Proceedings of Power Tech 2007, July 1-5, Lausanne

Transcription

1 Second Order Stochatic Dominance Portfolio Optimization for an Electric Energy Company M.-P. Cheong, Student Member, IEEE, G. B. Sheble, Fellow, IEEE, D. Berleant, Senior Member, IEEE and C.-C. Teoh, Student Member, IEEE, J.-P. Argaud, M. Dancre, L. Andrieu, and F. Baron Abtract Thi paper preent a framework of portfolio optimization for energy market from an electric energy company perpective. The obective of thi reearch i to determine the bet poible invetment plan by combining two potentially conflicting portfolio invetment goal. Firt, given the general characteritic of the generating aet and forecat of market variable, the deciion maker elect an efficient et of portfolio by optimizing the expected portfolio return. Secondly, an optimal portfolio i choen baed on company rik profile. Thi rik i controlled by guaranteeing that the portfolio model ha econd-order tochatic dominance (SSD) over the cumulative ditribution of a minimum tolerable reference ditribution. Deciion criteria are then applied to obtain an optimal and robut portfolio. The propoed approach i ued to determine the amount of optimal market hare value that maximize the expected value of the profit. Thi i performed by treating rik a a ditribution that repreent the minimum expected profit acceptable by the energy company. Reult how that different rik profile lead to different optimal portfolio. The optimal portfolio which give the highet expected profit may not have the bet robutne. Thi approach i alo applicable to problem characterized by other ource of epitemic uncertainty beide unknown dependencie. Index Term Portfolio optimization, econd-order tochatic dominance, interval analyi, and epitemic uncertainty. I. INTRODUCTION The pioneering work in portfolio optimization began with Manucript received April 5, 007. Thi work wa upported in part by Electricité de France. M.-P. Cheong i with Portland State Univerity, OR 9707 USA ( meipc@pdx.edu). G. B. Sheble i with Portland State Univerity, OR 9707 USA ( gbheble@pdx.edu). D. Berleant i with Univerity of Arkana at Little Rock, AR 704 USA ( berleant@gmail.com). C.-C. Teoh i with Portland State Univerity, OR 9707 USA ( ccteoh@pdx.edu). J.-P. Argaud i with Electricité de France, France ( eanphilippe.argaud@edf.fr). M. Dancre i with Electricité de France, France ( mathieu.dancre@edf.fr). L. Andrieu i with Electricité de France, France ( laetitia.andrieu@edf.fr). F. Baron i with Electricité de France, France ( francoi.baron@edf.fr). the propoition of mean variance model by Harry Markowitz [7] in 95. Since then, many related reearch work ha been focued on finance and economic. In the recent year, the adoption of uch concept ha increaed in the retructuring electric power ector. A more non-probabilitic factor affect the electricity market, not many reearch focued on olving the portfolio election problem when dependency relationhip between portfolio egment are unknown, in particular, when the market hare i repreented by a bounded variable with no known underlying ditribution. Becaue knowledge that upport invetment deciion i often limited by what information i available, inference under condition of epitemic uncertainty i important to do in order to upport optimized portfolio. Epitemic uncertainty in the context of thi paper refer to a lack of knowledge about the randomne of the deciion variable. In thi paper, uch uncertainty i quantified a bounded familie of probability ditribution. Applying bounded familie of ditribution to electric power bidding problem exemplifie what may be expected in engineering from applying uncertainty quantification. There i a growth of interet in uncertainty quantification by ournal pecial iue [-]. Application to problem in electric power [3-4] follow naturally from the inight and invetigation of other reearcher, who have found that uncertainty quantification ha applicability to power problem characterized by evere uncertainty. Prominent technique include interval [5-6] and fuzzy method [7-8]. The well recognized need for deciion in the preence of evere uncertainty, coupled with the grounding of our approach in the mathematically well-founded theory of probability, upport it ue in addreing important problem in electric power. Guidance can then be obtained regarding invetment deciion under condition of uncertainty in which tandard method would require extra, unutified aumption. The volatility of electricity price caued the uncertain return (profit or lo) for an electricity portfolio. Often time, tandard ditribution may not accurately fit the ditribution of return. Thi call for a methodology that doe not depend on the type of ditribution hape of the return but rather on the entire cumulative ditribution function (CDF) of the return. Thi paper addree thi problem by optimizing a portfolio while atifying the rik contraint which i repreented by the econd order tochatic dominance contraint. In the propoed approach, generating unit are modeled a a

2 erie of option. The return i repreented by park (ga) or dark (coal) pread depending on the fuel ued. There are generating unit for thi analyi. They include one nuclear unit and 3 thermal unit (coal-fired and ga-fired). A two-tep optimization approach i propoed to find the optimal financial portfolio for production. Thi problem i modeled from a viewpoint of a utility that ervice 4 different indutrie. The firt model aume that thi utility company i obligated to erve the 4 ector. The econd model relaxe thi contraint to allow the utility to ell to the forward market hould it ee profitable. The following ection decribe the portfolio optimization model. Section III define the robutne and tochatic dominance concept. Background ection II and III are borrowed and modified from a recently ubmitted ournal paper. The maor difference in thi paper a compared to the ubmitted ournal paper include the aumption the um of all weight in the portfolio doe not have to add up to, the return for each egment wa generated uing park and dark pread, and Anderon-Darling goodne-of-fit meaure are ued to fit ditribution for each return egment. Section IV decribe the interval algorithm. Section V and VI dicu concluion and future work. II. PORTFOLIO OPTIMIZATION MODEL Optimal portfolio are often identified by finding the weight of the portfolio egment uch that a mean-rik obective function i maximized [4]. Formally, the problem to be conidered i to find uch a portfolio given the contraint: ~ R = x( ) r( ) f R () lb x ( ) x( ) x( ) () R: return ditribution of the optimal portfolio : portfolio egment x( ): weight of egment in the portfolio r( ): return ditribution of optimal portfolio egment i R ~ : a given reference curve that repreent the minimum tolerable return ditribution ( rik limit ) The ymbol f deignate tochatic dominance of ith i order. A an additional contraint et, weight x( ), x( ), x( 3 ),, x( n ) ha to be within it upper and lower bound, x() lb and x() ub. Each weight may be required to be within ome interval in order to enforce a balance acro egment, a might be pecified by a company buine model contraint and invetment policie. Thi model alo aume that the firm ha no retriction in borrowing money when it ee profitable to optimally invet in a particular ector. The firt tep i to generate a et of optimal portfolio to earch within for the bet. A tandard approach baed on mean and rik and parameterized by rik poition [7] i ued. Let the deirability of a portfolio return random variable r be ub determined by the following function of a parameter z decribing the importance of rik: f(z,r) = mean(r) z *rik(r) (3) The following equation build on the concept of Eq. 3, and tate that given a rik poition z, optimal return ditribution i obtained by: y OPT( z) = up( µ zσ ) y Y y Y: et of portfolio y complying with contraint () and () µ y : expected return of a portfolio y σ y : variance of the return of a portfolio y z: degree of rik averion It i poible to account for propertie of portfolio variation beide variance [], but zσ y i neverthele widely ued to model rik poition. (4) III. STOCHASTIC DOMINANCE AND ROBUSTNESS Stochatic dominance contraint can be thought a a ranking tool with no aumption made on the hape of the return ditribution. It can be ued a a profit/rik indicator for any given portfolio. Thu, a given portfolio return ditribution can be teted for compliance with tochatic dominance contraint. By definition, a return of a portfolio X tochatically dominate another portfolio Y, to the firt order, if the following i true: X f Y P( X A) P( Y ) (4) A P( X A) : Cumulative ditribution function of X Similar to the firt order tochatic dominance, the condition for econd order tochatic dominance contraint i given by: A X f Y P( X B) db P( Y B) db (5) B SSD contraint i computed uing numerical integration or by umming area of trapezoid under the curve. The ize and number of trapezoid to um i determined by the tep ize choen for the integration proce. An optimal portfolio might or might not comply with an additional requirement that it ha tochatic dominance over a given reference return, R ~. The SSD contraint f enure the dominant portfolio i preferred by any rik avere player [3], and ha the additional virtue of being le contraining than the FSD contraint. Thu, SSD contraint i ued to be conitent with the degree of rik averion pecified in Eq. 3. Robutne i defined a the amount by which a portfolio dominate a reference curve (robutne would be negative if it

3 doe not dominate). By teting variou optimal portfolio for robutne, one with the highet robutne can be identified. Alternatively, one with the highet expected return that alo meet the SSD contraint could be found. In either cae, the trategy i to earch among a et of optimal portfolio provided by an under-contrained optimization problem for the one that i bet according to a econd criterion. SSD i computed by the minimum horizontal ditance between the integral of two cumulative ditribution. In other word, SSD meaure how much one curve for the integral of a ditribution can be moved toward another one along the x- axi before the two curve touch. SSD formalize the amount of eparation between the integral of two ditribution. The amount of epitemic uncertainty i defined by α. The ue of α follow the convention in Info-Gap Theory [9]. In thi paper, maximum of α value i obtained by calling Eq. 6. Thi value determine the robutne of the optimal portfolio. max F α new = αfleft + ( α) Fbet gue F reference F new : New curve obtained with max α. Thi i the α curve hown in Fig. 6. F left : Left envelope baed on the addition of 4 portfolio egment, given that their dependencie are unknown. Thi refer to the left curve in Fig. 6. F bet-gue : The curve baed on the addition of 4 independent portfolio egment. Thi i the bet-gue curve hown in Fig. 6. F reference : Thi i the minimum tolerable rik level. Thi i the reference curve in Fig. 6. IV. INTERVAL ALGORITHM The unknown dependency among the portfolio egment i computed uing Statool [9] that ue the Ditribution Envelope Determination (DEnv) algorithm. Thi algorithm wa developed by Berleant et al [0]. Thi ection briefly introduce the algorithm. Firt, a dicrete oint ditribution tableau i contructed. Each input i dicretized by repreenting it probability denity function (PDF) with interval (ee figure below). Figure. Dicretization proce. dicretization Thee dicretized input form the marginal of the oint ditribution tableau, which et the contraint for each interior cell. Each interior cell i repreented by a range for it probability ma ditribution. Next, the algorithm find the bound on the cumulative probability of the derived ditribution. The left envelope i obtained by the derived ditribution that rie the fatet and the right envelope i (6) obtained by the derived ditribution that rie the lowet. Thi i performed iteratively uing linear programming to find the maximum and minimum value to form the left and right envelope. Fig. illutrate the computation graphically. The operation i the addition of two random variable, X+Y=Z. X and Y are hown in PDF form and Z i repreented in both PDF and CDF form. X PDF + Y PDF = Figure. Illutration of the addition of two random variable, X and Y, given their dependency i unknown. V. ANALYSIS WITH INDUSTRIAL CUSTOMERS Z PDF Z CDF The portfolio optimization model accept two maor clae of input: (i) the deciion variable, which are the market egment and their profit ditribution, and (ii) a reference ditribution. The reference ditribution refer to the cumulative ditribution of a minimum tolerable return. Although a claic approach i to optimize the expected portfolio return ubect to a rik averion factor, thi model ha been modified to optimize baed on a minimum tolerable reference ditribution, the concept of econd-order tochatic dominance and Information-Gap Deciion Theory, and market hare contraint for different cutomer. The deciion variable here refer to portfolio egment, which are the market hare for cutomer within the following market: the indutry market ( ), middle market ( ), ma market ( 3 ), and ditribution market ( 4 ). Each market hare ha an allowable range for level of invetment determined by upper and lower limit. Suppoe that the deciion variable for each market i repreented by x i. Then the market hare i contrained by the following limit: x [ 0.7,.0]; x [0.7,.0]; x3 [0.6,0.8]; x4 [0.8,.0] Each number x i repreent a percentage of the entire European demand in market i. For example, in the indutry market ( ), the total European demand i 0767 MW/day, o the market hare limit for thi egment are [0.7*0767, *0767] = [ , 0767] MW/day. Therefore, there i no contraint indicating that the um of all market hare ha to add up to. Thi i true auming that the European firm i able to ell hort and would be able to borrow money when needed. Notice that thi i different than mot portfolio problem becaue it i commonly aumed that all the weight are fraction of the total budget that ha to add up to. Different cutomer pay different price to the company.

4 The price charged to each market i given by: 3 4 = 37 = + = 55 = : Daily pot price of electricity The price above are ued to calculate the revenue for each market. For example, the revenue for indutry market egment i in range [37*7,536.90, 37*0,767] = [78,865.3, 398,379]. The cot of production i baed on the average cot for all unit if they are committed to ell. The unit are committed baed on the leat cot dipatch order. There are 4 generating unit. If the 3 unit with the cheapet production cot can meet the demand for that day, then only 3 unit are committed and the cot of production i the average cot of the 3 unit time the demand for that day. Average cot i ued becaue there i no one-to-one mapping from a pecific unit output to a pecific market. In other word, if the ga and nuclear unit are committed to ell to the indutry market ( ) and middle market ( ) egment, there i no information on whether i getting it electricity from the ga generating unit or the nuclear generating unit. Profit ditribution baed on average cot of production for each egment i generated to obtain the expected return ditribution. Since the demand data for the entire year exhibit eaonality, the profit ditribution exhibit eaonality a well (ee Fig. 3). The Anderon-Darling tatitic (A ) i a tet that compare the fit of an oberved CDF to an expected CDF. Formally, for n number of obervation, it i defined a: A n (i ) (ln F( X + i = i ) ln( F( X n + = n n F(X i ): CDF of the ordered data, X i )) (7) Thi i a one-ided tet the critical value depend on the pecific ditribution that i being teted. The hypothei for the ditribution to follow a pecific hape i reected the tet tatitic i greater than the critical value. From Table, the percentage pread with repect to the correponding mean value i on the order of 0%. Although they are in thouand, the value are relatively mall if compared to the mean value. Demand Sector TABLE THE DISTRIBUTION FIT FOR EACH DEMAND SECTOR Ditribution Mean Standard Deviation Deviation Beta 3, , % Beta 368, , % 3 Student T 4, , % 4 Beta 9, , % Fig. 4 how the optimal value for market hare for each demand egment for different degree of rik averion (z = 0 to z = 5). The market hare converge to their lower bound when z approache 5. (Note: Although z<0 i not generally ued for the degree of rik averion, the market hare for all negative z-value would be at their upper bound becaue the variance for the demand ector are high enough that the obective function would benefit from their conequent rikine.) Market Share v. Z Value. Market Share Figure 3. Annual return ditribution for indutry market ( ), middle market ( ), ma market ( 3), and ditribution market ( 4). Total return, which i the ummation of all return are alo hown. To ue the yearly data without accounting for eaonality would produce inaccurate reult. Since thi paper focu i not on aduting the data for eaonality, a naphot of one month data i ued for the analyi, pecifically the month of July data. July data wa choen becaue it i typical of a peak period in the ummer. Thi data i ued a an input to the portfolio optimization model together with an arbitrarily elected reference ditribution. Table how the ditribution for each demand ector baed on the Anderon-Darling goodne-of-fit meaure [] z-value x x x3 x4 Figure 4. Market hare for different rik averion value (z value) for indutry market (x ), middle market (x ), ma market (x 3), and ditribution market (x 4).

5 Market Share v. Z Value. Market Share z-value x x x3 x4 Figure 5. Market hare for different z value from 0 to for the indutry market (x ), middle market (x ), ma market (x 3), and ditribution market (x 4). However, the optimal weight change for very mall poitive value of z. Fig. 5 illutrate how the weight for each demand ector move from their upper limit to their lower limit from z = 0 to z = The weight change over a mall range of z-value. Since the reference ditribution wa arbitrarily choen, a few value for z with different weight are elected to illutrate how one can ue the SSD and alpha metric to elect the z-value that give the bet reult. The following table give the optimal value for x, x, x 3, x 4, SSD, and max α, and the expected return for z = 0, , , , 0.003, and through 5. TABLE THE OPTIMAL MARKET SHARES FOR EACH DEMAND SECTOR GIVEN Z z x x x 3 x TABLE 3 THE OPTIMAL SSD, MAX ALPHA AND EXPECTED PORTFOLIO RETURN z SSD Max α µ 0 86, ,00, , ,64, , ,04, , , , , , ,40.00 Under the pecified aumption and the reference ditribution ued, the bet portfolio occur when z i 0. The weight for the middle market (x ) decreae firt followed by the indutry market (x ), ditribution market (x 4 ) and ma market (x 3 ). A we will dicu later, thee reult are due to the fact that there i no carcity. Fig. 6 how the left envelope of the unknown dependency curve, the bet-gue curve, the reference curve and the max alpha curve (α=.4) for z=0. Figure 6. Left-envelope curve, bet-gue curve, reference curve, and the max α (.4) curve for z=0. The left envelope curve i the curve generated by the addition of all profit egment without making any aumption about their dependency relationhip. Since the left envelope provide a wort cae bound for portfolio performance under uncertainty due to unpecified dependencie among egment, a high value of i aigned to α a the meaure of uncertainty expreed by the left envelope, ignoring the right envelope henceforth. The bet-gue curve i the curve uing the optimal weight for z=0, the weight are all at their upper limit. The reference curve i an arbitrary curve. The max alpha curve define the robutne meaure of the portfolio. VI. FUTURE WORK Other contraint uch a reource minimum capacity, ramp up rate and maintenance (or poible outage) uch that the firm cannot ell the committed amount of generation, are not included in thi model. If thee are included, the reult could be far more beneficial to a firm. In addition, tartup cot and the forced outage rate for each unit hould be included in order to be more complete. Another extenion to thi work would be to look into identifying the coherent rik meaure to ue in the SSD contraint. Seaonality of the data ha to be accounted for and more work need to be done to adut for thi eaonal trend. Multi-period hould be a natural extenion to thi paper a portfolio problem for a company hould be olved in a multiperiod manner. ACKNOWLEDGMENT The author would like to thank the ponor, in particular, J.-P. Argaud, M. Dancre, L. Andrieu, and F. Baron for their upport and contructive comment in the completion of thi reearch proect. REFERENCES [] Berleant, D., ed., Special iue on dependable reaoning about uncertainty, Reliable Computing, 003, 9 (6). [] Helton, J., and W.L. Oberkampf, ed., Special iue on epitemic uncertainty, Reliability Engineering and Sytem Safety, 004, 85 (-3). [3] Sheblé, G. and D. Berleant, Bounding the compoite value at rik for energy ervice company operation with DEnv, an interval-baed

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EDF Statitic for Goodne of Fit and Some Comparion, Journal of the American Statitical Aociation 69 (974), pp and Finance in 00. Hi reearch interet include artificial life technique for optimization; proce modeling of power ytem operation and market; proce modeling of dynamic deciion and optimization upport ytem; real option analyi of aet valuation; probabilitic analyi of contract compliance and ancillary ervice requirement; data mining of market and of health care ytem uing artificial life technique, upply chain management in the energy indutrie, bidding trategie baed on economic microanalyi, time erie forecating, and financial engineering. D. Berleant i an Aociate Profeor in the Dept. of Information Science, Univerity of Arkana at Little Rock. He received hi Ph.D. from the Univerity of Texa at Autin in 99. Hi reearch interet focu on imprecie probabilitie, technology foreight, and bioinformatic. Chin-Chuen Teoh (S 07) i currently puruing a Ph.D. degree in the Department of Electrical and Computer Engineering at Portland State Univerity, Portland Oregon. He attended Iowa State Univerity, receiving hi M.S. degree in Electrical Engineering in 004 and M.S. degree in Economic in 006. Hi reearch interet include application of financial tool for energy ytem expanion, and real option analyi of aet valuation. J.-P. Argaud i Reearcher in the R&D Diviion of Electricité de France (EDF). He received hi "Maîtrie" in Mathematic in 989, hi Engineer degree from Ecole Centrale de Pari (french "Grande Ecole") in 99 and hi PhD from the Univerity of Pari Dauphine in 995. Hi reearch interet focue on dicrete and continuou optimization, including data aimilation, rik management and application of finance for electricity. M. Dancre i a Rik Analyt in the Dept. of Energy Market Rik Control at EDF. He received hi Ph.D. from the Univerity of Computer Science at Aix-Mareille (France) in 999. Hi reearch interet focue on rik aement and policy, energy portfolio management, energy market price modeling. L. Andrieu i a reearch engineer with the Dept. of Optimization, Simulation, Rik and Statitic, R&D Diviion, Electricité de France. She received her Ph.D. from the Ecole Nationale de Pont et Chauée (France) in 004. Her reearch interet focu on financial engineering application (portfolio optimization), rik management (rik in energy market) and tochatic optimization (motly on probabilitic contrained optimization and tochatic gradient algorithm). M.-P. Cheong received her B.S. and M.S. in Computer Engineering from Iowa State Univerity in year 00 and 004. She alo received her M.S. in Economic in 006 from Iowa State Univerity. She i currently puruing Ph.D. in Electrical Engineering at Portland State Univerity. Her reearch interet include portfolio optimization and data mining technique applied to the dynamic electricity market. Gerald B. Sheblé (F 98) i the Maeeh Profeor at Portland State Univerity. He attended Purdue Univerity, receiving BSEE (97) and MSEE (974). He received hi Ph.D. in 985 from Virginia Tech. Dr. Sheblé received hi MBA from the Univerity of Iowa, pecializing in Economic