On-line supplement to: SMART: A Stochastic Multiscale Model for the Analysis of Energy Resources, Technology

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1 On-line supplement to: SMART: A Stochastic Multiscale Model for e Analysis of Energy Resources, Technology and Policy This online supplement provides a more detailed version of e model, followed by derivations of e derivatives needed to fit e value function approximations. A.1 The full energy model We describe e various elements of e model using a notational style developed specifically for stochastic, dynamic resource allocation problems. We en model two types of decisions in our model: e dispatch decisions to supply energy to e various sources of demand which will be performed on an hourly basis, and e capacity acquisition decisions to add new capacity to e conversion plants (or to build new plants for a newly discovered technology) which will be done every year. We note at dispatch decisions include decisions of how much to use, how much to store, and how much to widraw from storage. We let T denote e set of once-a-year time-periods where we add capacity and H e set of hourly periods wiin a year, where H = H and T = T. In is section, we provide e notation needed to model all e elements of a dynamic system, which is organized along five fundamental dimensions: 1) system state variables, 2) decision variables, 3) exogenous information (random variables), 4) transition function and 5) objective function. In section 5 of e paper, we use is modeling framework to formulate a deterministic linear programming model, and a stochastic optimization problem. System state variables We divide e state variable into e resource state (energy investments), storage (e amount of energy held in storage), and information about a range of system parameters. Energy investments and storage We let a represent a generic attribute vector. For example, e attribute vector of a conversion plant could be represented using a = a 1 a 2 a 3 = Type Location Age The attribute Type denotes e type of generation technology such as nuclear, gas turbine, photovoltaic or hydro-power. Age is typically in units of years. Location can be expressed at different levels of detail. For example, we may specify e location as a high-level region such as e central valley in California or at e more detailed level of a zip code. We conduct our numerical experiments only for a spatially aggregate model, but our maematical model is much more general an is, and e algorimic strategy scales fairly easily to more complex problems.. 31

2 It is useful to define e following attribute vector spaces: A = The set of all attribute vectors describing all possible investments in energy resources. A Res = The set of all attribute vectors describing resource nodes, where raw energy resources enter e network (coal, oil, wind, etc.). A Conv = The set of all attribute vectors describing conversion nodes, where raw resources are converted to electricity or oer usable forms of energy. A Stor = The set of all attribute vectors describing storage nodes. A Dem = The set of all attribute vectors describing demand nodes. A = The set of all attribute vectors describing all oer types of nodes, which include collection and distribution nodes used to simplify e linear program, = A \ (A Res A Conv A Stor A Dem ). We define e connectivity of e energy network using A(a) = The set of attribute vectors of nodes downstream in e dispatch network at are directly connected to a node wi attribute vector a. A(a) = The set of attribute vectors of nodes upstream in e dispatch network at are directly connected to a node wi attribute vector a. Using ese attributes, e resource state variables can be described using R a = The total capacity in megawatts (MW) wi attribute vector a A available in hour h of year t. R = The resource state vector in hour h of year t = (R a ) a A. y a = The amount of energy in storage for a node wi attribute vector a A Stor in hour h of year t. y = (y a ) a A Stor. Exogenously varying quantities In our model, energy demand is treated as exogenous, alough we recognize at an important dimension of e energy problem involves demand management. We model demands similarly to how we model energy resources. We let a A Dem be e attributes of demand, which will include e type of energy demand (electricity, natural gas, oil for heating, gasoline for transportation) and could also include location (in a model which captures where demand is occurring). This allows us to model demand as D a = Demand wi attribute a A Dem at hour h in year t. D = (D a ) a A. 32

3 Oer exogenously varying parameters include p = The amount of precipitation occurring in hour h, year t. ρ = The information state in hour h of time t, which can include e state of technology (ρ T ech ), climate (ρ Clim ), market (ρ Mkt ) and exogenous energy supply (ρ exo ). = (ρ T ech, ρ Clim, ρ Mkt, ρ exo ). The vector ρ exo = (ρexo a ) a A Conv is used to model intermittent energy at conversion facilities. If e resource a is nonintermittent (coal, nuclear, natural gas), en ρ exo a = 1. If it is an intermittent supply such as wind or solar, we would have ρ exo a < 1 whenever e exogenous supply of energy does not allow us to use e technology (such as a wind turbine) to its fullest capacity. We finally define e system state variable using S = The state of e system in hour h of year t = (R, y, D, p, ρ ). The system state could include additional variables. For example, if ere is a constraint imposed on e total CO 2 emitted since e beginning of e time horizon, en e state could have e added dimension of e current level of CO 2 emissions. Decision variables Here we describe e different types of decisions and e constraints ey have to observe at a point in time. x cap a = The total capacity wi attribute vector a A Conv added to e system in hour h of year t. x cap = (x cap a ) a A Conv.,aa = The flow from a node wi attribute vector a to a node wi attribute vector a during hour h of year t. = (,aa ) a A,a A. The vector of decisions is given compactly by x = The vector of decisions in hour h of year t = (x cap, xdisp ). The presentation of deterministic and stochastic optimization models which will determine how e decisions are made are given in section 5 of e main paper. We note here at, alough we have modeled all decisions at e hourly level, real-life capacity addition decisions are seldom made more an once a year. This does not require us to change any of e notation at has already been presented, but we assume at for all intermediate hours 0 < h H, x cap = 0. On e oer hand, e dispatch decisions for distributing power to e various sources of demand have to be made every hour h H. 33

4 These decisions are governed by a series of constraints at apply to decisions made at a point in time. For all conversion nodes, e output is constrained by e installed capacity at at node, given by,aa ρ exo ar a, a A Conv. (14) a A(a) Here, R a gives e installed capacity, while ρ exo a captures e effect of intermittent sources such as wind or solar. Conversion nodes are e primary source of constraint in e supply of energy. In our numerical work, we assume at supplies at e resource nodes (which captures e inputs of coal, natural gas, and so on) are unbounded, but oerwise we can write a A(a) a A(a),aa R a, a A Res. (15) Here, e units of R a can be in barrels of oil, cubic feet of natural gas or tons of biomass. Unit conversions are handled in e flow conservation constraints, which are given by θ a a,a a,aa = 0, a A \ A Stor, (16) y a + a A(a) θ a a,a a a A(a) a A(a),aa 0. a A Stor. (17) Equation (17) captures our ability to draw down from e amount at we have stored. θ a a captures changes in units (barrels of oil, cubic feet of gas and gallons of water to megawatts) as well as losses due to transmission and storage. We note at e sum over a A(a) includes flows from storage, and e sum over a A(a) includes flows at remain in storage. θ aa, for a A stor, would capture losses of energy held in storage. We note at equations (16)-(17) represent constraints at a point in time. Later we present e equations at govern how storage levels evolve over time. For all demand nodes, e total input should match e total demand,,a a = ˆD a, where a A Dem, (18) a A(a) where we assume at ere is some expensive, exogenous source (such as imported electricity) to handle situations when all oer sources are not sufficient to meet demand. Finally, we require 0. (19) Note at x cap disp is unconstrained in sign, as negative capacity implies retirement. Let X be e feasible region defined by equations (14) - (19). Exogenous information 34

5 Exogenous information is expressed in e form of random variables governed by probability distributions or an exogenously provided file of scenarios. All random variables are denoted by hats (as in ˆD). ˆR = Exogenous changes in e resource state in hour h of year t (due, for example, to equipment failures or weaer damage). ˆD a = Change in demand wi attribute vector a A Dem in hour h of year t. ˆρ = Exogenous changes in any parameters governing technology, markets, climate and exogenous energy sources, = (ˆρ T ech, ˆρ Mkt, ˆρ Clim, ˆρ exo ). ˆp = The change in precipitation in hour h of year t. We summarize all random variables in a single variable W t (wiout a hat), as follows: W = A vector of all exogenous information regarding parameters appearing in hour h of year t which can include data on weaer, demands, technological parameters and market information, ( = ˆR, ˆD ), ˆρ, ˆp. The availability of technologies of each type would be an element of ˆρ T ech. Examples of components of ˆρ Clim might be e level of allowable emissions out to year T, and e total quantities and hourly patterns of wind (which is a specific example of e exogenous production factor data, ρ exo a ). Interest rates, exogenous resource prices, and resource availability, would be included in ˆρ Mkt. Alough all of ese have been indexed at e hourly level, we note at each of ese could be arriving on different time-scales. For example, information regarding climate data might be available to be used by e system at anywhere from an hourly to monly frequency. Prices might change on a daily basis, while technological developments could take as long as a year to multiple years to impact e system. Later, we need a more formal vocabulary to describe e stochastic process driving our system. Let Ω be e set of all possible realizations of e sequence (W 11,..., W 1H, W 21,..., W 2H,..., W T H ) where T is e number of years and H is e number of time periods in a year. We let W (ω) be a sample realization of e random variables in W when we are following sample pa ω. To complete our formalism, let F be e sigma-algebra on Ω, wi filtrations F F t,h+1 and F th F t+1,0, where F is e sigma-algebra generated by e information up rough year t, hour h. We use is notation below to describe our algorim and establish properties of e decision function. Transition function The state of e system evolves as a result of e decisions taken as well as random events, which is written as, S t,h+1 = S M (S, x, W t,h+1 ), h = 0, 1,..., H 1, S t+1,0 = S M (S th, x th, W t+1,0 ). 35

6 Elements of e system state may transition in different ways. For example, new capacity may be added to e system and old ones may be retired, giving us R t,h+1,a = R a + x cap a + ˆR t,h+1,a, a A Conv, (20) where x cap a is positive for capacity additions and negative when capacities are retired. ˆRt,h+1,a involves changes in e resource state as a result of unforeseen events such as Hurricane Katrina. Changes in demand, precipitation and e parameter vector ρ t are all handled similarly using ˆD t,h+1 = D + ˆD t,h+1, (21) ˆp t,h+1 = p + ˆp t,h+1, (22) ρ t,h+1 = ρ + ˆρ t,h+1. (23) Let y a be energy stored in hour h of year t associated for storage location a A Stor. In our numerical work, we model e water reservoir as e only form of storage, but our model is much more general. We can represent e transition equation for storage facilities using y t,h+1,a = y a + a A(a) θ a a,a a a A(a),aa + ˆp t,h+1,a, a A Stor (24) where ˆp t,h+1,a captures exogenous precipitation into e reservoir wi attribute a. A slightly modified equation is used when h = H. We note at e only flow between time periods occurs rough storage nodes. All oer flows of energy occur wiin a time period. Cost functions The costs consist of e costs for adding new capacity, e costs related to energy dispatch which cover e physical movement of energy resources and transmission of electrical energy, and costs for purchasing fuel from e various energy sources. c cap a = The capital cost for adding a unit of capacity wi attribute vector a A Conv in hour h of year t. c disp,aa = The cost of unit flow from node wi attribute vector a A to node wi attribute vector a A in hour h of year t (for example, e transportation cost of moving biomass). c f a = Unit cost of fuel corresponding to resource node wi attribute vector a A Res in hour h of year t. c p a = Unit operating cost of a conversion plant wi attribute vector a A Conv in hour h of year t (assumed to be proportional to e flow). 36

7 Total capacity and dispatch costs are given by C cap (S, x cap ) = The total capital costs in hour h of year t, = c cap a xcap a. a A (S, ) = The total costs resulting from e dispatch of energy in hour h of year t, = c disp,aa,aa + c f a xdisp,aa + a A C disp a A a A Res a A a A Conv a A c p a xdisp,aa. We note at x cap a = 0 for hours oer an h = 0. ccap a is e total purchase cost of e capacity being invested when h = 0. We may express e total cost function in hour h of year t using, C (S, x ) = C cap (S, x cap ) + Cdisp (S, A.2 Calculating e derivative ˆv hydro We begin by computing e marginal value of additional energy in a single, scalar storage which, for our numerical experiments, was represented as a water reservoir. We do is by computing e marginal impact of having an additional unit of water in storage at a point in time. This calculation is en used in e derivation of e derivative for additional capacity, which impacts an entire year. The objective function at hour h is given by F(S π ) = C (S, X) π C (S, X π ) h >h t >t = C (S, X) π + V(S x ), x C disp (S, X) π C cap (S, X) π + ). C t h(s t h, Xt π h) (25) h H hydro V (y) x cap + V (Rx ). (26) Note at C cap (S, X π ) = 0 for all h > 0 since we assume at capacity addition decisions are made only at { e beginning } of each year. To update V hydro (y x ), we need to differentiate F π (S ) wi respect to y h H,t T which is given by ˆv hydro = F π (S ) y. We note at we differentiate around e pre-decision state y, but en use is to update V t,h 1 x (yx t,h 1 ) around e previous post-decision state yx t,h 1. We may reasonably claim at R x y 0 (27) which captures e belief at an incremental change in e amount of energy in storage will have virtually no impact on our investments in energy resources. For example, an 37

8 incremental increase in e amount stored in a reservoir will not have a measurable impact on e number of wind turbines we would purchase next year. This approximation gives us ˆv hydro = F π = (S ) (28) y ) (S, X) π hydro + V (y) x cap + V (Rx ) ( y ( y ( y C disp C disp C disp ) cap (S, X) π hydro + V (y) x V + (Rx ) R x ) (S, X) π hydro + V (y) x R x y where we use e approximation, R x / y 0. The derivative (29) is approximated using a finite difference. Let S + be e resource state where y is incremented by 1. We en approximate e derivative using ˆv hydro F disp,π (S + ) F disp,π (S ), where F disp,π (S ) = C disp (S, X π hydro ) + V (y x ), denotes e dispatch optimization component of e objective function. This numerical derivative is exceptionally fast to compute since it involves a trivial reoptimization of a small linear program. It is important to emphasize at e calculation of ˆv hydro captures not only e hourly variability of wind, but also e uncertainty, since e value of additional stored energy does not require knowing e future about wind. The marginal value is also computed in e context of a storage facility. A.3 Calculating e derivative ˆv cap The procedure for updating { cap V (Rx )} is somewhat more involved. We assume at h H,t T investment decisions are made once each year, taking effect in a future year. Our goal is to calculate e marginal value of an additional unit of energy conversion capacity from a decision made in a previous year. Changing e resource vector at e beginning of e year changes e resource vector for e entire year rough equation (14). Also, since we only make capacity decisions once cap each year, we do not compute V (Rx ) for all h, but raer only for h = 0. For is reason, we have to capture e marginal value of additional capacity on bo e yearly investment problem (equation (11) of e main paper) as well as all e hourly dispatch problems. We start wi Ft0 π given by equation (25), alough we will use e approximation in (26). Our goal is to compute ˆv cap (29) t0a = F t0(s π t0 ). (30) R t0a F π t0 can be written as a function of R t0,..., R,..., R t,h 1, recognizing at R = R t0 for 0 h < H. R impacts e dispatch decision at hour h, as well as e post-decision storage 38

9 level y x which impacts future problems. In addition, changing R t0 has an impact on R t+1,0, which impacts decisions in future years. To capture ese interactions, we write dft0(s π t0 ) = ( Ccap t0 (S t0, Xt0) π + F π ) t+1,0 R t+1,0 Ft0 π R. (31) dr t0a R t0a R t+1,0 R t0 R R t0 0 h<h We can compute e marginal value of additional resources for e capacity subproblem at hour h = 0, given by equation (11) of e main paper, by using eier e dual variable for equation (12) in e main paper, or a finite difference. We redefine S + t0 here to be S t0 wi R t0 incremented by 1. The finite difference is en given by ˆv t0a + = F cap,π t0 (S t0) + F cap,π t0 (S t0 ). This approximates e two terms in e set of pareneses on e right hand side of (31). We next observe at F π t0 = F π R R since changing R has no impact on e problem for earlier hours, and R R t0 = 1. The derivative F π / R involves finding e derivative of F π (S ) approximated by (26), which we compute using e dual variable for equation (14). Let ν disp a be e dual variable for (14). Since e right hand side of (14) is ρ exo a R a, e marginal impact of increasing R a on e dispatch optimization component of e objective function is given by F π (S ) = νdisp a. R a ρ exo a We finally pull ese calculations togeer to find e marginal value of an additional unit of energy resource at e beginning of e year using ˆv cap t0a h H ν disp a ρ exo a We use ˆv cap t0a to update e value function approximation A.4 The complete ADP algorim + ˆv + t0a. (32) V cap t 1,0a. We close our presentation wi a complete summary of e ADP algorim, given in figure 7. 39

10 cap,0 Step 0: Set e iteration counter, n = 1. Initialize e value function approximations { V t0 (Rt0) x = 0} t T hydro,0 and { V (y x )} h H,t T. Initialize e state S0a. 0 Set t = 0. Step 1: Solve e annual problem of capacity addition: We let x cap,n t0 ( t0 (Sn t0) = max C cap (St0, n x cap cap,n 1 x cap 0 t0 ) + V t0 (Rt0) x ) (33) F cap t0 denote e argument at solves equation (33) and ˆv +,n t0 cap = [F (Sn t0, Rt0a n + 1) F cap of numerical derivatives where ˆv +,n t0a Step 2: Set e hour, h = 0. Step 2a: Observe W t (ω n ) = ( ˆR t, ˆD t, ˆρ t, ˆp t ). t0 = (ˆv +,n t0a ) a A, e vector t0 (Sn t0, Rt0a] n a A. Step 2b: Solve e hourly optimization model (equation (10)) to obtain F disp (Sn ). Let ν disp,n (= (ν disp,n a ) a A ) be e vector of dual values corresponding to e flow conservation constraints represented by equation (3) and let ˆv hydro,n be e numerical derivative [ F disp (Sn, yn + 1) F disp (Sn, yn ) ]. Step 2c: Sample e incoming precipitation data, ˆp t,h+1 in order to compute e new reservoir level using equation (11). Step 2d: Set h = h + 1. If h < H, go to step 2. Oerwise, continue. Step 2e: Update e value function approximations for hydro-storage, V ( ) hydro,n t,h 1 (yt,h 1) x U V hydro,n 1 V, yt,h 1, x ˆv hydro,n h. t,h 1 Step 2f: Compute e hourly state transitions: St,h+1 n = S M ( S, n x n, Wt,h+1 n ) Step 3: Compute e combined dual values (see (32) in e online supplement). ˆv cap,n ta = ˆv +,n t0a + h H ν disp,n a ˆρ exo a a A. Step 4: Update e value function approximations for e capacity acquisitions, V ( ) cap,n t 1,0 (Rx t 1,0) U V cap,n 1 V t 1,0, R x,n t. Step 5: Compute e state transitions: St,h+1 n = S M ( S, n x n, Wt,h+1 n ) t 1,0, ˆvcap,n S n t+1,0 = S M ( S n th, x n th, W n t+1,0). Step 6: Set t = t + 1. If t < T, go to step 1. Oerwise, continue. Step 7: Set n = n + 1. If n < N, set t = 0 and go to step 1. Oerwise, return e value functions { V cap,n t0 (Rt0) } { x t T and V hydro,n (y }h H,t T x ), which define e policy for adding capacity over e years t T. Figure 7: The ADP algorim to solve e energy resource management problem. 40