Dynamic Economic Strategies of Breeder Reactor Commitments

Gottinger, Hans W. Article Dynamic Economic Strategies of Breeder Reactor Commitments Zeitschrift für Wirtschaftsund Sozialwissenschaften (ZWS) - Vierteljahresschrift der Gesellschaft für Wirtschaftsund Sozialwissenschaften, Verein für Socialpolitik Provided in Cooperation with: Duncker & Humblot, Berlin Suggested Citation: Gottinger, Hans W. (1985) : Dynamic Economic Strategies of Breeder Reactor Commitments, Zeitschrift für Wirtschaftsund Sozialwissenschaften (ZWS) - Vierteljahresschrift der Gesellschaft für Wirtschaftsund Sozialwissenschaften, Verein für Socialpolitik, ISSN 0342-1783, Duncker & Humblot, Berlin, Vol. 105, Iss. 1, pp. 1-15, https://doi.org/10.3790/schm.105.1.1 This Version is available at: https://hdl.handle.net/10419/291595 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Dynamic Economic Strategies of Breeder Reactor Commitments By Hans W. Gottinger This approach allows the user to analyze the effect of policy decisions upon the cost of electricity. The forward dynamic programming approach is adapted to an economic assessment of a nuclear reactor mix to allow many terminal conditions to be analyzed. In particular, when the commitment history can be traced for an optimal path, the life of each unit may be computed and a retirement cost may be attributed to each terminal state to reflect the loss incurred by termination of a Light Water Reactor due to unavailability of fuel before the planned lifetime of the plant has been fulfilled. 1. Introduction In a previous essay, Gottinger (1982), we have provided a cost-benefit analysis of the Light Water Breeder Reactor (LWBR) System based on a static assessment of fuel cycle costs. From the point of view of given nuclear energy choices, in this essay we look at dynamic aspects of phasing in and phasing out nuclear technologies in a way that optimize decisions for commitments in these technologies. Such commitment strategies could constitute a rational approach to intertemporal choice of technologies in the case of uncertainty, Hammond (1976). A comprehensive dynamic analysis for breeder reactor commitments in the case of the Liquid Metal Fast Breeder Reactor (LMFBR) has previously been given by A. S. Manne (1974). In contrast to the LMFBR we must consider the LWBR as a 'conceptual reactor'. In view of the hypothetical resource use situation described by Gottinger (1982), the question is whether an advanced converter on a Thorium-232/U-233 fuel cycle (e.g. the prebreeder) produces U-233 at a great enough rate to make the LWBR system a more favorable technology than the Light Water Reactor (LWR). Since Thorium-232 is several times more abundant than Uranium pursuing the Th-U-233 fuel cycle has a multiplying effect on available resources. The fuel cycle cost assumed for the breeder should be specified with some care. While the breeder has a lower fuel cycle cost than either the 1 Zeitschrift ftir Wirtschaftsund Sozialwissenschaften 1985/1 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.105.1.1 | Generated on 2023-04-04 12:07:10 2 Hans W. Gottinger LWR or the prebreeder, the cost of reprocessing is a significant contributor to the breeder fuel cycle cost. Under what scenarios then will the LWER possess favorable economic properties? It would appear that the main distinction between the LWBR and the LWR is the lifetime constraint which applies to the LWR. In an environment of severely limited uranium availability, the penalty associated with the early termination of LWR's as generating units may be sufficient to justify the added front-end expense of additional fuel cycle cost to establish a non-terminable nuclear electric economy. 2. A Dynamic Cost Model Consider a LWR energy system composed of three reactor types. Define the pertinent parameters as follows: Nj (t) is the number of reactors of type j, j = 1,2,3 in place at time i, Vj is the cost coefficient per reactor of type Cj (i) is the cost at time t due to reactors of type j, ej is the enrichment requirement for a generating unit of type E is the total capacity of enriching plants. Constraint: Assume that enrichment capacity is saturated. That is, that sufficient reactors are built so that no further enriching capacity is available. (1) UWje^E 3=1 Type 3 is assigned to the light water breeder; thus es = 0 and (2) S ^ffle^ 2 ZV;. (t) e, = E j=i j=i Objective Function: The discounted total cost Ctatai, is assumed to be minimized, written (3) Minimize Qt0 tal = / e - 2 ^ (t) dt o j = l where oc is the discount rate. This is in the tradition of formulating the intertemporal allocation problem.1 Consider now the breeder-prebreeder relationship on the basis of the Thorium-232/U-233 fuel cycle. Assume no retirement of prebreeders. i Hafele (1975). OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.105.1.1 | Generated on 2023-04-04 12:07:10 Dynamic Economic Strategies of Breeder Reactor Commitments The number of breeders which may be supported at the activation point is some function of the number of prebreeder-years which have accumulated. Assuming a constant y kg U-233 per prebreeder unit, the number of breeders which may be supported is (4) N3 (í) = ~~~~ y f 1N2(t)dt a 0 where R is the U-233 requirement, in kg, for one breeder unit. The upper limit is t — 1, assuming one year for processing and fabrication of U-233 fuel. Now (i) may be assumed to have the form (r) = / [1 — e~at], which is a convenient but also realistic assumption on the saturation process up to the enrichment capacity. This function is easily integrated to yield (5) / 1N2(r)dr = / * /[1 - e-«] dr =// * dr - ff V^dr 0 / 1N2(r)dr = /(i-l)-/ 0 — Q - ax a t - 1 + e-a(t-1) a Now from the constraint equation f ^fit-D+^le-a«-U-l] 0 ß (6) Substituting these relationships into the objective function, (7) CtoUl = / e-« U- [E - f (1 - e°t) e2] vt + f (1 - e-"t) v2 o I (8) Ctoui = /-f"»! (f)e~* dí + / 0 el 1 + f (1 - e-at) v2 (í) + y vs {t) (í - 1 + ± e-a (í-l) - -ij e-*t dt Now it is seen that if f is constant, the objective function may be written (9) Ct0M = f^-v1{t)e-*tdt + fTfe-*t o o --f-v1 (t)(i-e-«9 + (1 _ e-at) v2 (Í) + vs (Í) (t - 1 + — e-a (i-1) - — R \ Q> & t (10) C^-IjiTJ+ZI^T) dt OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.105.1.1 | Generated on 2023-04-04 12:07:10 4 Hans W. Gottinger When regarded as a function of the objective function is minimized by choosing / = 1 if h (T) is negative, or by selecting / = 0 if h (T) is positive. Now in general the commitment pattern for prebreeders is not selected in advance. Therefore, it is desired that the time-dependent mix of reactors [Ni (£), N® (f), IV3 (t)] be choosen to minimize the objective function independently of any particular algorithm. The relationship between the number of breeders and the prebreeder history is given, from equation (4), by (11) N3(t)=-^N2(t-l) ti where the dot denotes differentiation with respect to time. From the constraint equation, (12) N1(i)e1 + -y-JV3(i + l)e2 = E (13) 1^(0=-^-E-yNs(t + l)e2 Substituting this in the cost functional (14) Ptotoi-/«--E (f) - -yW3 (i + 1) e2) Uj (i) + -y Nz (t + 1) v2 (t) + N3 (t) u3 (i) dt R Writing y (t) = — N3 (t) and expanding y (t + 1) in a Taylor series about y (i), equation (14) becomes, after truncating, (15) Ctotal = / e-tl±- [E (t) - y (t) - y (*)] vt (t) 0 I ei + ife (0 lv (*) + vlffl v*»)} ^ (For the sake of realism t may refer to any suitable time unit matching the actual 'fuel cycle turnover time'.) Now the costs vi (f), (i) and vs (f) are dependent upon the commitment history, including capital investment, since UsOs-price may be modelled as a function of cumulative consumption. An effective method for the attack of problems where constraints limit the range of admissible functions is that of dynamic programming. In the case of prebreeder commitment, such an approach is tractable since the ability of the nuclear industry to support the U-233-Th fuel OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.105.1.1 | Generated on 2023-04-04 12:07:10 Dynamic Economic Strategies of Breeder Reactor Commitments 5 cycle is limited by lack of reprocessing facilities. Large capital requirements for construction of reprocessing plants would normally be assumed to act as a limiting influence upon the rate of growth of reprocessing facilities. As a means of allowing for consideration of limitations upon production capability, the Dynamic Programming formulation of the minimization problem allows the input of arbitrary limits upon the system mix. These constraints reflect assumptions upon the capacity of the nuclear service sector to meet needs arising from various proportions of light water reactors, prebreeders, and breeders. Alternatively, they could reflect policies adopted by either the Government or industry groups to support light water breeder reactor development. This method of attack yields an effective approach to the minimization problem defined by equations (1) and (3). 3. Dynamic Formulation for Reactor Plant Selection It is desired to determine the optimal path for reactor commitment based upon time-dependent values for capital cost, operating cost, and fuel cycle cost. The stage variable in this analysis is time; the state is the vector consisting of the reactors of various types. (16) x = [XlfX2,Xz] where Xn is the commitment of reactor type n. The control u applied at time t is the vector of plant additions. For convenience, the elements of the x and u vectors shall be taken to be GW (e) of nuclear-electric additions. The analysis may be expanded to include plants of other types such as coal-fired and natural gas facilities. We adopt constraints to help reduce the computing time of the problem at hand. In the first case, we are limited by uranium ore availability. Thus, the number of uranium-consuming reactors is limited. That is, the yearly consumption of uranium must be consistent with the capacity of the industry to provide it. A second constraint concerns availability of enriching services. The separative work required in any year must be less than or equal to available or projected available capacity. Fuel fabrication capability may also be employed as a constraint. However, it is generally taken to be a simplifying assumption that fabrication capability is not a limiting restriction. It is probably more precise to say that enriching services are a more restrictive constraint than is the availability of fuel fabrication facilities. Reprocessing availability will be a constraint for some reactor types but not for others. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.105.1.1 | Generated on 2023-04-04 12:07:10 6 Hans W. Gottinger The standard Light Water Reactors, the PWR and the BWR, are not entirely dependent upon a closed loop fuel cycle for their operation. On the other hand, the Light Water Breeder System is dependent upon reprocessing capability, and it should be observed that a rational decision-maker would not commit his company's resources to a breeder unless he were certain of reprocessing availability. The arguments may be extended to include various measures of social cost and adverse environmental effects as elements of cost computation or constraint formulation. The only requirement in such a case is that one has quantifiable relationships between costs and benefits, or more precisely, between the measurable impacts and the resulting component of cost. Such an undertaking is frequently impossible when dealing with matters relating to social costs, and even when such attempts are made, the results may not be accepted by the other scholars in the area. As a simplification for a dynamic programming approach, it is the case here that the social costs are taken to be equal for all reactor types. We wish to minimize the total discounted cost to society due to the installation of nuclear-electric power. If we denote by v (f) the total cost of power to society at time i, we may use Bellman's principle cf optimality to establish the iterative equation of Dynamic Programming (17) v (t) = Min [k (r, t) + a (r) A t + v(r + A t)] Reducing this to the discrete form (18) v (f) = Min [7c (t) + a(t)A t + exp (- <xt)v(t + A t)] where k (t) is the cost of adding new units a (f) is the output rate of the current units. The discount factor is the reciprocal of (1 + i), where i is the interest rate. For quantized t and unit value of A t, e-« equals (1 + i)-1. This equation says that, if one knows the ideal combination of units to generate power from year t + 1 to T, the terminal year in the analysis, then it is desired to determine the ideal combination of units which takes the system from the beginning of the final year. Now it is to be observed that the function v {t + A t) is a real-valued function of the vector x, as we are ultimately attempting to determine the values of the state variable x and the control variable k as well as that of the minimum cost. Constraints upon the problem may assume a number of forms, and in the simple analysis to be performed initially it shall be taken that the constraints be formulated in terms of state-vector quantities rather than control vector quantities. Denoting the enrichment constraint by OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.105.1.1 | Generated on 2023-04-04 12:07:10 Dynamic Economic Strategies of Breeder Reactor Commitments 7 e (f), a real quantity, and the average enrichment requirement for each reactor by the vector E, the first constraint equation may be written E' X <e (£) where the prime denotes transpose. Similarly, if the vector u denotes U3O8 production required to support each reactor type, the formulation LT X <,u (t) may be used to specify the uranium production constraint. The basic equation of the reactor deployment model is (19) x (t + 1) = x (t) + k (t) which states that the capacity distribution at time t + 1 is that at time t plus any additions which might occur between t and t + 1. There is one further constraint, an equality constraint, which states that the total capacity available in period t must be equal to the demand schedule d (t). For n plant types (20) 2 Xi(t) = d(t) i = 1 Now the control vector k (i) may assume an unlimited number of representations, since the mix of reactor types may be continuously represented. 4. Exemplification of the Dynamic Programming Approach A stage-increment technique can be developed to solve the optimization problem utilizing techniques of Dynamic Programming. The fundamental equation suggests that we may apply each control to all existing states in order to define the states at the succeeding stage. Additional assumptions are required to reduce the number of states kept during the search for an optimum in the dynamic programming algorithm. Two approaches were combined to yield a tractable solution. The first is a tunnel constraint, the other is related to the quantization of the states themselves. Where a calculated state is identical to an existing state at some stage t, the principle of optimality requires that the state having minimum cost be retained. A state is assumed to be a slowly-varying function of its parameters, it may be assumed the a nearby state is reflective of the same properties as a given state x. It must be borne in mind that the state x is an ordered n-tuple which, for the case of the LWBR system, includes as one of its components the number of prebreeder-years. Therefore, two states which are close to each other in particular have the same or nearly the same number of prebreeder-years, and hence have generated approximately the same quantity of U-233. The metric utilized to establish closeness is the sum of the absolute values of the deviations. In general, OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.105.1.1 | Generated on 2023-04-04 12:07:10 8 Hans W. Gottinger one might use d (x, y) = 2 I xi — Hi I as the distance function for the i = l evaluation of closeness of two states. For the system consisting of the 3 LWBR and the LWR, the equation reduces to d(x, y) = 2 \xi — Ui\- i = i Given e > 0 and a point y, one may say that x is in a neighborhood of y if d (x, y) < e. For the dynamic programming problem, all quantities are integers and therefore the neighborhood should be specified in terms of an integer. The substitution criterion therefore becomes d (x, y)<K where K is an integer. For initial dynamic programming studies, K has been taken to be equal to 2. This approximation may result in propagated error, since the algorithm is that if d (xj, xn) < K, j = 1,..., n — 1 then the state with minimum cost is retained and the other one is discarded. This is a sequence dependent procedure. Observe, for example, that the sequence of triples (12,10,11), (12,10,12), (12, 10,13), (12,10,14) will result in the storing of the single state (12, 10, 14), while the same four states in the order (12,10,11), (12,10,14), (12,10,13), (12,10,12) will result in the retention of states (12,10,14) and (12,10,12). Even with the imposition of constraints and simplifying approximations, the procedure of finding an optimal trajectory over a suitable planning period, say twenty years, is a formidable task. The problem may then be scaled to a manageable size in order to yield, in a reasonable amount of time, a solution which will bear some resemblance to the optimal path for the more elaborate form. For example, a stage may be taken to be two years instead of one, and power additions may be assumed to be in increments of two GW (e) rather than one. It would be desirable to investigate the effect of policy constraints upon the cost of power, or more appropriately, the cost of nuclear-electric power. It is not necessary to utilize absolute cost values; relative costs may be used to investigate the economic properties of the reactor system. 5. Computational Method It is necessary to determine whether there will be a net benefit to society deriving from the existence of the light water breeder reactor system. It has previously been noted that the research and development cost for the LWBR system are anticipated to be small relative to those for a reactor type not in production. Thus, it is quite valid to compare the light water breeder system with the light water reactor system, since it may be assumed that production capacity and operational characteristics are similar in the two cases. It is desired that quantification be made of the discounted cost differential between light water reactors and the light water breeder system. To determine the effect of this cost differential, a Dynamic Programming approach is employed. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.105.1.1 | Generated on 2023-04-04 12:07:10 Dynamic Economic Strategies of Breeder Reactor Commitments energy-generating cost calculation as a variational problem. The implementation of these concepts in computational form is done via dynamic programming. An algorithm is developed in which the number of the reactor types and power requirements are assumed and the optimal plant commitment schedules are generated for any set of hypothesized economic conditions. The application of this algorithm to the system containing Light Water Reactor, Prebreeders and Breeders is made and costs are generated. Zusammenfassung In Erweiterung einer bisher durchgeführten statischen Kosten-NutzenAnalyse auf den Fall einer dynamischen Ressourcenallokation für ein nukleares Energiesystem wird in diesem Beitrag ein Ansatz vorgeschlagen, der die Berechnung einer optimalen Reaktorkombination als die Lösung eines Variationsproblems mit Hilfe der dynamischen Optimierung ermöglicht. References Conaes (Committee on Nuclear and Alternative Energy Systems) (1980), Energy in Transition 1985 - 2010. National Research Council. San Francisco, Gottinger, H. W. (1982), The Economics of Breeder Reactors within a Nuclear Energy Regime. Angewandte Systemanalyse 3 (4), 167 - 175. Häfele, W. (1975), Objective Functions. II AS A Working Paper. Schloß Laxenburg, Austria, 75 - 125. Hammond, P. J. (1976), Changing Tastes and Coherent Dynamic Choice. Review of Economic Studies 43, 159 - 173. Manne, A. S. (1974), Waiting for the Breeder, Symposium on Exhaustible Resources. Review of Economic Studies 41, 47 - 65. National Uranium Resource Evaluation (1976), Preliminary Report, Report GJo-III (76). U.S. Energy Research and Development Adm. (ERDA). Grand Junction, Colo. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.105.1.1 | Generated on 2023-04-04 12:07:10