Model

This section provides a summary of the model stre3am. A more comprehensive description can be found under the STREAM/docs/model_d/ directory and typing make in a terminal emulator^[1].

Model basic idea¶

Let us consider a single plant or asset, and let $x_{ij} \in \mathbb{R}^{\mathtt{nv}}$ represent a vector of measurable quantities at a particular period $i$, subperiod $j$, e.g.\ the fuel consumption, or the variable O & M cost, etc. Further, assume the ordered sets $\mathcal{P} = \{1, 2, \dots, \mathtt{n\_periods}\}$ and $\mathcal{P}_1 = \{1,2,\dots,\mathtt{n\_subperiods}\}$ representing the periods and subperiods. Then, assuming every quantity of the plant is related linearly to one another, the following system characterizes the asset,

where $A_{ij}$ is an nvar by (nv=nvar+$d$) matrix ($d>0$ degrees of freedom), and $b_{ij}$ is a nvar vector of coefficients. The equation above can be used to determine the state of the plant given an arbitrary value of the $d$ degrees of freedom (e.g. the active capacity).

Therefore, a plant in stre3am can have varying states according to the arbitrary values of the $d$ degrees of freedom over time. A change in the technological makeup of the plant has the potential to change the relationship between state and their degrees of freedom. This is reflected in Equation (1) through the adoption characteristic matrices/vectors $A_{ij}$ and $b_{ij}$ for each technology adoption in the plant. Therefore, a plant with several possible technological makeup (e.g.\ plant retrofits) has the following system,

where $\mathcal{K}$ represents the (generic) technology ordered set, which in the current implementation of stre3am can refer to retrofits or new plants. The key aspect of the problem behind stre3am is that for the plant at most one technological makeup must be true (e.g. selected) at a given time. This decision is different from the other measurable quantities insofar as its discrete nature. Let $y_{ijk} \in \{0,1\}$ be 1 if technology $k$ is active and 0 otherwise. This binary variable is used to present the decision of a particular technology at a particular time in the following form,

and,

where $\nu_{ijk}$ is the disaggregated state variable for technology $k$, and $x^{\text{lb}}$ and $x^{\text{ub}}$ are the lower- and upper-bound of the state vector. These equations link the value of $y_{ijk}$ to the state vector and the respective matrix and vector $A_{ijk}$ and $b_{ijk}$. An alternative view of this system can be obtained by introducing a Boolean variable $Y_{ijk} \in \{\text{True, False} \}$ analogous to $y_{ijk}$, and the following equation,

where $\underline{\vee}$ represents the exclusive OR (XOR) operator. The previous equation are known as disjunctive constraints, and this particular instance assumes a single index $k$ has $Y_{ijk}$ equal True, and all else set to False. Let us assume that $k=1$ corresponds to the incumbent technology, i.e. the technology given to the plant before the first time-slice. Then, the possible values the Boolean variables over the whole horizon can be restricted through the following statements:

• A transition to a non-incumbent technology is True once for the whole horizon.

  $$Y_{ijk} \implies Y_{i, j+1, k} \; \forall i \in \mathcal{P}, \forall j \in \mathcal{P}_1 \setminus \left\{|\mathcal{P}_1|\right\}, \forall k \in \mathcal{K} \setminus \left\{1\right\},$$

  (7)

and,

• A transition from incumbent is True once for the whole horizon.

and,

These logic propositions collectively written here as $\Omega(Y) = \text{True}$. These propositions can also be written in terms of binary variables and algebraic equations, viz.,

Which collectively are written here as:

The decisions expressed in the previous set of equations can either have binary ($\{0, 1\}$), or Boolean ($\{\text{True, False}\}$), as well as continuous. Similar methodology can be applied to the following decisions within stre3am,

Objective function¶

The model uses the minimization of the Net Present Value (NPV) as objective function. Nevertheless, users can define custom objective functions in the same way as any JuMP model. The NPV in stre3am has several components, most noticeable the discount factor $\delta_{ij}$, which can be calculated from the assumed interest rate and year. The objective function considers the following items for every period and subperiod tuple ($(i,j)\in\mathcal{P}\times\mathcal{P}_1$),

1. Capital cost installments (CI)
2. Fixed operations and maintenance (fO&M)
3. Variable operations and maintenance (vO&M)
4. Fuel cost (FC)
5. Electricity cost (EC)
6. Feedstock cost (FdC)
7. CCS cost (CCSC)
8. Plant retirement (PR)
9. Last period loan factor (LLF).

The terms in the previous equation are related to the rest of variables from the model linearly.

Constraints¶

As mentioned in the model basis section, the model can be expressed either with disjunctions, logic statements, and Boolean (and continuous) variables^[2], or with algebraic equations and binary variables, i.e. MILP.

Either case, the decisions mentioned in Table 1, have their own set of Boolean/binary variables. Let $Y_{ij}^e, Y_{ij}^r, Y_{ij}^o, Y_{ij}^n \in \{\text{True, False}\}$^[3] represent the Boolean variables for expansion, retrofit, online, and new plant. Thus, the constraints of the model can be laid out as follows,

Expansion

Plant retrofit

Plant retirement

New plant^[4]

Logic propositions

The previous equations describe the changes on the plant through technological, capacity, and operational decisions. It is also possible to have continuous variables constrained on temporal basis, for example demand at every period and subperiod pair. This can be represented through the following equation,

^[5]

for some matrix and right-hand-side $B_{ij}$ and $p_ij$.

Finally, it is noted that Equations (17) through (21) can be represented using binary variables and algebraic equations as follows,

So far these equations describe the model for a single plant/asset. Typically, one might find that industries have several plants/assets, each with their unique characteristics, i.e. A_{ij}, b_{ij}, costs, etc. Let $\mathcal{L}\in\left\{1, 2, 3, ..., \mathtt{n\_loc}\right\}$ represent the set of locations. The multi-plant location problem is a natural extension of the single location problem shown thus far.

Model building actions¶

Given the input sets and data for a particular instance, building the model is done through a series of function calls that first build an unlinked model for each period, subperiod, and location. This is done through the following function,

m = createBlockMod(s.P, s.L, p, s)

Through this call the model m is generated. This call however, does not create either inter-period or inter-location constraints.

The linking with inter-period and inter-location is done as follows:

attachPeriodBlock(m, p, s)
attachLocationBlock(m, p, s)

This creates a linked model, which can be completed with the objective function (NPV minimization) creation:

attachFullObjectiveBlock(m, p, s)

The resulting m is an instance of a JuMP object with objective and constraints

Concluding remarks¶

It is encouraged to refer to the stre3am manuscript where the complete model is laid out. This page is a brief summary of some core aspects of the model.