Optimal Energy Commitments with Storage and Intermittent Supply


 Gerard Adams
 2 years ago
 Views:
Transcription
1 Submitte to Operations Research manuscript OPRE Optimal Energy Commitments with Storage an Intermittent Supply Jae Ho Kim Department of Electrical Engineering, Princeton University, Princeton, NJ 08540, Warren B. Powell Department of Operations Research an Financial Engineering, Princeton University, Princeton, NJ 08540, We formulate an solve the problem of making avance energy commitments for win farms in the presence of a storage evice with conversion losses, meanreverting price process, an an autoregressive energy generation process from win. We erive an optimal commitment policy uner the assumption that win energy is uniformly istribute. Then, the stationary istribution of the storage level corresponing to the optimal policy is obtaine, from which the economic value of the storage as the relative increase in the expecte revenue ue to the existence of storage is obtaine. Key wors : Markov Decision Process, Dynamic Programming, Energy History : This paper was first submitte on September 4, 2009 an has been revise an resubmitte on May 27, 2010, an on January 6, Introuction The emphasis on renewables, such as the goal set by the Department of Energy to have 20 percent of electric power from win by 2030, has raise the importance of efficiently managing win, an unerstaning the factors that affect the cost of using win. Currently, win energy accounts for a small fraction in the market an the gri operators allow the win energy proucers to eliver any amount of energy they prouce at a given time. However, as the share of win energy in the market grows, such a policy will become impractical, an gri operators will nee to make commitments on the amount of win energy that will be elivere in avance. Unfortunately, making commitments is complicate by the inherent uncertainty of win. This uncertainty can be mitigate by the presence of storage, which also introuces the imension of losses ue to the conversion neee to store an retrieve energy. 1
2 2 Article submitte to Operations Research; manuscript no. OPRE We aress the problem of making a commitment at time t to eliver energy from win uring the time interval t, t + 1). The moel is most easily applie in the hourahea market, although it can be use in an approximate fashion in the ayahea market. Energy storage has long been recognize as an important technology for smoothing the variability of win Castronuovo an Lopez 2004), Korpaas, et al. 2003), GarciaGonzalez, et al 2008), Brunetto an Tina 2007), Ibrahim, et al. 2008)). We assume that we store energy when the available energy from win excees the commitments we have mae, but we may incur significant conversion losses. The problem has cosmetic similarities with classical inventory problems storing prouct to meet eman), but with some funamental ifferences. Inventory problems are typically trying to control the supply of prouct to meet an exogenous eman Axsäter 2000), Zipkin 2000)). In our problem, we have exogenous supply energy generate from win) to meet eman by making avance commitments. This problem is similar to the reservoir management problem which is characterize by ranom rainfall see Nanalal an Bogari 2007) for an excellent review of ynamic programming moels for reservoir management). Our problem is istinguishe by the nee to make avance commitments, along with our interest in a simple, analytical solution that can be use in economic moels. In this paper, we erive an optimal policy for making energy commitments from win in the presence of an energy storage evice. We then use this policy to stuy the economics of storage capacity in this setting. Given the richness of the application, we analyze a stylize version of the problem, which allows us to erive the optimal policy in a simple, analytic form. Our moel captures some important imensions of the real problem such as the storage capacity constraints, storage conversion losses an a meanreverting process for realtime electricity prices. At the same time, we make a number of simplifying assumptions. Some of these inclue: We assume that we are a small player in a large market, making it possible to sell all of the energy we prouce as long as we make avance commitments. In aition, we assume that if the energy from win plus what is available in storage) falls below our commitment, that we can make up the entire shortfall using the current spot price.
3 Article submitte to Operations Research; manuscript no. OPRE If the moel is applie in a ayahea market, we ignore the ability to make hourahea ajustments. We capture storage capacity an conversion losses, but we otherwise ignore the physics of energy storage, such as the relationship between the rate of storage an storage capacity, an the impact of full ischarges on battery life. We assume meanreverting electricity prices an stationarity in the errors in win forecasts. Our analytical moel assumes that win follows a uniform istribution, although we then quantify this error in experimental comparisons using actual win patterns. More realistic moels require an algorithmic solution. A goal of our research is a simple policy that can be use in economic moels without requiring the complex machinery of stochastic optimization algorithms. The goal of a win farm operator is to maximize the cumulative profit over time by computing the amount of electricity to commit to sell uring the time interval t, t + 1) at each time t. Brown an Matos 2008), Brunetto an Tina 2007), Castronuovo an Lopez 2004), an Korpaas, et al. 2003) attempt to solve the problem by solving a eterministic optimization problem given a particular sample path over a finite horizon an then averaging the results over the sample paths. The sample paths are rawn from a fixe T + 1)imensional istribution escribing the electricity generate from the win farm uring the time interval t, t + 1) for each t = 0, 1,..., T. However, this approach oes not prouce a vali, amissible policy. In practice, we nee a policy that allows the win farm operator to compute at time t the amount of electricity to commit to sell uring the time interval t, t + 1) base on the state of the environment at time t. The objective of this paper is to fin such a policy an analyze it. The contributions of this paper are as follows. 1) We erive an analytical expression for an optimal policy, an the value of storage, for a stylize moel of an energy storage process in the presence of intermittent generation requiring avance commitments. 2) We establish assumptions on the electricity price an the istribution of win, size of the storage, an the ecision epoch intervals that allow us to erive the optimal policy for energy commitment in a close form an
4 4 Article submitte to Operations Research; manuscript no. OPRE explain the implications of those assumptions. 3) Uner those assumptions, we erive the optimal policy for avance energy commitment in a simple, analytical form, when we have a storage with an arbitrary rountrip efficiency, an when electricity prices are meanreverting. The optimal policy obtaine uner such assumptions resembles the optimal policy for the wellknown newsvenor problem Khouja 1999), Petruzzi an Daa 1999)). 4) We obtain the stationary istribution of the storage level corresponing to the optimal policy, from which we fin the economic value of the storage as the relative increase in the expecte revenue ue to the existence of storage. 5) We test our policy using win energy generate from truncate Gaussian istributions an emonstrate that the error introuce by assuming a uniform istribution for win is reasonably small. This paper is organize as follows. In 2, we moel the win energy storage problem as an MDP with continuousstate an control variables. In 3, we present our assumptions an the structural properties of the optimal value function of the MDP. In 4, the optimal policy for the infinite horizon problem for a storage with general rountrip efficiency is obtaine. Then, the stationary istribution of the storage level corresponing to the optimal policy is obtaine, from which the economic value of the storage as the relative increase in revenue ue to existence of storage, is erive. In 5, we compute the economic value of storage using the win spee ata obtaine from the North American Lan Data Assimilation System NLDAS) project Cosgrove, et al. 2003)), an the electricity price ata provie by a utility company. In 6, we summarize our conclusions. 2. The moel Operating a win farm epens on two markets: the electricity spot market an the regulating market. We sell to the spot market an pay a penalty when we fail to meet our commitment. The gri operator buys energy from the regulating market when we fail to meet our commitment. In the spot market, the energy proucers make their commitments to eliver sell) electricity in avance while the regulating market is a marketplace for reserve energy in which the proucers have the ability to sell electricity on a shorter notice than the spot market Korpaas, et al. 2003),
5 Article submitte to Operations Research; manuscript no. OPRE MacKerron an Pearson 2000), Morthorst 2003)). As a win farm operator, when the electricity prouction excees our expectation an we have an excess amount of electricity left over after fulfilling the contractual commitment, we store the excess amount. On the other han, when the electricity prouction falls too short to meet the contractual commitment, we have to pay a premium, a penalty for failing to meet the commitment, while the proucers in the regulating market make up for the gap. Therefore, if we commit too much, we can actually lose money. We have revenues from our sale on the spot market an costs from tapping into the regulating market when we fail to meet our commitment for elivery on the spot market see Chapter 16 of MacKerron an Pearson 2000) for a etaile exposition of the market system). At each time t, the market participants submit their bi for the supply an eman for electricity that must be elivere uring the time interval t, t + 1). The market overseer collects the biing information an etermines the spot market an the regulating market price for the time interval t, t + 1) shortly after the participants submit their bis. Therefore, as a win farm operator, we o not know what the prices will be when we are making our commitments. We make the following assumptions. First, we assume that at each time t, we have a probability istribution of the electricity we will generate uring the time interval t, t + 1). Secon, we assume that we are a small participant in the market such that the market can always absorb our supply an the effect of our biing on the expecte spot market an the regulating market prices of the electricity is negligible. Then, the prices can be treate as exogenous variables an we only nee to etermine the amount of electricity to commit to sell. Thir, we assume that the spot market price of the electricity is meanreverting an the ratio of the expecte spot market price over the expecte regulating market price is always less than the rountrip efficiency of our storage with the iscount factor. Otherwise, the cost of using the storage, which can be measure by the conversion loss, will be greater than the expecte cost of tapping into the reserve energy in the regulating market, negating the purpose of using a storage evice in the first place. The thir assumption is crucial in maintaining the concavity of the optimization problem.
6 6 Article submitte to Operations Research; manuscript no. OPRE System Parameters R max = upper limit on the storage. unit: storage energy capacity unit) = coefficient use to convert the generate electricity to potential energy in the storage. unit: storage unit / electricity unit) ρ E = coefficient use to convert the potential energy in the storage to electricity. unit: electricity unit / storage unit) Note that 0 < ρ E < 1, where ρ E enotes the rountrip efficiency. Throughout this paper, 1 ρ E is referre to as the conversion loss from storage. ρ E is aroun for most of the existing storage systems Sioshanshi, et al. 2009)). µ p = mean of the spot market price of the electricity. unit: ollar / electricity unit) σ p = stanar eviation of the change in spot market price of the electricity. unit: ollar / electricity unit) κ = meanreversion parameter for the spot market price of the electricity. κ is proportional to the expecte frequency at which the spot market price crosses the mean per unit time. unit: 1 / time unit) τ = time interval between ecision epochs. m = slope of the penalty cost for overcommitment. b = intercept of the penalty cost for overcommitment. unit: ollar / electricity unit) That is, when the spot market price of the electricity is p t, the penalty for overcommitment is mp t + b. µ Y = mean of the electricity generate from the win farm per unit time. unit: electricity unit / time unit) σ Y = stanar eviation per unit time of the electricity generate from the win farm. unit: electricity unit / time unit) γ = iscount factor in the MDP moel. 0 < γ < 1.
7 Article submitte to Operations Research; manuscript no. OPRE State Variables Let t N + be a iscrete time inex corresponing to the ecision epoch. The actual time corresponing to the time inex t is t τ. R t = storage level at time t. 0 R t R max, t. Y t = electricity generate from the win turbines uring the time interval t 1, t). Y t 0, t. p t = spot market price for electricity elivere uring the time interval t 1, t). p t 0, t. W t = Y t ), p 1 t t t) = exogenous state of the system. S t = R t, W t ) = state of the system at time t Decision Action) Variable x t = amount of electricity we commit to sell on the spot market uring the time interval t, t + 1) etermine by signing the contract at time t. x t 0. Since we are making an avance commitment, x t is not constraine by R t. The lack of an upper boun on x t inicates that we are a small player in the market an hence there will always be enough eman in the market to absorb our supply as long as we are making an avance commitment Exogenous Process ŷ t = noise that captures the ranom evolution of Y t. Specifically, M 1 Y t+1 = µ Y τ + α i Y t i µ Y τ) + ŷ t+1, 1) i=0 for some orer M an coefficients α i for 0 i M 1. ŷ t ) t 1 an Y t ) t 1 must be proportional to τ. p t = noise that captures the ranom evolution of p t. Specifically, we use a iscretetime version of the OrnsteinUhlenbeck process: p t+1 p t = κ µ p p t ) τ + p t+1. Let Ω be the set of all possible outcomes an let F be a σalgebra on the set, with filtrations F t generate by the information given up to time t : F t = σ S 0, x 0, Y 1, S 1, x 1, Y 2, S 2, x 2,..., Y t, S t, x t ).
8 8 Article submitte to Operations Research; manuscript no. OPRE P is the probability measure on the measure space Ω, F). Throughout this paper, a variable with subscript t is unknown ranom) at time t 1 an becomes known eterministic) at time t. In other wors, a variable with subscript t is F t measurable. We have efine the state of our system at time t as all variables that are F t measurable an neee to compute our ecision at time t Storage Transition Function R max, if R t + Y t+1 x t ) R max. R t + Y t+1 x t ), if x t < Y t+1, R t + Y t+1 x t ) < R max. R t+1 = R t 1 ρ E x t Y t+1 ), if Y t+1 x t < ρ E R t + Y t+1. 0, if x t ρ E R t + Y t+1. If Y t+1 excees the commitment x t, we store the excess amount Y t+1 x t with a conversion factor,. If Y t+1 is less than x t, the potential energy in the storage must be converte into electricity with a conversion factor, ρ E, to fulfill the gap, x t Y t+1. If the amount of electricity generate uring the time interval t, t + 1) plus the electricity that can be obtaine by converting the potential energy in the storage is not enough to cover the contractual commitment, we eplete our storage an we have to pay for the gap. It is important to note the ifference between the storage transition function shown above an the transition functions that generally appear in traitional inventory management an resource allocation problems Axsäter 2000), Zipkin 2000)). Unlike many of the transition functions that appear in traitional problems, here R t+1 is not a concave or convex function of x t or R t, which makes the concavity of the optimization problem not obvious. 2) 2.6. Contribution Revenue) Function The profit we make uring the time interval t, t + 1) is given by { p Ĉ t+1 = t+1 x t, if x t < ρ E R t + Y t+1. p t+1 x t mp t+1 + b) x t ρ E R t + Y t+1 )], if x t ρ E R t + Y t+1. p t+1 x t is the profit we earn by elivering x t amount of electricity to the market uring the time interval t, t + 1), an mp t+1 + b is the penalty we pay in the case of overcommitment. Assume m γ ρ E an b γ ρ E µ p. 3)
9 Article submitte to Operations Research; manuscript no. OPRE Then, the cost of using the storage, which can be measure by the conversion loss, is less than the cost of overcommitment. Otherwise, for the purpose of maximizing the revenue, there will be no reason to use a storage in the first place. concavity of the stochastic optimization problem. This affine penalty is sufficient to ensure the Note that these lower bouns on the penalty factor are unfavorable assumptions  they make the environment in which we operate more averse an lea to a more conservative policy. If we have to operate in an environment where the above assumptions o not hol, the optimal policy erive in this paper uner the above assumptions may not be optimal in maximizing revenue, but it shoul still be robust with limite risk  we lose less money than expecte in the case of overcommitment. Define where ] CS t, x t ) := E Ĉt+1 S t, x t = µ p + 1 κ τ) p t µ p )] x t m F t y)y b 0 y x t ρ E R t F t y) = P Y t+1 y F t ]. 0 y x t ρ E R t F t y)y, 4) C ) is known as the contribution, or the rewar function. See 1 in the ecompanion to this paper for the erivation of 4) Objective Function Let Π be the set of all policies. A policy is an F t measurable function X π S t ) that escribes the mapping from the state at time t, S t, to the ecision at time t, x t. For each π Π, let T ] G π t S t ) := E γ t t CS t, X π S t )) S t, 0 t T, t =t where 0 < γ < 1 is the iscount factor an T inicates the en of the horizon. The objective, then, is to fin an optimal policy π = π that satisfies for all 0 t T. G π t S t ) = sup G π t S t ), π Π
10 10 Article submitte to Operations Research; manuscript no. OPRE Main Assumptions an Structural Result The main contribution of this paper is the close form representation of the optimal policy for avance intermittent energy commitments that also allows us to express the value of the energy storage in a close form. In orer to achieve the results, we nee assumptions on the probability istribution of the spot market electricity price an win energy, limit on the storage size, an the ecision epoch intervals Electricity Price an Win Energy First, we assume that p t ) t 0 an ŷ t ) t 1 are inepenent in Ω, F, P). It is wellknown that the price of the electricity mainly epens on the eman as well as the main source of energy that is controllable; for example, electricity generate from coal plants. It is fairly reasonable to assume that the fluctuation in the electricity price is not significantly influence by the fluctuation in the uncontrollable an unpreictable energy supply from our win farm, especially if we are a small player in the market. In most cases, intermittent energy plays a minor role in the electricity markets, anyway. Next, assume p t ) t 0 are i.i. with istribution N 0, σ 2 p). Then, pt ) t 0 is a stanar meanreverting process an E p t+n F t ] = µ p + 1 κ τ) n p t µ p ), n, t N +. 5) It is common to use a meanreverting process to moel electricity prices, as shown in Eyelan an Wolyniec 2003).Similarly, assume ŷ t ) t 1 are 0mean an i.i. with stanar eviation σ Y τ. Then, in most cases, the istributions of ŷ t ) t 1 are assume to be truncate Gaussian with mean 0 an stanar eviation σ Y τ. However, in this paper, we assume that ŷ t ) t 1 are uniformly istribute with mean 0 with stanar eviation σ Y τ. Assuming that ŷ t ) t 1 are uniformly istribute allows us to explicitly compute various expectations that are neee to erive the optimal policy in a close form. Since a truncate Gaussian istribution is boune, as long as we match the mean an the variance, a uniform istribution can be a statistically robust substitute
11 Article submitte to Operations Research; manuscript no. OPRE for the truncate Gaussian istribution in the context of optimizing a value function. This fact is emonstrate in 5 where we conuct numerical experiments in which we apply the optimal policy erive uner the assumption of uniformly istribute ŷ t ) t 1 to the ata generate from truncate Gaussian istributions. Then, given F t, Y t+1 U θ t, θ t + β), where β := 2 3σ Y τ an M 1 θ t := µ Y τ + α i Y t i µ Y τ) β, t. 6) 2 The cumulative ensity function CDF) of Y t+1 compute at time t is given by 0, if y < θ t y θ F t y) = P Y t+1 y F t ] = t if θ β t y θ t + β 1, if y > θ t + β i=0 The expecte contribution function C ) is not inexe by t because the CDF F t ) is etermine by θ t, which is a eterministic function of S t. The expecte contribution is completely etermine by S t an x t. However, it is important to note that θ t an β o not necessarily have to be efine as shown above. The results obtaine in this paper are applicable as long as we use a forecasting moel that preicts that the electricity prouce uring the time interval t, t + 1) is uniformly istribute given F t Size of the Storage Next, we nee an assumption on the size of the storage. Since the electricity price is meanreverting, if we have an infinitely large storage, a naive policy that stores the energy when the expecte spot market price is less than some fixe price an commits to sell the energy in storage plus the energy we are certain to prouce when the spot market price is greater than some fixe price, will be a riskless arbitrage policy. Arbitrage here means that there is zero probability of losing money ue to overcommitment or losing energy ue to the storage being full. There is always a significant conversion loss. Such a case is comparable to traing a stock whose price is meanreverting. In reality, a storage with reasonably goo rountrip efficiency that can be charge an ischarge
12 12 Article submitte to Operations Research; manuscript no. OPRE in a short amount of time will be expensive to buil an maintain, an we nee an intelligent way of etermining the appropriate size of the storage. We propose that the size of the storage be etermine in comparison to ν, given by: ] σ Y m 1 ν := κ 2 3 min m, b. 7) b + ρ E γµ p It is obvious that as the penalty factors m an b become larger, we nee to allow for a larger storage since our commitment level will be more conservative an we will en up storing more energy. Also, if the rountrip efficiency of the storage ρ E is small, we must allow for a larger storage in orer to compensate for the energy that will be lost in conversion. Next, since γµ p is the iscounte expecte spot market price of the electricity, if γµ p is small, we nee to allow for a larger storage since our commitment level will be more conservative. What makes ν interesting is the term σ Y /κ. Recall that κ is proportional to the expecte number of times the price crosses the mean per unit time. Then, 1/κ is proportional to the expecte amount of time between two consecutive crossings. Therefore, σ Y /κ is proportional to the volatility in the win energy that is prouce while the spot market price completes a cycle. Since R max etermines our ability to accumulate energy while the price moves, we must allow for a larger storage when σ Y /κ gets larger. If R max =, we can implement an arbitrage policy, as explaine above. If R max ν, we must implement a more active, risktaking policy that consiers the movement of the price towars the mean but not count on the price reaching a esirable level within a esirable amount of time. The mile regime in which ν < R max < will eman the most complicate policy that mixes risktaking with arbitrage. Fining the optimal policy in this mile regime will be an interesting research topic, but it is beyon the scope of this paper. For this paper, we assume ] σ Y m 1 R max κ 2 3 min m, b. 8) b + ρ E γµ p 8) is necessary in orer to erive 10) shown in the next section, which in turn is necessary to prove lemma 18) that is use to erive the marginal value function in a close form. However,
13 Article submitte to Operations Research; manuscript no. OPRE even though 8) is impose for mathematical convenience, numbers come out reasonable, as shown in 5. If we use real ata to obtain σ Y, κ, µ p, ρ E an use m an b that satisfy 3), if we let R max σ Y = 1.7, for example, 8) is satisfie. That is, we can have the size of R max in the same orer of magnitue of the stanar eviation in win energy. Having a storage of limite size allows us to obtain the optimal policy in a close form an provie us with various insights, as is shown in 4. Moreover, before investing a significant amount of capital to buil a large storage, it is reasonable to assume that win farm operators will start with a small storage, stuy its effects, an then subsequently make the investment for aitional storage. This paper erives the optimal policy for energy commitment an the corresponing value of the storage when the storage is small. As will be sown in 4, the optimal policy uner the assumption 8) will still epen on the mean of the electricity price an how far the price is away from the mean. However, the optimal policy will be base on the premise that the storage is not large enough to allow us to avoi the risk of overcommitment by waiting for the price to rise without facing the risk of losing energy ue to the storage being full. Thus, 8) forces us to always balance the risk of overcommitment an the risk of unercommitment. We not only want to avoi paying the penalty for overcommitment, but we also want to avoi committing too little an lose energy ue to conversion an the storage being full. Suppose we have a large storage evice an 8) is violate, but we choose to implement the policy erive in this paper that is optimal uner the assumption of small storage, anyway. Then, the cost of overcommitment will not change, but the risk of unercommitment will be smaller than expecte because we are less likely to lose energy ue to the storage being full. Therefore, the optimal policy erive in this paper will still be robust when the assumption 8) oes not hol Decision Epoch Interval Finally, we nee an assumption on how often we make our commitment ecisions. We can rearrange the terms from 8) to obtain:
14 14 Article submitte to Operations Research; manuscript no. OPRE max R max m ρ E γ) 2 3 m 1) σ Y R max ρ E γκ, R maxb 2 3 σ Y b R max ρ E γκµ p ] 1 κ. We assume that the time interval τ between our ecision epochs satisfies the following: max R max m ρ E γ) 2 3 m 1) σ Y R max ρ E γκ, R maxb 2 3 σ Y b R max ρ E γκµ p ] τ 1 κ. 9) 9) ensures that the price always moves towar the mean in expectation, but oes not overshoot an move pass the mean in expectation. The lower boun can be rearrange to be written as m 1 R max β min m ρ E γ 1 κ τ), b b + ρ E γκ τµ p ]. 10) Since we have a limit on the size of our storage as shown in our assumption 8), if τ is too large, the amount of electricity that is prouce between our ecisions can be too large an we are likely to lose energy ue to the storage being full. 9) gives us a reasonable ecision epoch time interval τ Structural Results In this section, we show some structural results of the value function. Let V π t S t ) be a function that satisfies V π T S T ) = CS T, X π T S T )), V π t S t ) = CS t, X π t S t )) + γe V π t+1s t+1 ) S t ], 0 t T 1. Then, V π t S t ) = G π t S t ), 0 t T. For 0 t T, let V t S t ) satisfy the following: V T S T ) = max x R + CS T, x), V t S t ) = max x R + {CS t, x) + γe V t+1 S t+1 ) S t, x]}, 0 t T 1. V t S t ) is known as the value function. Accoring to Puterman 1994), V t S t ) = G π t S t ), 0 t T. Denote V x t S t, x) := E V t+1 S t+1 ) S t, x], 0 t T.
15 Article submitte to Operations Research; manuscript no. OPRE The augmente value function V x t S t, x) is an example of a Qfactor. Let Then, x t := arg max {CS t, x) + γv x t S t, x)} = X π S t ), 0 t T. x R + V t S t ) = max x R + {CS t, x) + γv x t S t, x)} = CS t, x t ) + γv x t S t, x t ), 0 t T. At the en of the horizon, we can show that R T V T S T ) = ρ E µ p + 1 κ τ) p T µ p )] 11) an hence 2 R 2 T V T S T ) = 0. 12) See 2 in the ecompanion to this paper for the erivation of 11) an 12). Now that we have efine the value function, we present its structure. The structural results are mainly attributable to the storage transition function an the contribution function, an they follow from three of the aforementione assumptions: ŷ t ) t 1 an p t ) t 1 are inepenent, p t ) t 1 is meanreverting as shown in 5), an Then, 0 t T 1, we have: m γ an b γ µ p. ρ E ρ E Structural Result 1. CS t, x) + γv x t S t, x) is a concave function of R t, x). Structural Result 2. The optimal ecision x t is positive an finite an Structural Result 3. x CS t, x t ) + γ x V x t S t, x t ) = 0. 13) V t S t ) = ρ E µ p + 1 κ τ) p t µ p )] 14) R t R t+1 +γe ρ E x + R ) ] t+1 V t+1 S t+1 ) S t, x t. R t R t+1
16 16 Article submitte to Operations Research; manuscript no. OPRE Structural Result 4. V t S t ) is a concave function of R t. Structural Result 5. ρ E µ p + 1 κ τ) p t µ p )] V t S t ) 1 µ p + 1 κ τ) p t µ p )]. 15) R t See 2 in the ecompanion to this paper for the proof of the above results. In Structural Result 3, which shows the recursive relationship between the marginal value functions, the meaning of the term ρ E µ p + 1 κ τ) p t µ p )] is obvious; if we ha an extra R t amount of energy in storage, we can commit to sell it an gain R t ρ E µ p + 1 κ τ) p t µ p )] in expecte revenue. However, the secon term requires some analysis. From 2), we know that R t+1 ρ E x + R { t+1 1 ρe ρ x=x R t = R, if x t < Y t+1, R t + Y t+1 x t ) < R max. t 0, otherwise. escribes the conversion loss that occurs when we use the energy that is put into the storage when we generate more electricity than we nee to satisfy the commitment. Therefore, the term R t+1 E ρ E x + R ) ] t+1 V t+1 S t+1 ) S t, x t R t R t+1 can be seen as the expecte portion of the marginal future value function that is save by not having to go through the process of energy conversion. 4. Main Result  Infinite Horizon Analysis In this section, we erive the marginal value function an the corresponing optimal policy for avance energy commitment that maximizes the expecte revenue in the infinite horizon case. However, while we can obtain the value of always having a storage as shown in this paper, it is important to note that the cost of always having storage is not the cost of installing the storage
17 Article submitte to Operations Research; manuscript no. OPRE once in the beginning. Batteries have finite lifetime, an we might have to reinstall them every ten years, for example. We let T an rop the inex t from the value function: Then, V S t ) satisfies T ] V S t ) = lim E γ t t CS t, X π S t )) S t. T t =t V S t ) = max x R + {CS t, x) + γe V S t+1 ) S t, x]} = CS t, x t ) + γv x S t, x t ). Since the structural properties shown in the previous section hols true for all T, V S t ) maintains those structural properties. In 4.1, we erive the optimal policy using the main assumptions state in 2 an the structural results shown in 3. We first state: Theorem 1. The optimal policy, when the electricity generate from the win farm is uniformly istribute from θ t to θ t + β, is given by x t = X π S t ) = ρ E R t + θ t + µ pk 1 + p t µ p ) 1 κ τ) K 2 m µ p + p t µ p ) 1 κ τ)] + b β 16) where an K 1 = 1 γ ρ E 1 ρ E ρ E K 2 = 1 γ 1 κ τ) 1 ρ E exp γ 1 ρ E ) 1 β ] ) R max 1, exp γ 1 κ τ) 1 ρ E ) 1 β ] ) R max 1. Before proving 16), we first analyze its components. Since ρ E R t is the amount of electricity that can be prouce by converting the energy in storage an θ t is the amount of electricity that is certain to be prouce, ρ E R t + θ t can be seen as the riskless term. Since there is a limit on the size of the storage an we lose energy if the storage is full, we always want to commit to sell at least ρ E R t + θ t. The issue is then how much more to commit relative to this base level. Overcommitment is costly because the expecte penalty always excees the expecte spot price. Unercommitment is costly for two reasons. First, excess prouction must be store an storage
18 18 Article submitte to Operations Research; manuscript no. OPRE is not free since the rountrip efficiency is less than 1. Secon, since there is a limit on the amount of energy you can store, R max, if we commit too little an prouce too much we lose the prouction that cannot be store. So the optimal extra commitment over the base level must balance the cost of overcommitment an unercommitment. β is the uncertainty in the electricity prouction, an committing µ p K 1 + p t µ p ) 1 κ τ) K 2 m µ p + p t µ p ) 1 κ τ)] + b 17) fraction of β achieves the balance between the cost of overcommitment an the cost of unercommitment. Note that the solution to a typical newsvenor problem states that the venor shoul always try to satisfy a fixe fraction of the ranom eman Khouja 1999), Petruzzi an Daa 1999)). However, in our case, the fraction is a function of the price because we can speculate on the movement of the price that is meanreverting. In 4.2, we obtain the stationary istribution of the storage level corresponing to the optimal policy. In 4.3, we erive the economic value of the storage as the relative increase in average revenue ue to the existence of the storage Optimal Policy In this section, we prove the optimal policy 16) by first eriving the marginal value function. From Structural Result 2, we know that the optimal ecision x t must satisfy x CS t, x t ) + x V x S t, x t ) = ] x CS t, x Rt+1 t ) + γe V S t+1 ) S t, x t x R t+1 = 0. Therefore, in orer to compute x t, we only nee to know the erivative of V S t+1 ) with respect to R t+1, an we o not nee to know V S t+1 ) itself. To erive lemma: R t+1 V S t+1 ), we nee the following Lemma 1. x t + R max R t θ t + β, t. 18)
19 Article submitte to Operations Research; manuscript no. OPRE Proof: See 3 in the ecompanion to this paper. The proof utilizes the inequality 10). We know that θ t + β x t is the maximum amount of excess electricity that can be left over after fulfilling the commitment. Suppose that the inequality 18) oes not hol. Then, θ t + β x t ) R max R t, inicating that there is always enough room left in the storage to accommoate all of the excess electricity, implying that there is no risk of unercommitment at all. However, we have restricte the size of the storage as shown in 10) precisely to avoi such a situation. We know that the optimal policy ought to balance the risk of unercommitment an the risk of overcommitment. The above lemma allows us to compute R t V S t ) from which we can erive the optimal policy. We first state: Theorem 2. V S t ) = ρ E µ p exp γ 1 ρ E ) 1 R t β +ρ E p t µ p ) 1 κ τ) exp )] Rmax R t γ 1 κ t) 1 ρ E ) 1 β Rmax R t )]. 19) Proof : Here, we show a conense version of the proof by omitting various algebraic steps. See 4 in the ecompanion to this paper for a etaile proof. We prove the theorem by using backwar inuction in the finite horizon setting an letting T go to infinity. First, we make the inuction hypothesis that R T i V T i S T i ) = ρ E µ p +ρ E p T i µ p ) 1 κ τ) for some i 0, an prove that i j=0 i j=0 1 γ 1 ρ E ) 1 j! β 1 j! )] j Rmax R T i 20) γ 1 κ t) 1 ρ E ) 1 β )] Rmax R T i i+1 1 V T i+1) S T i+1) ) = ρ E µ p γ 1 ρ E ) 1 )] j Rmax R T i R T i+1) j! β j=0 ) i+1 1 +ρ E pt i+1) µ p 1 κ τ) γ 1 κ t) 1 ρ E ) 1 )] j Rmax R T i+1). j! β From 11), we know that j=0 R T V T S T ) = ρ E µ p + ρ E p T µ p ) 1 κ τ).
20 20 Article submitte to Operations Research; manuscript no. OPRE Therefore, the expression for R T i V T i S T i ) shown above is true for i = 0. From 2), we can show that = R t+1 ρ E x + R t+1 R t { 1 ρ E ) 1 j! ) 1 j! R max R t Rmax R t+1 Y t+1 x t ) 0, otherwise. ) j x=x t ] j, if x t < Y t+1, R t + Y t+1 x t ) < R max.. Next, by 18), Then, we can show E R t+1 ρ E x + R ) t+1 1 R t j! From Structural Result 3, R T i+1) V T i+1) S T i+1) ) f t y) = 1 β, x t y x t + R max R t. Rmax R t+1 ) j S t, x t = ρ E µp + 1 κ τ) p T i+1) µ p )) + γe ρ E R T i i+1 = ρ E µ p j=0 1 γ 1 ρ E ) 1 j! β ) i+1 +ρ E pt i+1) µ p 1 κ τ) ] = 1 ρ E ) 1 β x + R T i )] j Rmax R T i+1) j=0 1 j! R T i+1) γ 1 κ τ) 1 ρ E ) 1 β Therefore, 20) is true for i 0. Next, substitute t for T i + 1). Then, T t 1 V t S t ) = ρ E µ p γ 1 ρ E ) 1 R t j! β j=0 T t +ρ E p t µ p ) 1 κ τ) t T. If we let T go to infinity, R t V S t ) = lim T = ρ E µ p exp V t S t ) R t j=0 γ 1 ρ E ) 1 β +ρ E p t µ p ) 1 κ τ) exp )] j Rmax R t 1 γ 1 κ τ) 1 ρ E ) 1 j! β )] Rmax R t γ 1 κ τ) 1 ρ E ) 1 β 1 j + 1)! ) j+1 Rmax R t. ) ] V T i S T i ) S T i+1), x T i+1) R T i )] j Rmax R T i+1). )] j Rmax R T i+1), Rmax R t )], t.
21 Article submitte to Operations Research; manuscript no. OPRE To compute the optimal ecision x t at time t, all we nee to know is R t+1 V S t+1 ). Since we now know what Proof of 16): From 19), R t+1 V S t+1 ) is, we are reay to prove 16). Rt+1 E x = µ p ρ E 1 ρ E ] V S t+1 ) S t, x R t+1 exp γ 1 ρ E ) 1 β p t µ p ) 1 κ τ) 2 ρ E 1 ρ E exp R max ] ) 1 γ 1 κ τ) 1 ρ E ) 1 β ] ) R max 1, 21) x ρ E R t + θ t. To see the erivation of 21), see 5 in the ecompanion to this paper. Then, from Structural Result 2, we know that the optimal ecision x t must satisfy x CS t, x t ) + γ x V x S t, x t ) = p t,t+1 mp t,t+1 + b) F t x t ρ E R t ) + γe = 0, Rt+1 x ] V S t+1 ) S t, x t R t+1 where p t,t+1 := E p t+1 F t ] = µ p + 1 κ τ) p t µ p ). Therefore, mp t,t+1 + b) 1 β x t ρ E R t θ t ) ] Rt+1 = p t,t+1 + γe V S t+1 ) S t, x t x R t+1 ρ E = µ p + p t µ p ) 1 κ τ) γµ p exp 1 ρ E γ p t µ p ) 1 κ τ) 2 ρ E exp 1 ρ E = µ p K 1 + p t µ p ) 1 κ τ) K 2. γ 1 ρ E ) 1 β R max γ 1 κ τ) 1 ρ E ) 1 β ] R max ) 1 ] 1 ) Then, x t = ρ E R t + θ t + µ pk 1 + p t µ p ) 1 κ τ) K 2 m µ p + p t µ p ) 1 κ τ)] + b β.
22 22 Article submitte to Operations Research; manuscript no. OPRE Note that both K 1 an K 2 increases when ρ E, R max, or γ is reuce. We know that the optimal amount shoul naturally epen on the penalty, the rountrip efficiency, the maximum storage limit, an the iscount factor as follows. First, it shoul ecrease with increasing penalty, as being short incurs the penalty. Secon, it shoul increase with reuce rountrip efficiency, as being long implies paying to store losing energy). Thir, it shoul ecrease with increasing maximum storage. If our storage capacity is greater, we lose less of the energy we o not sell, an we can affor to be more conservative an commit less. Fourth, it shoul increase with ecreasing iscount factor, because the value of what we store now to use in the future ecreases with the iscount factor. Next, suppose storage evices with sufficient capacities become ubiquitous in the future an hence electricity becomes a very liqui asset just like stocks. Then, arbitraguers taking avantage of preictable patterns such as meanreversion will cause the electricity prices to behave more an more like a martingale. Corollary 1. If κ = 0, implying that p t ) t 0 is a martingale, then K 1 = K 2 an x t = ρ E R t + θ t + p t mp t + b K 1β. 22) If the price is a martingale, it is stochastically constant an we cannot speculate on the future movement of the price. Then, the fraction is just irectly proportional to the ratio between the current expecte spot market price an the penalty price Stationary Distribution of the Storage Level Now that we have the optimal policy 16), we want to assess the expecte value of storage corresponing to the policy. In orer to obtain a closeform expression for the expecte value of storage, we must analyze the ynamics of our system at the steaystate an erive the stationary istribution of the storage level. Denote Z t := µ pk 1 + p t µ p ) 1 κ τ) K 2 m µ p + p t µ p ) 1 κ τ)] + b, t.
23 Article submitte to Operations Research; manuscript no. OPRE From 2), we know that R t+1 is a function of R t, x t, Y t+1 ). Since Y t+1 is a function of θ t an x t is a function of R t, Z t, θ t ) as shown in 16), we can think of R t+1 as a function of R t, Z t, θ t ). However, because θ t is the amount of electricity that we are certain to prouce an commit, we know that R t+1 in fact oes not epen on θ t. Thus, R t+1 is a function of R t, Z t ). Therefore, if the ranom process Z t ) t 0 is stationary ergoic, the process R t ) t 0 will reach a steaystate. Since Z t ) t 1 is riven by p t ) t 1, we first nee to know the istribution of p t ) t 1 in steay state. Here we use the term steaystate to refer to the unconitional process. Proposition 1. At steaystate, p t N Proof: We know that 23) is true if an only if 23) implies Suppose 23) is true. Then, p t+1 N 1 κ τ) p t µ p ) N ) σp 2 µ p,. 23) 1 1 κ τ) 2 ) σp 2 µ p,. 1 1 κ τ) 2 Since p t+1 is inepenent from p t an p t+1 N 0, σ 2 p), 1 κ τ) p t µ p ) + p t+1 N 0, 1 κ τ)2 σ 2 p 1 1 κ τ) 2 ) ) σp 2 0,. 1 1 κ τ) 2. Then, p t+1 = µ p + 1 κ τ) p t µ p ) + p t+1 N ) σp 2 µ p, 1 1 κ τ) 2 Since Z t is a eterministic function of p t, Z t ) t 1 reaches steaystate when p t ) t 1 reaches steaystate. Although realtime electricity spot prices can be negative ue to tax subsiies, it will be extremely rare for the ayahea forecast price to be negative. Thus, in practice, we may assume that the price is always going to be positive. The first an secon moments of Z t at steaystate given p t 0 is ] µp K 1 + ε µ p ) 1 κ τ) K 2 Z 1 := E m µ p + ε µ p ) 1 κ τ)] + b ε 0, 24)
24 24 Article submitte to Operations Research; manuscript no. OPRE an where Also, efine an Z 2 := E ) 2 µp K 1 + ε µ p ) 1 κ τ) K 2 ε 0] m µ p + ε µ p ) 1 κ τ)] + b ε N ) σp 2 µ p,. 1 1 κ τ) 2 ] µ p + ε µ p ) 1 κ τ) Z 1 := E m µ p + ε µ p ) 1 κ τ)] + b ε 0, Z 2 := E ) 2 µ p + ε µ p ) 1 κ τ) ε 0], m µ p + ε µ p ) 1 κ τ)] + b 25) corresponing to the case where R max = 0, which makes K 1 = K 2 = 1. Z 1,Z 2, Z 1, an Z 2 can be easily compute via MonteCarlo simulation using sample realizations of ε greater than zero. Proposition 2. Then, the stationary istribution of R t corresponing to the steaystate is f Rt r) = r P R t r] = Z 1 δ r) + Z 1 + ρ ) ] Rρ E 1 ρr ρ E ) 1 ρe ) exp r 1 {0 r Rmax} 1 ρ E β β ρ E + Z 1 1 ρ E ) ) ]) 1 ρe ) exp R max δ r R max ), 26) 1 ρ E β where δ ) enotes the Diracelta function. Proof: Here, we show a conense version of the proof by omitting various algebraic steps. See 6 in the ecompanion to this paper for a etaile proof. From 2) an 16), we can show that P R t+1 = 0 R t ] = P R t+1 = 0 ] = Z 1 an P R t+1 = R max R t ] = 1 Z 1 R max β + 1 ρ E ) R t, β in the steaystate. Also, from 2), we can show that { ρe if 0 < u < R β t f Rt+1 R t u R t ) = 1 if R. β t u < R max
25 Article submitte to Operations Research; manuscript no. OPRE Therefore, we can write the conitional probability ensity function as f Rt+1 R t u R t = r) = Z 1 δ u) + ρ E β 1 {0 u<r} + 1 β 1 {r u R max} + 1 Z 1 R max β + 1 ρ ) E ) r δ u R max ), β where δ ) enotes the Diracelta function. Since P R t = 0] = Z 1 in the steaystate, we know that the stationary istribution can be written as R max f Rt r) = Z 1 δ r) + g r) 1 {0 r Rmax} + 1 Z 1 g r) r δ r R max ), for some function g r). By efinition, the stationary istribution must satisfy f Rt+1 u) = R max r=0 f Rt+1,R t u, r) r = R max r=0 r=0 f Rt+1 R t u R t = r) f Rt r) r = f Rt u). By computing the integral an matching the terms, we can show that gu) = Z 1 1 ρ E ) β + ρ E β + 1 ρ E ) β Taking the erivative with respect to u on both sie gives Then, we can show that gr) = Z 1 + g u) = 1 ρ E ) gu). β ρ E 1 ρ E ) 1 ρr ρ E ) β u r=0 g r) r. ] 1 ρe ) exp r β an 1 Z 1 R max r=0 1 g r) r = 1 ρ E 1 ρ E + Z 1 1 ρ E ) ) ]) 1 ρe ) exp R max. β The stationary istribution 26) shows that if the rountrip efficiency is lower, the probability of hitting the capacity limit R max is lower while the probability of epleting the storage is higher, as expecte.
26 26 Article submitte to Operations Research; manuscript no. OPRE Economic Value of the Storage Using the stationary istribution of the storage level obtaine in the previous section, we can compute the following: Corollary 2. In steaystate, E R t ] = R max Z ρ E ρ E 1 ρ E ) ρr β 1 ρ E ] ) 1 ρr ρ E ) exp R max 1 β 27) an the expecte revenue in steaystate is C Rmax SS : = E C S t, x t )] = µ p ρ E E R t ] + µ p E θ t ] + µ p Z 1 β mµ p + b) β 2 Z 2 = µ pρ E R max µ p β Z 1 + ρ ) ] ) Rρ E ρe 1 ρr ρ E ) exp R max 1 1 ρ E 1 ρ E 1 ρ E β +µ p β Z 1 m Z ) + µ p µ Y bβ Z ) See 7 in the ecompanion to this paper for the erivation of 27) an 28). We know that K 1 = K 2 = 1 if R max = 0. Therefore, from 28), if we o not have a storage an R max = R t = 0, t, the expecte revenue at steaystate woul be is C 0 SS := µ p β Z 1 m Z ) + µ p µ Y bβ Z 2 2. Then, the relative increase in the expecte revenue in steaystate ue to the existence of storage ψ : = CRmax SS CSS 0 CSS { 0 ρe R max = 1 ρ E β + Z 1 Z ) Z 1 + ρ E ) ρe 1 ρ E m + b ) Z 2 µ Z ) } 2 / p 1 ρ { E exp 1 ρ E ) R ] max β Z 1 m Z µ Y β b Z2 µ p 2 ) 1 }. 29) 5. Numerical Results In the previous section, we have erive the optimal commitment policy an the corresponing value of the storage assuming that the forecast of electricity generate from the win farm is uniformly istribute. However, the hourly win spee ata obtaine from the North American Lan
27 Article submitte to Operations Research; manuscript no. OPRE W W W W W W N N N N/A N N/A Table 1 Mean of the cube of the spee of the win in January 2000 Data Assimilation System NLDAS) project shows that when forecasting the cube of the spee of the win, a truncate Gaussian istribution fits the ata better than a uniform istribution. In this section, we simulate the win energy process using a truncate Gaussian istribution an compare the relative increase in revenue ue to the existence of storage compute numerically by implementing our policy 16) to the one compute theoretically from the equation 29). From the NLDAS project, we extracte win spee ata from 22 locations across the Unite States. Since the win characteristics vary throughout the year ue to seasonal effects, it is common to assume that the win process is timeinvariant over a one month perio but not beyon that Ettoumi, et al. 2003)). Therefore, we use separate moel parameters an corresponing policies for each month. We foun that the thirorer correlation is very small compare to the first an secon orer correlation, an represent Y t ) t 1, the energy generate from our win farm, as a seconorer AR process: Y t+1 = µ Y + α 0 Y t µ Y ) + α 1 Y t 1 µ Y ) + ŷ t+1, 30) for some µ Y, α 0 an α 1. When we implement our policy 16), we assume ŷ t ) t 1 is i.i. with istribution U β 2, β 2 ), for some β. β is compute by matching β 2 12 to the variance of the resiual in the AR process, ŷ t ) t>1. µ Y s in m 3 /s 3 ) for the selecte 22 locations compute using the January 2000 ata, for example, are given in Table 1 an β s in m 3 /s 3 ) are given in Table 2. From Table 1 an Table 2, we can see that µ Y s an β s are comparable in magnitue, implying that win energy prouction is highly volatile. After we compute µ Y, α 0,α 1 an β using the NLDAS ata, we generate win energy processes ) Y t ) t 1 from equation 30) where ŷ t ) t>1 is i.i. an ŷ t N, t. However, when we are 0, β2 12 computing our commitment from our policy 16), we assume ŷ t U β 2, β 2 ). From 6),
28 28 Article submitte to Operations Research; manuscript no. OPRE W W W W W W N N N N/A N N/A Table 2 Sprea of the cube of the spee of the win in January W W W W W W N N N N/A N N/A Table 3 Relative increase in revenue compute by implementing our policy 16) θ t = µ Y + α 0 Y t µ Y ) + α 1 Y t 1 µ Y ) β 2. Next, we fit the hourly spot market price provie by a utility company to the process p t+1 = µ p + 1 κ τ) p t µ p ) + p t+1, 31) where an p t ) t 0 are i.i. with istribution N 0, σ 2 p). In our experiments, we use ρe = 0.75, γ = 0.99, µ p = 49.9, σ p = 47.46, τ = 1, κ = , m = 1.6, b = 67.5, an Rmax β Z 1 =.1912, Z 2 =.0467, Z 1 =.3270, an Z 2 = This gives = 0.5. Then, ) µy ψ = / β ) We implemente our policy 16) 100 times by generating the prices from 31) an the win energy process from 30) using the coefficients µ, α 0,α 1 an β. Then, we compute the relative increase in revenue ue to the existence of storage for each implementation of our policy an foun the average of those values over the 100 experiments. Next, we compute the relative increase in revenue irectly from equation 29). The relative increase in revenue compute by implementing our policy 16), average over 36 months, is given in Table 3. The relative increase in revenue compute from 29) an hence 32), is given in Table 4. From the above tables, we can see that the relative increase in revenue obtaine through a sample run implementing our policy 16) is comparable to the relative increase in revenue compute
Hull, Chapter 11 + Sections 17.1 and 17.2 Additional reference: John Cox and Mark Rubinstein, Options Markets, Chapter 5
Binomial Moel Hull, Chapter 11 + ections 17.1 an 17.2 Aitional reference: John Cox an Mark Rubinstein, Options Markets, Chapter 5 1. OnePerio Binomial Moel Creating synthetic options (replicating options)
More informationRisk Management for Derivatives
Risk Management or Derivatives he Greeks are coming the Greeks are coming! Managing risk is important to a large number o iniviuals an institutions he most unamental aspect o business is a process where
More informationAn intertemporal model of the real exchange rate, stock market, and international debt dynamics: policy simulations
This page may be remove to conceal the ientities of the authors An intertemporal moel of the real exchange rate, stock market, an international ebt ynamics: policy simulations Saziye Gazioglu an W. Davi
More informationOptimal Control Policy of a Production and Inventory System for multiproduct in Segmented Market
RATIO MATHEMATICA 25 (2013), 29 46 ISSN:15927415 Optimal Control Policy of a Prouction an Inventory System for multiprouct in Segmente Market Kuleep Chauhary, Yogener Singh, P. C. Jha Department of Operational
More informationJON HOLTAN. if P&C Insurance Ltd., Oslo, Norway ABSTRACT
OPTIMAL INSURANCE COVERAGE UNDER BONUSMALUS CONTRACTS BY JON HOLTAN if P&C Insurance Lt., Oslo, Norway ABSTRACT The paper analyses the questions: Shoul or shoul not an iniviual buy insurance? An if so,
More informationA Generalization of Sauer s Lemma to Classes of LargeMargin Functions
A Generalization of Sauer s Lemma to Classes of LargeMargin Functions Joel Ratsaby University College Lonon Gower Street, Lonon WC1E 6BT, Unite Kingom J.Ratsaby@cs.ucl.ac.uk, WWW home page: http://www.cs.ucl.ac.uk/staff/j.ratsaby/
More informationCh 10. Arithmetic Average Options and Asian Opitons
Ch 10. Arithmetic Average Options an Asian Opitons I. Asian Option an the Analytic Pricing Formula II. Binomial Tree Moel to Price Average Options III. Combination of Arithmetic Average an Reset Options
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS Contents 1. Moment generating functions 2. Sum of a ranom number of ranom variables 3. Transforms
More informationState of Louisiana Office of Information Technology. Change Management Plan
State of Louisiana Office of Information Technology Change Management Plan Table of Contents Change Management Overview Change Management Plan Key Consierations Organizational Transition Stages Change
More informationPurpose of the Experiments. Principles and Error Analysis. ε 0 is the dielectric constant,ε 0. ε r. = 8.854 10 12 F/m is the permittivity of
Experiments with Parallel Plate Capacitors to Evaluate the Capacitance Calculation an Gauss Law in Electricity, an to Measure the Dielectric Constants of a Few Soli an Liqui Samples Table of Contents Purpose
More informationMath 230.01, Fall 2012: HW 1 Solutions
Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The
More informationMODELLING OF TWO STRATEGIES IN INVENTORY CONTROL SYSTEM WITH RANDOM LEAD TIME AND DEMAND
art I. robobabilystic Moels Computer Moelling an New echnologies 27 Vol. No. 23 ransport an elecommunication Institute omonosova iga V9 atvia MOEING OF WO AEGIE IN INVENOY CONO YEM WIH ANOM EA IME AN
More informationSensitivity Analysis of Nonlinear Performance with Probability Distortion
Preprints of the 19th Worl Congress The International Feeration of Automatic Control Cape Town, South Africa. August 2429, 214 Sensitivity Analysis of Nonlinear Performance with Probability Distortion
More informationWeek 4  Linear Demand and Supply Curves
Week 4  Linear Deman an Supply Curves November 26, 2007 1 Suppose that we have a linear eman curve efine by the expression X D = A bp X an a linear supply curve given by X S = C + P X, where b > 0 an
More information10.2 Systems of Linear Equations: Matrices
SECTION 0.2 Systems of Linear Equations: Matrices 7 0.2 Systems of Linear Equations: Matrices OBJECTIVES Write the Augmente Matrix of a System of Linear Equations 2 Write the System from the Augmente Matrix
More informationRisk Adjustment for Poker Players
Risk Ajustment for Poker Players William Chin DePaul University, Chicago, Illinois Marc Ingenoso Conger Asset Management LLC, Chicago, Illinois September, 2006 Introuction In this article we consier risk
More informationThe oneyear nonlife insurance risk
The oneyear nonlife insurance risk Ohlsson, Esbjörn & Lauzeningks, Jan Abstract With few exceptions, the literature on nonlife insurance reserve risk has been evote to the ultimo risk, the risk in the
More informationCURRENCY OPTION PRICING II
Jones Grauate School Rice University Masa Watanabe INTERNATIONAL FINANCE MGMT 657 Calibrating the Binomial Tree to Volatility BlackScholes Moel for Currency Options Properties of the BS Moel Option Sensitivity
More informationAnswers to the Practice Problems for Test 2
Answers to the Practice Problems for Test 2 Davi Murphy. Fin f (x) if it is known that x [f(2x)] = x2. By the chain rule, x [f(2x)] = f (2x) 2, so 2f (2x) = x 2. Hence f (2x) = x 2 /2, but the lefthan
More informationChapter 2 Review of Classical Action Principles
Chapter Review of Classical Action Principles This section grew out of lectures given by Schwinger at UCLA aroun 1974, which were substantially transforme into Chap. 8 of Classical Electroynamics (Schwinger
More informationEnterprise Resource Planning
Enterprise Resource Planning MPC 6 th Eition Chapter 1a McGrawHill/Irwin Copyright 2011 by The McGrawHill Companies, Inc. All rights reserve. Enterprise Resource Planning A comprehensive software approach
More informationMathematics Review for Economists
Mathematics Review for Economists by John E. Floy University of Toronto May 9, 2013 This ocument presents a review of very basic mathematics for use by stuents who plan to stuy economics in grauate school
More informationDigital barrier option contract with exponential random time
IMA Journal of Applie Mathematics Avance Access publishe June 9, IMA Journal of Applie Mathematics ) Page of 9 oi:.93/imamat/hxs3 Digital barrier option contract with exponential ranom time Doobae Jun
More informationConsumer Referrals. Maria Arbatskaya and Hideo Konishi. October 28, 2014
Consumer Referrals Maria Arbatskaya an Hieo Konishi October 28, 2014 Abstract In many inustries, rms rewar their customers for making referrals. We analyze the optimal policy mix of price, avertising intensity,
More informationA Theory of Exchange Rates and the Term Structure of Interest Rates
Review of Development Economics, 17(1), 74 87, 013 DOI:10.1111/roe.1016 A Theory of Exchange Rates an the Term Structure of Interest Rates HyoungSeok Lim an Masao Ogaki* Abstract This paper efines the
More informationUCLA STAT 13 Introduction to Statistical Methods for the Life and Health Sciences. Chapter 9 Paired Data. Paired data. Paired data
UCLA STAT 3 Introuction to Statistical Methos for the Life an Health Sciences Instructor: Ivo Dinov, Asst. Prof. of Statistics an Neurology Chapter 9 Paire Data Teaching Assistants: Jacquelina Dacosta
More informationWeb Appendices to Selling to Overcon dent Consumers
Web Appenices to Selling to Overcon ent Consumers Michael D. Grubb MIT Sloan School of Management Cambrige, MA 02142 mgrubbmit.eu www.mit.eu/~mgrubb May 2, 2008 B Option Pricing Intuition This appenix
More informationChapter 10 Market Power: Monopoly and Monosony Monopoly: market with only one seller. Demand function, TR, AR, and MR
ECON191 (Spring 2011) 27 & 29.4.2011 (Tutorial 9) Chapter 10 Market Power: Monopoly an Monosony Monopoly: market with only one seller a Deman function, TR, AR, an P p=a bq ( q) AR( q) for all q except
More informationChapter 4: Elasticity
Chapter : Elasticity Elasticity of eman: It measures the responsiveness of quantity emane (or eman) with respect to changes in its own price (or income or the price of some other commoity). Why is Elasticity
More informationThroughputScheduler: Learning to Schedule on Heterogeneous Hadoop Clusters
ThroughputScheuler: Learning to Scheule on Heterogeneous Haoop Clusters Shehar Gupta, Christian Fritz, Bob Price, Roger Hoover, an Johan e Kleer Palo Alto Research Center, Palo Alto, CA, USA {sgupta, cfritz,
More informationOptimal Control Of Production Inventory Systems With Deteriorating Items And Dynamic Costs
Applie Mathematics ENotes, 8(2008), 194202 c ISSN 16072510 Available free at mirror sites of http://www.math.nthu.eu.tw/ amen/ Optimal Control Of Prouction Inventory Systems With Deteriorating Items
More informationThe Quick Calculus Tutorial
The Quick Calculus Tutorial This text is a quick introuction into Calculus ieas an techniques. It is esigne to help you if you take the Calculus base course Physics 211 at the same time with Calculus I,
More informationWeb Appendices of Selling to Overcon dent Consumers
Web Appenices of Selling to Overcon ent Consumers Michael D. Grubb A Option Pricing Intuition This appenix provies aitional intuition base on option pricing for the result in Proposition 2. Consier the
More informationStock Market Value Prediction Using Neural Networks
Stock Market Value Preiction Using Neural Networks Mahi Pakaman Naeini IT & Computer Engineering Department Islamic Aza University Paran Branch email: m.pakaman@ece.ut.ac.ir Hamireza Taremian Engineering
More informationIf you have ever spoken with your grandparents about what their lives were like
CHAPTER 7 Economic Growth I: Capital Accumulation an Population Growth The question of growth is nothing new but a new isguise for an ageol issue, one which has always intrigue an preoccupie economics:
More informationMSc. Econ: MATHEMATICAL STATISTICS, 1995 MAXIMUMLIKELIHOOD ESTIMATION
MAXIMUMLIKELIHOOD ESTIMATION The General Theory of ML Estimation In orer to erive an ML estimator, we are boun to make an assumption about the functional form of the istribution which generates the
More information_Mankiw7e_CH07.qxp 3/2/09 9:40 PM Page 189 PART III. Growth Theory: The Economy in the Very Long Run
189220_Mankiw7e_CH07.qxp 3/2/09 9:40 PM Page 189 PART III Growth Theory: The Economy in the Very Long Run 189220_Mankiw7e_CH07.qxp 3/2/09 9:40 PM Page 190 189220_Mankiw7e_CH07.qxp 3/2/09 9:40 PM Page
More informationModelling and Resolving Software Dependencies
June 15, 2005 Abstract Many Linux istributions an other moern operating systems feature the explicit eclaration of (often complex) epenency relationships between the pieces of software
More information6.3 Microbial growth in a chemostat
6.3 Microbial growth in a chemostat The chemostat is a wielyuse apparatus use in the stuy of microbial physiology an ecology. In such a chemostat also known as continuousflow culture), microbes such
More informationSafety Stock or Excess Capacity: Tradeoffs under Supply Risk
Safety Stock or Excess Capacity: Traeoffs uner Supply Risk Aahaar Chaturvei Victor MartínezeAlbéniz IESE Business School, University of Navarra Av. Pearson, 08034 Barcelona, Spain achaturvei@iese.eu
More informationOption Pricing for Inventory Management and Control
Option Pricing for Inventory Management an Control Bryant Angelos, McKay Heasley, an Jeffrey Humpherys Abstract We explore the use of option contracts as a means of managing an controlling inventories
More informationCHAPTER 5 : CALCULUS
Dr Roger Ni (Queen Mary, University of Lonon)  5. CHAPTER 5 : CALCULUS Differentiation Introuction to Differentiation Calculus is a branch of mathematics which concerns itself with change. Irrespective
More informationAPPLICATION OF CALCULUS IN COMMERCE AND ECONOMICS
Application of Calculus in Commerce an Economics 41 APPLICATION OF CALCULUS IN COMMERCE AND ECONOMICS æ We have learnt in calculus that when 'y' is a function of '', the erivative of y w.r.to i.e. y ö
More informationGoal Programming. Multicriteria Decision Problems. Goal Programming Goal Programming: Formulation
Goal Programming Multicriteria Decision Problems Goal Programming Goal Programming: Formulation Goal Programming?? How o you fin optimal solutions to the following? Multiple criterion for measuring performance
More informationData Center Power System Reliability Beyond the 9 s: A Practical Approach
Data Center Power System Reliability Beyon the 9 s: A Practical Approach Bill Brown, P.E., Square D Critical Power Competency Center. Abstract Reliability has always been the focus of missioncritical
More informationProfessional Level Options Module, Paper P4(SGP)
Answers Professional Level Options Moule, Paper P4(SGP) Avance Financial Management (Singapore) December 2007 Answers Tutorial note: These moel answers are consierably longer an more etaile than woul be
More information19.2. First Order Differential Equations. Introduction. Prerequisites. Learning Outcomes
First Orer Differential Equations 19.2 Introuction Separation of variables is a technique commonly use to solve first orer orinary ifferential equations. It is socalle because we rearrange the equation
More informationOn Adaboost and Optimal Betting Strategies
On Aaboost an Optimal Betting Strategies Pasquale Malacaria 1 an Fabrizio Smerali 1 1 School of Electronic Engineering an Computer Science, Queen Mary University of Lonon, Lonon, UK Abstract We explore
More informationA Data Placement Strategy in Scientific Cloud Workflows
A Data Placement Strategy in Scientific Clou Workflows Dong Yuan, Yun Yang, Xiao Liu, Jinjun Chen Faculty of Information an Communication Technologies, Swinburne University of Technology Hawthorn, Melbourne,
More informationSearch Advertising Based Promotion Strategies for Online Retailers
Search Avertising Base Promotion Strategies for Online Retailers Amit Mehra The Inian School of Business yeraba, Inia Amit Mehra@isb.eu ABSTRACT Web site aresses of small on line retailers are often unknown
More informationECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE
ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.
More informationLECTURE 15: AMERICAN OPTIONS
LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These
More informationOpen World Face Recognition with Credibility and Confidence Measures
Open Worl Face Recognition with Creibility an Confience Measures Fayin Li an Harry Wechsler Department of Computer Science George Mason University Fairfax, VA 22030 {fli, wechsler}@cs.gmu.eu Abstract.
More informationLecture L253D Rigid Body Kinematics
J. Peraire, S. Winall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L253D Rigi Boy Kinematics In this lecture, we consier the motion of a 3D rigi boy. We shall see that in the general threeimensional
More informationA Comparison of Performance Measures for Online Algorithms
A Comparison of Performance Measures for Online Algorithms Joan Boyar 1, Sany Irani 2, an Kim S. Larsen 1 1 Department of Mathematics an Computer Science, University of Southern Denmark, Campusvej 55,
More information2r 1. Definition (Degree Measure). Let G be a rgraph of order n and average degree d. Let S V (G). The degree measure µ(s) of S is defined by,
Theorem Simple Containers Theorem) Let G be a simple, rgraph of average egree an of orer n Let 0 < δ < If is large enough, then there exists a collection of sets C PV G)) satisfying: i) for every inepenent
More informationDetecting Possibly Fraudulent or ErrorProne Survey Data Using Benford s Law
Detecting Possibly Frauulent or ErrorProne Survey Data Using Benfor s Law Davi Swanson, Moon Jung Cho, John Eltinge U.S. Bureau of Labor Statistics 2 Massachusetts Ave., NE, Room 3650, Washington, DC
More informationStochastic Inventory Control
Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the
More informationSprings, Shocks and your Suspension
rings, Shocks an your Suspension y Doc Hathaway, H&S Prototype an Design, C. Unerstaning how your springs an shocks move as your race car moves through its range of motions is one of the basics you must
More informationA New Evaluation Measure for Information Retrieval Systems
A New Evaluation Measure for Information Retrieval Systems Martin Mehlitz martin.mehlitz@ailabor.e Christian Bauckhage Deutsche Telekom Laboratories christian.bauckhage@telekom.e Jérôme Kunegis jerome.kunegis@ailabor.e
More informationFactoring Dickson polynomials over finite fields
Factoring Dickson polynomials over finite fiels Manjul Bhargava Department of Mathematics, Princeton University. Princeton NJ 08544 manjul@math.princeton.eu Michael Zieve Department of Mathematics, University
More informationApplications of Global Positioning System in Traffic Studies. Yi Jiang 1
Applications of Global Positioning System in Traffic Stuies Yi Jiang 1 Introuction A Global Positioning System (GPS) evice was use in this stuy to measure traffic characteristics at highway intersections
More informationGame Theoretic Modeling of Cooperation among Service Providers in Mobile Cloud Computing Environments
2012 IEEE Wireless Communications an Networking Conference: Services, Applications, an Business Game Theoretic Moeling of Cooperation among Service Proviers in Mobile Clou Computing Environments Dusit
More informationISSN: 22773754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 12, June 2014
ISSN: 77754 ISO 900:008 Certifie International Journal of Engineering an Innovative echnology (IJEI) Volume, Issue, June 04 Manufacturing process with isruption uner Quaratic Deman for Deteriorating Inventory
More informationProduct Differentiation for SoftwareasaService Providers
University of Augsburg Prof. Dr. Hans Ulrich Buhl Research Center Finance & Information Management Department of Information Systems Engineering & Financial Management Discussion Paper WI99 Prouct Differentiation
More information1 HighDimensional Space
Contents HighDimensional Space. Properties of HighDimensional Space..................... 4. The HighDimensional Sphere......................... 5.. The Sphere an the Cube in Higher Dimensions...........
More informationCalculating Viscous Flow: Velocity Profiles in Rivers and Pipes
previous inex next Calculating Viscous Flow: Velocity Profiles in Rivers an Pipes Michael Fowler, UVa 9/8/1 Introuction In this lecture, we ll erive the velocity istribution for two examples of laminar
More informationA New Pricing Model for Competitive Telecommunications Services Using Congestion Discounts
A New Pricing Moel for Competitive Telecommunications Services Using Congestion Discounts N. Keon an G. Ananalingam Department of Systems Engineering University of Pennsylvania Philaelphia, PA 191046315
More information4. Important theorems in quantum mechanics
TFY4215 Kjemisk fysikk og kvantemekanikk  Tillegg 4 1 TILLEGG 4 4. Important theorems in quantum mechanics Before attacking threeimensional potentials in the next chapter, we shall in chapter 4 of this
More informationThe Derivative of ln x. d dx EXAMPLE 3.1. Differentiate the function f(x) x ln x. EXAMPLE 3.2
Chapter 4 Section 3 Differentiation of Logarithmic an Eponential Functions 331 3 RADIOLOGY Differentiation of Logarithmic an Eponential Functions 61. The raioactive isotope gallium67 ( 67 Ga), use in
More informationLagrangian and Hamiltonian Mechanics
Lagrangian an Hamiltonian Mechanics D.G. Simpson, Ph.D. Department of Physical Sciences an Engineering Prince George s Community College December 5, 007 Introuction In this course we have been stuying
More informationThe Inefficiency of Marginal cost pricing on roads
The Inefficiency of Marginal cost pricing on roas Sofia GrahnVoornevel Sweish National Roa an Transport Research Institute VTI CTS Working Paper 4:6 stract The economic principle of roa pricing is that
More informationThe most common model to support workforce management of telephone call centers is
Designing a Call Center with Impatient Customers O. Garnett A. Manelbaum M. Reiman Davison Faculty of Inustrial Engineering an Management, Technion, Haifa 32000, Israel Davison Faculty of Inustrial Engineering
More informationTraffic Delay Studies at Signalized Intersections with Global Positioning System Devices
Traffic Delay Stuies at Signalize Intersections with Global Positioning System Devices THIS FEATURE PRESENTS METHODS FOR APPLYING GPS DEVICES TO STUDY TRAFFIC DELAYS AT SIGNALIZED INTERSECTIONS. MOST OF
More informationExponential Functions: Differentiation and Integration. The Natural Exponential Function
46_54.q //4 :59 PM Page 5 5 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Section 5.4 f () = e f() = ln The inverse function of the natural logarithmic function is the natural eponential
More informationSensor Network Localization from Local Connectivity : Performance Analysis for the MDSMAP Algorithm
Sensor Network Localization from Local Connectivity : Performance Analysis for the MDSMAP Algorithm Sewoong Oh an Anrea Montanari Electrical Engineering an Statistics Department Stanfor University, Stanfor,
More informationLecture 8: Expanders and Applications
Lecture 8: Expaners an Applications Topics in Complexity Theory an Pseuoranomness (Spring 013) Rutgers University Swastik Kopparty Scribes: Amey Bhangale, Mrinal Kumar 1 Overview In this lecture, we will
More informationINFLUENCE OF GPS TECHNOLOGY ON COST CONTROL AND MAINTENANCE OF VEHICLES
1 st Logistics International Conference Belgrae, Serbia 2830 November 2013 INFLUENCE OF GPS TECHNOLOGY ON COST CONTROL AND MAINTENANCE OF VEHICLES Goran N. Raoičić * University of Niš, Faculty of Mechanical
More informationTopics in Macroeconomics 2 Econ 2004
Topics in Macroeconomics 2 Econ 2004 Practice Questions for Master Class 2 on Monay, March 2n, 2009 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
More informationPHY101 Electricity and Magnetism I Course Summary
TOPIC 1 ELECTROSTTICS PHY11 Electricity an Magnetism I Course Summary Coulomb s Law The magnitue of the force between two point charges is irectly proportional to the prouct of the charges an inversely
More informationGiven three vectors A, B, andc. We list three products with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B);
1.1.4. Prouct of three vectors. Given three vectors A, B, anc. We list three proucts with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B); a 1 a 2 a 3 (A B) C = b 1 b 2 b 3 c 1 c 2 c 3 where the
More informationAchieving quality audio testing for mobile phones
Test & Measurement Achieving quality auio testing for mobile phones The auio capabilities of a cellular hanset provie the funamental interface between the user an the raio transceiver. Just as RF testing
More informationCS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options
CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common
More information(We assume that x 2 IR n with n > m f g are twice continuously ierentiable functions with Lipschitz secon erivatives. The Lagrangian function `(x y) i
An Analysis of Newton's Metho for Equivalent Karush{Kuhn{Tucker Systems Lus N. Vicente January 25, 999 Abstract In this paper we analyze the application of Newton's metho to the solution of systems of
More informationnparameter families of curves
1 nparameter families of curves For purposes of this iscussion, a curve will mean any equation involving x, y, an no other variables. Some examples of curves are x 2 + (y 3) 2 = 9 circle with raius 3,
More informationCALCULATION INSTRUCTIONS
Energy Saving Guarantee Contract ppenix 8 CLCULTION INSTRUCTIONS Calculation Instructions for the Determination of the Energy Costs aseline, the nnual mounts of Savings an the Remuneration 1 asics ll prices
More informationInverse Trig Functions
Inverse Trig Functions c A Math Support Center Capsule February, 009 Introuction Just as trig functions arise in many applications, so o the inverse trig functions. What may be most surprising is that
More informationHardness Evaluation of Polytetrafluoroethylene Products
ECNDT 2006  Poster 111 Harness Evaluation of Polytetrafluoroethylene Proucts T.A.KODINTSEVA, A.M.KASHKAROV, V.A.KALOSHIN NPO Energomash Khimky, Russia A.P.KREN, V.A.RUDNITSKY, Institute of Applie Physics
More informationChapter 11: Feedback and PID Control Theory
Chapter 11: Feeback an ID Control Theory Chapter 11: Feeback an ID Control Theory I. Introuction Feeback is a mechanism for regulating a physical system so that it maintains a certain state. Feeback works
More informationIPA Derivatives for MaketoStock ProductionInventory Systems With Lost Sales
IPA Derivatives for MaketoStock ProuctionInventory Systems With Lost Sales Yao Zhao Benjamin Melame Rutgers University Rutgers Business School Newark an New Brunswick Department of MSIS 94 Rockafeller
More informationDifferentiability of Exponential Functions
Differentiability of Exponential Functions Philip M. Anselone an John W. Lee Philip Anselone (panselone@actionnet.net) receive his Ph.D. from Oregon State in 1957. After a few years at Johns Hopkins an
More informationStochastic Modeling of MEMS Inertial Sensors
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 10, No Sofia 010 Stochastic Moeling of MEMS Inertial Sensors Petko Petkov, Tsonyo Slavov Department of Automatics, Technical
More informationDi usion on Social Networks. Current Version: June 6, 2006 Appeared in: Économie Publique, Numéro 16, pp 316, 2005/1.
Di usion on Social Networks Matthew O. Jackson y Caltech Leeat Yariv z Caltech Current Version: June 6, 2006 Appeare in: Économie Publique, Numéro 16, pp 316, 2005/1. Abstract. We analyze a moel of i
More informationThe Elastic Capacitor and its Unusual Properties
1 The Elastic Capacitor an its Unusual Properties Michael B. Partensky, Department of Chemistry, Braneis University, Waltham, MA 453 partensky@attbi.com The elastic capacitor (EC) moel was first introuce
More informationM147 Practice Problems for Exam 2
M47 Practice Problems for Exam Exam will cover sections 4., 4.4, 4.5, 4.6, 4.7, 4.8, 5., an 5.. Calculators will not be allowe on the exam. The first ten problems on the exam will be multiple choice. Work
More informationChoice under Uncertainty
Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory
More informationLecture 13: Differentiation Derivatives of Trigonometric Functions
Lecture 13: Differentiation Derivatives of Trigonometric Functions Derivatives of the Basic Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the
More informationThe Impact of Forecasting Methods on Bullwhip Effect in Supply Chain Management
The Imact of Forecasting Methos on Bullwhi Effect in Suly Chain Management HX Sun, YT Ren Deartment of Inustrial an Systems Engineering, National University of Singaore, Singaore Schoo of Mechanical an
More informationUniversal Gravity Based on the Electric Universe Model
Universal Gravity Base on the Electric Universe Moel By Frerik Nygaar frerik_nygaar@hotmail.com Nov 015, Porto Portugal. Introuction While most people are aware of both Newton's Universal Law of Gravity
More information9.3. Diffraction and Interference of Water Waves
Diffraction an Interference of Water Waves 9.3 Have you ever notice how people relaxing at the seashore spen so much of their time watching the ocean waves moving over the water, as they break repeately
More information