# Model formulation¶

This section details the mathematical formulation of the different components. For each component, a link to the actual implementing function in the Calliope code is given.

## Objective function (cost minimization)¶

The default objective function minimizes cost:

$min: z = \sum_y (weight(y) \times \sum_x cost(y, x, k=k_{m}))$

where $$k_{m}$$ is the monetary cost class.

Alternative objective functions can be used by setting the objective in the model configuration (see Model-wide settings).

weight(y) is 1 by default, but can be adjusted to change the relative weighting of costs of different technologies in the objective, by setting weight on any technology (see Technology).

## Basic constraints¶

### Node resource¶

Defines the following variables:

• rs: resource to/from storage (+ production, - consumption)
• r_area: resource collector area
• rbs: secondary resource to storage (+ production)

It also defines the constraint c_rs. This constraint defines the available resource for a node, $$r_{avail}$$:

$r_{avail}(y, x, t) = r_{scale}(y, x) \times r_{area}(y, x) \times r_{eff}(y)$

The c_rs constraint also decides how the resource and storage are linked.

If the option constraints.force_r is set to true, then

$r_{s}(y, x, t) = r_{avail}(y, x, t)$

If that option is not set, and the technology inherits from the supply or unmet_demand base technologies,

$r_{s}(y, x, t) \leq r_{avail}(y, x, t)$

Finally, if it inherits from the demand technology,

$r_{s}(y, x, t) \geq r_{avail}(y, x, t)$

Note

For the case of storage technologies, $$r{s}$$ is forced to $$0$$ for internal reasons, while for transmission technologies, it is unconstrained. This is irrelevant when defining models and defining a resource for either storage or transmission technologies has no effect.

### Node energy balance¶

Defines the following variables:

• s: storage level
• es_prod: energy from storage to carrier
• es_con: energy from carrier to storage

It also defines three constraints, which are discussed in turn:

• c_s_balance_pc: energy balance for supply, demand, and storage technologies
• c_s_balance_transmission: energy balance for transmission technologies
• c_s_balance_conversion: energy balance for conversion technologies

#### Supply/demand/storage balance¶

A node that allows storage and either supply or demand is the most complex case, with the balancing equation

$s(y, x, t) = s_{minusone} + r_{s}(y, x, t) + r_{bs}(y, x, t) - e_{prod} - e_{con}$

$$e_{prod}$$ is defined as $$es_{prod}(c, y, x, t) \times e_{eff}(y, x, t)$$.

$$e_{con}$$ is defined as $$\frac{es_{con}(c, y, x, t)}{e_{eff}(y, x, t)}$$, or as $$0$$ if $$e_{eff}(y, x, t)$$ is $$0$$.

$$r_{bs}(y, x, t)$$ is the secondary resource and is always set to zero unless the technology explicitly defines a secondary resource.

$$s(y, x, t)$$ is the storage level at time $$t$$.

$$s_{minusone}$$ describes the state of storage at the previous timestep. $$s_{minusone} = s_{init}(y, x)$$ at time $$t=0$$. Else,

$s_{minusone} = (1 - s_{loss}) \times timeres(t-1) \times s(y, x, t-1)$

Note

In operation mode, s_init is carried over from the previous optimization period.

If no storage is allowed, the balancing equation simplifies to

$r_{s}(y, x, t) + r_{bs}(y, x, t) = e_{prod} + e_{con}$

#### Transmission balance¶

Transmission technologies are internally expanded into two technologies per transmission link, of the form technology_name:destination.

For example, if the technology hvdc is defined and connects region_1 to region_2, the framework will internally create a technology called hvdc:region_2 which exists in region_1 to connect it to region_2, and a technology called hvdc:region_1 which exists in region_2 to connect it to region_1.

The balancing for transmission technologies is given by

$es_{prod}(c, y, x, t) = -1 \times es_{con}(c, y_{remote}, x_{remote}, t) \times e_{eff}(y, x, t) \times e_{eff,perdistance}(y, x)$

Here, $$x_{remote}, y_{remote}$$ are x and y at the remote end of the transmission technology. For example, for (y, x) = ('hvdc:region_2', 'region_1'), the remotes would be ('hvdc:region_1', 'region_2').

$$es_{prod}(c, y, x, t)$$ for c='power', y='hvdc:region_2', x='region_1' would be the import of power from region_2 to region_1, via a hvdc connection, at time t.

This also shows that transmission technologies can have both a static or time-dependent efficiency (line loss), $$e_{eff}(y, x, t)$$, and a distance-dependent efficiency, $$e_{eff,perdistance}(y, x)$$.

For more detail on distance-dependent configuration see Model configuration.

#### Conversion balance¶

The conversion balance is given by

$es_{prod}(c_{prod}, y, x, t) = -1 \times es_{con}(c_{source}, y, x, t) \times e_{eff}(y, x, t)$

The principle is similar to that of the transmission balance. The production of carrier $$c_{prod}$$ (the carrier option set for the conversion technology) is driven by the consumption of carrier $$c_{source}$$ (the source_carrier option set for the conversion technology).

### Node build constraints¶

Defines the following variables:

• s_cap: installed storage capacity
• r_cap: installed resource to/from storage conversion capacity
• e_cap: installed storage to/from grid conversion capacity (gross)
• e_cap_net: installed storage to/from grid conversion capacity (net)
• rb_cap: installed secondary resource conversion capacity

Built capacity is managed by six constraints.

c_s_cap constrains the built storage capacity by $$s_{cap}(y, x) \leq s_{cap,max}(y, xi)$$. If y.constraints.use_s_time is true at location x, then y.constraints.s_time.max and y.constraints.e_cap.max are used to to compute s_cap.max at reference efficiency. If y.constraints.s_cap.equals is set for location x or the model is running in operational mode, the inequality in the equation above is turned into an equality constraint.

c_r_cap constrains the built resource conversion capacity by $$r_{cap}(y, x) \leq r_{cap,max}(y, x)$$. If the model is running in operational mode, the inequality in the equation above is turned into an equality constraint.

c_r_area constrains the resource conversion area by $$r_{area}(y, x) \leq r_{area,max}(y, x)$$. By default, y.constraints.r_area.max is set to false, and in that case, $$r_{area}(y, x)$$ is forced to $$1.0$$. If the model is running in operational mode, the inequality in the equation above is turned into an equality constraint. Finally, if y.constraints.r_area_per_e_cap is given, then the equation $$r_{area}(y, x) = e_{cap}(y, x) * r\_area\_per\_cap$$ applies instead.

c_e_cap constrains the carrier conversion capacity. If a technology y is not allowed at a location x, $$e_{cap}(y, x) = 0$$ is forced. Else, $$e_{cap}(y, x) \leq e_{cap,max}(y, x) \times e\_cap\_scale$$ applies. y.constraints.e_cap_scale defaults to 1.0 but can be set on a per-technology, per-location basis if necessary. Finally, if y.constraints.e_cap.equals is set for location x or the model is running in operational mode, the inequality in the equation above is turned into an equality constraint.

The c_e_cap_gross_net constraint is relevant only if y.constraints.c_eff is set to anything other than 1.0 (the default). In that case, $$e_{cap}(y, x) \times c_{eff} = e_{cap,net}(y, x)$$ computes the net installed carrier conversion capacity.

The final constraint, c_rb_cap, manages the secondary resource conversion capacity by $$rb_{cap}(y, x) \leq rb_{cap,max}(y, x)$$. If y.constraints.rb_cap.equals is set for location x or the model is running in operational mode, the inequality in the equation above is turned into an equality constraint. There is an additional relevant option, y.constraints.rb_cap_follows, which can be overridden on a per-location basis. It can be set either to r_cap or e_cap, and if set, sets c_rb_cap to track one of these, ie, $$rb_{cap,max} = r_{cap}(y, x)$$ (analogously for e_cap), and also turns the constraint into an equality constraint.

### Node operational constraints¶

Provided by: calliope.constraints.base.node_constraints_operational()

This component ensures that nodes remain within their operational limits, by constraining rs, es, s, and rbs.

$$r_{s}(y, x, t)$$ is constrained to remain within $$r_{cap}(y, x)$$, with the two constraints c_rs_max_upper and c_rs_max_lower:

$r_{s}(y, x, t) \leq timeres(t) \times r_{cap}(y, x)$$r_{s}(y, x, t) \geq -1 \times timeres(t) \times r_{cap}(y, x)$

$$e_{s}(c, y, x, t)$$ is constrained by three constraints, c_es_prod_max, c_es_prod_min, and c_es_con_max:

$e_{s,prod}(c, y, x, y) \leq timeres(t) \times e_{cap}(y, x)$

if c is the carrier of y, else $$e_{s,prod}(c, y, x, y) = 0$$.

If e_cap_min_use is defined, the minimum output is constrained by

$e_{s,prod}(c, y, x, y) \geq timeres(t) \times e_{cap}(y, x) \times e_{cap,minuse}$

For technologies where y.constraints.e_con is true (it defaults to false), and for conversion technologies,

$e_{s,con}(c, y, x, y) \geq -1 \times timeres(t) \times e_{cap}(y, x)$

and $$e_{s,con}(c, y, x, y) = 0$$ otherwise.

The constraint c_s_max ensures that storage cannot exceed its maximum size by

$s(y, x, t) \leq s_{cap}(y, x)$

And finally, c_rbs_max constrains the secondary resource by

$rb_{s}(y, x, t) \leq timeres(t) \times rb_{cap}(y, x)$

There is an additional check if y.constraints.rb_startup_only is true. In this case, $$rb_{s}(y, x, t) = 0$$ unless the current timestep is still within the startup time set in the startup_time_bounds model-wide setting. This can be useful to prevent undesired edge effects from occurring in the model.

### Transmission constraints¶

This component provides a single constraint, c_transmission_capacity, which forces $$e_{cap}$$ to be symmetric for transmission nodes. For example, for for a given transmission line between $$x_1$$ and $$x_2$$, using the technology hvdc:

$e_{cap}(hvdc:x_2, x_1) = e_{cap}(hvdc:x_1, x_2)$

### Node parasitics¶

Defines the following variables:

• ec_prod: storage to carrier after parasitics (positive, production)
• ec_con: carrier to storage after parasitics (negative, consumption)

There are two constraints, c_ec_prod and c_ec_con, which constrain ec by

$ec_{prod}(c, y, x, t) = es_{prod}(c, y, x, t) \times c_{eff}(y, x)$$ec_{con}(c, y, x, t) = \frac{es_{con}(c, y, x, t)}{c_{eff}(y, x)}$

For conversion and transmission technologies, the second equation reads $$ec_{con}(c, y, x, t) = es_{con}(c, y, x, t)$$ so that the internal losses are applied only once.

The two variables ec_prod and ec_con are only defined in the model for technologies where c_eff is not 1.0.

Note

When reading the model solution, Calliope automatically manages the es and ec variables. In the solution, every technology has an ec variable, which is simply set to es wherever it was not defined, to make the solution consistent.

### Node costs¶

Provided by: calliope.constraints.base.node_costs()

Defines the following variables:

• cost: total costs
• cost_con: construction costs
• cost_op_fixed: fixed operation costs
• cost_op_var: variable operation costs
• cost_op_fuel: primary resource fuel costs
• cost_op_rb: secondary resource fuel costs

These equations compute costs per node.

The depreciation rate for each cost class k is calculated as

$d(y, k) = \frac{1}{plant\_life(y)}$

if the interest rate $$i$$ is $$0$$, else

$d(y, k) = \frac{i \times (1 + i(y, k))^{plant\_life(k)}}{(1 + i(y, k))^{plant\_life(k)} - 1}$

Costs are split into construction and operational and maintenance (O&M) costs. The total costs are computed in c_cost by

$cost(y, x, k) = cost_{con}(y, x, k) + cost_{op,fixed}(y, x, k) + cost_{op,var}(y, x, k) + cost_{op,fuel}(y, x, k) + cost_{op,rb}(y, x, k)$

The construction costs are computed in c_cost_con by

$\begin{split}cost_{con}(y, x, k) &= d(y, k) \times \frac{\sum\limits_t timeres(t)}{8760} \\ & \times (cost_{s\_cap}(y, k) \times s_{cap}(y, x) \\ & + cost_{r\_cap}(y, k) \times r_{cap}(y, x) \\ & + cost_{r\_area}(y, k) \times r_{area}(y, x) \\ & + cost_{e\_cap}(y, k) \times e_{cap}(y, x)) \\ & + cost_{rb\_cap}(y, k) \times rb_{cap}(y, x))\end{split}$

The costs are as defined in the model definition, e.g. e.g. $$cost_{r\_cap}(y, k)$$ corresponds to y.costs.k.r_cap.

For transmission technologies, $$cost_{e\_cap}(y, k)$$ is computed differently, to include the per-distance costs:

$cost_{e\_cap,transmission}(y, k) = \frac{cost_{e\_cap}(y, k) + cost_{e\_cap,perdistance}(y, k)}{2}$

This implies that for transmission technologies, the cost of construction is split equally across the two locations connected by the technology.

The O&M costs are computed in four separate constraints, cost_op_fixed, cost_op_var, cost_op_fuel, and cost_op_rb, by

$\begin{split}cost_{op,fixed}(y, x, k) &= cost_{om\_frac}(y, k) \times cost_{con}(y, x, k) \\ & + cost_{om\_fixed}(y, k) \times e_{cap}(y, x) \\ & \times \frac{\sum\limits_t timeres(t)}{8760}\end{split}$
$cost_{op,var}(y, x, k) = cost_{om\_var}(y, k) \times \sum_t e_{prod}(c, y, x, t)$$cost_{op,fuel}(y, x, k) = \frac{cost_{om\_fuel}(y, k) \times \sum_t r_{s}(y, x, t)}{r_{eff}(y, x)}$$cost_{op,rb}(y, x, k) = \frac{cost_{om\_rb}(y, k) \times \sum_t r_{bs}(y, x, t)}{rb_{eff}(y, x)}$

### Model balancing constraints¶

Provided by: calliope.constraints.base.model_constraints()

Model-wide balancing constraints are constructed for nodes that have children. They differentiate between:

• c = power
• All other c

In the first case, the following balancing equation applies:

$\sum_{y, x \in X_{i}} ec_{prod}(c=c_{p}, y, x, t) + \sum_{y, x \in X_{i}} ec_{con}(c=c_{p}, y, x, t) = 0 \qquad\forall i, t$

$$i$$ are the level 0 locations, and $$X_{i}$$ is the set of level 1 locations ($$x$$) within the given level 0 location, together with that location itself. $$c$$ is the carrier, and $$c_{p}$$ the carrier for power.

For c other than power, the balancing equation is as above, but with a $$\geq$$ inequality, and the corresponding change to $$c$$.

Note

The actual balancing constraint is implemented such that es and ec are used in the sum as appropriate for each technology.

## Planning constraints¶

These constraints are loaded automatically, but only when running in planning mode.

### System margin¶

This is a simplified capacity margin constraint, requiring the capacity to supply a given carrier in the time step with the highest demand for that carrier to be above the demand in that timestep by at least the given fraction:

$\sum_y \sum_x es_{prod}(c, y, x, t_{max,c}) \times (1 + m_{c}) \leq timeres(t) \times \sum_{y_{c}} \sum_x (e_{cap}(y, x) / e_{eff,ref}(y, x))$

where $$y_{c}$$ is the subset of y that delivers the carrier c and $$m_{c}$$ is the system margin for that carrier.

For each carrier (with the name carrier_name), Calliope attempts to read the model-wide option system_margin.carrier_name, only applying this constraint if a setting exists.

## Optional constraints¶

Optional constraints are included with Calliope but not loaded by default (see the configuration section for instructions on how to load them in a model).

These optional constraints can be used both in planning and operational modes.

### Ramping¶

Constrains the rate at which plants can adjust their output, for technologies that define constraints.e_ramping:

$diff = \frac{es_{prod}(c, y, x, t) + es_{con}(c, y, x, t)}{timeres(t)} - \frac{es_{prod}(c, y, x, t-1) + es_{con}(c, y, x, t-1)}{timeres(t-1)}$$max\_ramping\_rate = e_{ramping} \times e_{cap}(y, x)$$diff \leq max\_ramping\_rate$$diff \geq -1 \times max\_ramping\_rate$

### Group fractions¶

This component provides the ability to constrain groups of technologies to provide a certain fraction of total output, a certain fraction of total capacity, or a certain fraction of peak power demand. See Parents and groups in the configuration section for further details on how to set up groups of technologies.

The settings for the group fraction constraints are read from the model-wide configuration, in a group_fraction setting, as follows:

group_fraction:
capacity:
renewables: ['>=', 0.8]


This is a minimal example that forces at least 80% of the installed capacity to be renewables. To activate the output group constraint, the output setting underneath group_fraction can be set in the same way, or demand_power_peak to activate the fraction of peak power demand group constraint.

For the above example, the c_group_fraction_capacity constraint sets up an equation of the form

$\sum_{y^*} \sum_x e_{cap}(y, x) \geq fraction \times \sum_y \sum_x e_{cap}(y, x)$

Here, $$y^*$$ is the subset of $$y$$ given by the specified group, in this example, renewables. $$fraction$$ is the fraction specified, in this example, $$0.8$$. The relation between the right-hand side and the left-hand side, $$\geq$$, is determined by the setting given, >=, which can be ==, <=, or >=.

If more than one group is listed under capacity, several analogous constraints are set up.

Similarly, c_group_fraction_output sets up constraints in the form of

$\sum_{y^*} \sum_x \sum_t es_{prod}(c, y, x, t) \geq fraction \times \sum_y \sum_x \sum_t es_{prod}(c, y, x, t)$

Finally, c_group_fraction_demand_power_peak sets up constraints in the form of

$\sum_{y^*} \sum_x e_{cap}(y, x) \geq fraction \times (-1 - m_{c}) \times peak$$peak = \frac{\sum_x r(y_d, x, t_{peak}) \times r_{scale}(y_d, x)}{timeres(t_{peak})}$

This assumes the existence of a technology, demand_power, which defines a demand (negative resource). $$y_d$$ is demand_power. $$m_{c}$$ is the capacity margin defined for the carrier c in the model-wide settings (see System margin). $$t_{peak}$$ is the timestep where $$r(y_d, x, t)$$ is maximal.

Whether any of these equations are equalities, greater-or-equal-than inequalities, or lesser-or-equal-than inequalities, is determined by whether >=, <=, or == is given in their respective settings.

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