BenOpt-HEAT-documentation.md

+# BenOpt-Heat model
+
+The <font color='blue'>**blue lines**</font> identify the decision variables, and <font color='red'>**red sentences**</font> are our comments.
+
+## 1. Objective function
+
+In the following paragraphs, blue items specify variables and black items are parameters.
+* <font color='blue'>$`\pi^{Bio}_{t, i, j, b} = `$ production of output from solid biomass technologies $`i`$ at time $`t`$ for sub-sector $`j`$ and biomass product $`b`$ $`(GJ)`$</font>
+* <font color='blue'>$`\pi^{Gas}_{t, i, j, b} = `$ production of output from natural gas/biogas/coal technologies $`i`$ at time $`t`$ for sub-sector $`j`$ and biomass product $`b`$ $`(GJ)`$</font>
+* <font color='blue'>$`\pi^{nonBio}_{t, i, j} = `$ production of output from non biomass, renewable technologies $`i`$ at time $`t`$ and for sub-sector $`j`$ $`(GJ)`$</font>
+* <font color='blue'>$`\pi^{Beh}_{t, i, j, c} = `$ production of output within the consumer clusters $`c`$ of technology $`i`$ at time $`t`$ and for sub-sector $`j`$ $`(GJ)`$</font>
+* <font color='blue'>$`n^{cap}_{t, i, m, j} = `$ installed/ existing number of technologies at time $`t`$ for technology $`i`$ for technology module $`m`$ and for sub-sector $`j`$</font>
+* <font color='blue'>$`n^{prodBeh}_{t, i, j, c} = `$ endogenously installed number of technologies $`i`$ at time $`t`$ for sub-sector $`j`$ in the consumer cluster $`c`$</font>
+* $`vc_{t, i, j, b} = `$ variable cost of technology $`i`$ at time $`t`$ for sub-sector $`j`$ and biomass product $`b`$ - OPEX including feedstock inputs ($`€/GJ`$) 
+* $`inv_{t, i, m, j} = `$ investment cost at time $`t`$ of technology $`i`$ and technology module $`m`$  for sub-sector $`j`$ ($`€`$) 
+* $`vc^{Beh}_{t, i, j, c} = `$ intangible variable cost of technology $`i`$ at time $`t`$ for sub-sector $`j`$ in the consumer cluster $`c`$ ($`€/GJ`$) 
+* $`inv^{Beh}_{t, i, j, c} = `$ intangible investment cost at time $`t`$ of technology $`i`$ for sub-sector $`j`$ in the consumer cluster $`c`$ ($`€`$) 
+
+
+$`min\, cost = \overbrace{\sum_{t, i, j}(vc_{t, i, j, 1} \times pi^{nonBio}_{t, i, j}) + \sum_{t, i, j, b}(vc_{t, i, j, b} \times pi^{Bio}_{t, i, j, b}) + \sum_{t, i, j, b}(vc_{t, i, j, b} \times pi^{Gas}_{t, i, j, b})}^\text{production costs} + \underbrace{\sum_{t, i, m, j}(inv_{t, i, m, j} \times n^{cap}_{t, i, m, j})}_\text{investment cost} + \overbrace{\sum_{t, i, j, c}(vc^{Beh}_{t, i, j, c} \times pi^{Beh}_{t, i, j, c}) + \sum_{t, i, j, c}(inv^{Beh}_{t, i, j, c} \times n^{prodBeh}_{t, i, j, c})}^\text{intangible cost}`$
+
+
+
+## 2. Constraints
+
+### 2.1 Demand:
+
+Heat demand per sub-sector needs to be fullfilled
+
+* <font color='blue'>$`\pi_{t, i, j} = `$ production of output from technologies $`i`$ at time $`t`$ for sub-sector $`j`$ $`(GJ)`$</font>
+* <font color='blue'>$`n^{prod}_{t, i, j} = `$ number of technologies $`i`$ producing heat at time $`t`$ for sub-sector $`j`$</font>
+* $`d_{t, j} = `$ heat demand per sub-sector $`(GJ)`$
+* $`d^{cap}_{t, j} = `$ heat demand per unit/house/heating system $`(GJ)`$
+
+
+$`d_{t, j} = \sum_{i}(\pi_{t, i, j}) \; \forall (t,j) \in (T,J)`$
+
+$`d_{t, j} = \sum_{i}(n^{prod}_{t, i, j} \times d^{cap}_{t, j}) \; \forall (t,j) \in (T,J)`$
+
+
+### 2.2 Capacity: 
+
+The following formulas regulate the capacity expansion, overcapacity, capacity producing heat, decomission of the initial stock and the decomission of expanded stock.
+
+* <font color='blue'>$`n^{cap}_{t, i, m, j} = `$ installed/ existing number of technologies $`i`$ at time $`t`$ for technology module $`m`$ and for sub-sector $`j`$</font>
+* <font color='blue'>$`n^{prod}_{t, i, j} = `$ number of technologies $`i`$ producing heat at time $`t`$ for sub-sector $`j`$</font>
+* <font color='blue'>$`n^{prodBeh}_{t, i, j, c} = `$ endogenously installed number of technologies $`i`$ at time $`t`$ for sub-sector $`j`$ in the consumer cluster $`c`$</font>
+* <font color='blue'>$`n^{cap1}_{t, i, m, j} = `$ number of existing units used for production at time $`t`$ for technology $`i`$ for technology module $`m`$ and for sub-sector $`j`$</font>
+* <font color='blue'>$`n^{cap2}_{t, i, m, j} = `$ overcapacity of existing units at time $`t`$ for technology $`i`$ for technology module $`m`$ and for sub-sector $`j`$</font>
+* <font color='blue'>$`n^{ext}_{t, i, m, j} = `$ number of units extended at time $`t`$ for technology $`i`$ for technology module $`m`$ and for sub-sector $`j`$</font>
+* <font color='blue'>$`n^{dec}_{t, i, m, j} = `$ sum of all unit reductions at time $`t`$ for technology $`i`$ for technology module $`m`$ and for sub-sector $`j`$</font>
+* <font color='blue'>$`n^{xdec}_{t, i, m, j} = `$ number of units of $`n^{ext}`$ that reached their lifetime at time $`t`$ for technology $`i`$ for technology module $`m`$ and for sub-sector $`j`$</font>
+* $`n^{sdec}_{t, i, m, j} = `$ yearly decrease of initial stock of units at time $`t`$ for technology $`i`$ for technology module $`m`$ and for sub-sector $`j`$
+* $`life_{i, m, j} = `$ lifetime of technology module $`m`$ of technology $`i`$ in sub-sector $`j`$ $`(a)`$
+
+
+$`n^{cap}_{t+1, i, m, j} = n^{cap}_{t, i, m, j} +  n^{ext}_{t+1, i, m, j} - n^{dec}_{t+1, i, m, j} \; \forall (t,i,m,j) \in (T,I,M,J)`$
+
+$`n^{cap}_{t, i, m, j} = n^{cap1}_{t, i, m, j} +  n^{cap2}_{t, i, m, j} \; \forall (t,i,m,j) \in (T,I,M,J)`$
+    
+$`n^{cap2}_{t, i, m, j} = n^{cap}_{t, i, m, j} -  n^{prod}_{t, i, j} \; \forall (t,i,m,j) \in (T,I,M,J)`$
+
+$`n^{cap2}_{t+1, i, 1, j} \geq n^{cap2}_{t, i, 1, j} -  n^{sdec}_{t+1, i, 1, j} \; \forall (t,i,j) \in (T,I,J)`$
+    
+$`\sum_{i, j}(n^{cap2}_{t, i, 1, j}) \leq 0.01 \times \sum_{i, j}(n^{prod}_{t, i, j}) \; \forall t \in T`$
+    
+$`n^{dec}_{t, i, m, j} = n^{sdec}_{t, i, m, j} + n^{xdec}_{t, i, m, j} \; \forall (t,i,m,j) \in (T,I,M,J)`$
+
+$`n^{xdec}_{t+life_{i, m, j}, i, m, j} = n^{ext}_{t, i, m, j} \; \forall (t,i,m,j) \in (T,I,M,J)`$
+    
+$`n^{prod}_{t, i, j} \leq n^{cap1}_{t, i, m, j} \; \forall (t,i,m,j) \in (T,I,M,J)`$
+    
+$`n^{prod}_{t, i, j} = n^{cap1}_{t, i, 1, j} \; \forall (t,i,j) \in (T,I,J)`$
+
+### 2.3 Heat production:
+    
+This regulates the share of production of the hybrid heat production concepts.
+
+* $`pm^{Bio}_{t, i, j} = `$ solid biomass share per technology $`i`$ at time $`t`$ and for sub-sector $`j`$ $`(\%)`$
+* $`pm^{Gas}_{t, i, j} = `$ natural gas/biogas/coal share per technology $`i`$ at time $`t`$ and for sub-sector $`j`$ $`(\%)`$
+* $`pm^{nonBio}_{t, i, j} = `$ non biomass, renewable share per technology $`i`$ at time $`t`$ and for sub-sector $`j`$ $`(\%)`$
+
+$`n^{prod}_{t, i, j} \times pm^{Bio}_{t, i, j} \times d^{cap}_{t, j} = \sum_{b}(\pi^{Bio}_{t, i, j, b}) \; \forall (t,i,j) \in (T,I,J)`$
+
+$`n^{prod}_{t, i, j} \times pm^{Gas}_{t, i, j} \times d^{cap}_{t, j} = \sum_{b}(\pi^{Gas}_{t, i, j, b}) \; \forall (t,i,j) \in (T,I,J)`$
+
+$`n^{prod}_{t, i, j} \times pm^{nonBio}_{t, i, j} \times d^{cap}_{t, j} = \pi^{nonBio}_{t, i, j} \; \forall (t,i,j) \in (T,I,J)`$
+
+$`\pi_{t, i, j} = \sum_{b}(\pi^{Bio}_{t, i, j, b}) + \sum_{b}(\pi^{Gas}_{t, i, j, b}) + \pi^{nonBio}_{t, i, j} \; \forall (t,i,j) \in (T,I,J)`$
+
+    
+### 2.4 Biomass consumption
+
+* <font color='blue'>$`bc_{t, i, j, b} = `$ actual consumed biomass in the technology $`i`$ at time $`t`$ in sub-sector $`j`$ and from biomass product $`b`$ $`(GJ)`$</font>
+* $`ef^{Bio}_{t, i, j} = `$ conversion efficiency of solid biomass technologies $`i`$ at time $`t`$ in sub-sector $`j`$ $`(\%)`$
+* $`ef^{Gas}_{t, i, j} = `$ conversion efficiency of natural gas/biogas/coal technologies $`i`$ at time $`t`$ in sub-sector $`j`$ $`(\%)`$
+* $`ef^{Methan}_{t, b} = `$ conversion efficiency of biomethane feed-in plant at time $`t`$ for biomass product $`b`$ $`(\%)`$
+* $`ba_{t, bm} = `$ available biomass from residues $`bmwaste`$ $`(GJ)`$ and available land for cultivation $`(ha)`$ for time $`t`$ and biomass type $`bm`$
+* $`ba^{maxw}_{t} = `$ maximal allowed biomass usage from waste $`(\%)`$ at time $`t`$ = represents the biomass pre-allocation for heating
+
+Consumed biomass equals the heat consumption divided by degree of efficiency.
+
+$`bc_{t, i, j, b} = \frac{\pi^{Bio}_{t, i, j, b}}{ef^{Bio}_{t, i, j}} + \frac{\pi^{Gas}_{t, i, j, b}}{ef^{Gas}_{t, i, j} \times ef^{Methan}_{t, b}} \; \forall (t,i,j,b) \in (T,I,J,B)`$
+
+Technology 'Gas boiler+Log wood stove+ST' can use different biomass products for different components. This equation regulates this issue.
+
+$`\sum_{b}(bc_{t, 2, j, b})=\sum_{b^{Gas}}(\frac{\pi^{Gas}_{t, 2, j, b^{Gas}}}{ef^{Gas}_{t, 2, j} \times ef^{Methan}_{t, b^{Gas}}}) + \sum_{b^{Scheit}}(\frac{\pi^{Bio}_{t, 2, j, b^{Scheit}}}{ef^{Bio}_{t, 2, j}}) \; \forall (t,j) \in (T,J)`$
+
+Technology 'Waste CHP plant+Wood chip boiler' can use different biomass products for different components. This equation regulates this issue.
+
+$`\sum_{b}(bc_{t, 28, j, b})=\frac{\pi^{Gas}_{t, 28, j, 26}}{ef^{Gas}_{t, 28, j}} + \sum_{b^{woodChip}}(\frac{\pi^{Bio}_{t, 28, j, b^{woodChip}}}{ef^{Bio}_{t, 28, j}}) \; \forall (t,j) \in (T,J)`$
+
+Technology 'Coal CHP plant with 5% wood chips' can use different biomass products for different components. This equation regulates this issue.
+
+$`\sum_{b}(bc_{t, 26, j, b})=\frac{\pi^{Gas}_{t, 26, j, 25}}{ef^{Gas}_{t, 26, j}} + \sum_{b^{woodChip2}}(\frac{\pi^{Bio}_{t, 26, j, b^{woodChip2}}}{ef^{Bio}_{t, 26, j}}) \; \forall (t,j) \in (T,J)`$
+
+Depending on the scenario, the consumed biomass from residues is limited to a certain degree of percentage (pre-allocation of biomass to the heat sector).
+
+$`\sum_{bm^{waste}(bm)}(ba_{t, bm} \times ba^{maxw}_{t}) \geq \sum_{i,j,b^{waste}(b)}(bc_{t, i, j, b}) \; \forall t \in T`$
+
+### 2.5 Conversion of biomass types (potential) to biomass products (price)
+
+* <font color='blue'>$`bu_{t, b, bm} = `$ actual converted biomass from biotype $`bm`$ to biomass product $`b`$ at time $`t`$ $`(GJ)`$</font>
+* $`ba^{maxc}_{t} = `$ maximal allowed biomass usage from cultivation at time $`t`$ = represents the biomass/available land pre-allocation for heating $`(\%)`$
+* $`yield_{t, b} = `$ yield of cultivation products $`b`$ at time $`t`$ $`(GJ/ha)`$
+
+Which residue biomass types can be used for which biomass products.
+
+$`ba_{t, bm^{waste}} \geq  \sum_{b}(bu_{t,b,bm^{waste}}) \; \forall (t,bm) \in (T,BM^{waste})`$
+
+Limitation of energy crops land potential to biomass products.
+
+$`ba_{t, 13} \times ba^{maxc}_{t} \geq  \sum_{b^{cult}(b)}(\frac{bu_{t,b,13}}{yield_{t, b}})  \; \forall t \in T`$
+
+Which fossil biomass types can be used for which biomass products.
+
+$`ba_{t, 14} \geq  \sum_{b}(bu_{t,b,14})  \; \forall t \in T`$
+
+Actual converted amount of biomass from biotype to biomass products equals sum of consumed biomass over the technologies.
+
+$`\sum_{bm}(bu_{t, b, bm}) = \sum_{i, j}(bc_{t,i,j,b})  \; \forall (t,b) \in (T,B)`$
+
+### 2.6 Set portfolio for energy crops in the starting year(s)
+
+* $`culstart_{b} = `$ Crop cultivation portfolio in the first year(s) (past) of cultivation products $`b`$ $`(ha)`$
+
+Set portfolio for digestible energy crops in the starting year(s):
+
+$`bu_{1, b^{culst}, 13} = yield_{1, b^{culst}} \times culstart_{b^{culst}}  \; \forall b \in B^{culst}`$
+
+Set portfolio of short roation coppice (KUP) in the starting year(s).
+
+$`\sum_{1,b^{kup}, bm}(bu_{1, b^{kup}, bm}) = yield_{1, 12} \times culstart_{12}  \; \forall b \in B^{kup}`$
+
+Set portfolio of Miscanthus (mis) in the starting year(s).
+
+$`\sum_{1,b^{mis}, bm}(bu_{1, b^{mis}, bm}) = yield_{1, 15} \times culstart_{15}  \; \forall b \in B^{mis}`$
+
+
+### 2.7 Set maximal yearly expansion of energy crops
+
+Maximal yearly expansion of digestible energy crops.
+
+$`bu_{t+1, b^{culst}, 13} \leq 2 \times bu_{t,b^{culst},13}  \; \forall (t,b) \in (T,B^{culst})`$
+
+Maximal yearly expansion of short rotation coppice (KUP).
+
+$`\sum_{b^{kup}}(bu_{t+1, b^{kup}, 13}) \leq 2 \times \sum_{b^{kup}}(bu_{t,b^{kup},13})  \; \forall (t,b) \in (T,B^{kup})`$
+
+Maximal yearly expansion of Miscanthus.
+
+$`\sum_{b^{mis}}(bu_{t+1, b^{,mis}, 13}) \leq 2 \times \sum_{b^{mis}}(bu_{t,b^{mis},13})  \; \forall (t,b) \in (T,B^{mis})`$
+
+### 2.8 Greenhouse gas emissions (GHG)
+
+* <font color='blue'>$`ghgf_{t, i, j, b} = `$ actual feedstock GHG emissions at time $`t`$ for technology $`i`$, sub-sector $`j`$ and biomass product $`b`$ $`(t)`$</font>
+* <font color='blue'>$`ghgt_{t, i, j} = `$ actual technology GHG emissions at time $`t`$ for technology $`i`$ and sub-sector $`j`$ $`(t)`$</font>
+* $`ghgr_{t, i, j} = `$ GHG emission factor per technology $`i`$ excluding feedstock emissions for time $`t`$ and sub-sector $`j`$ $`(t/GJ)`$
+* $`ghgfeed_{b} = `$ GHG emission factor per biomass product $`b`$ $`(t/GJ)`$
+* $`alloc_{i, j} = `$ allocation factor of CHP emissions to the heat sector for technology $`i`$ and sub-sector $`j`$
+* $`ghgmax_{t} = `$ GHG emission budget for time $`t`$ $`(t)`$
+
+$`ghgf_{t, i, j, b} = alloc_{i, j} \times ghgfeed_{b} \times bc_{t, i, j, b} \; \forall (t,i,j,b) \in (T,I,J,B)`$
+
+$`ghgt_{t, i, j} = alloc_{i, j} \times ghgr_{t, i, j} \times \pi_{t, i, j} \; \forall (t,i,j) \in (T,I,J)`$
+
+$`ghgmax_{t} \geq \sum_{i, j, b}(ghgf_{t, i, j, b}) + \sum_{i, j}(ghgt_{t, i, j}) \; \forall t \in T`$
+
+
+### 2.9 Consumer behavior
+
+A few sub-sectors were further divided into consumer segments to represent consumer behavior by applying intangible costs. This further seperation is defined here. E.g., the demand within the comsumer clusters needs to equal the production in the clusters, the sum of production in clusters equals the production in the sub-sector.
+
+* $`dBeh_{t, j, c} = `$ heat demand in the consumer clusters $`c`$ for time $`t`$ and sub-sector $`j`$ $`(GJ)`$
+
+$`dBeh_{t, j^{clus}, c} = \sum_{i}(\pi^{Beh}_{t, i, j^{clus}, c}) \; \forall (t,j,c) \in (T,J^{clus},C)`$
+
+$`\sum_{c}(\pi^{Beh}_{t, i, j^{clus}, c}) = \pi_{t, i, j^{clus}} \; \forall (t,i,j) \in (T,I,J^{clus})`$
+
+$`n^{prodBeh}_{t, i, j^{clus}, c} \times d^{cap}_{t, jclus} = \pi^{Beh}_{t, i, j^{clus}, c} \; \forall (t,i,j,c) \in (T,I,J^{clus},C)`$
+
+$`\sum_{c}(n^{prodBeh}_{t, i, j^{clus}, c}) = n^{prod}_{t, i, j^{clus}} \; \forall (t,i,j) \in (T,I,J^{clus})`$
+
+## 3. Constraints - fixed variables
+
+### 3.1 Fixed values for starting year:
+
+$`n^{start}_{i, j} =`$ number of units/plants in starting year.
+
+No overcapacity allowed in the first year.
+
+$`n^{cap2}_{1, i, m, j} = 0`$
+
+No expansion of units allowed in the first year.
+
+$`n^{ext}_{1, i, m, j} = 0`$
+
+No decomissioning of uints allowed in the first year.
+
+$`n^{dec}_{1, i, m, j} = 0`$
+
+No decomissioning of expanded plants in the first year.
+
+$`n^{xdec}_{1, i, m, j} = 0`$
+
+Number of units in starting year.
+
+$`n^{cap1}_{1, i, m, j} = n^{start}_{i, j}`$
+
+### 3.2 Dependecies/ possible pathways:
+
+* $`MT_{i,j}=`$ defines which technologies can be used in which sub-sectors.
+* $`TB_{i,b}=`$ defines which biomass products can be used in which technologies.
+* $`BB_{bm,b}=`$ defines which biomass types can be used for which biomass products.
+
+$`\pi_{t, i, j} = 0 \; \forall MT_{i,j}\neq 1`$
+
+$`bc_{t, i, j, b} = 0 \; \forall TB_{i,b}\neq 1`$
+
+$`bc_{t, i, j, 14} = 0 \; \forall (t=2020..2034, i=18..20, j) \in (T,I,J)`$
+
+$`bu_{t, b, bm} = 0 \; \forall BB_{bm,b}\neq 1`$
+
+### 3.3 Overcapacity restrictions:
+
+Sets nxdec=0 before any lifetime reduction of next happens
+
+$`n^{xdec}_{t, i, m, j} = 0 \; \forall t \leq life_{i,m,j}`$
+
+During the decrease of the initial stock, overcapacity is forbidden; despite in district heating market (i=24) and market 14 in the 95\% case
+
+$`n^{cap2}_{t, i, 1, j} = 0 \; \forall t \leq life_{i,1,j} + 1, \forall i \neq 24, \forall j \neq 14`$
+
+Overcapacity is only allowed for gas boiler/coal...
+
+$`n^{cap2}_{t, i, 1, j} = 0 \; \forall i \neq (1,24,25,30,36,40,46)`$
+
+
+### 3.4 Technology specific restrictions:
+
+Set waste CHP constant.
+
+$`n^{prod}_{t, 28, 15} = n^{start}_{28, 15}`$
+
+Set leach boiler constant.
+
+$`n^{prod}_{t, 48, 16} = n^{start}_{48, 16}`$
+