Contract selection problem in singapore electricity market

CONTRACT SELECTION PROBLEM IN SINGAPORE ELECTRICITY MARKET CHEN LIQIN NATIONAL UNIVERISITY OF SINGAPORE 2011 CONTRACT SELECTION PROBLEM IN SINGAPORE ELECTRICITY MARKET CHEN LIQIN (B.Eng, SJTU) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF INDUSTRIAL AND SYSTEMS ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2011 Acknowledgements This thesis would not have been possible without the help from people who keep supporting me throughout the journey of my graduate studies I would like to take this opportunity to express my gratitude and appreciation to all of them I owe my deepest gratitude to my main supervisor, Prof Tang Loon Ching, for his continuous guidance and cordial encouragement, without which I would have never achieved any accomplishment herein I would also like to thank Dr Chen Nan, who has provided insightful advices in my research I am also very grateful to other professors and my colleagues in ISE department of NUS for their kindly support and help during my graduate studies They are Dr Hung Hui-Chih, Prof Goh Thong Ngee, A/Prof Lee Loo Hey, A/Prof Poh Kim Leng, Fu Yinghui, Mu Aoran, Wang Qiang, Li Juxin, and Zhou Qi Last but not least, I would like to thank my parents and Mr Wu Huihui for their unfailing trust They have always believed in me, and I could never reward them enough for their continuing support iii Table of Contents Acknowledgements iii Table of Contents iv Summary viii List of Tables x List of Figures xi List of Notations xiv Chapter 1: 1.1 Introduction Background of Singapore electricity market 1.1.1 Market structure 1.1.2 Available contracts 1.2 Motivations of the study 1.3 Objectives and scopes 1.4 Organization of the thesis Chapter 2: 2.1 Literature Review The application of portfolio theory in the electricity sector 2.1.1 Power portfolio allocation problem 2.1.2 Electricity contract selection problem 12 2.2 Fuel oil price models and model selection criteria 16 2.3 Multi-period stochastic program and scenario generation 18 Chapter 3: Single-Period Contract Selection Problem 20 iv 3.1 Introduction and problem description 20 3.2 Decision framework 21 3.3 Problem formulation 22 3.3.1 Formulation of contracts 22 3.3.2 Single-period contract selection model 29 3.4 Numerical analysis 34 3.4.1 Efficient frontier 35 3.4.2 Risk aversion coefficient 37 3.4.3 Load profile 38 3.4.4 Fuel oil price volatility 40 3.4.5 Pool price volatility 41 3.5 Limitations and conclusions 42 Chapter 4: Multi-period Contract Selection Problem 44 4.1 Decision framework 44 4.2 Problem formulation 48 4.3 Oil price forecasting model selection 50 4.3.1 Oil price forecasting models 50 4.3.2 Model selection criteria 52 4.3.3 Model selection procedure 53 4.4 Characterizing Other Uncertainties 56 4.4.1 Forward fuel oil prices 56 v 4.4.2 Load profile 56 4.4.3 Pool prices 58 4.5 Scenario tree construction 60 4.5.1 Scenario tree construction procedure 60 4.5.2 Scenario generation 61 4.5.3 Scenario reduction 62 4.6 Numerical analysis 64 4.6.1 Efficient frontier 65 4.6.2 Significance level 66 4.6.3 Load volatility 67 4.6.4 Fuel oil price volatility 67 4.7 Limitations and conclusions 70 Chapter 5: Decision Support System for Consumers 72 5.1 Motivation 72 5.2 The decision support system 73 5.2.1 The framework of the decision support system 73 5.2.2 The relationship of electricity costs and fuel oil prices 76 5.2.3 The optimization model 79 5.3 An illustrative example 80 5.3.1 Input information 80 5.3.2 Forecasts based on most recent data 82 vi 5.3.3 5.4 Forecasts based on given scenarios 82 Limitations and conclusions 85 Chapter 6: Conclusions and Future Research 87 6.1 Main findings and contributions 87 6.2 Suggestions for future research 89 References 90 Appendix A: Model Selection Criteria 94 Appendix B: Preliminary Data Analysis 98 vii Summary Deregulation of Singapore electricity market has benefited electricity consumers in many aspects In addition to lower electricity prices, consumers are entitled to select from various types of electricity contracts offered by energy suppliers, which may have different risk characteristics This thesis investigates the optimal decisionmaking among these contracts for consumers in Singapore electricity market, with the objective to minimize total electricity costs taking into account the potential risks consumers are exposed to A preliminary study is firstly conducted to investigate the risk characteristics of available contracts, employing Mean-Variance framework from Modern Portfolio Theory Through numerical experiments, the impacts of different input parameters are investigated To extend the previous single-period model to multi-period cases, a stochastic programming model is proposed, employing Conditional Value-at-Risk as the risk measure of electricity costs Scenario generation and reduction techniques are employed to resolve the stochasticity of fuel oil prices and other uncertainties, through which the stochastic programming model can then be approximated by a deterministic linear programming model and solved Based on the proposed multi-period integer programming model, a practical tool operating in spreadsheet is developed to aid the decision making process This tool can provide optimal solutions to the deterministic programming model based on viii forecasted oil prices or under given scenarios, thus it can serve as a useful reference for consumers with subjective perspectives of oil price trends ix List of Tables Table 3-1 The price form and total electricity cost of available contracts 26 Table 3-2 The expected value of the total electricity cost 27 Table 3-3 The variance of the total electricity cost 27 Table 3-4 Covariance matrix 28 Table 3-5 Input parameters 35 Table 4-1 AICc for time series models 54 Table 4-2 Out-of-sample (12-mth ahead) model comparison for monthly data 55 Table 4-3 Out-of-sample (4-quarter ahead) model comparison for quarterly data 56 Table B-1 Ljung-Box-Pierce test result for monthly return of HSFO spot price 100 Table B-2 Ljung-Box-Pierce test result for quarterly return of HSFO spot price .100 Table B-3 LM Eagle’s test result for monthly return of crude oil 101 Table B-4 LM Eagle’s test result for quarterly return of crude oil 101 x Chapter 6: Conclusions and Future Research Chapter 6: Conclusions and Future Research The main purpose of this thesis is to develop a mathematical model to help contestable consumers in Singapore electricity market select optimal contract portfolio to minimize the total expected electricity cost and the associated risk In this chapter, the main findings and contributions of this study are summarized, and the implications are discussed Subsequently, suggestions for possible directions to extend this study in future research are provided 6.1 Main findings and contributions In a deregulated electricity market, consumers are faced with the decision making problem of selecting electricity contracts In this study, we have proposed a portfolio approach to the contract selection problem of contestable consumers in Singapore electricity market, which can minimize the total electricity cost and accommodate the risk preferences of consumers In the preliminary study, Mean-Variance framework has been chosen to model the single period contract selection problem Multi-period contract selection problem has been formulated as a stochastic programming model, adopting Mean-CVaR framework For the reason that fuel oil price is the dominant factor for electricity prices in Singapore electricity market, mean reversion process is chosen to model the fuel oil price movements Scenario generation and reduction techniques have been applied to resolve the uncertainties, and relax the stochastic programming model into a deterministic linear programming model Numerical experiments have been conducted for both single-period model and multi-period model to investigate the impacts of different input parameters on the optimal solutions 87 Chapter 6: Conclusions and Future Research In addition, a practical decision support system that operates in spreadsheet has been developed to provide a useful reference for consumers The main findings and contributions of this study include: (1) This thesis is the first study to take consumers’ perspective and the first attempt to address the contract selection problem in Singapore electricity market It provides consumers with a quantitative contract portfolio solution instead of a single contract solution so as to minimize the total expected cost and risk jointly (2) In terms of methodology, this study differs from the previous studies in taking fuel oil price movements into account, as it establishes the link between fuel oil prices and electricity prices and reflects the real market situation The adoption of CVaR as the risk measure in the multi-period model also reflects the concern of the consumers to the largest extent, which is not only the average cost they are faced, but also the cost in the worst case scenarios (3) Numerical experiments have been conducted to investigate the impacts of different input parameters, such as the load volatility, and the parameters of mean reversion model for fuel oil price movements It can be interpreted from the result that, consumers with more volatile load profile need to choose less risky contracts The major reason is that to achieve the trade-off between the risk and the cost, consumers subject to a higher risk need to sacrifice the potential savings in the cost in order to hedge against the risk they are exposed to 88 Chapter 6: Conclusions and Future Research (4) Another implication from the result is that, if a consumer is able to shift certain amount of electricity consumption from peak periods to off-peak periods, it will benefit both the consumer and the retailer and result in a winwin situation This motivates the future study of load management or demand management, which mainly deals with optimizing the cost savings from load shifting (5) The developed spreadsheet decision support system can be easily used, which can serve as a useful reference for decision makers Consumers can also forecast their electricity cost based on their subjective belief of the fuel oil price movements using the decision support system 6.2 Suggestions for future research There are some possible directions to extend this study in future work, as listed below: (1) It is obvious that the quality of the forecasts of electricity costs highly depends on the quality of the fuel oil price forecasting model A robust modelling approach may be adopted to mitigate the sensitivity to errors (2) As can be seen from the sensitivity analysis results, a flatter load profile leads to a smaller total expected cost This provides consumers an incentive to shift partial load from peak periods to off-peak periods, or to store certain amount of electricity in off-peak periods and to use it in peak periods These approaches of cost savings may be considered jointly with the contract selection problem to minimize the overall total electricity cost 89 References References Akaike, H 1973 Information theory and an extension of the maximum likelihood principle Second International Symposium on Information Theory 267-281 Artzner, P., Delbaen, F., Eber, J.M., Heath, D 1999 Coherent measures of risk Mathematical Finance (3) 203-228 Azevedo, F., Vale, Z., Oliveira, P., Khodr, H 2010 A long-term risk management tool for electricity markets using swarm intelligence Electric Power Systems Research 80 (4) 380-389 Bar-Lev, D., Katz, S 1976 A portfolio approach to fossil fuel procurement in the electric utility industry The Journal of Finance 31 (3) 933-947 Bjorgan, R., Liu, C.C., Lawarree, J 1999 Financial risk management in a competitive electricity market Power Systems, IEEE Transactions on 14 (4) 1285-1291 Bollerslev, T 1986 Generalized autoregressive conditional heteroskedasticity Journal of Econometrics 31 (3) 307-327 Box, G.E.P., Jenkins, G.M 1994 Time Series Analysis: Forecasting and Control Prentice Hall PTR Carrion, M., Philpott, A.B., Conejo, A.J., Arroyo, J.M 2007 A stochastic programming approach to electric energy procurement for large consumers Power Systems, IEEE Transactions on 22 (2) 744-754 Ceppi, S., Gatti, N 2010 An algorithmic game theory study of wholesale electricity markets based on central auction Integrated Computer-Aided Engineering 17 (4) 273-290 Chang, Y 2007 The new electricity market of Singapore: regulatory framework, market power and competition Energy Policy 35 (1) 403-412 Chang, Y., Park, C 2007 Electricity market structure, electricity price, and its volatility Economics Letters 95 (2) 192-197 Conejo, A.J., Fernandez-Gonzalez, J.J., Alguacil, N 2005 Energy procurement for large consumers in electricity markets Generation, Transmission and Distribution, IEE Proceedings-3 152 357-364 Diebold, F., Barro, R., Marthinsen, J., Carbaugh, R., Yarbrough, B., Yarbrough, R., Mourdoukoutas, P., Kaufman, B., Hotchkiss, J., Hyclak, T 2004 Elements of Forecasting Thomson Dupacová, J., Growe-Kuska, N., Romisch, W 2002 Scenario reduction in stochastic programming Mathematical Programming 95 (3) 493-512 Eichhorn, A., Romisch, W 2006 Mean-risk optimization models for electricity portfolio management 2006 International Conference on Probabilistic Methods Applied to Power Systems 262-268 Eichhorn, A., Romisch, W 2008 Dynamic risk management in electricity portfolio optimization via polyhedral risk functionals Power and Energy Society General Meeting - Conversion and Delivery of Electrical Energy in the 21st Century, 2008 IEEE 1-8 Elliott, R.J., Lyle, M.R., Miao, H 2010 A model for energy pricing with stochastic emission costs Energy Economics 32 (4) 838-847 Fabozzi, F.J., Kolm, P.N., Pachamanova, D., Focardi, S.M 2007 Robust Portfolio Optimization and Management Wiley 90 References Fleten, S.E., Wallace, S.W., Ziemba, W.T 1997 Portfolio management in a deregulated hydropower based electricity market Proceedings of Hydropower 197-203 Fleten, S.E., Wallace, S.W., Ziemba, W.T 2002 Hedging electricity portfolios via stochastic programming IMA Volumes in Mathematics and its Applications 128 71-94 Frey, G., Manera, M., Markandya, A., Scarpa, E 2009 Econometric models for oil price forecasting: a critical survey CESifo Forum 10 (1) 29-44 Gan, D., Bourcier, D.V 2002 A single-period auction game model for Modeling oligopolistic competition in pool-based electricity markets 2002 IEEE Power Engineering Society Winter Meeting Conference Proceedings 101-106 Giacometti, R., Vespucci, M.T., Bertocchi, M.I., Barone-Adesi, G 2010 A stochastic model for hedging electricity portfolio for an hydro-energy producer Working paper Gross, R., Blyth, W., Heptonstall, P 2010 Risks, revenues and investment in electricity generation: why policy needs to look beyond costs Energy Economics 32 (4) 796-804 Guastaroba, G., Mansini, R., Speranza, M 2009 On the effectiveness of scenario generation techniques in single-period portfolio optimization European Journal of Operational Research 192 (2) 500-511 Haneveld, W.K.K., van der Vlerk, M.H 2001 Optimizing electricity distribution using two-stage integer recourse models Stochastic Optimization: Algorithms and Applications 137-154 Heitsch, H., Romisch, W 2005 Generation of multivariate scenario trees to model stochasticity in power management Power Tech, 2005 IEEE Russia 1-7 Hochreiter, R., Pflug, G., Wozabal, D 2006 Multi-stage stochastic electricity portfolio optimization in liberalized energy markets System Modeling and Optimization 199 219-226 Huisman, R., Mahieu, R., Schlichter, F 2009 Electricity portfolio management: Optimal peak/off-peak allocations Energy Economics 31 (1) 169-174 Hyman, L.S 2010 Restructuring electricity policy and financial models Energy Economics 32 (4) 751-757 Kaut, M., Wallace, S.W 2003 Evaluation of scenario-generation methods for stochastic programming Stochastic Programming E-Print Series 14 (1) Kettunen, J., Salo, A., Bunn, D.W 2008 Dynamic risk management of electricity contracts with contingent portfolio programming Submitted to Management Science Kettunen, J., Salo, A., Bunn, D.W 2010 Optimization of electricity retailer's contract portfolio subject to risk preferences Ieee Transactions on Power Systems 25 (1) 117-128 Kirschen, D.S 2003 Demand-side view of electricity markets Ieee Transactions on Power Systems 18 (2) Kleindorfer, P., Li, L 2005 Multi-period VaR-constrained portfolio optimization with applications to the electric power sector Energy Journal 26 (1) 1-26 Kubat, J 2006 Electricity market game Proceedings of the 24th International Conference on Mathematical Methods in Economics 2006 307-311 Kullback, S., Leibler, R.A 1951 On information and sufficiency Annals of Mathematical Statistics 22 79-87 91 References Latorre, J.M., Cerisola, S., Ramos, A 2007 Clustering algorithms for scenario tree generation: Application to natural hydro inflows European Journal of Operational Research 181 (3) 1339-1353 Lehmann, E.L., Casella, G 1998 Theory of Point Estimation Springer Verlag Li, J., Liu, J.Y., He, M., Xiang, Y 2009 A Game-based study of regional virtual electricity trade of small hydro-power 2009 International Conference on Sustainable Power Generation and Supply (4) 348-353 Liang, T.P., Huang, J.S 2000 Framework for applying intelligent agents to support electronic trading Decision Support Systems 28 (4) 13 Liu, M., Wu, F.F 2007 Portfolio optimization in electricity markets Electric Power Systems Research 77 (8) 1000-1009 Markowitz, H 1991 Portfolio Selection: Efficient Diversification of Investments Wiley Murty, K.G 1978 Linear and Combinatorial Programming John Wiley & Sons Inc Nourai, A., Schafer, C 2009 Changing the electricity game Ieee Power & Energy Magazine (4) 42-47 Øksendal, B.K 2003 Stochastic Differential Equations: an Introduction with Applications Springer Verlag Oliveira, F.A., Paiva, A.P., Lima, J.W.M., Balestrassi, P.P., Mendes, R.R.A 2011 Portfolio optimization using mixture design of experiments: scheduling trades within electricity markets Energy Economics 33 (1) 24-32 Praca, I.R., Ramos, C., Vale, Z., Cordeiro, M., Mascern 2003 A multiagent system that simulates competitive electricity markets IEEE Intelligent Systems 18 (6) 54-61 Rockafellar, R., Uryasev, S 2002 Conditional value-at-risk for general loss distributions Journal of Banking & Finance 26 (7) 1443-1471 Romisch, W 2009 Scenario reduction techniques in stochastic programming Stochastic Algorithms: Foundations and Applications 1-14 Sanchez, J.J., Barquin, J., Centeno, E., Lopez-Pena, A 2008 A multidisciplinary approach to model long-term investments in electricity generation: combining system dynamics, credit risk theory and game theory 2008 Ieee Power & Energy Society General Meeting (11) 3406-3413 Schwarz, G 1978 Estimating the dimension of a model Annals of Statistics (4) 461-470 Sen, S., Yu, L., Genc, T 2006 A stochastic programming approach to power portfolio optimization Operations Research 54 (1) 55-72 Shafiee, S., Topal, E 2010 A long-term view of worldwide fossil fuel prices Applied Energy 87 (3) 988-1000 Shapiro, A., Dentcheva, D., Ruszczy ski, A 2009 Lectures on Stochastic Programming: Modeling and Theory Society for Industrial Mathematics Stoft, S 2002 Power Systems Economics: Designing Markets for Electricity John Wiley & Sons Sueyoshi, T., Tadiparthi, G.R 2008 An agent-based decision support system for wholesale electricity market Decision Support Systems 44 (2) 425-446 Sugiura, N 1978 Further analysts of the data by Akaike's information criterion and the finite corrections Communications in Statistics-Theory and Methods (1) 13-26 Wu, J., Guan, X.H., Gao, F., Sun, G.J 2008 A game theoretic model for equilibrium pricing of futures electricity contracts 2008 Ieee Power & Energy Society General Meeting (11) 3608-3614 92 References Ye, M., Zyren, J., Shore, J 2005 A monthly crude oil spot price forecasting model using relative inventories International Journal of Forecasting 21 (3) 491501 Zhang, Q., Wang, X., Wang, J 2010 Electricity Purchasing and Selling Risk Decision for Power Supplier Under Real-time Pricing Automation of Electric Power Systems 34 (3) 22-43 93 Appendix A: Model Selection Criteria Appendix A: Model Selection Criteria (1) Error-based model selection criteria: Mean Square Error (MSE) of an estimator is one of many ways to quantify the difference between an estimator and the true value of the quantity being estimated The square root of MSE is Root Mean Square Error (RMSE), which has the same unit as the quantity being estimated Thus RMSE is more straightforward than MSE n MSE   (s  sˆ ) i 1 i i n RMSE  MSE n : no of observations sˆi : forecasted/estimated value in period i (A.1) si : actual value in period i RMSE is closely related to two other statistics: residual sum of squares (RSS) and R (Diebold et al., 2004) n RSS   ( si  sˆi ) i 1 n R  1  (s  sˆ ) i 1 n i  (s  s ) i 1 i (A.2) i It can be seen that selecting the model with the smallest RMSE is equivalent to selecting the model with the smallest RSS and highest R However, there is a problem associated with selecting models using RMSE When more explanatory variables are added to a model, the in-sample RMSE will fall down or at most remain the same Such effects are called in-sample over-fitting, which reveals that including 94 Appendix A: Model Selection Criteria more variables in a forecasting model won’t necessarily improve its out-of-sample forecasting performance, although it will improve the model’s fit on historical data Hence, RMSE is a biased estimator of out-of-sample 1-step-ahead prediction error variance, and the size of the bias increase with the number of variables included in the model (Diebold et al., 2004) Besides, RMSE is sensitive to occasional large error since the squaring process gives disproportionate weight to very large errors; in that case, other measures such as Mean Absolute Error (MAE) or Mean Absolute Percentage Error (MAPE) may be more relevant MAE is also commonly adopted, and is usually slightly smaller than RMSE It may be easier to understand than RMSE for some users MAE  n  | yˆi  yi | n i 1 (A.3) MAPE is often useful, since it is expressed in the percentage form, which is more straightforward without the knowledge of the magnitude of the evaluated data MAPE  n n | i 1 ( sˆi  si ) | si (A.4) However, it has its own drawbacks in practical application For example, if any of the observed value is zero, there is a division by zero (2) Information-based Criteria 95 Appendix A: Model Selection Criteria Kullback-Leibler information is the basis for this approach (Kullback and Leibler, 1951); K-L information between models f and g is defined for continuous functions as the integral in (A.5) I ( f , g )   f ( x) ln f ( x) dx g(x |  ) (A.5) The notation I ( f , g ) denotes the “information loss when g is used to approximate f” The purpose is to find an approximating model that loses as little information as possible, which is equivalent to minimize I ( f , g ) over g (Kullback and Leibler, 1951) The formal calculation of K-L distance requires knowing the true distribution f as well as all the parameters in the approximating models Thus, K-L distance cannot be calculated in real problems Therefore, relative K-L distance is used as equivalence to the absolute K-L distance Akaike’s seminal paper proposed a rigorous way to estimate the K-L information based on the empirical MLE and suggested to use it as a fundamental basis for model selection (Akaike, 1973) Akaike Information Criteria (AIC) is proposed as an approximately un-biased estimator of the applied K-L information AIC  2ln L  2k ln L : the maximized value of log-likelihood function k: (A.6) the number of parameters In the special case of least squares estimation with normally distributed errors, AIC can be expressed as a simple function of the residual sum of squares (RSS) If all the models in the set assume normally distributed errors with a constant variance, then AIC can be computed from least squares regression statistics as in (A.7): 96 Appendix A: Model Selection Criteria AIC  n ln(ˆ )  2k where ˆ  ˆ  i n (the MLE of  ) (A.7) However, it is found that AIC may perform poorly in the case of too many parameters and too small sample sizes To solve this problem, a second order variant of AIC (AICc) is derived (Akaike, 1973): 2k (k  1) n  k 1 n : the sample size AICc  AIC  (A.8) Another commonly employed information-based criterion is Bayesian Information Criteria (BIC) (Sugiura, 1978) BIC  2ln L  k  ln n (A.9) It has assumptions that an exactly “true model” exists, that it is one of the candidate models being considered, and that the model selection goal is to select the true model (Schwarz, 1978) Due to these assumptions, in some circumstances, it may be more relevant to employ AIC and AICc since there may not exist a “true model” 97 Appendix B: Preliminary Data Analysis Appendix B: Preliminary Data Analysis A preliminary investigation of the historical data of fuel oil prices is firstly conducted The available data set is the monthly HSFO spot prices from Jan 1990 to Sep 2008 The following graphs show the movements of HSFO prices and the returns (Figure B-1, Figure B-2) In general, it exhibits an upward trend, followed by a sharp dive from the June of 2008 (Figure B-1) Figure B-1 HSFO monthly spot price (in USD/MT) from Jan 1990 to Sep 2008 Figure B-2 Monthly return of HSFO Price (in USD/MT) from Feb 2003 to Sep 2008 98 Appendix B: Preliminary Data Analysis (a) Seasonal Effect Analysis In Figure B-3, the monthly variation within year and yearly variation within month are shown There is no evidence of month-of-year effect, whereas a yearly upper trend is observed Similar conclusions can be drawn for the quarterly data in Figure B-4 Figure B-3 Monthly variation within year and yearly variation within month Figure B-4 Quarterly variation within year and yearly variation within quarter 99 Appendix B: Preliminary Data Analysis (b) Ljung-Box-Pierce test To examine whether there exists autocorrelations in the return series, we perform Ljung-Box-Pierce test (i.e portmanteau test) The null hypothesis is there is no auto correlation, and the test statistic is as follows If the sample value of Q exceeds the critical value of chi-square distribution with s degree of freedom, at least one value of r is statistically different from zero at the specified significance level s Q  T (T  2) rk2 / (T  k ) k 1 T : total number of observations (A.10) s: number of coefficients to test autocorrelation rk : autocorrelation coefficient for lag k The data is tested for up to 1, 6, and 12 lags at a 0.05 significance level, and the result is as follows in Table B-1 and Table B-2 The test shows that the autocorrelation for monthly return is statistically significant; whereas the autocorrelation for quarterly return is not so significant Table B-1 Ljung-Box-Pierce test result for monthly return of HSFO spot price Lag (s) 12 H 1 pValue 0.0003 0.0000 0.0000 Statistic 13.0558 30.3516 51.0969 CriticalValue 3.8415 12.5916 21.0261 Table B-2 Ljung-Box-Pierce test result for quarterly return of HSFO spot price Lag (s) 12 H pValue 0.7454 0.0269 0.0688 Statistic 0.1055 14.2573 19.9092 CriticalValue 3.8415 12.5916 21.0261 100 Appendix B: Preliminary Data Analysis (c) Lagrange Multiplier’s test for ARCH effect We also perform Engle’s Lagrange Multiplier’s test to examine whether there exists ARCH effect The null hypothesis of LM test is there is no ARCH effect Under the null hypothesis, the test statistic shown as below is asymptotically Chi-Square distributed E  TR (A.11) T : total number of observations R : the sample multiple correlation coefficient The test result is shown as below in Table B-3 and Table B-4 (under the significance level of 0.05), which is in support of ARCH effect for the monthly data; however, the ARCH effect for the quarterly data is not significant Table B-3 LM Eagle’s test result for monthly return of crude oil Lag (s) 12 H 1 pValue 0.0132 0.0000 0.0000 Statistic 6.1366 31.6845 84.2166 CriticalValue 3.8415 12.5916 21.0261 Table B-4 LM Eagle’s test result for quarterly return of crude oil Lag (s) 12 H pValue Statistic CriticalValue 0.000 0.000 0.000 0.5833 0.7114 0.8563 0.3009 3.7434 7.0192 3.8415 12.5916 21.0261 101 .. .CONTRACT SELECTION PROBLEM IN SINGAPORE ELECTRICITY MARKET CHEN LIQIN (B.Eng, SJTU) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF INDUSTRIAL AND SYSTEMS ENGINEERING... the contract or contract portfolio in the beginning of decision making process, and maintains the contract throughout the contractual period 21 Chapter 3: Single-Period Contract Selection Problem. .. Chapter 3: Single-Period Contract Selection Problem Chapter 3: Single-Period Contract Selection Problem In this chapter, single-period contract selection problem is investigated MeanVariance