# Help: THERMOS Supply Model Formalism and Calculations

THERMOS includes a supply optimisation model. This part of THERMOS tries to determine a set of choices about how to build an energy center which can meet a certain heat demand at least cost.

It is coupled to the network optimisation model through the network model's choice of which buildings to connect.

## 1 Decision variables

The supply optimiser's job is to make a choice about how to meet a given demand for heat, at minimum cost.

The primary decisions available to the supply optimiser are:

• What plant to purchase?

This is effectively equipment that produces heat at a certain unit cost. In the user interface a 'menu' of supply options can be defined; for each of these, the optimiser can choose what capacity to purchase.

• What heat storage to purchase?

This is equipment which can be charged up with heat at one time during a day, and discharged later, with some loss. Again, as a user you can define a menu of storage options, from which the optimiser can choose what to purchase.

Along with these decisions the supply optimiser must make operational decisions

• In each interval of time, how much heat to generate from each item of plant, and
• How much heat to store into each item of storage

## 2 Constructing a heat demand profile

Two outputs from THERMOS' network model are:

1. A list of buildings that are connected to the heat network.

Each building has a peak demand (kW" and an annual demand (kWh/yr).

2. A peak demand and annual demand for the network's supply point

These demands are not detailed enough for supply-side modelling, which depends on prices that vary on an hourly or half-hourly basis across the year. More detail of this is given below, but for now the important point is that the supply model's decisions are over a large number of time intervals representing a typical operating year.

For example, if modelling the year as 5 representative "day types", each having 48 half-hourly intervals, there are 240 half-houly heat demand intervals that need to be modelled. However, we only have two facts (peak and average demand) about heat at the supply point, and at each building.

Because of this, THERMOS needs a way to convert these summary statistics into a load profile for the supply location which says how much heat is required in each modelled interval.

This is done in two stages:

1. First, for each building, we contruct a load profile for the building by deforming a profile shape for that building so that it has the right peak demand and annual demand.

The profile shape itself is a user input describing a relative heat demand in interval.

A profile shape like this can be adjusted easily to have a given peak value - we just have to divide each interval's value by the maximum in any interval (effectively normalising the profile shape), and then multiply by the desired peak. However, this operation will probably not give the desired annual demand; in truth there are an unlimited number of possible profiles that fit a given annual demand and peak demand, but we need some way to construct one that looks realistic.

In THERMOS this is done by compressing or stretching the normalised profile shape, so that it has the desired annual demand once it's multiplied by the desired peak. The compression or stretch is the same for each interval, and is determined by a "flattening" parameter α: in a given interval, if the normalized value is $$x$$, the deformed value is $$x^α$$.

Since the values in the normalized shape range from 0 to 1, no choice of α can move any of the intervals outside that range. An α of 1 makes no change to the shape; an α of 2 makes the shape "peakier" (since all of the sub-peak intervals are squared, and being less than 1 this reduces them); an α of 0.5 makes the shape flatter (each interval is square-rooted). In the limit, an α of infinity makes every point except the peak have value 0, and an alpha of 1 makes every point have value 1.

The required value of α is determined numerically, allowing the construction of a per-building load profile which has a peak & average demand that reflect the values seen by the network model.

2. Next, for the supply, we sum all the load profiles for the buildings, and then repeat the profile deformation process so that the resulting peak and average demand equal the values predicted by the network model for the supply point.

This is not just the sum of all the building profiles because of the load diversity effect, which flattens the peak a bit1.

This final reshaped load profile gives a heat demand on the supply for each interval, in each representative day being modelled.

## 3 Supply cost

Now we have a demand profile we can think about the cost of producing heat to meet that profile. The costs considered are:

1. Capital and operating costs for plant & storage
2. Fuel and emissions costs for plant
3. Grid export for CHP plant
4. Curtailment costs for any unmet demand

### 3.1 Capital and operating costs

Plant and storage definitions each have capital cost terms, broken down as a fixed cost, cost per kW capacity, and cost per kWh.

The capital cost for plant is

$\text{fixed cost} + [ \text{cost per kW} \times \max_{t \in T} \text{output}_t ] + [ \text{cost per kWh} \times \sum_{t \in T} \text{output}_t \times \text{weight}_t ]$

Where $$\text{weight}_t$$ is the number of kWh per year that interval $$t$$ would contribute at an output of 1 kW; for example, an interval representing half an hour, in a day that occurs 50 days per year, would have a weight of 25. $$T$$ is the set of all the intervals modelled in the profile.

The capital cost for storage is similar, except that the cost per kWh is in terms of kWh storage capacity rather than kWh output:

$\text{fixed cost} + [ \text{cost per kW} \times \max_{t \in T} \text{flow}_t ] + [ \text{cost per kWh} \times \max_{t \in T} \text{charge}_t ]$

Where $$\text{flow}_t$$ is the charging rate during interval $$t$$ (so the maximum of this is the capacity of the connection between the store and the plant), and $$\text{charge}_t$$ is the number of kWh of heat that are in the store at time $$t$$ (so the maximum of this is the capacity of store required).

Both plant and storage have a given lifetime - if the plant or store must be replaced during the accounting period of the optimisation, the capital costs are paid again each time the lifetime is reached.

Operating costs are only calculated for the plant, and follow the same formula as the plant capital cost formula; the resulting cost is incurred every year the plant operates.

### 3.2 Fuel and emissions costs

Fuel costs are incurred by plant when it operates; for each plant, in every time interval $$t$$ there is a fuel price per kWh entered by the user. Each plant also has a heat production efficiency. The annual fuel cost for the plant is then given by

$\sum_{t\in T} \frac{\text{output}_t}{\text{efficiency}} \times \text{price}_t \times \text{weight}_t$

Emissions costs are similar, except with an additional time-varying emissions factor for the fuel:

$\sum_{t\in T, e \in E} \frac{\text{output}_t}{\text{efficiency}} \times \text{factor}_{t,e} \times \text{weight}_t \times \text{unit cost}_e$

where $$e$$ is an emissions type (CO2, NOX, …) from the set of modelled emissions types $$E$$

### 3.3 Grid export

Some plant is marked as CHP, and has a power efficiency as well as a heat efficiency. For these types of plant, an additional revenue (or potentially cost) is incurred when the plant runs, as it produces power to sell to the grid.

The value for this is analogous to the fuel cost, but with a different price and efficiency.

An important consideration for both CHP and electrically powered plant is substation headroom. In THERMOS, substation capacity is modelled as a hard constraint on how much power the substation can deliver or accept in any time interval.

Each substation also has a user-input timeseries of pre-existing demand, which determines the headroom (spare capacity); in every modelled interval the sum of pre-existing demand and new demand net of any CHP generation must not exceed the substation's capacity.

### 3.5 Curtailment

Curtailment gives the optimiser the possibility to undersupply heat in a given interval. It is included mostly so that the problem cannot become infeasible. Each kWh of heat un-supplied due to curtailment incurs a very high cost, so the optimiser should only choose to undersupply if the maximum possible plant capacity is insufficient, or if the cost of producing heat is exceedingly large.

## 4 Formal description

The problem is defined over the following sets and parameters. All costs are adjusted for replacement lifetime, accounting period and discount rate before doing any optimisation, so operating and capital costs are combined, and fuel prices and emissions costs are combined.

Time intervals
$$T$$, usually indexed by $$t$$
Weight
$$w_t$$, a parameter indicating how many hours per year interval $$t$$ stands for
Heat demand
$$h_t$$, the heat demand in kW in interval $$t$$
Fuel cost
$$Price_{t,p}$$ the cost per kWh of fuel for plant $$p$$ in time $$t$$
$$Load_{t,s}$$, the existing demand on substation $$s$$ in time $$t$$ in kW
Grid offer
$$Grid_{t}$$, the price offered by the grid per kWh of electricity in time $$t$$
Plant
$$P$$, usually indexed by $$p$$
Capacity
$$MaxCap_p$$, the maximum output in kW from a plant of type $$p$$
Heat Efficiency
$$eH_{p}$$, the rate of heat production per fuel for plant $$p$$
Power Efficiency
$$eP_{p}$$, the rate of electricity production per fuel for plant $$p$$, if it is a CHP
Substation
$$Sub_{p}$$, one of the substations in $$N$$, if the plant is connected to a substation we are interested in
Cost
$$CFix_p$$, $$CCap_p$$, $$COut_p$$ as fixed, per-kw, and per-kwh capital and operating costs for $$p$$.
Storage
$$S$$, usually indexed by $$s$$
Capacity
$$MaxCapF_s$$, the maximum connection size in kW, and $$MaxCapS_s$$ the maximum storage size in kWh
Cycle efficiency
$$e_{s}$$, the proportion of heat output per unit of heat stored.
Cost
$$CFix_s$$, $$CCapF_s$$, $$CCapS_s$$ as fixed, per-kw, and per-kwh capital costs for $$s$$.
Substation
$$N$$, indexed by $$n$$
Capacity
$$Cap_n$$, the substation forward capacity in kW, and $$R_n$$, the ratio of reverse capacity to forward
$$L_{n, t}$$, the prior load on the substation in kW in time $$t$$

### 4.1 Objective

The decisions to be made are then

• $$Purchase_p$$, whether to buy plant $$p$$ (binary)
• $$Cap_p$$, the capacity of that plant to buy
• $$Purchase_s$$, whether to buy storage $$s$$ (binary)
• $$CapF_s$$, the flow capacity to buy for $$s$$
• $$CapS_s$$, the storage capacity to buy for $$s$$
• $$Output_{p, t}$$, the heat output from plant $$p$$ in interval $$t$$
• $$FlowIn_{s, t}$$ and $$FlowOut_{s, t}$$ the heat flow in and out of store $$s$$ in time $$t$$
• $$Curtailment_{t}$$, the curtailment in time $$t$$.

The objective is to minimize

$TotalCost = PlantCost + StoreCost + FuelCost + CurtailmentCost - GridRevenue$

Where

$PlantCost = \sum_p Purchase_p \times CFix_p + Cap_p \times CCap_p + (\sum_t w_t \times COut_p \times Output_{p, t})$

and

$StoreCost = \sum_s Purchase_s \times CFix_s + CapF_s \times CCapF_s + CapS_s \times CCapS_s$

and

$FuelCost = \sum_{t, p} w_t \times Price_{t, p} \times Output_{p, t} / eH_{p}$

and

$CurtailmentCost = \sum_t w_t \times Curtailment_t \times C$

where $$C$$ is a big number

and

$GridRevenue = \sum_{t, p} w_t \times Grid_{t} \times eP_{p} \times Output_{p, t} / eH_{p}$

summing over CHP plants only

### 4.2 Constraints

To make the optimiser pay for what it uses, the objective is solved subject to:

Satisfied demand

$$\forall t : \sum_p Output_{p, t} + \sum_s (e_s \times FlowOut_{s, t} - FlowIn_{s, t}) + Curtailment_t \geq h_t$$

Ensures that in each time slice, at least as much heat is produced as is needed. Storage cycle efficiency is applied here, through the multiplication with $$e_s$$.

This can be made an equality, if overproduction of heat to sell power is not allowed.

Indicators

$$\forall p : Cap_p \leq Purchase_p \times MaxCap_p$$ and $$\forall s : CapF_s \leq Purchase_s \times MaxCapF_s$$

These constraints ensure that the fixed costs are paid if any storage or plant capacity is used.

Plant capacity sufficient
$$\forall p, t: Output_{p, t} \leq Cap_p$$
Store flow sufficient
$$\forall s, t: FlowIn_{s, t} \leq CapF_s$$, and $$\forall s, t: FlowOut_{s, t} \times e_s \leq CapF_s$$
Store size sufficient
$$\forall s, t: Charge_{s, t} \leq CapC_s$$
Store charge balance

$$\forall s, t: Charge_{s, t} = Charge_{s, t-1} + H_t \times (FlowIn_{s, t-1} - FlowOut{s, t-1})$$

Where $$H_t$$ is the number of hours time-slice $$t$$ represents in the day, and $$t-1$$ wraps around within the day at the start (so the charge state at the end of the day comes back to the start of the day).

Because of this, charge within the store cannot be passed between representative day types, or stored for more than a single day.

$$\forall t, n : \sum_{p | Sub_p = n} -R_n \times Cap_n \leq L_{n, t} - eP_p \times Output_n/eH_p \leq Cap_n$$

Where $$eP_p = -1$$ if the plant is not a CHP but is electrically powered.

## 5 Future work

It would be useful to support two further features

1. Binary output variables; some plant either runs or doesn't
2. Limited switch-ons; some plant should not be turned on and off too frequently

## Footnotes:

1

Perhaps the modelling for this should be improved - for example, load diversity might be better represented by some smoothing kernel.