Random Reactor Request-Based Exchange

This section describes the resource exchange generation for the case where reactors are ordering fuel from suppliers. The primary goal is to discern what are the possible options that parameterize a given instance of such an exchange in order to test the underlying formulation under different scalings of these parameters.

A request group represents a reactor fuel request where items (nodes) in that request satisfy the same demand. For example, a fast reactor may make a request for assemblies for its radial blanket. Such a request could include mostly natural uranium, but a subset may be replaceable with thorium.

A variety of parameters are required to described an instance of a resource exchange where reactors request fuel. The sum of these parameters define the state of the exchange, which is comprised of both supply and request groups and individual nodes within those groups. A parameter either represents a constant value for each group it parameterizes or it represents an average value, distribution, and related information, and the group-specific value is sampled for each run of that ^instance^ of the exchange. As may be expected, average values are used in cases where it does not make sense for the system state in question to be modeled as constant across the entire system. For example, since request nodes roughly model the number of assemblies being ordered by a reactor, it makes more sense for this to be an average value rather than a static value. By using an average value, the model provides insight into cases where most reactors order ^around^ that many assemblies. In general, constant values are preferred, in order to reduce the overall number of runs required to achieve representable analytics.

It should be noted that representing a resource exchange is inherently stochastic. Take for instance a resource exchange where not every possible arc exists (i.e., there is a supplier of a commodity that is not connected to a consumer), the degree of which is measured by the connection probabilty parameter. In such cases, which suppliers and consumers must be chosen arbitrarily, an inherently random process.

number of commodities

The number of commodities associated with the exchange.

Distribution Candidacy

None

Theoretical Basis

The number of commodities is a fundamental parameter for resource exchange.

number of requesters

The number of request groups, where a group defines an overall request that will fuel a reactor. Some portion of that request may be satisfied by more than one commodity.

Distribution Candidacy

None

Theoretical Basis

The number of requesters is a fundamental parameter for resource exchange.

assemblies per request

How many items to include in the request group. Nominally, a small number (1, 2) corresponds to a ^batch^-type fueling system whereas a large number corresponds to an assembly-type fueling system. Generally, each assembly corresponds to a node in the exchange graph. If an assembly can be satisfied by multiple commodities, multiple exchange nodes will be added while the subsequent demand constraint remains the same.

It is likely that this parameter will be sampled bimodally – in small numbers (i.e., 1 - 3) to model batch-type reactors and in large numbers (40 - 65). The large number scaling is indicative of a range for an AP1000 operating in a 4-batch mode (i.e., 157 / 4) to a ^Typical XL Plant^ running in 3-batch mode (i.e., 193 / 3) (see http://www.nrc.gov/reactors/new-reactors/design-cert/ap1000/dcd/Tier%202/Chapter%204/4-1_r14.pdf).

Distribution Candidacy

Possibly a integral distribution around an average.

Theoretical Basis

In general, the number of assemblies per request is directly proportional to the total number of request nodes. Therefore, scaling all requests simultaneous simply results in a larger problem size. Medium-scale problem sizes are handled by medium-scale number of assemblies.

total request value

Total request values are modeling full reactor requests. A basic assumption will be made that each assembly-type object will be equal in size and can be represented as having a quantity of unity.

Distribution Candidacy

None

Theoretical Basis

Perturbation effects based on variable assembly quantities manifest themselves as variable unit coefficients on the primary request constraint. This effect is handled by the addition of request constraints with distribution-sampled unit coefficients.

multicommodity zone fraction

The fraction of requests that can be met with more than one commodity. This expands the number of request nodes by (fraction * assemblies * (commodities - 1)).

Distribution Candidacy

Possibly a (0, 1) distribution around an average.

Theoretical Basis

Any non-trivial reactor-request resource exchange will involve demand that can be met by more than one commodity. Because the total number of request nodes is directly proportional to this parameter, a first attempt will be made with integral values. If a distribution is used, it will simply serve as a way to investigate problem sizes within the integral value bounds.

commodities in multicommodity zone

The number of commodities that satisfy the multicommodity fraction of assemblies.

Distribution Candidacy

Possibly a integral distribution around an average.

Theoretical Basis

It is possible for more than one commodity to satisfy a request, and conceivable that an arbitrary number of commodities could satisfy a request.

exclusion probability

The probability that a given request group will be comprised of exclusive requests. A basic assumption is made that for a given collection of requests (i.e., request group), a reactor will want each node to either be exclusively satisfied or not. In other words, either a request group models quantized assemblies or it does not.

Distribution Candidacy

Possibly a distribution around an average, but unlikely.

Theoretical Basis

This parameter is directly related to the assembly modeling fidelity required by a given reactor model. A value of 0 implies minimum fidelity, a value of 1 implies maximum fidelity, and it is conceiveable with module mixing that this level of fidelity may exist on a spectrum.

number of demand constraints

The average number of additional demand constraints that a supply group adds to the solver. This will follow the number of supply constraints with one additional constraint to mirror the default mass-flow demand constraint.

Distribution Candidacy

Possibly an integral distribution around an average.

If the average value is used, an integral, truncated distribution that peaks at the average will be sampled.

Theoretical Basis

A given requester may have multiple filters on their requests (e.g., mass and plutonium content).

demand constraint values

Demand constraint values are closely related to the request values. The default demand constraint value is equal to a given request group’s total request. Accordingly, additional demand constraint values will either be equal to or approximately equal to the mass-flow demand constraint value, the effect of which will be investigated.

Distribution Candidacy

Possibly a distribution around the total request value for the given request group.

Theoretical Basis

Two assumptions are made for additional demand constraints: that the constraint values are proportional to the total mass of the demand (e.g. plutonium content is proportional to the total mass for plutonium-based commodities, or reactivity is proportional to the total mass for reactor fuel), and that variability amongst suppliers occurs implying that unit capacities can be sampled around unity (see below).

number of suppliers

The number of suppliers. A supplier may supply more than one commodity. By definition, there must be at least one supplier per commodity. If there are more suppliers than commodities, the additional suppliers are randomly assigned base commodities.

Distribution Candidacy

None

Theoretical Basis

The number of suppliers is a fundamental parameter for resource exchange.

fraction of multi-commodities suppliers

The fraction of suppliers that supply more than one commodity.

Distribution Candidacy

Possibly a distribution around an average.

Theoretical Basis

An example might include a fast reactor fuel supplier that supplies multiple types of fast reactor fuel defined as different commodities.

number of commodities per supplier

The average number of commodities that a multicommodity supplier supplies.

Distribution Candidacy

Primarily two cases of interest exist. The first assumes a relatively even distribution of suppliers per commodity. The second assumes that the distribution peaks at some commodity, while some are minimally satisfied. The former case will be investigated first.

Theoretical Basis

A supplier may offer more than one commodity that share constraint values, e.g., a fast reactor fuel supplier may offer two types of fast reactor fuel which are istopically similar but treated as separate commodities.

number of supply constraints

The number of additional supply constraints that a supply group adds to the solver. Values are integral and span [1, 3].

Two primary issues exist:

• what should the upper bound be (is there a good reason why it shouldn’t be 3?)
• whether the number of additional constraints should be a static or average value

See the discussion regarding supply constraint values.

Distribution Candidacy

Possibly an integral distribution around an average.

If the average value is used, an integral, truncated distribution that peaks at the average will be sampled.

Theoretical Basis

A given supplier may have multiple constraints on their supply, for example process and existing inventory constraints.

supply constraint values

Supply constraint values drive the mass flow with respect to whether flow comes from actual suppliers or the ^faux^ suppliers (in order to guarantee a feasible solution). The total amount needed to fully supply for a commodity can be known after the demands for that commodity are formulated. Accordingly, an optimal solution that does not involve faux suppliers can be achieved by setting the supply constraint value for each supplier equal to that total demand.

A number of effects will be investigated for both unit capacities of unity and random values:

• a single supply constraint per supplier set at sufficient supply
• a single supply constraint per supplier set at less-than sufficient supply (e.g., 1/4, 1/2, 3/4 of sufficient supply)
• multiple supply constraints, one of which is set at less than sufficient supply
• multiple supply constraints, more than one of which is set at less than sufficient supply

The goal of these experiements is to determine the effect of constrained supply on the solution of the multicommodity formulation and the effect of more than one constraint enforcing that effect. It is assumed that the effect of less-than sufficient constraints will manifest with the first such constraint but will not increase with subsequent additions.

Distribution Candidacy

Described above.

Theoretical Basis

Suppliers can be constrained by more than one non-related constraint (e.g., process constraints and inventory constraints), which suggests the above approach is reasonable to investigate such cases.

unit capacity/demand coefficient values

The unit capacity coefficients are designed to model the amount of capacity (or demand) consumed by satisfying a unit of a request, i.e., if a unit of the proposed resource flows along the arc in quesiton.

Distribution Candidacy

A distribution around unity will is used. At present a uniform distribution from (0, 2] is used.

Theoretical Basis

Because the coefficients will depend on the process being modeled, e.g., enrichment, separations, etc., the actual translation function that produces such coefficients can be wide ranging. Similarly, demand translation functions can be wide ranging. However, a basic assumption is made that the relation between the unit capacity coefficients and capacitating value is approximately the same for demand constraints and supply constraints in the same ^class^ (i.e., full, half, quarter of required supply).

In other words, two arbitrarily chosen supply or demand constraints look approximately the same if each is normalized.

preference coefficient values

Because preferences are a relative value, a simple (0, 1) uniform distribution is used for each preference assignment. A possible improvement would be to sample preferences in the same neighborhood for each supplier/consumer group pair.

Distribution Candidacy

A (0, 1) uniform distribution will be used with possible clustering around requester/supplier pairs.

Theoretical Basis

A simulation’s entities can have an arbitrary process for providing preference values, and a known use case includes preferring region-region or institution-institution trades. The former implies using a simple uniform distribution and the latter implies using a clustered distribution as described.

connection probability

A measure of the probability that a request node and supply node of the same commodity will be connected. A probabiliy of 0 indicates that the graph is minimally connected (i.e., each request node has exactly one arc to it) whereas a probability of 1 indicates that the graph is maximally connected (all possible connections are made).

Distribution Candidacy

Possibly a distribution around an average, but unlikely.

Theoretical Basis

Not all possible connections are required to be accounted for, and reducing possible connections reduces problem size.