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Optimal allocation of load variability for hydro systems : a stochastic dynamic programming approach Mazariegos, Raquel 2006

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O P T I M A L A L L O C A T I O N O F L O A D V A R I A B I L I T Y F O R H Y D R O S Y S T E M S : A S T O C H A S T I C D Y N A M I C P R O G R A M M I N G A P P R O A C H . b y R A Q U E L M A Z A R I E G O S B.Sc. Un ive rs idad de las Amer icas-Puebla , 2002 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S ( C i v i l Engineer ing) T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A December 2006 © Raquel Mazariegos, 2006 Abstract The deregulat ion o f the electric industry has mot ivated the companies i nvo l ved i n electr ic i ty business to opt imize their system resources to remain compet i t ive , but without over look ing the tradit ional object ive o f p rov id ing customers w i th an eff ic ient and uninterrupted service. Th i s requirement, in addit ion to the ava i lab i l i ty o f detai led in format ion on every spec i f ic system component, has encouraged the system operators to use opt imizat ion techniques for system p lann ing and operat ion. One o f the most important funct ions that the system operator performs is the system E c o n o m i c D ispatch . E c o n o m i c D ispa tch guarantees that the electric load demand is al located among the avai lable generation system units in the most economica l and eff ic ient way possib le . A detai led study and analysis o f the factors in f luenc ing the performance o f the generating system found that up to now o ld mode ls and methods d id not address and acknowledge the within-the-hour uncertainty i n load predict ion despite the fact that it is considered to be one o f its m a i n and most important inputs. Research on load forecasting have showed that the electr ic load forecast ing techniques have been developed to the point where the magnitudes o f errors have been opt imized and further reduction o f the error w i l l be hard to achieve. Thus , the opt imizat ion opportunity for E c o n o m i c D ispatch does not reside on spending more resources on ach iev ing a more accurate load forecast, but on the fact that within-the-hour load uncertainty can be modeled and exploi ted to prov ide quantitative in format ion for dec is ion m a k i n g support. Th i s research work uses a dynamic p rogramming a lgor i thm to allocate uncertainty i n load forecast. The first method is a tradit ional approach that on l y considers the product ion costs for the economic dispatch dec is ion , and the second one takes into account the seasonal var iab i l i t y i n the load forecast error to find the most l i k e l y load increment to allocate and then prov ide the best dispatch scheme for the three largest plants i n the B C Hyd ro ' s generating system. i i Table of Contents Abstract i i Tab le o f Contents i i i L i s t o f Tables v L i s t o f F igures v i i Acknowledgements i x 1 Introduction • 1 1.1 The post-deregulation environment o f the electric industry 1 1.2 Load var iab i l i ty and its impact i n the system's opt imal operation 3 1.3 The load forecast ing exercise and the load forecast error 6 1.4 The goal o f this research 10 1.5 Organizat ion o f the thesis 11 2 Background 13 2.1 Literature Rev i ew 13 2.1.1 Opt im iza t ion techniques for economic dispatch 13 2.1.2 Stochastic Cont ro l Theory and load uncertainty 22 3 Opt imiza t ion and dec is ion mak ing processes at B C H y d r o 23 3.1 The B C H y d r o generation system 23 3.2 Automat i c Generat ion Cont ro l 27 3.3 The components o f the A G C system 28 4 Me thodo logy 35 4.1 D y n a m i c P rogramming 35 4.1.1 De f in i t i on o f the p rob lem 37 4.1.2 The mode l ing environment 41 4.1.3 D y n a m i c p rogramming mode l 41 4.2 Stochastic dynamic p rogramming 47 4.2.1 Data preparation 48 4.2.2 De f in i t i on o f the p rob lem 56 4.2.3 Stochastic Dynam i c p rogramming mode l 60 5 Results 60 5.1 Characteristics of the probability distributions of the load forecast error 60 5.2 The deterministic dynamic programming model 62 5.3 The stochastic dynamic programming model 67 6 Conclusion and recommendations 74 Bibliography 76 iv List of Tables Tab le 4.1 PI System data for the first hour o f December 2001 50 Tab le 4.2 Pre l iminary calculat ions 51 Tab le 4.3 Frequency and probabi l i ty analysis for the fu l l per iod 52 Tab le 4.4 N o r m a l and ramping per iod c lass i f icat ion 53 Tab le 4.5 Frequency analysis and probabi l i t ies for normal and ramping periods 53 Tab le 4.6 Probabi l i t y distr ibutions for winter 55 Tab le 4.7 Probabi l i t y distr ibut ions for spr ing and summer 56 Tab le 4.8 Probabi l i t y distr ibutions for fa l l .' 56 Tab le 5.1 Statistical analysis for the load forecast error 62 Tab le 5.2 H o u r l y Generat ion and increments 63 Tab le 5.3 H o u r l y Generat ion cost 64 Tab le 5.4 Results o f the dynamic p rogramming mode l 65 Tab le 5.5 Expected values for seasonal probabi l i t y distr ibutions (winter) 68 Tab le 5.6 Expected values for seasonal probabi l i t y distr ibutions (summer) 68 v 'Table 5.7'Expected values for seasonal probabi l i ty distr ibutions (fall) 68 Table 5.8 Expected load values 70 Table 5.9 Results o f the dynamic programming mode l for the expected loads 70 Table 5.10 Cost and benefit o f handl ing uncertainty 73 List of Figures F igure 1.1 L o a d variat ions dur ing a random day 3 F igure 1.2 L o a d variat ions dur ing a random spr ing day 4 F igure 1.3 L o a d variat ions dur ing a random summer day 4 F igure 1.4: M e a n cost o f inaccurate forecasts 9 F igure 2.1 F l o w slope curve w i th a convex increas ing characteristic 15 F igure 2.2 The scenario analysis scheme 21 F igure 3.1 Br i t i sh C o l u m b i a Imports and Expor ts page 24 F igure 3.2 M a p o f B C H y d r o Serv ice Areas 25 F igure 3.3 The Au tomat i c Generat ion Cont ro l System 31 F igure 3.4 Forecast ing and Schedu l ing Conceptua l D i ag ram 34 F igure 4.1 The dynamic p rogramming elements 36 F igure 4.2 The unexpected increment in load due to load forecast error 38 F igure 4.3 D y n a m i c p rogramming formulat ion o f the prob lem 40 F igure 4.4 Plant generation cost funct ion 42 v i i F igure 4.5 The stochastic dynamic p rogramming f ramework 48 F igure 4.6 Behav io r o f the load forecast error dur ing ramping per iods 49 F igure 4.7 Probab i l i t y d istr ibut ion for the fu l l per iod 54 F igure 4.8 Probab i l i t y distr ibut ion for the r amp ing per iod 54 F igure 4.9 P robab i l i t y d istr ibut ion for the norma l per iod 55 F igure 5.1 W in te r probabi l i ty distr ibutions for norma l and ramping periods 60 F igure 5.2 Spr ing probabi l i t y distr ibutions for norma l and ramping per iods 60 F igure 5.3 Summer probabi l i ty distr ibutions for norma l and ramping 61 F igure 5.4 F a l l probabi l i ty distr ibutions for norma l and ramping 61 v i i i Acknowledgements I wou ld l i ke to thank m y supervisor D r . Z i a d Shawwash for g i v i ng me the opportunity to work w i th h i m on the B C Hydro research project. H e gave me an example o f hard work and has been an inspirat ion on h o w to transform theory into practice, ideas into pract ical solutions. H e should be praised for his enormous contr ibutions to science and engineering. I also wou ld l i ke to thank D r . T o m S iu , who always mot ivated me and was so generous w i th his knowledge and t ime. The complet ion o f this thesis wou ld not have been possible without his advice, and m y t ime at B C H y d r o w o u l d not have taught me so m u c h without his guidance. Thanks also to A l a a A b d a l l a for a l l his help, i n h i m I found a true and m u c h treasured fr iend. I also thank Dr . Barbara Lence for her support, t ime, energy and guidance. A n d last but not least, I thank a l l the people at B C H y d r o who taught me so m u c h : W u n K i n Cheng , D o u g Rob inson , and C o l i n F ingler . F ina l l y , I wou ld l i ke to thank m y parents for encouraging and support ing me dur ing m y graduate work . Thank G o d for the gift o f l i fe . i x 1 Introduction 1.1 The post-deregulation environment of the electric industry. As the main supplier of electricity in British Columbia and the third largest electric power utility in Canada, the mission of BCHydro is "to provide integrated energy solutions to customers in an environmentally and socially responsible manner". The accomplishment of this objective has been done in stages of change, in which every component of the power system has evolved in a new culture of efficiency and innovation. Traditionally, the main concerns of any electric energy provider were to deliver reliable power supply and to maintain a sustainable growth rate. Around the world there are several electric utilities that are still in this early stage of development. However, B C Hydro has already surpassed this basic objective. Now-a-days, its power system has developed in size and complexity, and the attention is now mainly centered on the way that the utility plans, operates and enhances the performance of the system in the most efficient way. Less than ten years ago, the electric industry in the United States and Canada was operated by vertically integrated utilities (Bose & Christie, 1996). These utilities supplied power to costumers at regulated rates. The main concerns of those utilities were to deliver reliable power supply and to maintain a sustainable growth rate. This situation changed when a model for deregulation of the electric industry was proposed and approved, allowing a number of independent power producers to compete with the established utilities in a free market. The subsequent market dynamics that evolved have posed new challenges. Deregulation in the electricity industry has brought the need for the introduction of commercial interfaces between the functions of generation, transmission, distribution and retailing (Bose & Christie, 1996). The 1 behavior o f e lectr ic i ty differs f rom other commodi t ies , s ince an inventory cannot be bui l t to create p r i c i ng securities. Fo r this reason, to remain compet i t ive and achieve business excel lence, companies invo lved i n electr ic i ty trade were forced to strengthen their dec is ion m a k i n g f ramework i n order to guarantee the opt imal use o f the system resources and address r isks that have arisen. The B C H y d r o generation system is a hydro dominated system in wh i ch almost 9 0 % o f the electr ic output comes f rom hydroelectr ic plants, 7 . 5 % is generated by the Bur rard thermal generation station, and the remainder is suppl ied by purchases. Fo r hydro systems, the most prof i table schedule o f generation w i l l al locate the avai lable resources (i.e. water and fuel) i n such a w a y that product ion cost is m i n i m i z e d wh i l e sat isfy ing the constraints representing re l iab i l i ty , environmental requirements and addit ional f inanc ia l goals. It is important to note that water resources avai lable for hydroelectr ic power generation encompass important uncertainties. The stochastic behavior o f the avai lable water resources for hydro generation outl ines an addit ional chal lenge to the ones posed by markets and contracts. The most uncertain o f the constraint l imi ts are those that der ive f rom predicted energy demands, energy pr ices and forecasted water in f lows into reservoirs. Compan ies and researchers i nvo l ved i n the electr ic i ty trade have acknowledged that this uncertainty can s ign i fy either a huge opportunity or a source o f h igh r isk, and that it can affect a u t i l i t y ' s bottom l ine. The B C H y d r o ' s system is operated to m a x i m i z e the value o f the resources at the various levels o f p lanning for power supply operations that fo l lows a hierarchical approach for opt ima l operation o f the hydro system. Th i s approach is d i v ided into several computat iona l ly manageable levels. These are: 1) L o n g term p lanning: Cove r ing 1 to 4 years o f act iv i ty , 2) M e d i u m term p lanning: Cove r ing da i l y or week l y act iv i t ies, 3) Short term p lann ing : Cove r ing hour l y and sub-hourly activit ies. Fo r any o f the p lann ing levels ment ioned above, one o f the ma in inputs for the development o f every leve l 's operation strategy is the data prov ided by forecasts. Forecasts help ident i fy the needs that must be satisf ied i n a qual itat ive and quantitative way. S ince deregulat ion intensif ied the compet i t ion among power producers and have 2 forced them to operate i n an environment o f suppressed profit margins and w i th h igh vo la t i l i t y o f electr ic i ty pr ices, and have highl ighted the need to develop accurate load forecasting systems. Th i s research work is ma in l y concerned w i th short term p lann ing activit ies, part icular ly w i th short term load forecasting and opt imal loading o f hydroelectr ic plants. 1.2 Load variability and its impact on optimal operation of the system. In a perfect wor ld for electric ut i l i t ies, every customer wou ld use a constant amount o f power at a l l t imes o f the day, every day o f the year. Th is w o u l d make it easy for the companies responsible o f operating the electr ic i ty system to keep it running smoothly and at an economica l pr ice. Unfortunately , electr ic i ty use patterns vary broadly. F igures 1.1 to 1.3 show the behav ior o f electr ic i ty demand dur ing typica l days for three different seasons. The c o m m o n pattern for those three f igures is that demand for electr ic i ty is l ow after 10:00 p.m., when domest ic , commerc ia l and most industr ial customers are o f f work and resting. L o a d starts increasing at around 6:00 a.m. to reach the first peak o f the day, and then remains at a somewhat steady h igh unt i l some t ime after 6:00 p.m., where the absence o f natural l ight induces more demand for energy, wh i ch reaches its day m a x i m u m at around 7 :00 p.m. 7000 6500 6000 5500 50O0 4500 4000 1 I H 1 1 i IA\ K ] f - \ ! V \ I hi / i I 1 ' 1 v 400 300 200 + 100 0 -100 -200 -300 -400 4/16/02 4/17/02 4/17/02 4/17/02 4/17/02 4/17/02 4/18,02 *'18.«2 7:12 PM 12:00 AM 4:48 AM 9:36 AM 2:24 PM 7:12 PM 12:00 AM 4:48 AM - BCH_SrS_LoadAvg5rrm_MVy_00 PSOSE Load Forecast BCH_Sr'S_LoadHoi*ly_MWi_pO 5 mn drff F igure 1.1 Load variat ions dur ing a typical spr ing day. 3 7000 6500 6000 5900 5000 4500 4000 [ J , 1 It r , 1 y I I V J 1 * i -1 1 1 1 1 i f 300 200 100 o -1O0 -200 -300 81.TO2 84402 8,1402 81402 81402 81402 8/1&02 8/1502 7:12 P M 12:00 A M 4:48 A M 9:33 A M 2:24 P M 7:12 P M 12:00 A M 4:48 A M 5-Mn AVGIoad Houily load PSOSE Load Forecast 5 m i n d i f f Figure 1.2 Load variations during a typical summer day. 9000 8500 8000 7500 7000 6500 6000 5500 5000 t in L i wr*v_ d il • t 1 f --1 1 1 1 1 1 12O01 12*4/01 12/4701 12/4701 12/4701 12/4/01 12001 12S01 7:12 PM 1200/Wl 4;4B£M 9:36 >SM 224 PM 7:12 Ffvl 1200/M 4:48 PM 5-MnAVG -—Hour ly load PSOSE Load Forecast 5 Ml n diff Figure 1.3 Load variations during a typical winter day Despite of the similarities mentioned above, some differences in the load behavior during those days can also be observed. Those differences are caused by several weather sensitive and non-weather sensitive parameters (Rahman, 1990). Among the weather sensitive parameters temperature, wind direction and speed play a major role. 4 Non-weather sensitive parameters affect the var iat ion i n demand shown in F igures 1.1 to 1.3 as f o l l ows : 1) Season o f the year: E a c h season has a part icular effect, and it is m a i n l y dictated b y changes i n temperature. It can be seen that dur ing the summer and winter days (Figures 1.1 and 1.3), when air condi t ion ing or heating are needed, the load m i n i m u m and m a x i m u m values are higher than those dur ing the spr ing. 2) Seasonal load shape: There are shifts i n the load peak m a i n l y due to lack o f natural l ight and also variat ions i n temperature between day and night. Energy savings t ime induces a late peak demand for energy at night and also inf luences consuming patterns. F igure 1.3 shows the highest peak i n demand (8500 M w h ) o f a l l seasons i n this study. Demand is h igh due to a combinat ion o f lack o f natural l ight and co ld weather. 3) D a y o f the week: Seasonal load shapes are s imi la r for the weekday ' s type, except for the ho l idays . D u r i n g hol idays and weekends, energy is not used b y industr ia l or commerc ia l customers, w h i c h represent the stable part o f the demand. Steady operat ing machinery and bu i ld ings are o f f duty, and the load behavior is dictated b y domest ic demand. A t first glance, load behav ior could seem to f o l l ow a consistent pattern that w o u l d a l low the establishment o f statistical relationships to easi ly develop a straightforward and very precise forecasting method. However , a number o f studies have shown that load forecasting prec is ion tends to vary broadly dur ing weekends and ho l idays [see, e.g. Ranaweera et a l . , 1997; O. M o h a m m e d et a l . , 1995; and T . M . P e n g et a l . , 1992], and thus deve lop ing separate forecasting exercises for these part icular days becomes important. In addit ion, it can be seen i n F igures 1.1 to 1.1 that the discrepancy between the hour ly forecasted and actual load increases considerably i n the ramp-up and ramp-d o w n hours. The importance o f pay ing attention to var iat ion i n hour l y load has also been acknowledged b y several researchers [see e.g., Va l enzue l a et a l . , 2000] when deve lop ing cost product ion mode ls and load forecasting algorithms. 5 A c c o r d i n g to B u n n and Farmer (1985), the uncertainty, inherent i n load forecasting has a large impact on the system's performance, ma in l y because load predict ions are used i n three o f the functions that are crucia l to the opt imal operation o f the system: 1) Plant schedul ing: The process i n p lann ing i n advance the start-up and shut-down schedules o f the power plants, i n order to meet the var iat ion i n demand throughout the day and day b y day. 2) L o a d dispatching and system security assessment: The minute-to-minute a l locat ion o f loads to the generating units , to meet the demand at m i n i m u m cost. 3) A l l o c a t i o n o f reserve capacity : Add i t i ona l generating capacity used as a reserve to cater for uncertainty i n load behavior and the variations i n power plant capabi l i t ies. The schedul ing procedure requires predict ions up to about 24 hours ahead. Those schedules are then ref ined as more accurate load forecast in format ion becomes avai lable. In the case o f load d ispatching, short-term forecasts are necessary to calculate the f lows i n the transmission system and to enforce transmission security constraints. O n the other hand, the op t imum magnitude o f the reserve capacity is dependent upon the load forecast ing error. 1.3 The load forecasting exercise and the load forecast error. A power company is obl igated to match its o w n generation w i th the net load on the system on real t ime basis, w i th in the bounds o f preva i l ing security and re l iab i l i ty criteria. In a deregulated environment, the tradit ional load forecasting funct ion is now changing into a generation predic t ion exercise (Rahman, 1990). L o a d forecast ing re l ied on heurist ic forms o f knowledge , inc lud ing histor ical relationships and operator experience. U t i l i t i es stored real and predicted load prof i l e data us ing large, isolated spreadsheets and database systems. Computers w i th enough power to manage the huge amount o f data f rom w h i c h predict ive relat ionships cou ld be extracted were not avai lable unt i l recently, and now , as the power systems have grown i n complex i ty , addit ional needs have emerged as the tradit ional roles o f the customer and the ut i l i ty 6 have changed. Hence , the h istor ica l cause-effect relat ionships have started to be less rel iable for forecast ing the load. Nowadays , the best forecast ing technique must be intel l igent and sensit ive to changes i n the ut i l i t y environment: be adaptive, and able to dynamica l l y benefit from the experience o f system operators, in format ion about loca l phenomena (such as irregular demand or plant shutdown) and customer response to regulatory constraints or economic incent ives, w h i c h is also a factor in f luenc ing the load. In order to prov ide the input for plant schedul ing, load d ispatch ing and a l locat ion o f reserve capacity, several load forecasting techniques have been investigated. These techniques can be d i v ided into three areas o f appl icat ion ( Bunn and Farmer, 1985): 1 . L o n g term econometr ic forecasts for system p lann ing [see, e.g., K a n d i l et a l . ,2002; Kauhan i em i , 1991] ; 2. M e d i u m term forecasts for the schedul ing o f fuel supplies and maintenance programs [see, e.g., Jue C h a n et a l . , 2 0 0 1 ; Bo-Juen Chen , 2004 ; Tay l o r and B u r z z a , 2002;Tsekouras et a l . ,2006]; and 3. Short term forecasts for the day-to-day operat ion and schedul ing o f the power system [see, e.g., Dash et a l . , 1997; Shy J ier H u a m g and Kuang-Rong Sh ih , 2003 ; Rahman , 1990; K y u n g - B i n Song et a l . ,2005; J ia-Yo C h i a n g et a l . 2000] . In addit ion to this c lass i f i cat ion, forecast ing systems are also d i v ided into off -l ine and on-line predictors ( Bunn and Farmer, 1985). Off- l ine predictors are appl ied to the schedul ing o f generating plant f rom a few hours to few days ahead. On-l ine forecasts are used for the minute to minute operation o f the power system and for the economic load ing o f the generating units. A var iety o f methods, wh i ch inc lude the so-cal led s imi lar day approach, var ious regression mode ls , t ime series, neural networks, statistical learning a lgor i thms, f u z z y log ic , and expert systems, have been developed for short-term forecast ing (Fe inberg and Geneth l iou , 2005) . The ma in goal o f most o f the methods proposed i n the literature is to m i n i m i z e the forecasting errors. Power companies are l o o k i n g to reduce their purchase o f electr ic i ty i n real-time as these tend to be cost ly and they look for methods and 7 techniques to better forecast their customers ' demand for electr ic ity. Th is reduct ion i n forecasting error is on l y poss ib le i f these companies develop more accurate forecast ing techniques. However , results reported i n the literature show that the accuracy o f forecasts is i n the range o f 2 to 5 percent o f the actual load. Th i s leads to the quest ion, what is an acceptable level of accuracy? A c c o r d i n g to Ranaweera (1997), this depends on the characteristics o f the operating ut i l i ty . In a study developed by Hobbs et a l . (1999), 28 A m e r i c a n power producer companies were surveyed on the s igni f icance o f load forecast ing accuracy. Fou r o f those ut i l i t ies quanti f ied the commitment benefits and conc luded that a 1 % reduct ion i n mean absolute percentage error ( M A P E ) w o u l d translate into a $1.7, $28, $42 , or $143 annual benefit per peak M W o f demand (depending upon the ut i l i ty ) . The highest va lue represents approximate ly 0 . 1 5 % o f ut i l i ty 's variable generation costs and it results i n annual savings i n the order o f $ 7 . 6 M for a ut i l i ty w i th a peak load o f 35 G W . The use o f A N N S T L F (a neural network forecast ing approach) has lowered the M e a n Abso lu te Percentage Er ror ( M A P E ) b y 1.5 % for the same ut i l i ty . In another study ( Bunn and Farmer, 1985) in the U K , an increase o f 1 % i n forecasting error increased the operat ing costs about £10 m i l l i o n per year, i n 1984 prices. It has also been shown (Ortega-Vazquez et a l . , 2006) , that the impact o f the load forecast error is even b igger when the cost o f energy not served due to outages is taken into account. A c c o r d i n g to this study, a forecasting error o f 3 % has a da i l y economic impact o f $1651.70. W i thout a doubt, the more accurate the load forecast, the lower is the exposure to penalties due to underest imating supply requirements or to unnecessary supply costs due to over or under est imation o f the load to be met. The effect o f pred ic t ion errors may be evaluated in terms o f system economic and technical performance. A c c o r d i n g to B u n n and Farmer (1985), the increase in operating cost due to a d ispatching error is not usua l ly substantial, unless there is a s ignif icant difference between the incremental and detrimental costs o f the margina l generation plant. Hobbs et a l . (1999) descr ibed the effects o f load forecast ing error. I f predicted loads are h igher than the actual load : • Un i ts are unnecessari ly commit ted , increasing fuel costs and perhaps maintenance costs. 8 • Expens ive power is purchased and opportunit ies to sel l bu lk power are foregone. • Hydropower that w o u l d have been more valuable later is not produced at the opt imal t ime. • H i g h real-time prices are estimated, thereby decreasing potential sales. • Unnecessary interruptions might be implemented, thereby annoy ing customers and lower ing revenue. O n the contrary, i f predicted loads are l ower than the actual load : • System re l iab i l i ty is endangered when not enough resources are avai lable for meet ing security constraints. • To meet the unanticipated load increase, the ut i l i ty incurs cost for uneconomic generation and purchases i n spot power markets. • Power sales are made at a pr ice less than the value o f that power for the ut i l i ty . • Revenues might fa l l short o f the ut i l i t y ' s cost i f too l o w real-time prices are estimated. The var iat ion o f product ion cost as a funct ion o f M A P E is i l lustrated i n F igure 1.2. It can be seen that for smal l M A P E ' s under and over forecasts entail s imi la r penalt ies, but for med ium and large errors, costs are larger than expected and are more severe. MAPE F igure 1.4 M e a n cost o f inaccurate forecasts (Hobbs et a l . , 1999) 9 Research on load predict ion techniques have been carried out for several years and w i l l continue as increasing demand and compet i t ion w i l l demand even more forecast ing prec is ion . However , it w i l l a lways be imposs ib le to k n o w exact ly the values o f future demand. L o and W u (2003) indicated that the accuracy o f Short Te rm L o a d Forecast ing ( S T L F ) depends on f ive major factors: 1. The forecast technique chosen for different load prof i les , 2. The parameter estimate a lgor i thm for a suitable forecast mode l , 3. Inc lus ion o f important exogenous variables that have a h igh leve l o f correlat ion w i th load. 4. The character o f the load prof i le . 5. The lead-time o f forecasts. F r o m these factors it is recognizable that forecasters might be able to choose the best predict ion technique avai lable, but the exogenous variables ment ioned b y L o and W u (2003), such as weather and var iab i l i ty i n consumpt ion patterns, are beyond the control o f the system operator and w i l l pose constant uncertainty. 1.4 The goal of this research. Predominant ly , demand tends to be overestimated, as it is preferable b y the system operator to incur economic penalties rather than fac ing supply interruptions w i t h disastrous consequences. In this case, l i tt le can be done to reduce the effects o f load forecast ing errors. A very different chal lenge is the one posed b y underestimated demand. The results o f the study developed b y Hobbs et al . (1999) showed that for a M A P E equal to 5 .4% , 6 5 % o f the annual penalty is incurred dur ing underforecast days, when cost o f back-up power imposes more o f a penalty than the unnecessary start-up costs due to overforecasts. Th i s provides a very good opportunity for opt imizat ion , especia l ly tak ing into account that when the load is underestimated, system managers are usua l l y forced to allocate a random increment to the forecasted load i n an almost instantaneous way i n order to cater for the unexpected requirement. The m a i n quest ion to be asked is then: Which of the plants will meet this increment at the lowest possible cost? 10 The system operations engineers (shift engineers) at B C Hyd ro have been answer ing this question us ing heurist ic methods us ing the Au tomat i c Generat ion Cont ro l system ( A G C ) . The analysis proposed i n this thesis considers uncertainty for within-the-hour load forecast error, cost funct ions and other important factors to generate a tool to support the shift engineers' dec is ions i n this part icular task. Th i s research work has the f o l l o w i n g objectives: 1 T o obtain the probabi l i ty distr ibutions o f the load forecast errors for characteristic periods o f the day and year, to enable the shift engineers to select the distr ibut ion that most appropriately represents the current condit ions o f their analysis. 2 Investigate and develop a dynamic p rogramming mode l that allocates the unexpected load increment among the system plants i n the most economic way. 3 Investigate and develop a stochastic dynamic p rogramming mode l to consider w i th in the hour load forecast uncertainty to al locate the unexpected load increment among the system plants i n the most economic way. 4 Test and ver i fy the mode ls us ing real-time operational data. 5 Test a prototype o f the tool at B C H y d r o . 1.5 Organization of the thesis. The f o l l o w i n g chapter describes the B C Hyd ro ' s dec is ion mak ing environment and the way the proposed methodology fits into the ut i l i t y ' s opt imizat ion set o f tools used. Chapter 3 presents a descr ipt ion o f the activit ies performed b y the Generat ion Operat ions department and the P S O S E shift o f f i ce , who w i l l be the m a i n users o f the mode l developed i n this research. It is very important to understand the system operations environment and the different activit ies that are carried out b y the system operations engineers and the way they interact w i t h each other, i n order to learn about the users ' needs and then develop an eff ic ient and workable tool . A b r i e f summary o f the Au tomat i c Generat ion Cont ro l ( A G C ) system, the framework o f the mode l proposed, a long w i th an explanat ion on the factors determining the plant product ion 11 cost, one o f the key elements for opt imal a l locat ion o f load is prov ided i n Chapter 3. Chapter 4 prov ides a descr ipt ion o f the two methodologies used to address the p rob lem o f a l locat ing within-the-hour load uncertainty among the plants i n the system: dynamic p rogramming and stochastic dynamic p rogramming . The p rob lem is described i n terms o f those methodologies , i nc lud ing an explanat ion o f the opt imizat ion variables, parameters and calculat ions, a long w i th a descr ipt ion o f the statistical analysis performed to describe the stochastic behav ior o f the load forecast ing error. Chapter 5 illustrates the results o f the studies that were performed us ing these models . The study uses h istor ica l generation data and compares the actual system operation w i t h recommendat ions for the developed models i n this thesis. F i na l l y conc lus ions and recommendat ions are prov ided in Chapter 6. 12 2 Background 2.1 Literature Review 2.1.1 Optimization techniques for Economic Dispatch. The emergence o f power fu l and pract ical numer ica l opt imizat ion methods for power system engineering and operat ion has been one o f the more important areas o f technical development over the past twenty years. The value o f opt imizat ion use in power systems is considerable, not on ly i n economic terms, sav ing m i l l i ons o f dol lars annual ly , but also i n terms o f system re l iabi l i ty . W i t h the electric industry deregulat ion, opt imiza t ion techniques are not anymore an asset avai lable on ly to some, they have become essential, first hand tools in the da i l y operat ion o f any electric ut i l i ty . Opt imiza t ion concepts and algorithms were first introduced in the mid-sixt ies i n order to mathemat ica l ly fo rmal ize dec is ion-making w i th regard to the myr i ad o f objectives subject to technical and non-technical constraints ( M o m o h , 2001). A s more eff ic ient computat ional tools became avai lable, research activit ies have introduced more sophisticated methods to help improve the system f rom the general to the part icular : the large observed problems f rom the past have been broken down to smal ler , more detailed ones i n order to explore even the smallest opt imizat ion opportunity. E c o n o m i c dispatch ( ED ) o f power generating systems is an example o f such problems. One o f the ma in goals o f power system operation is dispatching the power outputs o f generators so as to m i n i m i z e total fuel cost wh i le sat isfy ing load demand and a set o f technical and security constraints. In the case o f a hydro system, improved generation and water resource management are achieved b y increasing the amount o f energy produced f rom the water avai lable for release. A c c o r d i n g to Demar t in i et a l . (1996), there are two basic categories o f the prob lem formulat ion: real-time and study E D s . Real-time E D is executed every few 13 minutes or after a change of, say, 1 % in the system load is observed. Th i s category o f E D has usua l l y more stringent performance and accuracy requirements. A typica l instance o f the E D study is found i n the Un i t Commi tmen t ( U C ) appl icat ion where a sequence o f E D problems corresponding to t ime o f the U C is to be solved. A l t hough in a study mode , the performance is st i l l a very important issue since the E D is to be executed at least as many t imes as there are t ime steps i n the U C p lann ing hor i zon (often 168 hours or more) . S ince an incremental formulat ion cannot be jus t i f i ed , there is a need to mode l more accurately the prob lem nonl inear i t ies, i n part icular those o f the cost funct ions and sp inn ing reserve. Extens ive research on this topic has produced a large number o f a lgor i thms for f i nd ing the most economica l way o f a l locat ing the required load demand between generating units, for example : Me r i t Order L o a d i n g [see e.g. Sheble, 1989], Incremental Cost [see e.g. R u x L . M . , 1993], Neu ra l Ne tworks [see e.g. Ruey-Hsum L i ang . , 1999], Newton-Raphson [see e.g. L i n et a l . , 1992], Dan tz ig-Wo l f Decompos i t i on [see e.g., Aganag i c & M o k h t a r i , 1997], and more recently Genet ic A l go r i t hms [see e.g., Sheble and B r i t t ig , 1995]. N o n e o f those methods has been def ined as the c lassical approach for economic dispatch, since the circumstances and the requirements o f a part icular situation have a s ignif icant inf luence on choos ing the appropriate mode l . In addi t ion, the results presented i n the literature are usua l ly not d i rect ly comparable to each other. A l t h o u g h the different approaches to solve the prob lem vary i n methodology, the def in i t ion o f the opt imizat ion p rob lem is usua l l y the same: it considers the cost o f p roduc ing electr ic i ty as a funct ion o f power generated g iven the generation capabi l i ty o f each unit and some restrictions on h o w the unit is operated. The speci f ic object ive o f the economic dispatch funct ion can be expressed as fo l lows (Aganagic & M o k h t a r i , 1997): 1. The object ive funct ion: n 2.1 14 where: Cj (pi) = a cost funct ion o f unit generation i, i = 1,.. . ,n. A n example o f a cost funct ion for hydro systems is g i ven b y R u x (1993), cons ider ing that the input to a hydroelectr ic generating unit is water released per unit t ime or f l ow : Q (unit o f volume/s), and the output is electr ical energy, P ( M W ) . The derivat ive o f a unit's input-output curve (dQ/dP), also k n o w n as its f l ow slope, represents the incremental cost o f va ry ing the unit's generation leve l . Th i s can be seen in F igure 2.1, wh i ch represents two units in i t i a l l y operat ing at point A . Suppose that it is proposed to increase the output o f one unit to point B 2 , wh i ch w i l l require an increase i n the required f l ow b y ? Q 2 ; and to un load the other unit unt i l it reaches an output o f B l , wh i ch w i l l b r ing a reduct ion on flow b y ? Q l . The total M W output does not change, but i f ? Q2> ? Q l , then the change i n load a l locat ion is not opt imal since the product ion cost incurred is higher than the in i t ia l one. OUTPUT, P{MW) F igure 2.1 F l o w slope curve w i th a convex increas ing characteristic (Rux , 1993). The opt imizat ion problem has the f o l l o w i n g constraints: u i i » t 15 2. Un i t capacity l imi ts : Pl<Pi P. >P: i max 2.2 / min where: Pi = product ion power. 3. System constraints (demand supply balance) N 2.3 i= 4. Transmiss ion constraints: A c t i v e power f l ow along l ines may be restricted due to security considerations. 5. Sp inn ing reserve requirement: A required reserve prov ided b y generators and interruptible load resources that are already synchronized to the power system and can p i ck up load immediate ly dur ing fai lures o f generators ( Song & W a n g , 2005) . 6. Water constraints: Plant water releases m a y be scheduled on a month ly , week l y or da i l y schedule. Each hour, the al located plant generation must operate w i th in the estimated release l imi ts . Th i s s impl is t i c approach to economic dispatch has evo lved overt ime. Th i s research work and analyses o f the factors in f luenc ing its performance found that the o ld models fa i led to acknowledge the uncertainty i n load pred ic t ion , w h i c h poses the r i sk o f hav ing insuff ic ient committed capacity to compensate for unit fai lures and/or unanticipated load var iat ion. O n the other hand, the effect o f load uncertainty on unit commitment r isk and on product ion cost was studied by Zha i et al (1994) and J ia-Yo Chang et a l . (2000). B o t h studies showed that the pr imary effect o f load uncertainty is a h igh increase i n r isk, and that the r isk posed by load forecast ing uncertainty is m u c h more s igni f icant than the one that results f rom uncertainty i n uni ts ' ava i lab i l i ty . Uncerta inty i n load forecasting also poses r isk to domest ic electr ic i ty suppl iers, since they may purchase more or less energy than they could sell ( Lo & W u , 2003) . In addit ion, the r isk o f short 16 term power system operational p lann ing i n the presence o f load forecasting errors also exists. A study developed by Doug las et a l . (1998) used the Expected Cost o f uncertainty ( E C O U ) , obtained by means o f statistical dec is ion theory, to use it as a r isk indicator. They have found that the E C O U grows in a s igni f icant way as the lead forecasting t ime becomes larger. Now-a-days, on l y high-end forecast ing technology is be ing used by ut i l i t ies around the wor ld . The prob lem is not that the forecasting systems are not accurate enough. A l t hough there is a lways a chance o f imp rov i ng the performance o f those predict ion systems, the best way to deal w i th load forecast ing uncertainty is to understand its unavoidable existence and assign more resources to research l ines that study its effects i n the performance o f the system and develop proact ive, innovat ive methods to decrease them. Unfortunately , i n contrast w i th the enormous attention on load forecasting techniques, l i tt le has been reported about methodology a imed to develop an economic dispatch a lgor i thm that considers the consequences o f load forecast ing uncertainty. One o f the most important tasks o f economic dispatch is the p lann ing o f the operat ing reserve. Operat ing reserve provides an electric power system w i th the abi l i ty to respond to unanticipated load changes and sudden generation outages (Fotuhi et a l . , 1996). Operat ing reserve evaluat ion invo lves two different aspects. The first is unit commitment , in wh i ch the system operator decides w h i c h units should be committed to satisfy the operating criteria. The second aspect is associated w i th unit dispatch decis ions and the evaluation o f response capabi l i ty o f committed units. In a pract ical power system, the reserve cannot be u t i l i zed instantaneously and is restricted b y the ramp rate characteristic o f the commit ted generating units. A system may have a large amount o f sp inn ing reserve at a part icular t ime but the actual responding capabi l i ty m a y be quite inadequate for rel iable system operat ion. W h e n deve lop ing economic dispatch mode ls and descr ib ing their l imitat ions, a number o f authors have reported the effects o f load forecasting uncertainty and the need to create adaptive algori thms to manage its impacts. These f indings pose a complex challenge because, depending on the s ize o f the system and the preva i l ing circumstances, a large number o f variables may be invo lved . In addit ion, the commitment schedule should mainta in the balance between the power generated and the system demand under normal condit ions 17 (Tong & Shahidehpour, 1989). Demart in i et a l . (1996) also recognized the importance o f tak ing into account the load forecast errors when t ry ing to solve the E D prob lem us ing Lagrange mul t ip l ie rs . They found that enforc ing ramp-rate l imi ts b y us ing the penal izat ion g iven in past trials is successful on l y i n the unreal ist ic case when the actual and forecasted load behaviors are ident ical . T o satisfy the constraints when the load is different f rom the forecast, the Lagrange mul t ip l iers must be mod i f i ed to account for the load var iat ion, w h i c h must be done before execut ing the static dispatch. The use o f stochastic control theory for resource a l locat ion under uncertainty was explored by Ba r Sha lom, et a l . , (1974). T o exempl i f y this, they used the select ion o f diagnost ic experiments and the al locat ion o f repair crews to prevent fai lures in machines. The state o f the components was not kn o w n . They were considered as random variables w i t h a g iven a priori probabi l i t ies. Fo r a class o f resource a l locat ion problems under uncertainty, it was demonstrated that a s izable improvement i n performance and sav ing i n resources can be achieved when the a l locat ion is made us ing stochastic control methods that have the closed-loop property: at every stage the dec is ion is done us ing the past measurements on the resu l t o f the previous actions, and the statistics o f the future measurements. In this way the present dec is ion takes into account the value o f future informat ion. M o r e into the economic dispatch subject, E l-Hawary & M b a m a l u (1990) developed an opt imal f l o w mode l that deals w i th the hydro-thermal opt imal power dispatch prob lem account ing for the effects o f uncertainty i n pred ic t ing active power demand. The method relates the probabi l i ty d istr ibut ion o f the uncertain factors i n the system active power generation, bus voltages and phase angles to the probabi l i ty d istr ibut ion o f the errors result ing from the forecast o f the active power demand. It is assumed that random errors in power demand are norma l l y distr ibuted w i th zero mean and some variance. The iterative solut ion o f the reformulated p rob lem uses New ton ' s method and Powe l l ' s penalty funct ion approach. The object ive for an opt imal power f l ow scheme is to m i n i m i z e some system generation performance funct ion such as system losses, system thermal fuel cost or other object ive wh i l e respect ing the phys ica l load ing l imi ts imposed by the network conf igurat ion. 18 Electricity demand is time dependent and needs to be treated as a stochastic process. They proposed an objective function (total thermal fuel cost) which minimizes over the optimization interval: , T F i NS (2.4) . t=l Where: JS = Objective function related to thermal generation cost TF= Final time for optimization process NS= Number of thermal generators in the system a;, Bj, ?j = Fuel parameters for the thermal plants in the system P gi= Active generation at the active ith bus Considering that statistically the actual value of the active generation at the ith bus can be expressed as: P. = P +AP. gi gi gi where: Pgi = predicted value of the active power generation, ? Pgi =inherent predictor error, then the expected system thermal fuel cost can be expressed as: (2.5) Perturbations in the voltages angles and the phase angles due to perturbation in the actual power demand are considered as equality constraints and included in the optimization process. Inequality constraints are associated with reactive power dispatch. The hydraulic aspects of the system are evaluated considering that there is no 19 fuel cost associated w i th the operation o f a hydro unit . The hydro mode l for short per iod dispatch can then be expressed as fo l l ows : where: a;, Bj, ?j = K n o w n parameters for the hydro plants i n the system N H = N u m b e r o f avai lable hydro units Jq = Hyd rau l i c vo lume o f water object ive por t ion Subst i tut ing equation 2.5 into equation 2.7 to account for load forecasting uncertainty prov ides the express ion o f the expected hydrau l i c vo lume o f water i n the objective funct ion : In this equation, K represents a probabi l i t y constraint. T o solve the determinist ic constrained power system p rob lem, wh i ch make use o f a var iat ional calculus method that uses the Lagrange mul t ip l ie rs to augment the or ig ina l object ive funct ion by appending the equal ity constraints, i n combinat ion w i th the N e w t o n ' s method. Results f rom compar ing this stochastic approach w i th the determinist ic one shows that the former contributes to reduce power and energy losses. One o f the contr ibutions o f this research is that the authors were able to improve the ab i l i ty o f the dispatch operator to deal w i th uncertainty. Other references that deal w i th the effect o f load uncertainty in E c o n o m i c D ispatch was the work o f Takr i t i et.al (1996). In this paper, a method for so l v ing unit commitment problems cons ider ing the stochasticity o f the load i n the system was developed us ing the scenario analysis approach. Th i s methodo logy deems a number o f poss ib le load occurrence scenarios, and assigns a weight (i.e. probab i l i t y o f occurrence, PS) to each one o f them. The opt imal po l i c y must guarantee that when scenario s and s' (2.7) J « = f fc K \ ( « / + Pip* + r, K + APJ 't (2.8) 20 are undistinguishable at time period t, the decision made for scenario s must be the same as that of scenario s'. The scheme of scenario analysis is illustrated in Figure 2.2. Scenario .1 Decisions made for Scenarios 1 ami 2 mmi be Ute sa«*{j-Decisions made for Sceuarios 1, % i, mi 4 must Ue ihesamc. > — , >o— •• ,-, q ',, , o 3 Scenario 3 Moit Tag Wed Tltu Frj \Sa ( . Sou Dedsiows Wia4e t for S c M i a i w $ a n d 4 m u s t \ n Scenario 4 i m j s a m e . Fig 2.2 Scenario analysis (Takriti et al., 1996) The constraint was modeled by partitioning the scenario set at each time period into disjoint subsets that are called scenario bundles. Each scenario is a member of one bundle at a time. The objective function is to minimize the expected cost over all the possible scenarios, as outlined in the model listed in 2.9 below: min x,ii£Ss=l Ps Z " = , X , = , / / ' " M > < ) Zn 2.9 x','s >ds where: u't's - state of each unit, /, at a time period, t, for scenario s. x't's = power output level at which unit / operates during a period, t, for scenario s. f = cost function of operating a unit, i, at a power output level, x,. dst = total demand for electricity during period t, for scenario s. The problem was solved using a Lagrangian relaxation technique which was named progressive hedging, where every multiplier is associated with each one of the constraints, decomposing the problem into single-generator sub-problems. A typical weekly, stochastic unit commitment problem with 10 scenarios and a 168-hour 21 plann ing hor i zon required approximate ly one hundred progressive hedg ing iterations, and s ix hours o f C P U time. Fo r a typ ica l week, the use o f progressive hedg ing had an expected cost o f $19,978,000, w h i c h was lower than that o f the determinist ic strategy, $20,124,000, y i e ld ing a sav ing o f $146,000 per week. The results i l lustrate the potential gains that cou ld be accrued when load uncertainty is tak ing into considerat ion for E c o n o m i c D ispatch . 2.1.2 Stochastic control theory and load uncertainty. In addit ion to E D and unit commitment algorithms, another use o f stochastic theory was found in the long-term p lann ing o f pub l i c l y owned electr ic i ty d istr ibut ion networks (Kauhan iemi , 1991). The method developed take into account the uncertainties related to the growth o f loads and the dynamic nature o f the p rob lem, presenting the load growth as a random variable. The opt imiza t ion uses a number o f predef ined alternative network conf igurat ions, each o f w h i c h have its o w n set o f equipments insta l led i n the network as we l l as f i xed states o f switches and disconnectors. The strategy consisted o f the rules for the transitions f rom one network conf igurat ion to another. The results are given i n the fo rm o f a f l ex ib le investment strategy, der ived b y us ing stochastic dynamic p rogramming a lgor i thm. Stochastic control theory has also been used to study the prob lem o f opt imal energy purchase for three times sequential energy markets i n considerat ion o f demand informat ion rev is ions, formulat ing it as a stochastic dynamic p rogramming prob lem (Yan et. a l , 2000). The buyers are fac ing a sequential dec is ion m a k i n g p rob lem, where the dec is ion variables are the amounts o f energy to be purchased i n each market. The major uncertain factors i n the demand al locat ion process are pr ice and demand. W i t h the assumption o f non-decreasing purchase cost functions in energy markets they prov ide the opt imal demand a l locat ion condi t ion, deve lop ing two algor i thms i n f ind ing the opt imal demand a l locat ion i n different markets. 22 3 Optimization and decision making processes at BC Hydro. 3.1 The BC Hydro Generation System The need for opt imizat ion at B C H y d r o emerges f rom very spec i f ic technical and f inancia l reasons. B r i t i sh C o l u m b i a is expect ing an annual growth i n demand o f around 3 . 7 % . Th i s growth is not as signif icant as it is i n deve lop ing countries such as B r a z i l , w h i c h is expect ing a 7 % rise i n demand, but it is considered considerable for a an industr ia l economy. T o meet this growth, B C Hyd ro faces a number o f tradeoffs between costs, resources avai labi l i ty , re l iab i l i ty , and socia l and environmental impacts. Structural and Non-structural constraints, such as transmission total transfer capabi l i ty and tariffs, restrict the way i n w h i c h electr ic i ty can be generated, distr ibuted and traded. Moreover , although hydropower has except ional qualit ies among generation opt ions, a hydro dominated system carries a b i g number o f uncertainties that need to be addressed. F inanc ia l performance focuses on the returns to B C H y d r o ' s shareholders and the electr ic i ty rates pa id b y their customers. The economic va lue that B C H y d r o generates for the prov ince benefits customers and a l l B r i t i sh Co lumb ians . A m o n g the f inancia l mot ivat ions for the use o f opt imizat ion is the fact that B C H y d r o has entered the electr ic i ty trade business. A s a member o f the Northwestern Power P o o l (an associat ion o f generating ut i l i t ies serv ing the Northwestern Un i t ed States, B r i t i sh C o l u m b i a and A lber ta ) , B C H y d r o has established trade partnerships w i th a number o f A m e r i c a n electric ut i l i t ies , such as the Bonnev i l l e Power Admin i s t ra t ion , the U.S. A r m y Corps o f Engineers , the Bureau o f Rec lamat ion and a l l pub l i c and investor-owned power generating uti l i t ies. In addi t ion, the regional locat ion a l lows B C H y d r o generation and storage faci l i t ies to prov ide the prov ince w i th the abi l i ty to engage i n "energy b a n k i n g , " i.e., storing water i n reservoirs dur ing off-peak per iods for generation and export dur ing peak per iods (either da i l y or seasonal) when prices are higher, and depending on speci f ic t rading arrangements, l o w pr iced power can be imported i n off-peak periods. 23 International agreements introduce more variables and constraints to the system's optimal economic operation, especially considering that electricity is not a storable good and for this reason financial pricing securities cannot be created. Therefore, traders and operators need decision support systems that are capable to organize and present information when it is needed and to provide real-time recommendations for actions in the future. According to Shawwash et al. (2000), the electricity sold in 1998 totalled 56500 gigawatt-hours, of which 23.3% represented out-of-province electricity trade. The behaviour o f imports and exports in British Columbia can be observed in Figure 3.1. It is interesting to note that export sales peaked in 2000, with high demand brought about by the California electricity crisis in late 2000 and early 2001. Imports have increased since 1996, which reflects the opportunities under open access to import power when it is advantageous to do so and to benefit from low prices during off-peak periods, especially late at night. On a seasonal basis, imports tend to increase during the second quarter which enable the replenishment of B . C . ' s reservoirs from the spring run-off (Canadian Electricity Imports and Exports, 2003). GW.h 12000 i 1 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 (e) II Firm Exports Q Interruptibie Exports ™ ;* Imports Figure 3.1 British Columbia Imports and Exports (Canadian Electricity Imports and Exports, 2003) B C Hydro serves about 1.6 mil l ion customers over an area o f almost 95 mil l ion hectares. To achieve that, 18,286 kilometres of transmission lines and 55,254 24 ki lometres o f distr ibut ion l ines have been bui l t over the years. F igure 3.2 shows the distr ibut ion o f the ma in service areas. F igure 3.2 M a p o f B C H y d r o Serv ice Areas ( B C Hydro website). Even though the evolut ion o f computat ional tools has a l lowed the development o f models to prov ide methodologica l analysis on some processes that affect the system performance, a number o f issues are st i l l addressed wi th heuristic methods, and although these methods are based on experience, the lack o f a robust methodo logy might cause some variables and issues to be neglected and thus poses the r isk o f losses i n potential revenues. T o overcome this poss ib i l i t y and to be able to m a x i m i z e the use o f the avai lable resources and to m i n i m i z e the r isk invo lved , month ly , da i l y and hour ly 25 opt imizat ion hor izons and targets have been establ ished in almost every operat ing level at B C H y d r o . The opt imizat ion mode l developed in this thesis is a imed to serve as a dec is ion mak ing tool i n the Generat ion Operat ions department, ma in l y to support the economic dispatch activit ies that are carried out b y the P lann ing , Schedul ing and Operat ions Shift Engineers ( PSOSE ) . T o per form the p lann ing and operations functions generation operations fo l l ow the steps described be low: 1. Da ta co l lec t ion: Input in format ion such as weather forecasts, i nc lud ing precipi tat ion, expected runoff , temperature and w ind data is organized. L o a d and rate-of-flow (stream f low) forecasts are gathered and other non-power goals as we l l as maintenance objectives are rev iewed. Fo r load forecasting, B C H y d r o uses the Advanced A r t i f i c i a l Neura l Ne two rk Short Te rm L o a d Forecaster ( A A N S T L F ) , w i th a forecast ing accuracy o f between 2 and 5 % . 2. 30-day computer study: Nex t month system operations are s imulated to h ighl ight potential problems and to help the operation o f reservoirs us ing the informat ion gathered in the first step. 3. Pre-scheduling: The load for the next day is planned. N e x t day 's schedules are ver i f ied w i th other ut i l i t ies, ensuring that load and resources can be matched. 4. Schedul ing: H y d r o operations for the next few hours are prepared and the hour l y generation o f the B C H system and the independent power producers (IPP) are opt imized us ing Automat i c Generat ion Cont ro l ( A G C ) . Operat ions are moni tored minute b y minute to guarantee that the system operates w i t h i n the constraints. R ivers f lows are balanced w i th generation, us ing the L o a d Resource Ba lance ( L R B ) system. 5. D ispatch ing : T ransmiss ion and generation levels are moni tored second by second and the match ing o f generation and load is performed. D i spa tch ing is responsible for l ine sw i tch ing and substation operations. A l s o dur ing this step l ine and substation outages are scheduled. 6. Project schedul ing: The required power is generated f rom avai lable water. The hydroelectr ic plants can be placed into one o f three groups: Peace R i ve r , w i th 2 generation plants contr ibut ing wi th 3 7 % o f the total generation; C o l u m b i a 26 R i ve r , w i th 4 generation plants contr ibut ing w i th 4 5 % o f the total generation and the 23 smal l hydro plants that contribute w i th 1 5 % o f the total generation. In addi t ion to these activit ies, the shift o f f i ce works very c lose ly w i th P o w e r E x ' s (the B C H y d r o electr ic i ty trade subsidiary) real-time traders who sell and purchase electr ic i ty i n the spot power markets i n the U S and A lber ta , to ma in l y ident i fy P o w e r E x ' s trade l imi ts and to opt imize the system to m a x i m i z e export and import opportunit ies. They also manage fuel supply and gas use o f the Burrard Therma l Plant operation, and also, dur ing the occurrence o f extreme events, they are responsible for direct reservoir f lood rout ing operations. 3.2 Automatic Generation Control (AGC). The m a i n tool for load schedul ing, not on l y at B C H y d r o but also at almost every ut i l i ty i nvo l ved i n the electricity trade business, is A G C . The A G C program is responsible for the control o f generating resources to m i n i m i z e the cumulat ive instantaneous control area error (the instantaneous difference between net actual and scheduled interchange), tak ing into account the effects o f frequency bias i nc lud ing a correct ion for meter error. The control area represents a power system or a combinat ion o f power systems for wh i ch a c o m m o n generation control scheme is appl ied. The control area inc ludes the generating units, the connect ing transmission l ines, and the area tie l ines operated to supply a load. The most recent advancements in this area have proposed the appl icat ion o f concepts l i ke neural networks, f u z z y log ic , and genetic algorithms to tackle the di f f icu l t ies associated w i th the design o f A G C regulators for the power systems w i th nonl inear mode ls (Ibraheem et al . , 2005) . The A G C p rob lem has been studied for more than three decades and a rev iew o f the literature illustrates a history o f the methodologies that have been proposed over the years. Th i s section w i l l focus on the A G C system used at B C Hyd ro as the mode l developed i n this thesis work is c lose ly related to this topic. In addit ion to P S O S E , whose decis ions inf luence the way the A G C is operated, the system control centre is entrusted b y B C Hydro generation for its operation. The Operations P lann ing staff determines the opt imum plant/unit generation 27 schedule based on long , m e d i u m and short term management goals, load forecasts, i n f l ow forecasts, outage schedules and operat ing constraints. 3.3 Components of the AGC system: The A G C Subsystem o f B C Hyd ro is consists o f the f o l l o w i n g subroutines ( B C Hyd ro ) : 1) AGC Executive Program (AGCRUT): A s the ma in program o f the A G C system, it prov ides system t im ing and checks for system t ime change (for the change to da i l y sav ing t ime, for example) , executes scheduled t ime correct ion, detects upcoming changes i n the next hour 's interchange schedule a larming the user, sends out control pulses to units and set-points to plant control lers and reads and writes A G C data tables. It also cal ls the A G C I N I , R E F R N C and A C E C A L routines. 2) Initialization Routine (AGCINI): It in i t ia l izes the operating condit ions for the A G C program, reads A G C input data tables, moves data into common area buffers and writes updated tables and records to the database. 3) Scanned Data Processing Routine (REFRNC): Th i s routine d ig i ta l l y smoothes the prev ious ly revised input data: frequency, t ime error, forebay and tailrace elevations etc. It also compares the actual and scheduled frequency and interchange to calculate the corresponding deviat ions. I f the deviat ions exceed a user-defined l im i t , an alarm is set on. A n alarm is also set on when the output o f an onl ine unit is be low the user-defined l im i t . 4) ACE Calculation and Processing Routine (ACECAL): Th is process uses the smoothed quantities to calculate the A r e a Cont ro l E r ror ( A C E ) , and checks i f it is w i th in the established l imi ts . 5) Power Allocation Routine (PWRALL): It allocates the calculated error among the regulat ing plants and units to el iminate the generation-load mismatch 6) Generator Pulsing Routine (PLSGEN): Th i s routine converts the calculated error into pulses to the unit. It keeps track o f the pulses sent and determines the expected response. I f the unit fai ls to respond, the routine w i l l e l iminate it f rom A G C . 7) Reserve Monitoring Routine (RESMON): A l s o cal led b y A G C R U T , it computes reserves and obl igat ions every 5 minutes, i n order to guarantee that their va lue is i n accordance w i th the m i n i m u m operating cr i ter ia o f the Northwest Power 28 Poo l and sets an a larm i f the calculated values are be low those criteria. It calculates the operat ing reserve ob l igat ion ( O R O ) as 5 % o f the hydro generation plus 7 % o f the thermal generation. Sp inn ing reserve is calculated as one h a l f o f O R O . 8) Interchange Scheduling Program (ITSGRD): It stores al l activit ies for the previous day, current ly i n process and scheduled for up to 7 days into the future. It also calculates the net control lable interchange schedule and at the end o f each hour, it sends the schedule values to the arch iv ing database. 9) Hydro Generation Dispatch program (HGD): Th i s funct ion, avai lable i n real t ime and study modes , is the equivalent o f economic dispatch in a thermal system. It allocates the total desired hydro generation among the hydro plants. It also calculates the plant and unit base points and unit economic part ic ipat ion factors us ing unit characteristic data. The E c o n o m i c program consists on reassigning plant loads i n an economic fashion to account for deviat ions f r om load forecast, interchange schedules and anticipated system condit ions. It is i n this spec i f ic part o f A G C where the need for a robust tool that helps the P S O S E to al locate the load var iab i l i t y was ident i f ied. A t the moment , the a l locat ion o f the unforeseen increments is per formed to some extent b y A G C , according on l y to their normal ized regulat ion part ic ipat ion factors. These factors are computed i n proport ion to each uni t ' s capabi l i ty . The ef f i c iency program aims at m i n i m i z i n g water use at a speci f ic plant for the g iven plant load ing , subject to hydrau l i c and electric constraints such as m a x i m u m and m i n i m u m discharge for each unit and the uni ts ' inoperable operat ing zones. The H G D prob lem is formulated as an opt imiza t ion prob lem, where the object ive is to m a x i m i z e the plant e f f i c iency , and it can be def ined as fo l l ows : Minq(i) sum (q(i)) 3.1 where: i = l , . . .n N = number o f on-line generating units i n the plant q(i) = discharge through unit i Subject to the f o l l o w i n g constraint: 29 sum P(i) - PL 3.2 where: P L = M W load assigned to the plant P(i )= unit i M W output The unit output in M W is g iven as a funct ion o f the uni t ' s discharge and net head: where: H L ( i ) = head losses H G ( i ) = gross head A l t hough the object ive funct ion (3.1) is l inear, the constraints are nonl inear. T o ensure a feasible so lut ion, the nonl inear constraints are mode led as convex funct ions, so the so lut ion can be found app ly ing gradient methods. The results o f the opt imiza t ion process w i l l prov ide previous and current plant and unit discharges, as we l l as the correspondent energy part ic ipat ion factors and basepoints. Da ta va l idat ion routines: Some data va l idat ion procedures are carried out i n between the execut ion o f the A G C system components to ver i fy the data qual i ty. The B C H y d r o ' s A G C system operates in a real-time mode to prov ide for the control o f 90 generators (1993) and 24 interchange tie l ines, but w i l l be expanded to inc lude 170 units and up to 150 generation plants i n the future. It is executed per iod ica l l y or at user-speci f ied intervals. The scheme o f the system and the in format ion and processes f lows can be seen in F igure 3.3. 3.3 The net head H N ( i ) is g i ven by: HN(i) = HG(i) - HL(i) 3.4 30 AGC INI (Initialization routine) AGC DISPATCHER REFRNC k Time error and actual area frequency Actual tie line & generator power AGC DATA ACECAL PWRALL PLSGEN r W Net Schedule & Net Ramp ITSGRD AREA POWER SYSTEM Generator Base points and Economic participation factors. ITSEXV HGD Hydro Dispatcli Data F igure 3.3 The Au toma t i c Generat ion Cont ro l System (Landys & Gy r , 1993) The In i t ia l izat ion Rout ine ( A G C I N I ) starts as soon as the dispatcher acquires in format ion on the fu l l interchange load schedules, the generating uni ts ' operat ion mode, and the plant part ic ipat ion factors. A telemetry scan is per formed every two seconds to obtain real t ime unit data, such as the unit breaker status, gate l im i t , condenser status, the generating plant hour ly M w h , and unit A G C and on/off status. Data gathered through telemetry also inc lude frequency dev iat ion measurements, meteorologica l data (w ind speed, w ind direct ion, temperature and precipitat ion), plant forebay and tailwater elevations, and t ime deviat ion measurements. W h e n informat ion is gathered, the A G C system then calculates the A r e a Cont ro l Er ror ( A C E ) in M W us ing the A C E C A L funct ion. The A C E is a bas ic error s ignal , on wh i ch al l successive A G C functions are based on. The A C E represents the contro l led area's deviat ion f rom scheduled interchange and system frequency deviat ion. Negat i ve values o f A C E denote under-generation, and posi t ive values mean over-generation. 31 W h e n an area under or over generates, regulat ion and costs are imposed on other areas. Th i s creates inadvertent energy, t ime error, loss o f b i l l ab le load, further addit ional regulat ion and correct ion procedures resul t ing in benefits to some and economic loss to others (Henderson et a l , 1990). The output o f contro l led generators is then adjusted to balance these deviat ions and i f poss ib le , to correct earl ier accumulated errors. The A C E can be def ined as fo l lows : ACE = (Tl-T0)-lOBfft -F0)-SQ3BtTd)-Tob+C 3.1 Where : T i = net actual interchange i n M W . It is def ined as the algebraic sum o f the power on the area tie l ines. To = scheduled net interchange in M W . It is the mutua l l y pre-arranged intended net energy on the area tie l ines. To have a posi t ive scheduled interchange means that more energy is intended to be del ivered f rom, rather than received b y the area. B f = control area's f requency bias setting, considered to have a negative s ign. F i = Ac tua l system frequency in H z . Fo = Scheduled system frequency i n H z , usua l l y 60 H z . B t = control area's t ime error bias in MW/second . Ta = instantaneous value o f t ime error i n seconds S = a constant o f either 1 or 0 depending on the control area's accumulat ion o f interchange. T 0 b = dispatcher scheduled interchange energy i n M W , used to correct inadvertent accumulat ions. A negative accumulat ion o f inadvertent interchange w o u l d be corrected b y putt ing in a pos i t ive schedule, creating an A C E that w o u l d rise generation. C = Equipment error, a value i n M W used to compensate for equipment error, such as non scheduled losses, absence o f telemetry or ca l ibrat ion errors in the A G C algor i thm. It shal l not exceed an entered l im i t o f 0 to 100 M W and w i l l norma l l y be calculated every 1 to 10 minutes. 32 Each generation source i n A G C must be able to respond to two part ic ipat ion factors: the regulat ing part ic ipat ion factor wh i ch allocates the A C E among var ious units and the economic part ic ipat ion factor w h i c h allocates a required generation change among certain units economica l l y . The economic part ic ipat ion factor is determined b y Operations P lann ing , and A G C automatical ly calculates the regulat ing part ic ipat ion factors. These part ic ipat ion factors can be al located per plant or per unit basis. Wh ichever the case is , A G C w i l l d i v ide the part ic ipat ion requirements between units on a log ica l fashion to ensure the appropriate response f rom the plant. In order to opt imize the number o f runn ing generating units at a hydro plant and to obtain a snapshot o f their loads at any instant, the dispatcher uses a program cal led S P U C (Static Plant U n i t Commitment ) . The Forecast ing and Schedu l ing functions relate to A G C when a l l ow ing the dispatcher to authorize var ious schedules, forecasts and data received f rom the Operations P l ann ing Computer and to use this informat ion i n a number o f funct ions i n the energy management system. They inc lude the f o l l ow ing : • Short T e r m L o a d Forecast ing ( S T L F ) • Short T e r m Inf low Forecast ing (STIF) • Generat ion Schedul ing (GS ) , a set o f programs and interfaces used b y the dispatcher to in i t ia l ize and revise the schedules and forecasts received f rom Operat ion P lann ing , and to determine new hour l y plant generation schedules us ing manua l mod i f i ca t ion . • Static P lant Un i t Commi tment ( S P U C ) • Equ ipment Outage Schedu l ing ( EOS ) Bo th the S T L F and the ST I F are conf igured to automatica l ly adapt forecasts i n accordance to weather behaviour. Separate forecasts are presented for the integrated system, the Vancouve r Island sub-area, the C o l u m b i a R i v e r Sub-area, the West Kootenay Power system, the A l c a n system and three other load sub-areas. Fo r each o f those, the S T L F trends the actual load and calculates a forecast for the next 12 hours. Th i s is executed every 5 minutes to update the actual load, the revised real t ime forecast 33 load, the forecast error and the forecast error bias trend for a l l eight-load forecasts. A scheme o f the interactions between A G C , H G D , and S P U C is shown in F igure 3.4. OPERATION PLANNING INPUT DATA: LOAD FORECAST, INFLOW FORECAST. GENERATION SCHEDULES ETC.. OPERATING LIMITS AG DATA BASE INITIALIZE INPUT REVISE SCHEDULES AUTHORIZE SCHEDULES MONITOR SCHEDULES AND FORECAST SPUC AGC k HGD F igure 3.4 Forecast ing and Schedu l ing Conceptual D i ag ram (Landis & G y r , 1993). The outputs o f the G S funct ion are the ma in inputs to the A G C system. They inc lude the hour l y hydro unit summary (unit load ing , turbine f l ow and net head), the plant hour ly summary ( inc lud ing the number o f units on-line, the total generation, reserves, reservoirs elevations and tai lwater elevations. A s a f ina l note on A G C , it is important to k n o w that it imposes an economic burden on day-to-day operations, starting f rom the in i t ia l implementat ion and ongo ing maintenance costs (Henderson et a l . , 1990). In addit ion, a number o f factors have been ident i f ied as sources o f increased operat ing cost, such as hav ing narrower A G C ranges than the uni t ' s proper operat ing ranges, uneconomic unit commitment per formed on ly to.sat isfy the A G C requirements, units often not be ing at their economica l l y assigned dispatch level and fai lure o f units on manual control to carry their share o f the load. Fo r this reason, i n order to m i n i m i z e the cost o f system regulat ion, it is imperat ive to select the units to be placed on A G C careful ly . The dec is ion support system developed i n this thesis w i l l help the system operations engineers i n dec id ing on wh i ch units to dispatch g iven the within-the-hour load uncertainty and units ava i labi l i ty , and it complements and enhances the A G C control funct ion. 34 4 Methodology Th i s chapter discusses the two models that were deve loped to opt ima l l y al locate increments o f load due to load forecasting uncertainty. The models are prototypes that w i l l be incorporated to the P S O S E environment i n the future. D y n a m i c P rogramming and Stochastic D y n a m i c P rogramming are two w ide l y used mathematical p rogramming techniques. Mathemat ica l p rogramming is def ined as a mathematical representation a imed at p lann ing the best possible distr ibut ion o f limited resources among competing activities, under a set of constraints imposed by the nature of the problem being studied (Bradley et al . ,1977). The present chapter includes descriptions o f the methodologies used and the w a y they were adapted to solve this part icular p rob lem, the or ig in o f the input data and some comments on the future work that w i l l be carried out after the complet ion o f this research. 4.1 Dynamic Programming D y n a m i c p rogramming is an opt imizat ion approach that transforms a complex p rob lem into a sequence o f s impler problems, or i n other words, a mult istage dec is ion p rob lem is decomposed as a sequence o f single stage dec is ion problems. A n N-variable p rob lem is represented as a sequence o f N single var iable problems that are solved successively. In most o f the cases, these N sub-problems are easier to solve than the or ig ina l p rob lem ( M o m o h , 2001) . The essence o f dynamic p rogramming is R i cha rd Be l lman ' s p r inc ip le o f opt imal i ty : "An optimal policy has the property that whatever the initial state and the initial decisions are; the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision. " Th is bas ica l l y means that the dec is ion to take depends on on l y the current state and not on h o w this state was reached. Fo r dynamic p rogramming problems i n general, knowledge o f the current state o f the system includes a l l the informat ion about its 35 prev ious behaviour necessary for the determining the opt ima l po l i c y to fo l l ow . A process that has this characteristic is a Markovian process. I f the mul t ivar iab le opt imizat ion prob lem is decomposed into a series o f stages, then, at some point i n t ime at stage n, the process w i l l be i n some state s n . M a k i n g p o l i c y dec is ion x n moves the process to some state s n + i at stage n + 1. Under an opt imal po l i c y , the contr ibut ion to the objective funct ion wou ld be f* n+i(s n+i)- S ince the p o l i c y dec is ion x n also adds to the objective funct ion, then the contr ibut ion o f the stages n onward to the object ive funct ion w o u l d be f n ( s n , x n ) . The state at the next stage is complete ly determined by the state and p o l i c y dec is ion at the current stage. The elements o f a dynamic p rogramming scheme can be seen on F igure 4.1. Stage n Stage n+1 Contribution of x r value: ysirV F igure 4.1 The dynamic p rogramming elements (H i l l i e r and L ieberman) . The solut ion procedure begins b y f i nd ing the opt ima l p o l i c y for the last stage, w h i c h prescribes the opt imal po l i c y dec is ion for each o f the poss ib le states at that stage and then it moves backwards stage b y stage, each t ime f ind ing the opt imum po l i c y for that stage unt i l it f inds the opt imum po l i c y starting at the in i t ia l stage. There are several ways o f categor iz ing determinist ic dynamic p rogramming problems. One w a y o f do ing that is b y the fo rm o f the objective funct ion: the objective might be to m i n i m i z e or m a x i m i z e the sum o f the contr ibutions f rom the ind iv idua l stages. M o m o h (2001) proposes the f o l l ow ing categories: 1. Init ial value prob lem, when the value o f the in i t ia l state variable x n + i is prescr ibed. 2. F ina l value p rob lem, when the value o f the f inal state var iable x i is prescribed. 3. Boundary va lue p rob lem, when both the input and output variables are specif ied. 36 One o f the most c o m m o n problems encountered b y the users o f dynamic p rogramming is the curse o f d imensional i ty . The curse o f d imens iona l i ty causes schemes w i t h large number o f irrelevant states to be behave i n an undesirable way, ma in l y because the d imens ion o f the input space is h igh , and thus the network uses almost a l l its resources to represent irrelevant regions o f the space. A remedy is to preprocess the input i n the r ight way, for example b y sca l ing the components accord ing to their importance. Howeve r , even i f we have an a lgor i thm w h i c h is able to focus on important regions o f the input space, the h igher the d imens iona l i ty o f the input space, the more data may be needed to f ind out what is important and what is not. P r io r to the 1960 's and since the appl icat ion o f l inear p rog ramming to generator schedul ing, opt imiza t ion was conf ined to economic dispatch, not the " o n / o f f question o f unit commitment . The method used margina l generating costs, and the Lagrangian mul t ip l iers method was w ide l y used. In the early 1960 's , analyt ical models were implemented for unit commitment i n order to replace the heurist ic pr ior i ty l ist method wh i ch was then in use. It was i n this per iod that dynamic p rogramming took root as a unit commitment technique (De lson & Shahidehpour, 1992). It prov ided a combinator ia l procedure to enumerate and evaluate solut ions, reject ing immedia te ly many infer ior combinat ions . The computat ional burden was huge due to the nature o f the prob lem. Heur is t i c methods were used to l im i t the dynamic range o f the search. 4.1.1 Definition of the problem. A s it was ment ioned i n Chapter 3, wh i l e execut ing A G C there is the need to reassign unpredicted plant loads i n an economic fashion to account for deviat ions caused b y load forecast errors. A t present, P S O S E make use o f the plant economic part ic ipat ion factors and heurist ic rules to al locate unforeseen increment i n load among the system's plants. A snapshot o f the 5 minutes average load , the hour ly system load, P S O S E load forecast error and the 5 minutes dif ference (i.e. difference between the real and forecasted load) for a random day is prov ided i n F igure 4.2. The area w i th in the red circ le i l lustrates the condi t ions o f the prob lem. A t around 10:00 a.m., the 5 minutes average load has a va lue o f over 7000 M w h . It can also be seen that the forecasted load for that same t ime is around 6950 M w h . Therefore, an increment o f 50 M w h needs to 37 be generated i n addit ion to the 6950 M W H that were scheduled to compensate for the load forecast error. The way i n wh i ch this increment w i l l be distr ibuted among the plants i n the system can be opt imized wi th the models proposed in this thesis work. A s a first opt ion , a determinist ic dynamic p rogramming mode l was constructed. In this section, a descr ipt ion o f the ma in inputs and the formulat ion o f the dynamic p rogramming a lgor i thm are prov ided. —BCH_SYS_LoadAvg5min_MW_00 7100 «BCH_SYS_LoadHourly_MWh_00 PSOSE Load Forecast 7000 6900 6300 6700 6600 6500 L \ I \ \ \ K; >Mf-*\iJK \ V v \ / \ A I | i v[ y 5 Min diff 500 400 300 200 100 0 -100 -200 -300 •400 4-14-03 4:48 AM 4-14-03 7:12 AM 4-14-03 9:36 AM 4-14-03 12:00 PM 4-14-03 2:24 PM 4-14-03 4:48 PM 4-14-03 7:12 PM F igure 4.2: The unexpected increment in load due to load forecast error. The dynamic p rogramming algori thm for the a l locat ion o f load var iab i l i ty was formulated as fo l lows (see F igure 4.3): 1) The prob lem requires m a k i n g three interrelated decis ions, namely , how m u c h o f the total load (i.e. scheduled load plus increment) T should be al located among the plants. The plants among wh i ch the load can be allocated are the stages. T o s imp l i f y the problem formulat ion , three plants w i l l be considered: G M S , M C A and R E V . However , the mode l developed can be easi ly mod i f i ed to al locate the load increment among any number o f plants. N = number o f plants n = current stage 38 2) The decision variables for each stage are the fraction of the total increment to be allocated to the stage (plant) n. x n = decision variable for stage n 3) The states of the problem are represented by a scalar vector containing a range of increments in Mwh that can still be allocated at the remaining plants. s„,t = current state for stage n Thus, at stage 1 (GMS), where all three plants remain under consideration for allocations, the value of the first state is equal to the value of the total increment T that needs to be allocated among the plants. However, at stage 2 or 3 (MCA and REV) is just T minus the fraction of the interval already allocated at preceding stages, having for example the following sequence: si j = T, S2,i = T - x\ S3 j = T - X2 It is important to note that the dynamic programming algorithm is a recursive procedure of solving backward stage by stage. When stages 2 or 3 are being solved, allocations at the preceding stages shall not yet have been solved. Therefore, every possible state (i.e. every possible fraction of the total increment to allocate) must be considered. For example, Figure 4.3 shows the states considered at each stage in a typical setting. The two links shown depict one feasible and one unfeasible transition in states from stage 3 to stage 2. The numbers at the right of each state are the contributions to the measure of performance. The measure of performance is the cost of allocating a certain amount of load on each plant. The overall problem is to find a path that minimizes the sum of the numbers along the path. 4) If CJ(XJ) is the measure of performance from allocating x-t MWHr to plant n, then the objective is to choose xi , X2 and X3 so that: subject to: 3 n=\ 5) The contribution of stages n, n+l... ,N to the objective function can then be described as follows: 39 3 fn , *„ ) = C n , t ( X n ) + ™ ™ Z C - (Xi ) 4.3 Stages: N = 3 j ! Decision variables: x. T= 1300 Mwh — Feasible — Unfeasible Figure 4.3 Dynamic programming formulations for optimal allocation of load variability. where the minimum is taken over x n+i, - - .x3 such that 3 4.4 n=\ and Xj are nonnegative integers. In addition, 6) The recursive relationships can then be defined as f,*(s„,l) = minf„(s„,nx,l) 4.5 M s „ , i , * „ ) = c„,t ( * „ ) + fL * min /„ (snJ - xn) 4.6 40 4.1.2 The modeling environment. The models developed i n this thesis were developed us ing the student vers ion o f A M P L ( A Mathemat ica l P rog ramming Language) , wh i ch is an algebraic mode l l i ng language for mathematical p rogramming . The A M P L programming language uses algebric expressions wh i ch are very s imi lar to the standard algebraic notat ion. A M P L offers a comprehensive, power fu l and f lex ib le command environment for setting up and so l v ing l inear, nonl inear and integer p rogramming problems often encountered i n opt imizat ion . The key mode l l i ng environment features offered b y A M P L are: • Interactive command environment w i th batch f i les process ing options. D i s p l a y commands a l l ow the v i ew o f any mode l component or expression, b rows ing o n -screen or wr i t i ng to a f i le . • Separation o f mode l and data. W i t h this feature, models developed i n A M P L remain concise even as sets and data tables grow. M o d e l s may incorporate many k inds o f condit ions for va l id i ty o f the data. • Interfaces solvers inc lud ing C O N O P T , C P L E X , L A M P S , L A N C E L O T , L O Q O , L S G R G , M I N O S , O S L , S N O P T , and X A . 4.1.3 Dynamic programming model for optimal allocation of load variability. The dynamic p rogramming mode l developed for this thesis wo rk consists o f three m a i n f i les: 1. A data f i le conta in ing the inputs o f the mode l (dynallocate.dat). The extension " .da t " is necessary for A M P L to ident i fy the input f i le . 2. A run f i le conta in ing the core o f the mode l (dynallocate.run): the parameters, sets and variables def in i t ions, commands for the integration o f the input data f rom the data f i le , the execut ion o f the recursive backwards procedure and commands for the d isp lay o f the results. 3. A f i le conta in ing the results o f the mode l (dynallocate.out). One o f the inputs to both the determinist ic and the stochastic models deserves special attention: the cost funct ion. A typ ica l cost functions for the three plants are i l lustrated i n F igure 4.4: 41 Plant Generation Co ^  Fundi on^ $ 2250 Plant Generation, hlWHr F igure 4 . 4 Plant generation cost funct ion The cost funct ion was der ived f rom a relat ionship between ef f ic iency and the marg ina l value o f water, w h i c h then it converted to a measure o f $/m 3 : PlantGeneration _ HK GC = • Rres* 4 .7 Hk 3.6 Where : G C = Generat ion cost, in $/m 3 . Plant Generat ion = given i n M w h Hk = ef f ic iency relat ionship between f l ow and generation given i n Mwh/cms Rres = margina l value o f energy at the plant, g iven in $/Mwh W h e n calculat ing the cost funct ion it is very important to consider the units i n w h i c h generation and f l ow are g iven. The negative values observed in Figure 4 . 4 mean that generating energy in a plant w i th such a cost is more benef ic ia l than generating it i n a plant w i th posi t ive values. Fo r example , for a generation o f 1150 M W h , the opt ima l a l locat ion cost w o u l d be -707, at G M S . The negative cost corresponds to the dev iat ion o f the plant generation f rom the point where the plant wou ld be typ ica l l y set as dictated b y the long and short term models , for the particular unit avai labi l i ty . 42 A n example o f h o w the program components interact w i th each other is shown next, w i t h the formats needed to be entered i n the A M P L program. First the file conta in ing the data (dyna l loca te .da t ) : A . The stages o f the p rob lem set p l a n t := GMS MCA R E V ; B. The states for stage 1 ( G M S ) set s t a t e s [ ' GMS' ] : = 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950 2000 : C. The states for stage 2 ( M C A ) set s t a t e s ['MCA']:= 300 350 400 700 750 800 850 900 950 1200 1250 1300 ; D. The states for stage 2 ( R E V ) set s t a t e s [ ' R E V ' ] : = 400 450 500 550 800 850 900 950 1000 1050 1100 1300 1350 1400 ; E. Trans i t ion costs for stages 1 and 2 450 500 550 600 1000 1050 1100 1150 650 600 1150 650 700 1200 1250 750 set c o s t s l := -252 -290 -172 127 375 -128 322 881 340 51 181 54 1 67 196 413 -162 -281 735 616 512; F. Trans i t ion costs for stages 2 and 3 set costs2 := 260 55 13 396 895 664 435 223 441 1017 874 758 695 708 827 1083 1512 1588; 63 0 100 G . Va lues o f the current generation, the m a x i m u m and m i n i m u m generation capabi l i ty for each plant and the increment to the total load due to load forecast error: param current_gen := GMS 1150 MCA 500 R E V 800; param Max_Gen := GMS 2000 MCA 1300 R E V 1400; param Min_Gen := GMS 1000 MCA 300 R E V 400; param increment := 200; H . The costs associated w i th a l locat ing a fract ion o f the total load (first number) to the plants.: 43 param c o s t l ( t r REV : = 400 -252 450 -128 500 322 550 881 600 340 650 51 700 -162 750 -281 800 -290 850 -172 900 127 950 375 1000 181 1050 54 1100 1 1150 67 1200 196 1250 413 1300 735 1350 616 1400 512 param cost2 (tr) : MCA 300 260 350 55 400 13 450 396 500 895 550 664 600 435 650 223 700 63 750 0 800 100 850 441 900 1017 950 874 1000 758 1050 695 1100 708 1150 827 1200 1083 1250 1512 1300 1588 param c o s t 3 ( t r ) GMS 1000 -742 1050 -632 1100 -568 1150 -707 1200 -773 1250 -757 1300 -628 1350 -602 1400 -711 1450 -750 1500 -715 1550 -566 1600 -572 1650 -648 1700 -664 1750 -603 1800 -429 1850 -465 1900 -502 1950 -486 2000 -395 The mode l i t se l f is located in the d y n a l l o c a t e . r u n f i le that is explained next: A . Sets and parameters def in i t ions. Th i s is the way the program recognizes the data contained i n the input f i les. s e t p l a n t o r d e r e d ; s e t s t a t e s { p l a n t } o r d e r e d ; s e t c o s t s l o r d e r e d ; s e t c o s t s 2 o r d e r e d ; param i n c r e m e n t ; param c o s t l {j i n p l a n t , k i n s t a t e s [ j ] } ; param c o s t 2 {j i n p l a n t , k i n s t a t e s [ j ] } ; param c o s t 3 {j i n p l a n t , k i n s t a t e s [ j ] } ; param c u r r e n t _ g e n { p l a n t } ; param Min_Gen { p l a n t } ; param Max_Gen { p l a n t } ; param t o t _ r e q _ g e n e r = sum {j i n p l a n t } c u r r e n t _ g e n [ j ] + i n c r e m e n t ; param o p t _ c o s t l = min {k i n s t a t e s [ ' R E V ' ] } ( c o s t l [ ' R E V ' , k ] ) ; param membl = o r d ( o p t _ c o s t l , c o s t s l ) ; param g e n e r l = member(membl,states['REV'])> param gener2; param gener3; param memb2; param memb3; 45 param o p t _ c o s t 2 ; param o p t _ c o s t 3 ; param o p t _ c o s t 3 p ; param c o s t _ t r a n s 2 { k i n s t a t e s [ ' M C A 1 ] } = cost2['MCA',k] + o p t _ c o s t l ; param c o s t _ t r a n s 3 { k i n s t a t e s [ ' G M S ' ] , m i n s t a t e s [ ' M C A ' ] } ; param c o r r e s _ g e n e r _ s t a g e 2 { m i n s t a t e s [ ' M C A ' ] } ; param c o r r e s _ g e n e r _ s t a g e 3 { k i n s t a t e s [ ' G M S ' ] } ; B. Once the m a i n elements and variables for the mode l are def ined, the mode l w i l l ca l l for the input file: d a t a d y n a l l o c a t e . d a t ; C . The core o f the mode l is the implementat ion o f the backwards relat ionships expla ined i n equations 4.3 to 4.6. f o r { k i n s t a t e s [ ' G M S ' ] , m i n s t a t e s [ ' M C A ' ] } { i f k + m + g e n e r l <> t o t _ r e q _ g e n e r t h e n { l e t c o s t _ t r a n s 3 [ k , m ] := 100000} e l s e { l e t c o s t _ t r a n s 3 [ k , m ] := cost3['GMS',k] + c o s t _ t r a n s 2 [ m ] } } ; l e t o p t _ c o s t 3 p := min{k i n s t a t e s [ ' G M S ' ] , m i n st a t e s [ ' M C A ' ] } ( c o s t _ t r a n s 3 [ k , m ] ) ; l e t { m i n sta t e s [ ' M C A ' ] } c o r r e s _ g e n e r _ s t a g e 2 [ m ] : = min{k i n s t a t e s [ ' G M S ' ] } ( c o s t _ t r a n s 3 [ k , m ] ) ; l e t { k i n states['GMS']} c o r r e s _ g e n e r _ s t a g e 3 [ k ] : = min{m i n s t a t e s [ ' M C A ' ] } ( c o s t _ t r a n s 3 [ k , m ] ) ; f o r { k i n states['GMS'],m i n s t a t e s [ ' M C A ' ] } { i f corres__gener__stage2 [m] = o p t _ c o s t 3 p t h e n { l e t gener2:= m} } ; f o r { k i n states['GMS'],m i n s t a t e s [ ' M C A ' ] } { i f c o r r e s _ g e n e r _ s t a g e 3 [ k ] = o p t _ c o s t 3 p t h e n { l e t gener3:= k } } ; l e t memb2 := o r d ( g e n e r 2 , s t a t e s [ ' M C A ' ] ) ; l e t o p t _ c o s t 2 := member(memb2, c o s t s 2 ) ; l e t { m i n states['MCA']} o p t _ c o s t 3 := o p t _ c o s t 3 p - c o s t _ t r a n s 2 [ g e n e r 2 ] ; D. The last step is to order the d isp lay o f the results i n the dyna l loca te .ou t f i le : p r i n t 'Data f o r Stage 1 ( R e v e l s t o k e ) : ' > d y n a l l o c a t e . o u t ; p r i n t '' > d y n a l l o c a t e . o u t ; d i s p l a y t o t _ r e q _ g e n e r , o p t _ c o s t l , g e n e r l > d y n a l l o c a t e . o u t ; p r i n t 'Data f o r Stage 2 ( M i c a ) : ' > d y n a l l o c a t e . o u t ; p r i n t '' > d y n a l l o c a t e . o u t ; d i s p l a y opt__cost2, gener2 > d y n a l l o c a t e . o u t ; p r i n t 'Data f o r Stage 3 (GMS):' > d y n a l l o c a t e . o u t ; 46 The results o f the mode l are shown next. The scheduled load have a value o f 2450 M w h . W i t h the unforeseen increment brought b y the load forecast error, the total load to allocate among the three chosen plants is now 2650 M w h . F r o m the results, the best way to allocate that load among the three chosen plants is as fo l lows : tot_req_gener = 2 650 Data f o r Stage 1 (Revelstoke): o p t _ c o s t l = -290 generl = 800 Data f o r Stage 2 (Mica): opt_cost2 = 13 gener2 = 400 Data f o r Stage 3 (GMS): opt_cost3 = -750 gener3 = 1450 The computat ional t ime for 3 stages w i th 21 correspondent states is l itt le more than a second. A l t h o u g h the number o f generation plants i n the B C H y d r o ' s system is more than 30, the number o f plants under A G C is s igni f icant ly lower , and for this reason, this p rob lem is not under the threat o f confront ing the curse o f d imensional i ty . 4.2 Stochastic dynamic programming Stochastic dynamic p rogramming is a framework for mode l l i ng opt imizat ion problems that i nvo l ve uncertainty. These problems have several characteristics that are not present i n the absence o f uncertainty The two most important o f these are the need to consider r isk i n the p rob lem formulat ion and the poss ib i l i t y o f in format ion gathering dur ing the dec is ion process, since this in format ion may be used w i th advantage when m a k i n g future decis ions (Bertsekas, 1976). Th is methodology differs from determinist ic dynamic p rogramming in that the state at the next stage is not complete ly determined by the state and po l i c y dec is ion at the current stage. Instead, the next state is determined by a probabi l i ty distr ibut ion that describes the load increment. The basic ideas o f determining stages, states, decis ions, and recursive formulae st i l l ho ld : they s imp l y take on a s l ight ly different fo rm. A diagram showing the bas ic structure for stochastic dynamic p rogramming is shown i n F igure 4.5. 47 Stage n Stage n + 1 Contribution F igure 4.5 Stochastic dynamic p rogramming f ramework (H i l l i e r and L ieberman, 2005). The determinist ic mode l described i n section 4.1.2 solves the prob lem o f a l locat ing unpredicted loads to the generation plants i n the most economic way i f the uncertainty associated w i t h the load forecasting error is not considered. A s out l ined i n Chapter 1, the load forecast error has a stochastic nature, and its behaviour can be descr ibed w i th a probabi l i t y distr ibut ion. 4.2.1 Data preparation A frequency analysis was developed to prov ide the probabi l i t y distr ibut ion for the load forecast ing error. T o complete this analysis, data for the 5-minute average load and the hour l y load o f the B C H y d r o system, as we l l as the hour l y load forecast error for 2002 and 2003 , respect ively, were retrieved from the B C H y d r o Plant Information (PI) System and databases. T o be consistent w i th what was expla ined i n Chapter 1, the data gathered were organized i n four different spreadsheets, to develop an independent frequency analysis and obtain the correspondent probabi l i ty distr ibut ion o f the load forecast error for each one o f the seasons o f the year, as fo l l ows : 48 1) Winter : December , January and February 2) Spr ing: M a r c h , A p r i l , M a y 3) Summer : June, Ju ly , August 4) F a l l : September, October, November Th is breakdown was dec ided upon w i th the help o f the P S O S E . S imp le data analysis proved the expected var iab i l i ty o f the load throughout the seasons o f the year. In addi t ion , it was observed that the 5-min hour l y difference (i.e. the difference between the P S O S E load forecast and the real 5-min average load) is especia l ly s ignif icant dur ing the ramping periods o f every day, as it can be seen i n F igure 4.6, especia l ly i n the areas w i th in the red c irc les. Th i s f ind ing suggested that, in addit ion to obta in ing the probabi l i t y distr ibut ion for each season, a separate frequency analysis o f the data in norma l and ramping periods w i th in each season should also be performed. Th i s way, the user w i l l have the opt ion o f choos ing a probabi l i ty distr ibut ion according to the required season and t ime o f the day. 1 00 1 1/30/01 7:12 PM 1 2/1/01 1 2:00 AM 12/1/01 4 48 AM 12/1/01 9.36 AM 12/1/01 2.24 PM 12/1/01 7:12 PM 12/2/01 12 0 0 A M 1 2/2/01 4 , 4 8 A M •Se r i e s5 B C H 5 min ave load — B C H Hour ly L o a d — P S O S E L o a d F o r e c a s t Hourly D i f f e r e n c e F i g 4.6 Behav iour o f the load forecast error dur ing ramping periods. The spreadsheets conta in ing the frequency analysis to obtain the probabi l i ty distr ibut ion o f the load forecast error are organized as fo l l ows : 1) PI system data (Co lumns A to E) : The data retrieved f rom the B C H y d r o ' s PI system database (the hour l y load forecast error, the t ime, the 5 min-average load and the hour ly load) are stored in this part o f the spreadsheet. The users can 49 s imp ly copy and paste new data whenever it becomes avai lable. C o l u m n s C represents the minutes w i th in the hour, necessary for further ca lculat ions, as it w i l l be shown later. A B C D E PI SYSTEM DATA HOURLY 5-Min AVG Hourly LFE Time Minutes BCH_SYS_LoadAvg5min_MW_00 BCH_SYS_LoadHourly_MWh_00 46 12/1/01 12:00 AM 0 6167.249 6378.186 9 12/1/01 12:05 AM 5 6134.202 5982.653 41 12/1/01 12:10 AM 10 6112.238 5982.653 53 12/1/01 12:15 AM 15 6091.822 5982.653 30 12/1/01 12:20 AM 20 6074.713 5982.653 -18 12/1/01 12:25 AM 25 6044.111 5982.653 33 12/1/01 12:30 AM 30 6004.523 5982.653 -79 12/1/01 12:35 AM 35 5959.991 5982.653 -68 12/1/01 12:40 AM 40 5934.924 5982.653 -4 12/1/01 12:45 AM 45 5903.347 5982.653 36 12/1/01 12:50 AM 50 5880.202 5982.653 -99 12/1/01 12:55 AM 55 5849.797 5982.653 Table 4.1 PI System data for the first hour o f December 2001 . 2) P re l im inary calculat ions (Co lumns F to H ) : The first row should appear as N/A since every hour is considered to start at the 5 t h minute. a. P S O S E L o a d Forecast = hour ly load + hour l y load forecast error, b. 5 M i n d i f f = P S O S E L o a d Forecast - 5 M i n average load , c. H o u r l y Forecast dif ference = P S O S E L o a d Forecast^) - P S O S E L o a d Forecast ( t+i). The H o u r l y Forecast difference was calculated on l y for the interest o f the P S O S E , but it is not relevant for this analysis. 50 B | F G H Time l-imM:l E L / - I - : : - : : : : : : - : - : : - : - : : - : jPSOSE Load Forecast 5 Min cliff HOURLY FORECAST DIFFERENCE " 12/1/01 12:00 A M N/A N/A JN/A 12/1/01 12:05 A M 6029: -106) 12/1/01 12 :10AM 6029: -84 12/1/01 12:15 A M 6029 -63! 12/1/01 12;20 A M ! 6029 -46 12/1/01 12:25 A M 6029 12/1/01 12:30 A M 6029! 24 12/1/01 12:35 AM 6029| 69 12/1/01 12:40 A M 6029 94 12/1/01 12:45 AM 6029 125 12/1/01 12:50 A M 6029 148 12/1/01 12 55 A M 6029 179 12/1/01 1:00 A M ' 6029 227 278 Table 4.2 P re l iminary calculat ions. 3) Frequency analysis fu l l per iod (Co lumns K to N ) : A frequency analysis is performed for the complete set o f data o f the season, without di f ferent iat ing between ramping and normal periods. The ma in intent o f this analysis is to find the frequency o f occurrence o f the load forecast error ( C o l u m n I) w i t h i n for a g iven M w h intervals. The user can choose the size o f the interval and the spreadsheet w i l l perform the calculat ions. The probabi l i t ies are calculated as: F P = 4.7 1 0 0 * J ] F Where : Ft = Frequency o f load forecast error i n speci f ic interval SF= Tota l N u m b e r o f occurrences 51 K L M N Interv - I — Freq cum Freq Prob Error Full season : -1040 0 0 0 00 -880 0 0 6.00 j -720 o 0 0.00 -560 0 0 00 -400 2 2 0.05 -240 25 23 0.52 -80 446 421 9.53 80 3601 3155 71 44 240 4361 760 1721 400 4 4 1 1 50 1.13 560 4414 3 0.07 720 4415 1 1 0.02 880 4416 1 0.02 1040 4416 0 0.00 1'HOttAtfH.IHES :>;:;:::: I 1 Table 4.3 Frequency and probabi l i ty analysis for the fu l l per iod. 4) H o u r l y c lass i f icat ion into normal and ramping periods (Co lumns P to T ) : A n hour belongs to a ramping per iod i f the absolute value o f its correspondent hour ly forecast difference ( C o l u m n H ) is larger than a certain amount o f M w h , def ined b y the user, i n this study was set at 250 M w h . Th i s w i l l indicate that the load is increasing or decreasing i n a steep way. I f the hour l y forecast dif ference is sma l l , the hour at the t ime step w i l l be long to a norma l per iod , and its correspondent load forecasting error w i l l be displayed i n C o l u m n Q , as shown in Table 4.4. The load forecasting errors o f ramping hours are shown i n C o l u m n R. C o l u m n s S and T are part o f a counter to f ind out the number o f t imes that an hour is c lass i f ied into a norma l or a ramping per iod. 52 H O U R L Y F O R E C A S T D I F F E R E N C E 278 138 p 250 . . „ . . . . 0 . „ „ , R PERIOD C LA5SIF1CATION NORM RAMP 12/1/01 12:00 AM 1271/01 12 0 5 AM N/A N/A 12/1/01 12:10AM 12/1/01 12:15AM 12/1/01 12:20 AM 12/1/01 12:25 AM 12/1/01 12:30 AM 12/1/01 12:35 AM 12/1/01 12:40 AM 12/1/01 12:45 AM 12/1/01 12:50 AM 12/1/01 12:55 AM 12/1/01 1:00 AM 12/1/01 1:05 AM 46 12/1701 1:10 AM 12/1/01 1:15 AM 12/1/01 1:20 AM 1271/01 1 25 AM 12/1/01 1:30 AM 12/1/01 1:35 AM 12/1/01 1:40 AM 12/1/01 1:45 AM 12/1/01 1:50 AM 12/1/01 1:55 AM 12/1/01 2:00 AM 9 Table 4.4 Normal and ramping period classification. 5) Frequency analysis of normal and ramping periods (Columns W to A D ) : A frequency analysis for both the ramping and the normal periods was performed. The probabilities are calculated following equation 4.7. ilil?:Mlil!!I:!jIl ~ A C " " A D FREQUENCY ANALYSIS NORMAL AND RAMPING PERIODS I Interval NORM: Interval RAMP: ^* 160 ^ 600]Freq cum NOR Freq cum RAMPFreq NORM :Freq RAMF Prob NORM Prob RAMP | -1200 -1200! 0 0 0 I 0 0 0 -1040 -600! 0! 189! * 189 0 15.1563753 -380 0! 0! 608! 0 419 0 33.60064154 -720 I ( 0 1246! 0! 638! 0 51.1627907 -560 1200! 0! 1247! 0 1 0 0.080192462 -400 u < r 0! 1247! o! b] 0 0 -240 2400! 49! 1247! 49! bl' 1.595052083 0 -80 3000! 667 1247! 618 0 20.1171875 0 30 3600! 2171! 1247! 1504! 0! 48.95833333 0 240 4200: 3040! 1247 869! 0! 28.28776042 0 "" 400 4800! 3072! 1247 32 0 1.041666667 0 560 5400: 3 1247 0: 0! 0 0 720 6000! 3072! 1247 0 0 . . . . 0 0 PROBABILITIES 100 100! Table 4.5 Frequency analysis and probabilities for normal and ramping periods. 53 6) Graphs: Included i n the spreadsheets are the plots o f the probabi l i ty distr ibutions for the fu l l per iod (F igure 4.7), the ramping per iod (F igure 4.8) and the normal per iod (F igure 4.9). Probability of error (full period) F igure 4.7 P robab i l i t y d ist r ibut ion for the fu l l per iod. Probability of error (ramping period) -30-MWh 300 400 F igure 4.8 P robab i l i t y d istr ibut ion for the ramping per iod. 54 Probability of error (normal period) MWh Figure 4.9 Probability distribution for the normal period. 7) Ramping and normal hours counter (Columns B C to DB): As requested by the P S O S E , this counter allows the user to know the number of times that every hour is classified into a ramping or a normal period. The probability distributions obtained in the spreadsheets for the four seasons, with the correspondent normal and ramping periods are summarized in Tables 4.6 to 4.9. " ; T - . . . , ; V ; ' . • . v " ; •„; Ub.'JAN -I-EB FULL PERIOD NORMAL RAMPING Lo,id Pro I) Load Prob Load Pioh -300 0.000906 400 0.000704 -250 0.003812 -240 0.004755 -340 0.000352 -200 0.006353 -180 0.007926 -280 0.001408 -150 0.011436 -120 0.030571 -220 0.00528 -100 0.055273 M 0.106658 -160 0.01056 -50 0.120076 0 0.276268 -100 0.041183 0 0.217916 60 0.317708 40 0.158043 50 0.248412 120 0.165082 20 0.332277 100 0.183609 180 0.059556 80 0.280887 150 0.081321 240 0.018342 140 0.117916 200 0.034943 300 0.009058 200 0.035903 250 0.021601 360 0.002038 260 0.011264 300 0.007624 420 0 000226 320 0 004224 350 0.004447 480 0.000679 400 0.000635 540 0.000226 450 0.000635 500 0.001271 550 0.000635 1=1 Z=1 1=1 Table 4.6 Probability distributions for Winter. 55 MRP Al FULL PERIOD NORMAL RAMPING FULL PERIOD NORMAL RAMPING Lft.nl | Prob Load Prob Load | Prob Load | Prob Load | Prob Load 1 Prob 1 400 I 0.000226 400 0.000305 -300 ,0.001751 -300 ! 0.09058 -300 10.000923 -300 i 0.000935 i i " Taooi'585 280 0.002135" -200 0 007042 -240 Ta22644'9' -200 0 033076 I'm 10.011225 2li\ ;o :G4?d3 160 0.017078 IIJO 0 066901 -180 T0I75543" 100 To'"035066' :io j 0.096352 1 HO T'bl33288 1 • 0 0 377641 -120 "!2196558' •1 \ j430329 in ! 0.561272" 40 T u X / 1 ' 4 10 '0.'624276 li'O 0 411972 -60 ' 10.00906"' 100 ! 0.4737 110 '0.257-34" 50 i 0.507699 *IM» 0.119854 f i l l I 0.095951 0 12 200 ) 0.050446 /Mi I 0.038354 140 Tbl94746" 1 'II 0.0'f0369' .•Hi ! 0.02993 60 136.00543" 300 ! 0.005844 fi,n ; 0.001871 i 230 ! 0.040987 110 0.000915' inn ! 0.004401 120 I 12 88496" 400 ! 0.000308" 1 i< 10.001871 320 10.006793 '>i.n 77 0.000305 '.•H> 0 002641 180 : :~ : :o" i " 500 10.000308" 510 ! 0.000935 I 410 'To"o6l'5B5' .on ! 0.001761 240 I 0.883152" I 500 Ta000453" 300 * 0181159" ! 5<l0 Tb"!6fJ0679 360 J.U452y 420 ! 0.04529 480 i 0.022645 < j 1=1 I 1=1 I =^1 I *=1 1 1=1 1=1 Table 4.7 Probab i l i t y distr ibutions for Spr ing and Summer. FULL PERIOD NORMAL RAMPING Load Prob Load Prob Load Prob -500 0.022894 -500 j 0.000326 400 0.001604 -380 0.068681 410 i 0.000326 -280 0.000802 -260 0.389194 -320 10.001302 -160 0.028869 | -140 3.754579 -230 10.007813 40 0.222935 -20 32.02839 -140 i 0.033529 80 0.57498 100 54.64744 -50 j 0.172526 200 0.147554 I 220 8.012821 40 \ 0.522786 320 0.01684 340 0.938645 130 10.218099 440 0.00401 460 0.068681 220 ! 0.034505 560 0.000802 580 0.022894 310 j 0.007487 680 0.001604 700 0.045788 400 | 0.001302 1=1 1=1 1=1 Table 4.8 Probabi l i t y distr ibut ions for Fa l l . 4.2.2 Definition of the problem. The prob lem o f a l locat ing an unforeseen amount o f load due to forecast ing errors among the plants o f the B C H y d r o ' s generating system is typica l o f those that are observed at the beg inn ing o f a discrete t ime per iod to be i n a part icular state. A f t e r observat ion o f the state (Ross, 1983), an act ion must be chosen. T o c o m p l y w i th Be l lman ' s pr inc ip le o f opt imal i ty , and based on l y on the state at that t ime and the opt ion chosen, a reward is earned and the probabi l i ty distr ibut ion for the next state is determined. 56 In order to formulate such a problems, Bertsekas (1976) considers a system characterized by a set o f inputs or states (U) , a set o f uncertainties (W) , a set o f outputs (Y ) , and a system funct ion S. W h e n the input is a sequential funct ion, then the output can be observed to prov ide more in format ion on the uncertainties. I f an input u e U and an uncertain value produce the output y = S (u,w), then p is a po l i c y or dec is ion for the system i f for each w e W u = 7r[s(u,w)] 4.8 has a unique solut ion dependent on w, for u. I f ? is the set o f pol ic ies , and w is selected w i th a g iven probabi l i ty measure that may depend on p, then the prob lem can be formulated b y means o f a ut i l i ty funct ion, to f ind p e ? that max im izes : F(x) = E{U(u,w,y)} 4.9 w A n important subclass o f a l l po l ic ies is the class o f stationary po l i c ies . A p o l i c y is stationary i f the action it chooses at t ime t on ly depends on the state o f the process at t ime t (Ross, 1983). W h e n a stationary po l i c y is employed , as it w i l l be i n this case, then the sequence o f states forms a M a r k o v chain w i th transit ion probabi l i t ies : Py=Py(4S(u,w)]) 4.10 The transit ion probabi l i t ies represent al l the poss ib le outcomes, " s w i t c h i n g " the states on and off. They can be represented i n matr ix notat ion s imp ly b y descr ib ing the possible transit ions f rom one state to the other. I f S ( S i , S2, S 3 ) represents the set o f states at t ime t, and the transitions f rom one state to the other at t ime t+1 are str ict ly sequential (it is not va l id to move f rom S i to S 3 ) , then the fo l l ow ing equations can be der ived: sr =s:Pn+s'2pl2+s;o  4-u c '+ i_c" n , c< n + c » n 4.12 S'S=SlO + S'2p22+S;Pi3 ' 4.13 57 Then , the set o f equations can be presented in matr ix f o rm: 0 >= Pn Pn P22 P23 P32 P32, s: S2 Si 4.14 Th i s can be s imp l i f i ed as fo l lows : S"+L=PN" 4.15 Th is is k n o w n as the transition probabi l i t ies matr ix . The set o f S; consists o f a l l poss ib le outcomes. A n outcome is the state vector S that is equal to S l before and to N t + 1 after the transit ion. The vector S constitutes the sample space for this p rob lem, and thus the sum o f probabi l i t ies i n every co lumn o f the transit ion matr ix should be equal to 1. S ince the use o f M a r k o v i a n chains is be ing proposed to solve this p rob lem, then it is k n o w n that the result o f the transition depends on ly in the current state o f a system and does not depend on al l previous states. Th i s a l lows the use o f the same values o f p;j for the f o l l ow ing transit ions: SN+K=PKN" 4.16 In order to choose pol ic ies that are opt ima l , an opt imal i ty cr i ter ion must be def ined (Ross, 1983). Fo r this research, the total expected discounted return was selected as the opt imal i ty cr i ter ion, be ing the discount factor s, 0< s < l . I f the funct ion o f the expected discounted return earned when the p o l i c y p is chosen and the in i t ia l state is i can be expressed as: (=0 Where : R(S t ,d t ) = reward earned at t ime t d t = decis ion at t ime t E p = Expected va lue g iven that po l i c y p is used Y,R(Sltd,)a'\S0=i 4.17 58 Then a po l i c y is p* opt ima l i f : V^(i) = V(i) f o r a l l i = 0. 4.18 If p is any arbitrary po l i c y and p chooses a dec is ion d at t ime 0 w i th probabi l i ty P d , then (Ross, 1983): deD R(i,d) + YJPij(d)W7[(j) 4.19 Where W p(j) is the expected discounted return f rom t ime 1 onward, g iven that p o l i c y p is used. Howeve r , as we are us ing stationary po l i c ies , the situation at t ime j is the same as i f the process has started at state j , except that a l l the returns are n o w mul t ip l i ed b y s : WK(j)<oVU) 4.20 A n d thus (Ross, 1983), equation 4.19 becomes: ^(0<Z^ deD max dsD RM + o-^idWU) } 4.21 A n d k n o w i n g that the pol ic ies are arbitrary, the opt imal i ty equation becomes (Ross, 1983): Vn (i) < max R(i,d) + aJ]Pij(d)V(j) 4.22 There are a number o f methodologies used to solve this k i nd o f multi-stage, stochastic p rob lem, such as the method o f successive approximat ions, po l i c y iteration, and the use o f l inear programming. Th i s thesis work makes use o f the l inear p rogramming technique due to its e f f i c iency and ease o f use. 59 5 Results Th i s Chapter discusses the results o f der iv ing the probab i l i t y distr ibut ions and o f running the models developed to op t im ize the a l locat ion o f within-the-hour uncertainty i n load. 5.1 Characteristics of the probability distributions of the load forecast error. A s ment ioned i n Chapter 4, the probabi l i ty distr ibut ions o f the load forecast error were obtained for four seasons o f the year and for ramping and norma l per iods, as i l lustrated i n f igure 5.1-5.4. WINTER PROBABILITY DISTRIBUTIONS 0.12 0:1/-•0,08 ;r^0vQ4 ••/•/ 0:02-4 -0-— Normal — Rarnpind x. X , -400 -300 -200 -100 0 100 200 300 400 F i g . 5.1 W in te r probabi l i ty distr ibut ions for normal and ramping per iods. SPRING PROBABILITY DISTRIBUTIONS 0.140 •0.120/1 OrlljJ.1-•Ovoio-0-OCCi / /0:040" 0.020 0.000 — Normal — Ramping! -400 -300 -200 -100 0 100 200 300 400 F i g . 5.2 Spr ing probabi l i ty distr ibut ions for normal and ramping per iods. 60 SUMMER PROBABILITY DISTRIBUTIONS -Or-1-60--400 -300 -200 -100 0 100 200 300 400 F i g . 5.3 Summer probabi l i ty distr ibut ions for normal and ramping periods. FALL PROBABILITY DISTRIBUTIONS _O_-|.&0--400 -300 -200 -100 0 100 200 300 400 F i g . 5.4 Fa l l probabi l i ty distr ibut ions for normal and ramping periods. T o develop a measure o f the behav iour o f the dataset, and to carry out the analysis assuming normal i ty o f the distr ibut ions, it is useful to calculate the standard dev ia t ion and the third (skewness) and fourth (kurtosis) moments around the mean. S igni f icant skewness and kurtosis c lear ly indicate that data are not normal . The results for this statistical analysis are shown i n Table 5.1. 61 Normal Ramping Kurtosis Skewness Stand. Dev Kurtosis Skewness Stand. Dev Winter 2.110893 -0.165948 78.394312 1.974038 0.174279 91.965655 Spring 2.917905 0.129464 76.366507 3.734303 0.707555 92.364691 Summer 2.993202 0.075507 62.969009 2.126454 0.216633 77.870607 Fall 3.066186 -0.278136 79.440712 4.620802 0.611307 91.337258 Table 5.1 Statistical analysis for the load forecast error. A s it can be observed, the kurtosis i n a l l the cases is close enough to 3 (the expected value for a normal distr ibut ion) , so i n this respect the data can be considered to be normal . Th is can be corroborated b y observ ing the figures o f the probabi l i t y distr ibut ions: a clear peak around the mean and somewhat heavy tails. The case o f the W in te r is part icular ly interesting. Some studies on the load forecast error behaviour have shown that in periods o f h igh or irregular demand, such as the W in te r and dur ing the weekends, the load forecast error tends to increase i n value. Th i s can exp la in w h y the kurtosis is smal ler in this season compared to the others: a more discret ized occurrence o f load forecast error. The winter has one o f the highest standard deviat ions, w h i c h can be expla ined by the same l ine o f reasoning. O n the other hand, the measure o f skewness for al l cases is smal l and is c lear ly around 0, wh i ch suggests symmetry. The standard deviat ion o f the ramping per iod is a lways larger than the one o f the normal per iod, and this verif ies the fact that dur ing ramping periods, the load forecast error is larger. 5.2 The deterministic dynamic programming model. In order to analyse the results o f the mode l , the hour ly generation o f a number o f random hours for the three plants inc luded in this study ( G M S , M C A and R E V ) , were retr ieved from the PI system database at B C Hyd ro . The data provides a good representation for typica l normal and ramping hours. The data retrieved were then rounded to mul t ip le o f 10, since the states o f the mode l were discret ized at intervals o f 10 M w h . Fo r the purpose o f this analysis, plant shutdowns were not considered. Table 5.1 shows the per iod c lass i f icat ion (n for normal and r for ramp ing per iods) , hour l y generation for each one o f the plants and the difference (D) between the load o f 62 the hour o f interest, and the load o f the previous hour. The total required generation at each hour is shown i n the last co lumn o f the table. It is important to note that the total required generation for the hour o f interest is equal to the total required generation o f the prev ious hour plus the correspondent increment result ing f rom increase or decrease i n load, exc lud ing the uncertainty i n within-the-hour load. The hours that w i l l be studied are formatted i n bo ld . Hourly generation for hours selected for study. Hour ! Period ! R E V MCA GMS \ D \ Total Req Gen 1-1-03 5:00 AM ! n \ 750 1290 1590 i 3630 1-1 4)3 6:00 AM n 740 1280 1790 : 180 3810 1-1-03 7:00 AM n 740 970 2110 i 10 3820 1-1-03 8:00 AM n 750 1010 2080 ! 20 3840 1-1-03 1:00 PM n i 1270 1110 2490 ! j 4870 1-14)3 2:00 PM ii 1270 1100 2510 ! 10 4880 t-1 J03 3:00 PM ii 1270 1110 2660 ! 160 5040 1-14)3 4:00 PM i 1090 1190 2700 ! -60 4980 8-28-03 6:00 AM i r \ 1790 720 1240 | ! 3750 8-28-03 7:00 AM i 1600 1640 1770 \ 1260 5010 8-284)3 8:00 AM r 1600 1210 1920 \ -280 4730 8-28-03 9:00 PM ! n 1590 1600 2350 ! 460 5540 8-284)3 10:00 PM i 1590 1590 1640 ! -720 4820 10-8-03 1:00 PM ! n ! 870 1090 2150 ! 4110 1043-03 2:00 PM ii 670 1080 2140 i -220 3890 10-84)3 3:00 PM n 730 1090 2150 ! 80 3970 10434)3 4:00 PM ii 590 1090 2060 j -230 3740 10-8-03 7:00 PM ; n 1590 1030 2160 j j 4830 10434)3 8:00 PM n 1510 1270 2160 i no 4940 Table 5.2 H o u r l y generation and increments. Once the hours were sorted, the corresponding hour ly cost for each plant was then retrieved from the cost funct ion used i n the mode l as described i n Chapter 4, and the total hour l y generation cost was calculated. Th i s data w i l l be used as a reference to compare the economy o f the load a l locat ion suggested b y the mode l . Table 5.2 shows the hour l y generation cost for each plant and the total generation cost for the total hour ly load. The hours o f interest are formatted i n bo ld . 63 Hourly generation cost for hours selected for study. Hour REV cost MCA cost I GMS cost i Total cost I Total Req Gen 1-1-03 5:00 A M -459.96 -805.16 ; 314.62 | -950.50 | 3630 1-1-03 6:00 AM -525.23 238.47 i 459.96 I 746.71 ! 3810 1-14)3 7:00 A M -515.62 -212.67 j -249.96 | -978.24 j 3820 1-1-03 8:00 A M -525.23 -338.32 i -230.06 i -1093.60 j 3840 1-1-03 1:00 PM 125.10 76.49 ! -304.17 i -102.59 ! 4870 1-1-03 2:00 P M 253.09 486.94 i 167.73 j 496.12 4880 1-1-03 3:00 P M 253.09 489.08 1 976.51 j 740.52 5040 1-1-03 4:00 P M -338.11 -362.50 j 1413.95 ! 713.34 4980 8-28-03 6:00 A M -722.31 2209.06 | -483.12 | 1003.63 | 3750 3-234>3 7:00 A M i 224.84 600.34 I -531.17 ! 294.01 5010 8-284)3 8:00 A M I 224.84 -285.29 I 453.02 j -513.48 ! 4730 8-28-03 9:00 PM 1.17 314.16 \ 224.84 \ 540.16 5540 8-284)3 10:00 P M I 190.22 314.16 j -570.97 I -66.58 4820 . 10-8-03 1:00 PM I -261.19 -416.79 j -338.11 I -1016.09 ! 4110 10-8-03 2:00 P M ! -219.88 482.11 j -261.19 ! -963.19 ! 3890 10-8-03 3:00 P M j -330.97 473.19 ! -262.64 j -1066.79 ! 3970 10-8-03 4:00 P M j -501.54 482.11 j -261.19 i -1244.85 j 3740 10-8-03 7:00 PM i -258.75 314.16 j -334.28 I -278.86 j 4830 10-84)3 8:00 P M 16.70 76.49 j -258.75 ! -165.56 j 4940 Tab le 5.3 H o u r l y generation cost. The mode l executes the backwards mult istage a lgor i thm two t imes. Th i s is done to a l low the user to choose the plant that w i l l be the starting stage ( M C A or R E V ) . The user w i l l be able to decide between the two a l locat ion ways , either to m i n i m i z e cost or to avo id shedding too m u c h load f rom one plant. Fo r the purpose o f this analysis, the m i n i m u m total cost was the criteria used to choose the opt imal a l locat ion. The load ing o f the plants, as recommended b y the mode l is shown in Table 5.3, wh i ch also inc lude the op t im ized generation cost, the actual load and the actual generation cost. A compar ison between the actual cost o f generation and the load a l locat ion recommended b y the mode l is also inc luded. Savings are formatted i n bo ld and have a negative s ign. 64 REV MCA GMS Totals 1-1-03 6:00 AM Starting generation (h,..,) 750 1290 1590 | 3630 Optimized generation (ht) 1100 760 1950 | 3810 Actual Generation (ht) 740 1280 1790 3810 A Gen 180 Optimized cost -338.6 -810.38 -438.67 j -1587.7 Actual cost -525.23 238.47 -459.96 I -746.71 A Cost -840.94 REV MCA GMS Totals 1-1-03 7:00 AM Starting generation (ht..,) 739.7 1280 1790 3809.7 Optimized generation (ht) 770 1120 1930 3820 Actual Generation (ht) 740 970 2110 3820 A Gen 10 Optimized cost -530.51 -487 83 -449.77 ! -1468.1 Actual cost -515.62 -212.67 -249 96 -978.24 A Cost ^189.86 REV MCA GMS j Totals 1-1-03 8:00 AM Starting generation (h t 1) 740 970 2110 | 3820 Optimized generation (ht) 770 1120 1950 j 3840 Actual Generation (ht) 750 1010 2080.121 3840 A Gen 20 Optimized cost -530.51 -487.83 -438.67 j -1457 Actual cost -525.23 -338.32 -230.06 1-1093.60 A Cost -363.40 REV MCA GMS Totals 1-1-03 2:00 PM Starting generation (h t 1) 1270 1110 2490 4870 Optimized generation (ht) 1580 760 2540 4880 Actual Generation (ht) 1270 1100 2510.16 4880 A Gen 10 Optimized cost 158.51 -810.38 239.877 \ -411.99 Actual cost 253.09 -486.94 167.73 -66.12 A Cost -345.88 REV MCA GMS Totals 1-1-03 3:00 PM Starting generation (h t 1) 1270 1100 2510.16] 4880 Optimized generation (ht) 770 1650 2620 | 5040 Actual Generation (h,) 1270 1110 2660 | 5040 A Gen 160 Optimized cost -530.51 650.905 613.622 i 734.018 Actual cost 253.09 -489.08 976 51 i 740.52 A Cost -6.51 REV MCA GMS Totals 1-1-03 4:00 PM Starting generation (ht..,) 1270 1110 2660 5040 Optimized generation (ht) 1640 760 2580 4980 Actual Generation (ht) 1090 -60 1190 -810738' 2 l 6 2 J £ 2700 | 4980 A Gen Optimized cost 394.704 -338.11 .400.56 i -15.115 Actual cost 1413.95! 713.34 A Cost -728.46 Table 5.4 Results of the dynamic programming model. 65 REV MCA GMS Totals 8-28-03 7:00 AM Starting generation (h t 1) 1790 720 | 1240 I 3750 Optimized generation (ht) 1660 760 | 2590 I 5010 Actual Generation (ht) 1600 1640 I 1770 [ 5010 A Gen 1260 Optimized cost 498.024 -810.38 I 449.363 ! 137.008 Actual cost 224.84 600.34 -531.17 294.01 A Cost -157.00 REV MCA GMS Totals 8-28-03 8:00 AM Starting generation (h t 1) 1600 1640 1770 5010 Optimized generation (ht) 1520 | 760 2450 | 4730 Actual Generation (ht) 1600 | 1210.07 1920 | 4730 A Gen -280 Optimized cost 28.7318 -810.38 53.3166 i -728.33 Actual cost 224.84 -285.29 -453.02 ! -513.48 A Cost -214.86 REV MCA GMS Totals 8-28-03 10:00 PM Starting generation (h t 1) 1590 | 1600 | 2350 | 5540 Optimized generation (ht) 1550 j 760 I 2510 | 4820 Actual Generation (ht) 1590 | 1590 | 1640 | 4820 A Gen -720 ! ! ! Optimized cost 81.1012 i -810.38 ! 167.727 ! -561.55 Actual cost 190.22 ! 314.16 I -570.97 j -66.58 A Cost -494.97 ! REV MCA GMS Totals 10-8-03 2:00 PM Starting generation (ht..,) 870 1090.34| 2150 | 4110.34 Optimized generation (h,) 770 1140 | 1980 | 3890 Actual Generation (ht) 670 1080 12140.06) 3890 A Gen -220 Optimized cost -530.51 -472.87 ! -411.81 I -1415.2 Actual cost -219.88 -482.11 j -261.19 I -963.19 A Cost 452.00 REV MCA GMS Totals 10-8-03 3:00 PM Starting generation (h t 1) 670 1080 2140.06 3890.06 Optimized generation (ht) 770 780 2420 I 3970 Actual Generation (h,) 730 | 1090.41 2150 | 3970 A Gen 80 Optimized cost -530.51 ! -802.26 17.8601 ! -1314.9 Actual cost -330.97 \ -473.19 i -262.64 1-1066.79 A Cost -248.11 REV MCA GMS Totals 10-8-03 4:00 PM Starting generation (h t 1) 730 1090.41 | 2150 |3970.41 Optimized generation (ht) 1090 j 760 1890 I 3740 Actual Generation (ht) 590 j 1090 2060 I 3740 A Gen -230 ! Optimized cost -338.11 i -810.38 , -451 i -1599.5 Actual cost -501.54 j -482.11 I -261.19 I-1244.85 A Cost -354.64 \ Table 5.4 Results of the dynamic programming model (continued). 66 REV MCA GMS Totals Starting generation (h t 1) 1590 1080 2160 4830 Optimized generation (ht) 1620 760 2560 4940 10-8-03 3:00 PM Actual Generation (ht) 1510 1270 2160 4940 A Gen 110 ! j Optimized cost 303.417 -810.38 314.612 -192.35 Actual cost 16.70 76.49 -258.75 -165.56 A Cost -26.79 I Tab le 5.4 Results o f the dynamic p rogramming mode l (continued). It can be seen that the mode l prov ides a better economica l approach to plant load ing than the one that was performed at that t ime and thereby sav ing hundreds o f dol lars for most hours as shown in Tab le 5.3. 5.3 Stochastic dynamic programming model. A l though i n theory the l inear p rogramming approach to dynamic p rog ramming should be a good so lut ion method for the stochastic p rob lem, it d id not however behave in the most opt imal w a y for this case study and t ime l imi ta t ion d id not a l l ow to investigate the behav iour o f the a lgor i thm. A s stated b y Mos tag i r and U h a n (2004), s ince the l inear p rog ramming a lgor i thm was so lved hav ing a constraint for every state-decision pair, the p rob lem derives i n a large opt imizat ion prob lem. Further research w i l l be done on so l v ing the stochastic p rob lem us ing p o l i c y improvement or va lue iteration methods. However , a conservat ive approach to the stochastic p rogramming p rob lem can be made b y us ing the expected value o f the probabi l i t y distr ibutions for each season, to calculate poss ib le increment i n load that w o u l d need to be al located among the plants o f the system. Th is increment is shown in the co lumn D o f Tables 5.2 to 5.5. The increment i n c o l u m n D has two components: the f i rm load that was forecasted, and the uncertain load forecast error. The expected value o f a probabi l i t y d ist r ibut ion can be calculated as: = 5.1 67 The expected values for the probabi l i ty distr ibutions o f the norma l and ramping periods discret ized on intervals o f 100 M w h are shown i n Table 5.10: WINTER Normal Ramping L P(x) E(x) L P M E(x) -400 0.000704 -0.281591 -200 0.0101652 -2.03304 -300 0.000704 -0.211193 -100 0.066709 -6.6709 -200 0.0095037 -1.900739 0 0.3379924 0 -100 0.0485744 -4.857445 100 0.4320203 43.20203 0 0.374516 0 200 0.1162643 23.25286 100 0.4516015 45.16015 300 0.0292249 8.767471 200 0.0989088 19.78177 400 0.0050826 2.033037 300 0.0147835 4.435058 500 0.001906 0.952986 400 0.000704 0.281591 600 0.0006353 0.381194 Totals: 1 62.4076 1 69.88564 Rounded (E(x) 60 70 Table 5.5 Expected values for seasonal probabi l i ty distr ibut ions (winter) SUMMER Normal Ramping L P M E(x) L P M E M -300 0.000903 -0.27092 -300 0.000915 -0.2745 -200 0.00301 -0.60205 -200 0.008234 -1.6468 -100 0.034618 -3.46177 -100 0.065874 -6.5874 0 0.428356 0 0 0.410796 0 100 0.475316 47.53161 100 0.431839 43.184 200 0.051475 10.295 200 0.065874 13.175 300 0.005719 1.715834 300 0.013724 4.1171 400 0.000301 0.120409 400 0.002745 1.0979 500 0.000301 Totals: 1 55.32812 1 53.065 Rounded (E(x) 60 50 Table 5.6 Expected values for seasonal probabi l i ty distr ibut ions (summer) FALL Normal Ramping L P M E M L P M E M -300 0.003 -0.878906 -200 0.0112 -2.2454 -200 0.012 -2.473958 -100 0.0754 -7.5381 -100 0.067 -6.705729 0 0.3897 0 0 0.416 0 100 0.3929 39.294 100 0.427 42.675781 200 0.1075 21.492 200 0.062 12.304688 300 0.0144 4.3304 300 0.011 3.4179688 400 0.0064 2.5662 400 0.002 0.78125 500 0.0024 1.2029 Totals: 1 49.121094 1 59.102 Rounded (E(x) 50 60 Table 5.7 Expected values for seasonal probabi l i ty distr ibutions (fall) 68 Fo r this exercise, the same hours studied i n the determinist ic case were selected for the analysis. The probabi l i ty distr ibutions w i l l be scaled to reflect the increase in load due pure ly to load forecast uncertainty. Th i s is done b y subtracting the expected value o f the corresponding probabi l i t y distr ibut ion (values shown in Tables 5.4 to 5.6) to the increment i n load ( co lumn D i n Table 5.1). U s i n g the corresponding probabi l i ty distr ibut ion (seasonal and for a ramping or norma l per iod) , the actual hour generation values and the intervals o f the probabi l i ty distr ibutions were entered to the mode l as inputs. The interval o f the probabi l i ty distr ibut ion w i l l be the load increment to allocate among the three plants. Thus , after obta in ing the opt imal load a l locat ion for each plant f rom the mode l , the expected generation can be expressed as: E(G) = YJPiGi 5.2 where: E (G ) = Expected generation va lue Pi = Probab i l i t y for the increment G i = O p t i m i z e d generation (output f rom the model ) A n example o f the ca lculat ion o f the expected generation values is shown in Table 5.7, us ing the actual load o f January 1st, 2003 , at 5:00 A . M . The first row o f generation values (the actual plant load at the t ime) is entered to the mode l , a long w i th scaled intervals f rom the second co lumn. The mode l should be run for each interval . The expected values were then calculated as the product o f the opt imal loads and the correspondent probabi l i t ies , for this case, the normal per iod probabi l i t y distr ibut ion for the winter, shown in Tab le 5.4. The total expected generation value were then rounded to the next mul t ip le o f ten. The sum o f the expected generation values must be equal to the load generation o f the next hour, i n this case, 6:00 A . M . 69 [ Original Scaled REV (Mwh) MCA (Mwh) GMS (Mwh) Totals E M I Increment Increment 750 1290 1590 3630 REV MCA GMS i -400 -280 770 810 1770 3350 0.542063 0.570222 1.24604 I -300 -180 770 770 1910 3450 0.542063 0.542063 1.344597 I -200 -80 1100 760 1690 3550 10.45407 7.222809 16.06125 | -100 20 1120 760 1770 3650 54.40338 36.91658 85.97677 I o 120 1090 760 1900 3750 408.2225 284.6322 711.5804 ! 100 220 1100 760 1990 3850 496.7617 343.2172 898.6871 I 200 320 1080 760 2110 3950 106.8215 75.17071 208.6976 | 300 420 1100 760 2190 4050 16.26188 11.23548 32.37592 I 400 520 1470 760 1920 4150 1.034847 0.535023 1.351637 I 1= 1095.044 760.0422 1957.321 (Rounded E(x): 1090 760 1960 (Total generation for 6: 00 a.m. 3810 Tab le 5.8 Expected load values The results o f runn ing the mode l for the probable load increments suggested b y the probabi l i ty distr ibutions are summar ized i n Tab le 5.9. R E V M C A G M S Totals Starting generation (h t 1 ) 750 1290 1590 3630 Optimized generation (h t) 1090 760 1960 3810 1-1-03 6:00 AM Actual Generation (h t) 740 1280 1790 3810 A Gen 180 Optimized cost -338.11 -810.38 -431.02 -1579.5 Actual cost -525.23 238.47 -459.96 -746.71 A Cos t -S3 2.79 R E V M C A G M S Totals Starting generation ( h M ) 739.7 1280 1790 3809.7 Optimized generation (ht) 1100 760 1960 3820 1-1-03 7:00 AM Actual Generation (ht) 740 970 2110 3820 A Gen 10 Optimized cost -338.6 ''-8T0.38 ''-43T02''' -T580 Actual cost -515.62 Zlz!-2i.. A Cos t -601.75 R E V M C A G M S Totals Starting generation (ht..,) 740 970 2110 3820 Optimized generation (ht) 1080 760 2000 3840 1-1-03 8:00 AM Actual Generation (ht) 750 1010 2080.12 3840 A Gen 20 Optimized cost -530.51 -487.83 -438.67 -1457 Actual cost -525.23 -338.32 -230.06 -1093.60 A Cost -363.40 Table 5.9 Results o f the dynamic p rogramming mode l for the expected loads. 70 REV MCA GMS Totals 1-1-03 2:00 PM Starting generation (h,.t) 1270 | 1110 | 2490 4870 Optimized generation (h,) 1550 | 810 | 2520 4880 Actual Generation (ht) 1270 | 1100 j 2510.16 4880 A Gen 10 I Optimized cost 81.1012 ! -742.93 I 190.066 -471.764 Actual cost 253.I 3.94 I 167.73 66 12 A Cost -405.65 ; REV MCA GMS Totals 1-1-03 3:00 PM Starting generation (h t 1) 1270 | 1100 2510.16 4880 Optimized generation (h,) 1680 I 760 2600 5040 Actual Generation (ht) 1270 | 1110 2660 5040 A Gen 160 I Optimized cost 613.551 ! -810.38 509 224 312.397 Actual cost 253.09 ! -489.08 976.51 740.52 A Cost -428.13 REV MCA GMS Totals 1-1-03 4:00 PM Starting generation (h t 1) 1270 | 1110 ! 2660 5040 Optimized generation (ht) 1510 | 970 | 2500 4980 Actual Generation (h,) 1090 | 1190 | 2700 4980 A Gen -60 ! I Optimized cost 16.6984 j -212.67 i 145.647 -50.3225 Actual cost -338.11 ! -362.50 ! 1413.95 713.34 A Cost -763.66 i REV MCA GMS Totals Starting generation (h t 1) 1790 | 720 1240 3750 Optimized generation (ht) 1540 | 950 2520 5010 8-28-03 7:00 AM Actual Generation (ht) 1600 | 1640 1770 5010 A Gen 1260 i Optimized cost 60.8891 j 142 fit 1 gilOGG 108.342 Actual cost 224.84 ! A Cost -185.66 REV MCA GMS Totals 8-28-03 8:00 AM Starting generation (h t 1) 1600 | 1640 | 1770 5010 Optimized generation (h,) 1530 | 760 | 2440 4730 Actual Generation (h,) 1600 j 1210.07 ! 1920 4730 A Gen -280 i ! Optimized cost 43.5175 | -810.38 j 39.9117 -726.949 Actual cost 224.84 I -285.29 I -453.02 -513.48 A Cost -213.47 j I -Table 5.9 Results of the dynamic programming model for the expected loads. (continued) 71 REV | MCA GMS | Totals 8-28-03 10:00 PM Starting generation (h t 1) 1590 | 1600 2350 | 5540 Optimized generation (ht) 1550 | 770 2500 I 4820 Actual Generation (ht) 1590 I 1590 1640 | 4820 A Gen -720 | Optimized cost 81.1012 I -809.55 145.647 ! 582.797 Actual cost 190.22 I 314.16 -570.97 I -66.58 A Cost -516.21 ! REV MCA GMS Totals 10-8-03 2:00 PM Starting generation (h t 1) 870 j 1090.34 2150 I 4110 Optimized generation (ht) 1270 j 760 1860 | 3890 Actual Generation (h,) 670 I 1080 12140.061 3890 A Gen -220 | Optimized cost 253.094 j -810.38 | -430.43 (-987.714 Actual cost -219.88 ! -482.11 i -261.19 ! -963.19 A Cost -24.53 ; REV MCA GMS Totals 10-8-03 3:00 PM Starting generation (h t 1) 670 1080 I 2140.06! 3890 Optimized generation (ht) 1270 j 760 | 1860 | 3890 Actual Generation (ht) 730 | 1090.41 | 2150 I 3970 A Gen 0 | Optimized cost -284.67 ! -810.38 I -230.06 : 1325. II Actual cost -330.97 j -473.19 | -262.64 .-11.166.79 A Cost -258.31 i REV MCA GMS Totals Starting generation (h t 1) 730 1090.41 2150 3970 Optimized generation (ht) 590 1090 2060 3740 10-8-03 4:00 PM Actual Generation (ht) 650 760 2060 3470 A Gen -230 | Optimized cost -244.83 -810.38 -256.65 -1311.86 Actual cost -501.54 -482.11 -261.19 -1244.85 A Cost -67.01 ; REV MCA GMS Totals Starting generation (h t 1) 1590 1080 2160 4830 Optimized generation (ht) 1620 760 2560 4940 10-8-03 8:00 PM Actual Generation (ht) 1510 1270 2160 4940 A Gen 110 I Optimized cost 224.836 -789.77 272.004 -292.934 Actual cost 16.70 76.49 -258.75 -165.56 A Cost -127.38 ! Table 5.9 Results of the dynamic programming model for the expected loads (continued). It can be seen that the mode l output provides a less cost ly load a l locat ion for the expected generation values than those that were actual ly implemented o f the three plants. A g a i n , the benefit o f us ing the mode l to assign the load to the plants is reflected i n savings o f hundreds o f dol lars per hour, on the average. 5.4 C o s t o f h a n d l i n g unce r t a i n t y . The total costs o f the load a l locat ion scheme suggested b y the mode l for both the determinist ic and the probabi l i s t i c case are shown in Table 5.10: Cost Det - Prob Total hourly pen Deterministic Probabilistic A 3810 -1587.65 -1579.50 3820 -1468.10 -1580.00 -111.89 3840 -1457.01 -1528.39 -71.38 4880 -411.99 -471.76 -59.77 5040 734.02 312.40 -421.62 4980 -15.12 -50.32 -35.21 5010 137.01 108.34 -28.67 4730 -728.33 -726.95 4820 -561.55 -582.80 -21.25 3890 -1415.19 -987.71 1^ 42:747::;;; 3970 -1314.91 -1325.11 -10.20 3740 -1599.48 -1311.86 4940 -192.35 -292.93 -100.58 Tab le 5.10 Cost and benefit o f handl ing uncertainty. The costs obtained for the determinist ic and the probabi l i s t ic cases di f fer s l ight ly due to the cost o f hand l ing uncertainty, and because o f the savings that assessing that uncertainty can b r ing to the analysis. It can be seen that tak ing into account load uncertainty can be expensive, as indicated in the greyed out cel ls i n the table. Nevertheless, when uncertainty is inc luded i n the analysis, the result ing schedule w i l l be more robust as it w i l l pos i t ion the system to handle uncertain load demands w i th in the hour. 73 6 Conclusion and Recommendations The new competi t ive electr ic i ty market have triggered research areas that aims at ident i fy ing key operational processes than can be improved or ref ined i n order to make the most o f the avai lable resources. In addi t ion, recent computat ional capabi l i t ies have made opt imizat ion techniques avai lable to greater number o f potential users, and have demonstrated that their appl icat ion generates important benefits for the uti l i t ies that further promote their use. Fo r the part icular case o f this research work , dynamic p rogramming was chosen as the methodology to allocate unforeseen load increments among the three largest plants i n the B C Hyd ro system. A s indicated in literature surveyed, dynamic p rogramming reduces computat ional t ime and s impl i f ies the analysis o f multi-stage dec is ion problems. The a lgor i thm was tested us ing randomly selected hours, i n both ramping and normal per iods, and in different seasons o f the year. The actual load a l locat ion o f the plants was compared to the mode l opt imal result, and it was found that for every hour studied, the mode l provides a more economica l w a y o f a l locat ing the load among the plants, w i th potential savings o f hundreds o f dol lars on an hour l y basis. The computat ional t ime needed to obtain an opt imal load a l locat ion is very short, mak ing this mode l ideal for use on a real t ime basis, whenever it is needed. In order to consider the stochastic nature o f the load forecast,, the determinist ic mode l was used to prov ide the opt imal expected costs for three hours. In this case, the expected opt imal load al locations prov ided b y the mode l were also more cost effect ive than the actual plant loading. It is evident that dynamic p rogramming was an excel lent opt ion to solve the determinist ic prob lem i n this research work . The prob lem was also formulated as a stochastic dynamic p rogramming a lgor i thm, choos ing l inear p rog ramming as the approach to solve it. However , the l inear p rog ramming approach needs to be further ref ined and investigated and t ime l imitat ions d id not a l low fu l l explorat ion o f the technique. Further research needs to be done to solve the prob lem us ing p o l i c y improvement or value iteration methods, as we l l as the l inear p rogramming technique. 74 A s a conservative approach to the stochastic p rogramming approach, the dynamic p rogramming mode l was used w i th the correspondent probabi l i ty distr ibut ion to represent load intervals as the increments to allocate among the three plants o f the system. Th i s exercise prov ided insight on the behaviour o f the load error and also proved the effectiveness o f the mode l when running a larger number o f intervals. 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I E E E Transact ions on Power Systems, vo l . 9, issue 1, page(s):510 -517, Feb. 1994. 79 E lectronic References: A M P L http://www.ampl.com B C H y d r o http://www.bchydro.com Nor th A m e r i c a n Energy Standards Boa rd http://www.naesb.orR Northwest Power Poo l http://www.nwpp.org 

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