1 X S ¼1 U X t¼1 X h2h t c f þ t;;t f þ t;;t þ cft;;t c p h;;t xp h;;t þ cq h Q h;;t X f t;;t j2j! X c r k;h r k;j;hx j;h;;t k2k c F þ ;t Eþ ;t þ cf ;t E ;t!. ð18þ The production and trading contraint for cenario generated in period t * become hx ;t ; E ;t i2xðq ;t Þ; ¼ 1;...; S; ð19þ hf ;t ; E ;t i2f; ¼ 1;...; S. ð20þ However, olving (18) (20) a a ingle problem i not meaningful, becaue it would allow the trading proce to foreee the future in each cenario and yield infinite profit by peculative operation. Thi i of coure not poible in practice. Intead, we mut deign an optimization cheme that can be implemented alo in real life.
4.2. Decompoition and coordination approach 4.2.1. Decompoition of CHP production proce and trading proce From formula (18) (20) we can ee that the production and trading procee interact only through the emiion variable E. No matter how the trading proce i run, the production proce mut atify the heat demand and aim to maximize the production profit minu the emiion cot. Recall that the pot price for allowance i the bet etimate alo for the future price. Therefore, when optimizing the hourly production, the pot price i the expected marginal cot for the caued emiion. Thu, the hourly CHP production hould be optimized by introducing the allowance pot price a a penalty for the caued emiion. We add thi penalty on the fuel price baed on their pecific CO 2 emiion. The penalized fuel cot for fuel k in hour h for cenario generated at the beginning of trading period t * i then ~c r k;h;;t ¼ cr k;h þ cf t;;t g k; h 2 H t. ð21þ After thi, the CHP production for a cenario generated at t * can be optimized independently of the trading proce:! X X max c p h;;t xp h;;t þ cq h Q h;;t X X ~c k k;h;;t r k;j;hx j;h;;t ð22þ.t. t¼t A. Rong, R. Lahdelma / European Journal of Operational Reearch 176 (2007) 1874 1895 1883 h2h t j2j k2k hx ;t ; E ;t i 2 XðQ ;t Þ. ð23þ The olution of thi problem combined with the realized hitory for period 1;...; t 1 determine directly the cumulated emiion E ;t and the production profit during the planning horizon in cenario generated at period t *. 4.2.2. Coordination between production and emiion trading procee No matter how the production proce i run, the trading proce mut balance the allowance with the caued emiion after the end of the planning horizon, and try to do it in the mot cot-efficient way. In principle it i poible to trade allowance arbitrarily during the planning horizon. However, becaue the DM i rik-avere, we can retrict the trading proce ignificantly. A rik-avere DM prefer a certain outcome to an uncertain outcome with the ame expected value. Therefore the DM cannot expect to gain from buying or elling an exce of allowance in a peculative manner, becaue the current (known) allowance price i the bet etimate alo for the future (uncertain) price. If the future emiion are known accurately, but there i great uncertainty about the future allowance price, the DM hould trade allowance early to meet the emiion and reduce the rik of having to pay a higher price. In contrat, if the future allowance price i known accurately, but there i great uncertainty about the future emiion, the DM hould delay the trading in order to avoid the rik of aiming at the wrong target and having to re-balance the allowance again in ubequent trading period. The latter cae i particularly important when tranaction cot are involved in the trading. In practice, we need a trading cheme that adapt imultaneouly to different degree of uncertainty both in the future allowance price and amount of caued emiion. Such a cheme will compromie between early and delayed trading to balance the allowance with the caued emiion. The baic idea of the algorithm i to balance the allowance with the emiion that are etimated uing the cenario-baed production model. Becaue the cot of the emiion rather than the amount i relevant, we etimate the emiion weighted by the allowance price c f t;;t in different period and cenario. Thi technique conider imultaneouly the uncertainty both in the price and amount. To avoid elling and buying large quantitie of allowance in ubequent period due to fluctuation in emiion etimate, we trade allowance to reach a confidence interval E low t ; Eup t intead of meeting the (weighted) expected value lðe t Þ. The confidence interval can be determined either directly from the dicrete et of cenario, or baed on a uitable probability ditribution (uch a the normal ditribution) whoe parameter are etimated
1884 A. Rong, R. Lahdelma / European Journal of Operational Reearch 176 (2007) 1874 1895 baed on the cenario. In Algorithm 1 below, we apply the latter technique. Here rðe t Þ denote the weighted tandard deviation of the cumulated emiion in cenario during the planning horizon and the confidence factor n i the number of tandard deviation that correpond to ome confidence level (1 a). Algorithm 1. Heuritic for determining allowance trade in period t *. Step 1. Calculate the confidence limit for emiion cot. E low t E up t ¼ lðe t Þ nrðe t Þ; ¼ lðe t ÞþnrðE t Þ. Step 2. Determine the target for the cumulated allowance level at the end of period t *. F t ¼ max E low t ; min ff t 1; E up t g. Step 3. Determine the allowance trade f t in period t *. ¼ F t F t 1. f t Conidering the confidence interval for the emiion etimate i important when tranaction cot are involved. The confidence factor determine the tradeoff between the trading frequency and quick reaction to variable allowance price. A mall confidence factor will caue aggreive purchae and ale to follow random variation in the emiion etimate. Thi can incur exceive tranaction cot. However, a confidence factor that i too large will diable the trading activity. Generally, the confidence factor hould be related to the tranaction cot: the higher the tranaction cot, the larger the confidence factor hould be. 4.2.3. Dealing with penalty for exceive emiion Trading in the lat period i different from the previou period, becaue thi i the lat opportunity to decide how to balance the allowance with the overall emiion. Trading after the lat period i forced and depend on the tochatic outcome of the lat period. Therefore, it will not be ufficient to aim at a confidence interval for the emiion. Intead, the producer hould try to meet the emiion target a accurately a poible in different tochatic outcome, taking into account the penalty for exceive emiion and the concave hape of the utility function. In the cenario repreentation thi mean that the producer hould determine the trading level f which maximize the expected utility of the profit. By omitting the period index t * = from the notation, we can rewrite the objective function (18) at the lat period a max Uðf Þ¼ 1 S where z ðf Þ¼Z 0 X S ¼1 Uðz ðf ÞÞ; þ cf ; f c f þ ; f þ þ c F E c F þ E þ ð25þ i the overall profit in cenario. Here Z 0 i the part of the profit that doe not depend on f, i.e., the already realized profit in the previou period plu the production profit for the lat period in cenario. Let u examine the hape of the profit function. Baed on (10) (15) we can write the profit function a z ðf Þ¼Z 0 þ cf ; maxf0; f g c f þ ; maxf0; f gþc F maxf0; F 1 þ f E g c F þ maxf0; E F 1 f g. ð26þ We can ee that the profit in each cenario i a piecewie linear function with bending point at f = 0 and f = E F 1. Furthermore, the profit function i concave, becaue the purchae price for allowance i ð24þ
A. Rong, R. Lahdelma / European Journal of Operational Reearch 176 (2007) 1874 1895 1885 higher than the ale price, i.e., c f ; < cf þ ; and c F < c F þ. Becaue the utility function i an increaing concave function, it mean that alo U(z (f )) i concave. Then alo the expected utility (24) i a convex function of f, becaue it i computed a an average over the cenario utilitie. Auming that the utility function i mooth (ha a continuou derivative), then the expected utility function will be mooth except at the bending point (f = 0 and f = E F 1 ). The optimality condition (Taha, 1992) tate that the maximum of a concave, piecewie mooth function i either where the derivative become zero, or at a bending point where the derivative change it ign. The derivative i obtained imply a the average of the derivative of each cenario: k ¼ duðf Þ=df ¼ 1 S where k ¼ duðz ðf ÞÞ=df. X ¼1 k ¼ 0; ð27þ ð28þ The cenario-pecific derivative depend on the ign of the lat period trade and whether the emiion exceed or fall below the allowance: k ¼ U 0 ðz ðf ÞÞðc F c f þ ; Þ when f > 0; E > 0; ð29þ k ¼ U 0 ðz ðf ÞÞðc f þ þ ; cf Þ when f > 0; E þ > 0; ð30þ k ¼ U 0 ðz ðf ÞÞðc f ; cf Þ when f < 0; E > 0; ð31þ k ¼ U 0 ðz ðf ÞÞðc F þ c f ; Þ when f < 0; E þ > 0. ð32þ Here U 0 (Æ) i the derivative of U(Æ) with repect to z. Whether k(f ) ha a zero depend on the price coefficient in different cenario. Normally the penalty for exceive emiion i large, which mean that c F þ c f ; > 0in(32). Becaue U0 (Æ) i poitive, thi mean that k(f ) i poitive for large negative value of f.ifk(f ) obtain negative value for f!1then there will be a zero in the range f 2 ( 1, 1). In the oppoite cae, the optimal olution i to buy an infinite amount of allowance in the lat period at price c f ; þ and ell them at cf after the lat period. Thi olution i not very likely to happen in reality, becaue it would require better information about the future allowance price than the other actor on the market have. To guarantee a bounded olution and to avoid the peculative trading, we limit the value of f between f min, which i the larget value making E þ P 0 for all of generated cenario, and f max, which i the mallet value enuring E P 0 for all of generated cenario. If k(f ) doe not change ign in that range, then we ue the end point of the range a the olution. Otherwie we find the optimal olution to the lat period trading problem by a modified binary earch (Braard and Bratley, 1996) algorithm. The binary earch mut conider dicontinuitie in k(f ). The termination condition of the binary earch mut be relaxed to top with a olution where a ufficiently narrow range for f ha been found. The algorithm for finding f i preented below. Algorithm 2. Finding the optimal olution with the penalty cot for exceive emiion. Step 1. Determine the initial interval [f min, f max ] for the binary earch. f max ¼ maxfe F 1 g; f min ¼ minfe F 1 g.
1886 A. Rong, R. Lahdelma / European Journal of Operational Reearch 176 (2007) 1874 1895 Step 2. Search the optimal allowance trade f if (k(f max ) P 0) f = f max ele if (k(f min ) 6 0) f = f min ele f i found by binary earch to atify either k(f ) = 0 or the left and right value for the binary earch i cloe enough. Step 3. Determine allowance level F at the end of the planning horizon. F ¼ F 1 þ f. 4.3. Stochatic imulation and coordination algorithm Now we ummarize the olution approach for the integrated CHP production and emiion trading planning problem. Fig. 3 illutrate the coordination between the production and trading procee in the algorithm. A time advance, the production proce i run to atify variable heat demand and react to both variable power price and allowance price, which in turn, update the emiion etimate. In the trading proce, allowance trade i determined to repond to the changed emiion etimate and then the allowance level i updated. The pecific procedure for the algorithm are given below. The notation ued in the algorithm are the ame a thoe in (1) (20). Algorithm 3. Stochatic imulation and coordination algorithm for the integrated CHP production and trading planning problem. Step 1. Initialization: F 0 = hinitial allowance allocationi. Step 2. Determine allowance trade f t and cumulated allowance level F t in each period t *. for t * 1to Step 2.1. Generate cenario (time erie) fhq h;;t c p h;;t ; cf t;;ti; ¼ 1;...; Sg panning the planning horizon for tochatic parameter uch a heat demand, power price, and emiion allowance price. (The value of parameter for 1;...; t 1 in cenario coincide with realized hitory.) Step 2.2. For each cenario, olve the CHP production model with penalized fuel price (21) (23) and obtain caued emiion. Step 2.3. Determine allowance trade f t and cumulated allowance level F t if (t * < ) Determine f t and F t baed on Algorithm 1. Time advance update AP update HD PP production proce update EE trading proce update AL AT Fig. 3. Coordination between production proce and trading proce. AP: allowance price, HD: heat demand, PP: power price, EE: emiion etimate, AL: allowance level, AT: allowance trade.
A. Rong, R. Lahdelma / European Journal of Operational Reearch 176 (2007) 1874 1895 1887 ele Determine f t and F t baed on Algorithm 2. end for Step 3. Check the balance between the emiion level and allowance level at the end of the planning horizon. E þ ¼ maxf0; E F g; E ¼ maxf0; F Eg. Step 4. Compute the value of the objective function baed on (1), (9), (16) and (17). 5. Computational reult Next we tet numerically the propoed algorithm with weighted emiion etimation (WEA). We want to reach three goal with the tet run: (1) We ae the computational peed of the WEA algorithm. (2) We tet the effectivene of the WEA algorithm by comparing it againt two impler trading cheme. (3) We apply the WEA algorithm to evaluate the relative efficiency of the fuel witch and fuel mix trategie for fulfilling the emiion compliance. 5.1. Tet problem Since the major purpoe of the numerical experiment i to tet the performance of the propoed trading algorithm, we ue a imple production model, a decribed in Section 3. We aume that the production model i convex and no dynamic contraint are preent in the ytem. The complexity of the production model mainly affect the computational peed of the algorithm. We will dicu thi a little bit later. We conider the planning problem of a CHP producer with power generation capacity around 100 MW. The hourly CHP plant model i baed on a real-life power plant model and the characteritic extreme point of the CHP plant are generated baed on different fuel choice a decribed in Lahdelma and Rong (2005). The number of extreme point in the plant varie typically from 5 to 20. In our example, the number of extreme point i 14 and 5, repectively, when the plant operate in the fuel mix and ingle-fuel mode. We apply a one-year planning horizon and divide it into 52 weekly trading period. We have generated ix tet problem baed on the hitorical heat demand of a Finnih energy company in the pat ix year and hitorical power price in Nord Pool (The Nordic Power Exchange) (Nord Pool, 2004). We aume that the heat demand and power price vary around a time erie forecat model according to a multivariate normal ditribution. The forecat model and the variation are etimated baed on hitory data. A no hitory information about allowance trade wa available when thi work wa done, we generated the allowance price baed on Brownian motion, varying from 5 to 25 /ton CO 2. Thi ha turned out to be quite realitic, although even higher allowance price have occurred during the year 2005. To imulate yearly trading, we generated 20 intance of each of the ix tet problem by ampling hitory cenario from the aumed probability ditribution. Within each hitory cenario and at each of the 52 period, we generate 20 future cenario to repreent the future uncertaintie. Thu a total of 1040 future planning problem are olved over the entire planning horizon while olving each tet problem intance. The yearly trading imulation model i not only uitable for trategic analye for the producer, but it erve alo a a benchmark for the computational peed and effectivene of the trading algorithm.
1888 A. Rong, R. Lahdelma / European Journal of Operational Reearch 176 (2007) 1874 1895 5.2. Computational peed We have made the peed tet uing the fuel mix model, which i a relatively complex model with a large number of extreme point. The production model wa olved by Power Simplex (PS) (Lahdelma and Hakonen, 2003). To reduce the effect of random variation in CPU time meaurement, each tet problem wa run 10 time and the average CPU time wa computed. All tet run were performed in a 2.2 GHz Pentium 4 PC under the Window XP operating ytem. Table 1 give the CPU time (econd) for the yearly trading model by mean of WEA a well a the time for olving a ingle intance of the yearly production model without trading. When olving the production planning model (22) and (23) at time t * for one future cenario, we mut in effect olve the production model for (53 t * ) week. During the olution of the yearly trading model (with 20 hitory cenario and 20 future cenario) we therefore mut olve 20 20 (52 + 51 + + 1) = 551,200 weekly model. The olution time for the yearly trading model hould therefore be about 551,200/52 = 10,600 time larger than for the yearly production model. From Table 1 we can compute the average ratio of 8816, which i quite cloe to the theoretically derived value. The light advantage in the actual implementation i due to aved initialization overhead when olving a large number of imilar problem. By comparing the theoretical ratio with the empirical reult, we can conclude that the computational peed of the propoed algorithm i principally determined by the time for olving the production model. Time performance of algorithm in more complex etting can be etimated baed on the time increae factor oberved in the tet run with comparable model. The time increae factor i the ratio between the olution time for the complex model and for the imple etting. For example, in non-convex planning, Makkonen and Lahdelma (2006) reported a time increae factor of 70, and for problem with dynamic energy torage contraint Lahdelma and Makkonen (1996) reported a time increae factor of 13. In a combined etting uch time increae factor may be multiplied, reulting in olution time of 1 day for the nonconvex yearly trading model with energy torage contraint. Thi i barely reaonable in trategic analye uing the PS algorithm. With the ECON/ECOFF (Rong and Lahdelma, in pre) and EBB (Rong and Lahdelma, 2005c) algorithm we can expect much horter olution time. In on-line trading with the preented method, it i only neceary to olve at each period t * the production model for the remaining part of the year uing the 20 (or more) cenario and to determine the weekly allowance trading according to Algorithm 1 or 2 (lat period). In thi etting the longet olution time will be 20 time the olution time for the yearly production model. Thi mean that we can expect olution time of 0.2 econd in the implet etting and about 3 minute in the combined complex etting. 5.3. Effectivene of the algorithm To tet the effectivene of the propoed WEA method, two comparion are made. Firtly, we compare it againt the direct (non-weighted) etimation algorithm (DEA) in our early work (Rong et al., 2004). DEA Table 1 CPU time (econd) for yearly trading by WEA and for olving ingle-yearly model Model Yearly trading Yearly production model A_1 102.0 0.0116 A_2 98.4 0.0111 A_3 102.7 0.0116 A_4 98.3 0.0113 A_5 97.6 0.0109 A_6 97.4 0.0112 Average 99.4 0.0113
A. Rong, R. Lahdelma / European Journal of Operational Reearch 176 (2007) 1874 1895 1889 ue caued emiion directly a an etimate for controlling the allowance level. Secondly, we compare WEA againt an artificially determinitic trading algorithm (DTA). In DTA, the trading cot are etimated by c f ð1 dþðe F 0 Þ, where c f i the average allowance price over the entire planning horizon, d i the ratio of the tranaction cot to the allowance price, E i the caued emiion, and F 0 i the initial allowance allocation a decribed in Section 3. Thi comparion i to tet the trading efficiency of WEA. The objective function of the integrated CHP production and trading planning problem i to maximize the expected utility of the profit. The utility function play an important role in characterizing the rik attitude of the DM. However, the pecific value of utility function i not very meaningful. The following quantitie are ued to evaluate the performance of algorithm and the different emiion compliance option (e.g., fuel witch and fuel mix). (a) Turnover (TO) conit of the revenue from elling the produced heat and power plu the value of the initial allowance allocation (c f F 0 ). (b) Certainty equivalence of profit (CEP) i the profit that correpond to the expected utility that i maximized. (c) Profit-to-turnover ratio (PTTR) i 100 * CEP/TO %. To evaluate the effectivene of the algorithm, we compare the different algorithm in term of different fuel choice at different tranaction cot level of emiion trading. We have three fuel choice and three tranaction cot level: FH FL FM fuel with higher pecific CO 2 emiion (e.g., coal) fuel with lower pecific CO 2 emiion (e.g., natural ga) fuel mix in which there are contraint on the maximum amount of high-emiive and low-emiive fuel Three tranaction cot level are d = 0 (no tranaction cot), 5% (moderate tranaction cot) and 10% (high tranaction cot) repectively. T_L denote the tranaction cot level in the ubequent table reporting the reult. For effectivene comparion, we ue the cae with the higher emiion fuel a a reference. That i, the turnover of the fuel with higher emiion act a the common denominator when the PTTR for different fuel choice are calculated. The performance of WEA depend on the etting of the confidence interval in emiion etimation. The confidence interval i determined by the confidence factor n, which i choen heuritically. Generally, the higher the tranaction cot level, the higher the confidence factor hould be. Baed on experiment, fuel choice alo ha ome effect on n. Fuel with lower emiion (FL) react well to the higher confidence factor regardle of the tranaction cot level. Thi may be explained by the penalized fuel price (21). For FL, the fuel price i higher and the pecific CO 2 emiion i lower. That mean the emiion penalty cot (related to allowance price) account for a relatively mall portion in the penalized fuel price. Thu, FL i le enitive to the allowance price a compared with the fuel with higher emiion (FH). A a reult, a higher confidence factor i needed to guarantee a ufficient buffer even though when the tranaction cot are lower. Table 2 how the confidence factor etting for different fuel choice at different tranaction cot level. Table 3 how the PTTR difference between WEA and DTA for different fuel choice at different tranaction cot level. We can ee that the improvement of WEA over DEA i ignificant. Three factor contribute to the improvement. Weighted emiion etimation combined with appropriate choice of confidence factor in etimation can ugget a more favorable volume of allowance trading baed on the allowance price. Introduction of the optimization procedure in the lat trading period can reduce the penalty cot for exceive emiion at the end of the planning horizon. Fig. 4 illutrate the effect of the
1890 A. Rong, R. Lahdelma / European Journal of Operational Reearch 176 (2007) 1874 1895 Table 2 Confidence factor etting for different fuel choice at different tranaction cot level T_L Fuel choice FH FL FM 0% 0 3 0 5% 1 3 1 10% 1 3 2 Table 3 The PTTR difference between WEA and DEA for different fuel choice at different tranaction cot level (% point) T_L Problem Fuel choice FH FL FM 0% A_1 1.07 0.54 0.45 A_2 0.85 0.75 0.40 A_3 1.06 0.57 0.26 A_4 1.00 0.72 0.41 A_5 1.91 2.09 0.87 A_6 0.69 0.30 0.31 Average 1.10 0.83 0.45 5% A_1 2.59 1.17 2.24 A_2 3.40 1.82 2.68 A_3 2.71 1.60 2.05 A_4 1.56 1.08 1.88 A_5 2.79 2.56 2.71 A_6 1.25 0.67 1.45 Average 2.38 1.48 2.17 10% A_1 4.38 1.80 4.72 A_2 5.83 2.89 5.41 A_3 4.83 2.63 4.66 A_4 2.72 1.44 3.74 A_5 4.05 3.03 5.40 A_6 2.16 1.03 3.07 Average 4.00 2.14 4.5 weighted etimation method and optimal procedure on the trading cot when there are no tranaction cot. Fig. 5 illutrate the impact of confidence factor on the trading cot when tranaction cot are involved. The relatively active trading activity for WEA in Fig. 4 implie that WEA can react better to uncertainty of allowance price than DEA. We alo ee the lower trading cot for WEA at the end of the planning horizon due to the adoption of optimal procedure. (The penalized trading cot of exceive emiion for WEA and DEA are 0.058 and 0.45 million, repectively.) The total trading cot for WEA and DEA are 3.65 and 5.81 million, repectively. Fig. 5 how the profile of the trading proce ubject to tranaction cot. With tranaction cot, overly active trading activity i not encouraged. If the confidence factor i maller (WEA0, n = 0), the trading frequency i too high, which implie higher tranaction cot. On the other hand, if the confidence factor i higher (WEA2, n = 2), thi reduce the trading activity and the proce cannot react to allowance price variation well. In thi example, WEA1 (n = 1) i an appropriate choice and provide a good tradeoff between trading frequency and the reaction to the uncertainty of allowance price. A a reult, the total trading cot for WEA0, WEA1 and WEA2 are 6.04, 4.12 and 5.97 million, repectively.