Fast/Thermal Reactor Request-Based Exchange

The primary criticism agaisnt a random assignment of commodities, preference coefficients, and constraint coefficients is that using domain-specific models provides specific structure to a given formulation. This problem species is designed to target a specific set of cases that focus randomness in the domain-related values and defines domain-specific translations from those values to the associated coefficients.

The goal of this document is to provide a basis for the values used in modeling the request-based exchange and to explore the effects of increasing various kinds of fidelity on the general performance of the formulation.

Model Fidelity

Furthermore, this species is specifically targeted at investigating the effects of model fidelity on a given formulation. Eight different fidelity “levels” have been defined in three categories.

Category	Subcategory
Reactor	batch assembly
Fuel Cycle	Once-Through UOX + MOX F/Th Recycle UOX + MOX F/Th Recycle + Thorium F Recycle
Location	None Coarse Fine

Commodities

There are four possible commodities based on the fuel cycle fidelity modeled:

• enriched UOX
• fast MOX
• thermal MOX
• fast ThOX

Facilities

In order to allow for rapid instance generation, surrogate models of facilties must be used. Surrogate models simplify the decision making that would normally occur in agent archetypes. The goal of using surrogate models is provide instances generally domain-valid structure.

Material

All surrogate facility models require a notion of materials. Because simplicity is required, materials have two properties: a commodity and fissile enrichment. Certain commodities are fungible, e.g., fast and thermal plutonium. Fungible commodities are delineated by preference assignment and supplier process coefficients, both of which described in the following sections.

Reactors

Two types of reactors are used: thermal and fast. Thermal reactors are simplified models of AP-1000 reactors, and fast reactors are simplified models of BN-600 reactors.

Using the dimensions in the following table, one can estimate that the AP-1000 core volume is approximately 12.5 times larger than the BN-600 core. The remainder of this section will assume BN-600 reactor cores have size unity and AP-1000 cores have size 12.5, respectively, and that fuel density is approximately equivalent.

Active Core Dimensions

Reactor	Core Height (m)	Core Diameter (m)
AP1000	4.27	3.04
BN600	0.75	2.05

A further simplifying assumption is that both reactor types will reload \(\frac{1}{4}\) of their core at any given timestep (as has been assumbed for other BN-600 analyses).

Under these assumptions, each fast reactor will request 1 unit of fuel and each thermal reactor will request 12.5 units of fuel. Each may be further binned into smaller quantities to more accurately model assemblies.

As a rough approximation using the figures from active core size, and assuming that a single AP-1000 fuel assembly holds 450 kg of Uranium, a unit of fuel is roughly

\[\frac{450 \frac{kg}{assembly} * 157 assemblies * \frac{1}{4} core}{12.5 units} = ~1.4 \frac{tonnes}{fuel unit}\]

Again one fuel unit is approximately equal to a quarter of a BN-600 reactor core.

Thermal Reactors

Thermal reactors are capable of using either UOX or recycled MOX, and have preferences over the commodities as described below. Fissile isotope enrichments can vary from reactor to reator and from assembly to assembly within a reactor. Accordingly, a surrogate model of enrichment preference is used, randomly selective an enrichment within a viable range. Furthermore, because MOX fuel is backfilled by another istopically fertile material, it is assumed that a MOX request is approximately 7% of a UOX request. The MOX enrichment range is based off IAEA estimates.

Thermal Reactor Requst Surrogate Model Summary

Commodity	Enrichment Range	Relative Request Size
UOX	\([3.5, 5.5]\)	1
Th & F MOX	\([55, 65]\)	0.07

Fast Reactor

Fast reactors come in two flavors based on the fuel cycle being modeled: MOX-preferring reactors and ThOX-preferring reactors. Enrichment ranges are similarly based off plutonium fissile enrichment values in the above IAEA report. It is assumed that the Plutonium oxide in MOX takes up ~20% of the total mass.

Fast Reactor Requst Surrogate Model Summary

Commodity	Enrichment Range	Relative Request Size
UOX	\([15, 20]\)	1
Th & F MOX, ThOX	\([55, 65]\)	0.2

Commodity Preferences

Commodity-Preference Mapping for Reactor Types

Reactor Type	EUOX	Th MOX	F MOX	F ThOX
Thermal	0.5	1	0.1	N/A
F MOX	0.1	0.5	1	0.25
F ThOX	0.1	0.25	0.5	1

Questions

• What critiques are there regarding the commodity-preference mapping?
  • functional form effects (e.g., linear vs. exp) could be added
• What critiques are there regarding reactory enrichment generation?
  • start simple with one enrichment per reactor, a possible upgrade is to introduce 2 or 3 bins around an average enrichment to emulate enrichment zones

Supporting Facilities

The supporting facilities represent the separations and fuel fabrications processes for each fuel type. Supporting facilities are the suppliers in the reactor request case, and therefore must provide supply constraints. The supporting facility surrogate models have an inventory constraint and possibly a process constraint, depending on the fidelity level used.

Both constraints must have an associated conversion function, that takes a surrogate material, i.e., an enrichment and quantity.

UOX Supplier

The UOX supplier has basic parameters, e.g., feed and tails assays, can be safely assumed as follows

Parameter	Value
feed assay	0.711
tails assay	0.3

The conversion functions are also well known.

\[conv_{inv}(\epsilon, q) = NatU(\epsilon, q)\]\[conv_{proc}(\epsilon, q) = SWU(\epsilon, q)\]

MOX and ThOX Suppliers

Due to the lack of commercially viable, well documented fast reactor fuel suppliers, a simple linear surrogate model is assumed for an inventory constraint. There are many possible process surrogate models that could be used, such as heat production or radiotoxicity; however, each of these requires a detailed isotopic composition to be relevant. Per the current IAEA practice, and extrapolating the same effect for reprocessing U-233, a factor, \(f_{commod}\), of 100 is added for for Plutonium and Thorium-based commodities.

\[conv_{inv}(\epsilon, q) = \epsilon q\]\[\begin{split}f_{commod} = \begin{cases} 1,& \text{if UOX}\\ 100, & \text{otherwise} \end{cases}\end{split}\]\[conv_{proc}(\epsilon, q, commod) = q f_{commod}\]

Supplier Constraint RHS Values

Supporting facilities have a nominal throughput capacity. The proposed Eagle Rock Enrichment Plant purports to have a capacity of 3.3M SWU per year. From previous conversations with industry representatives, a reasonable size for a processing plant is 800 tonnes per year, which is similar to Rokkassho. With the factor of 100 discussed above, a 800 t U/ 8 t Pu facility could service on the order of 2-3 fast reactors or ~2 thermal reactors with 1/3 a request as MOX (in other words, an 8 t/yr plant cannot process 1/4 of a thermal core in one month).

Using the following assumptions

• enrichment facilities primarily service thermal reactors
• an exchange represents a monthly timestep
• requests are based on a single unit of fuel (rather than kilograms, etc.)

\[S_{proc, SWU} = \frac{3.3E6 WU}{12 \frac{month}{year}} = ~2.75e5 \frac{SWU}{month}\]\[S_{proc, recycle} = \frac{800 \frac{t}{year}}{12 \frac{month}{year}} = ~66.7 \frac{t}{month}\]

From the formulation point of view, interesting cases arise when either constraint is dominated by the other and when neither is dominant. Furthermore, instanes should be investigated in which supply is generally constrained and when it is not.

In order to accomplish these goals, the supply constraint values are formulated as follows

\[S_{proc}, given\]\[S_{inv} = S_{proc} r_{inv, proc} \frac{conv_{proc}(\bar{\epsilon}, 1)}{conv_{inv}(\bar{\epsilon}, 1)}\]

Parameters

\(r_{inv, proc}\) : the ratio of the inventory RHS to the process RHS

Fuel Cycles

More commodities are required to model more complex fuel cycles. Similarly, as more fungible commodities are added a given instance of the GFCTP becomes more complex. This species of the GFCTP can add fuel cycle, and therefore commodity, complexity in three steps.

Once Through

The least complex fuel cycle is the Once Through (OT) fuel cycle. Reactors request enriched uranium, and supporting facilities are represented by Enrichment Fuel Fabricators.

Parameters

None

Recycle

Next, a Recycle (R) scenario is considered. Thermal and fast reactors are included, and a ratio between the two is set as a parameter. Supporting facilities include Enrichment, Thermal, and Fast Fuel Fabricators. The amount of thermal reactors requests that can be satisfied by recycled fuel is set as a parameter. The fraction is capped at \(\frac{1}{3}\), in line with current French LWR refueling practices. In the low-fidelity reactor scenario, \(f_{mox}\) acts a probability that the batch request will be for thermal mox fuel.

Parameters

\(r_{t, f}\) : the ratio of thermal reactors to fast reactors

\(f_{mox} \in [0, \frac{1}{3}]\) : the fraction of thermal reactor requests that can be met with recycled fuel

\(r_{s, r}\) : the ratio of primary suppliers to their primary requesters

Recycle + Thorium

Finally, a fuel cycle with a thorium breeder reactor is modeled. Building on the R scenario, the Recycle + Thorium (RTh) adds an additional fast reactor model that prefers Thorium-based recycled fuel. The fraction of fast reactors that are Thorium-based is set as a parameter. Additionally, a Thorium Fast Fuel Fabricator is added to the pool of suppliers.

Parameters

\(r_{th, pu}\) : the ratio of Thorium to Plutonium-based fast reactors

Location Assignment

Location values can be assigned in either a coarse or fine fashion. In both cases, a location proxy is assigned uniformly, e.g., on \([0, 1]\). Locations are binned, representing regions. If coarse, only regional relationships are taken into account; if fine, regional relationships are taken into account as well as total proximity.

Once location values are assigned, they can then affect preferences. A surrogate model function is required, and one suggestion is

\[p_{l}(i, j) = \delta_{l} \frac{\exp(- | reg_{i} - reg_{j} | ) + \delta_{fine} \exp(- \| loc_{i} - loc_{j} \| )}{1 + \delta_{fine}}\]

Parameters

\(\delta_{l}\) : whether to include a location preference

\(\delta_{fine}\) : whether to include a fine location proxy

\(n_{reg}\) : the number of regions

Surrogate Models

\(p_{l}(i, j)\) : location-based preference

Preference Determination

Given that facilities have preference assignments based on commodity matching, \(p_c\), and, optionally, location, \(p_l\), a valid question is whether the formulation is affected by their relative magnitude. Therefore a final parameter is added to determine the total preference

\[p(i, j) = p_{c}(i, j) + r_{l, c} p_{l}(i, j)\]

Parameters

\(r_{l, c}\) : the importance ratio of location to commodity types

Parameter Summary

All of the parameters that can be set in a run control for this species are listed below:

Structured Request Species Parameters

Handle	Full Name	Possible Values
\(f_{rxtr}\)	reactor fidelity	\(\{0, 1\}\)
\(f_{fc}\)	fuel cycle fidelity	\(\{0, 1, 2\}\)
\(f_{loc}\)	location fidelity	\(\{0, 1, 2\}\)
\(n_{rxtr}\)	number of reactors	any
\(r_{t, f}\)	ratio of thermal reactors to fast reactors	\([0, \frac{1}{4}]\)
\(r_{th, pu}\)	ratio of Thorium to Plutonium-based fast reactors	\([0, 1]\)
\(r_{s, th}\)	ratio of primary suppliers to thermal reactors	\([0, \frac{1}{2}]\)
\(r_{s, mox, uox}\)	ratio of mox to uox thermal supplier	\([0, 1]\)
\(r_{s, mox}\)	ratio of primary suppliers to fast mox reactors	\([0, \frac{1}{2}]\)
\(r_{s, thox}\)	ratio of primary suppliers to fast thox reactors	\([0, \frac{1}{2}]\)
\(f_{mox}\)	fraction of thermal reactor requests that can be met with mox fuel	\([0, 1]\)
\(r_{inv, proc}\)	ratio of the inventory RHS to the process RHS	\(\{0.75, 1, 1.5\}\)
\(n_{reg}\)	number of regions	any
\(r_{l, c}\)	ratio of location to commodity preference	\([0, 2]\)