# Energy balance

## Question

What is energy balance and how is it modelled?

Summing up the amount of energy produced and subtracting the amount of energy consumed within a time period gives the energy balance. Since the electricity grid and district heat network lack significant storage mechanics, the balance has to be virtually zero over short periods. When considering the balance of a particular area (e.g. Helsinki), we can make the assumption that electricity can be imported and exported in international markets. The energy in the district heat network, however, has to be produced locally. This sets up the non-trivial problem of optimising production so that there are no significant deficits as well as minimising losses and maximising profits. This problem is solved (to some extent) by market forces in the real world.

In Opasnet, there are two different ways to calculate energy balance. Our most recent energy balance model uses linear programming tools to solve an optimum for the activity of a given set of production units in simulated instances created by the main model. The main model is responsible for the decision making aspects, while the energy balance optimisation only functions as an approximation of real world market mechanics. This version was used in Helsinki energy decision 2015.

The previous version was based on setting up a set of linear equations describing the inputs, outputs, and shares of different energy and plant processes. This approach is less flexible, because it does not use an optimising function and everything must be described as linear (or piecewise linear). However, this approach was successfully used in Energy balance in Kuopio and Energy balance in Suzhou.

This code is an example how the energy balance model is used in a city case. The data comes from Helsinki energy decision 2015.

 ```library(OpasnetUtils) library(ggplot2) openv.setN(0) # use medians instead of whole sampled distributions # Download case-specific data, in this case from Helsinki. objects.latest("Op_en7237", code_name = "intermediates") # [[Helsinki energy decision 2015]] # Download energy balance model and its parts: # EnergyConsumerDemandTotal # EnergyFlowHeat # EnergyFlowOther # EnergyNetworkDemand # EnergyNetworkOptim # fuelUse # EnergyNetworkCost objects.latest("Op_en5141", code_name = "EnergyNetworkOptim") # [[Energy balance]] oprint(summary(EvalOutput(EnergyNetworkOptim))) ```

## Rationale

### Energy balance with linear programming

The linear programming problem is set up as follows.

For each production unit: let xi be activity of the plant. Lets also have variables yj for deficits and excesses for each type of energy produced.

The objective function is the function we are optimising. Each production unit has a unit profit per activity denoted by ai which is determined by the amount of different input commodities (e.g. coal) per amount of different output commodities (i.e. electricity and heat) and their market prices. Also, lets say we want to make sure that district heat demand is always met when possible and have a large penalty factor for each unit of heat demand not met (1 M€ in the model). In addition, it must be noted that excess district heat becomes wasted so it counts as loss. Let these deficit and excess related losses be denoted by bj. The whole objective function then becomes: sum(xiai) + sum(yjbj).

The values of variables are constrained by equalities and inequalities: the sum of production of a commodity is equal to its demand minus deficit plus excess, activity is constrained by the maximum capacity and all variables are non-negative by definition. This can be efficiently solved by computers for each given instance. Production wind-up and wind-down is ignored, since time continuity is not considered. As a consequence fuel limits (e.g. diminishing hydropower capacity) are not modelled completely either.

Ovariables like EnergyNetworkOptim below are used in Helsinki energy decision 2015. Prices of fuels in heat production are used as direct inputs in the optimising.

 ```## This code is Op_en5141/EnergyNetworkOptim [[Energy balance]]. library(OpasnetUtils) EnergyConsumerDemandTotal <- Ovariable("EnergyConsumerDemandTotal", dependencies = data.frame( Name = c( "EnergyConsumerDemand" ) ), formula = function(...) { energy <- oapply(EnergyConsumerDemand, NULL, sum, c("Renovation", "Efficiency")) energy <- energy * 1e-6 # W to MW return(energy) } ) EnergyFlowHeat <- Ovariable("EnergyFlowHeat", dependencies = data.frame( Name = c( "EnergyConsumerDemandTotal", "fuelShares" ) ), formula = function(...) { EnergyFlowHeat <- EnergyConsumerDemandTotal[EnergyConsumerDemandTotal\$Consumable %in% c("Heating", "Hot water"),] * fuelShares return(EnergyFlowHeat) } ) EnergyFlowOther <- Ovariable("EnergyFlowOther", dependencies = data.frame( Name = c( "EnergyConsumerDemandTotal", "temperene", "nontemperene" ) ), formula = function(...) { EnergyFlowOther <- merge( EnergyConsumerDemandTotal[!EnergyConsumerDemandTotal\$Consumable %in% c("Heating", "Hot water"),], unique(OpasnetUtils::combine(temperene, nontemperene)[,c("Consumable", "Fuel")]) ) EnergyFlowOther@name <- EnergyConsumerDemandTotal@name return(EnergyFlowOther) } ) EnergyNetworkDemand <- Ovariable("EnergyNetworkDemand", dependencies = data.frame( Name = c( "EnergyFlowHeat", "EnergyFlowOther", "EnergyConsumerDemandTotal" ) ), formula = function(...) { demand <- OpasnetUtils::combine( EnergyFlowHeat[EnergyFlowHeat\$Fuel %in% c("Heat", "Electricity"), colnames(EnergyFlowHeat@output) != "Burner"], EnergyFlowOther[EnergyFlowOther\$Fuel %in% c("Cooling", "Electricity"),] ) # EnergyFlowOther should not have other flows #assign("EnergyFlowOther", NULL, .GlobalEnv) #assign("EnergyFlowHeat", EnergyFlowHeat[!EnergyFlowHeat\$Fuel %in% c("Heat", "Electricity"),], .GlobalEnv) # Total consumer demand also unnecessary? #assign("EnergyConsumerDemandTotal", NULL, .GlobalEnv) NAindex <- sapply(lapply(lapply(demand@output, unique), is.na), sum) NAindex <- names(NAindex)[NAindex != 0] demand@output <- fillna(demand@output , NAindex) demand <- oapply(demand, NULL, sum, c("Heating", "Consumable"))#, "Renovation", "Efficiency")) # Assumes no NA return(demand) } ) EnergyNetworkOptim <- Ovariable("EnergyNetworkOptim", dependencies = data.frame( Name = c( #"energy", # Energy supply and demand in the system #"nondynsupply", # Energy supply that is not optimised. #"requiredName", # Fuel name for the energy type that is balanced for inputs and outputs. #"fuelShares", # Shares of fuels for different heating types. "energyProcess", # Matrix showing inputs and outputs of the process of each plant. "fuelPrice", # Prices of fuels (by Time) #"timelylimit", # Maximum energy supply in the given conditions at a particular time point "plantParameters", # Capacities and costs of running each plant #"temperene", #"nontemperene", "EnergyNetworkDemand" ), Ident = c( #NA, #NA, #NA, #NA, NA, "Op_en4151/fuelPrice", #NA, NA, #"Op_en5488/temperene", #"Op_en5488/nontemperene", NA ) ), formula = function(...) { require(lpSolve) require(plyr) require(reshape2) # Prices are given as €/MWh while plant activity is given as MW while optimisation is done # with time resolution = 1 day optim_time_period_adj <- 24 # List non-interating indices exclude <- c("Fuel", "Plant", "Parameter", "Heating", "Consumable", "Burner") # Calculate unitprofit for plant activity energy flows unitp <- fuelPrice * energyProcess * optim_time_period_adj unitp <- oapply(unitp, NULL, sum, c("Fuel", "Burner")) colnames(unitp@output)[colnames(unitp@output) == "Result"] <- "UnitPrice" # Reshape crucial variables to reduce merging difficulty ePmargs <- colnames(energyProcess@output)[energyProcess@marginal] energyProcess@output <- dcast( energyProcess@output, as.formula(paste(paste(ePmargs[ePmargs != "Fuel"], collapse = "+"), "~ Fuel")), value.var = paste(energyProcess@name, "Result", sep = ""), fill = NA ) energyProcess@marginal <- colnames(energyProcess@output) %in% ePmargs pPmargs <- colnames(plantParameters@output)[plantParameters@marginal] plantParameters@output <- dcast( plantParameters@output, as.formula(paste(paste(pPmargs[pPmargs != "Parameter"], collapse = "+"), "~ Parameter")), value.var = paste(plantParameters@name, "Result", sep = ""), fill = NA ) plantParameters@marginal <- colnames(plantParameters@output) %in% pPmargs EnergyNetworkDemand\$Fuel <- paste(EnergyNetworkDemand\$Fuel, "Demand", sep = "") ENDmargs <- colnames(EnergyNetworkDemand@output)[EnergyNetworkDemand@marginal] EnergyNetworkDemand@output <- dcast( EnergyNetworkDemand@output, as.formula(paste(paste(ENDmargs[ENDmargs != "Fuel"], collapse = "+"), "~ Fuel")), value.var = paste(EnergyNetworkDemand@name, "Result", sep = ""), fill = NA ) EnergyNetworkDemand@marginal <- colnames(EnergyNetworkDemand@output) %in% ENDmargs fuelPrice\$Fuel <- paste(fuelPrice\$Fuel, "Price", sep = "") fPmargs <- colnames(fuelPrice@output)[fuelPrice@marginal] fuelPrice@output <- dcast( fuelPrice@output, as.formula(paste(paste(fPmargs[fPmargs != "Fuel"], collapse = "+"), "~ Fuel")), value.var = paste(fuelPrice@name, "Result", sep = ""), fill = 0 ) fuelPrice@marginal <- colnames(fuelPrice@output) %in% fPmargs # Duplicates a couple of values, but provides better performance vars <- merge(EnergyNetworkDemand, plantParameters) vars <- merge(vars, energyProcess) vars <- merge(vars, fuelPrice) vars <- merge(vars, unitp) # Split into iterations #vars <- dlply(vars@output, colnames(vars@output)[vars@marginal & !colnames(vars@output) %in% exclude], I) #out <- data.frame() include <- union(ePmargs, pPmargs) include <- union(include, ENDmargs) include <- union(include, fPmargs) include <- include[!include %in% exclude] # Optimisation iteration function optf <- function(varsi) { # Acquire relevant sections of variables with respect to index iteration demandi <- unlist(varsi[1,grepl("Demand\$", colnames(varsi))]) names(demandi) <- gsub("Demand\$", "", names(demandi)) fuelPricei <- unlist(varsi[1,grepl("Price\$", colnames(varsi))]) names(fuelPricei) <- gsub("Price\$", "", names(fuelPricei)) fuelPricei <- fuelPricei[names(fuelPricei) != "Unit"] plants <- as.character(unique(varsi[["Plant"]])) # Used for ordering parameters if (nrow(varsi) != length(plants)) stop("Wrong subset given to energy production optimizer! Each row not unique with respect to Plant.") lower <- varsi[["Min"]] upper <- varsi[["Max"]] opcost <- varsi[["Operation cost"]] unitpi <- varsi[["UnitPrice"]] #commodities <- as.character(unique(energyProcessi\$Fuel)) # Used for ordering parameters # Constraint matrix # Total number of variables: # * plant activity for each plant # * (commodity flow total) # * commodity deficit and excess with respect to demand # * opeartion costs nvars <- length(plants) + length(demandi) * 2 + 1 # Rows: # * demand rows and their constraints # * (commodity flow rows) # * plant activity constraints # * operation costs row nrows <- 3 * length(demandi) + 2 * length(plants) + 1 M <- matrix( 0, nrows, nvars ) sign <- rep(">=", nrows) constant <- rep(0, nrows) for (j in 1:length(demandi)) { # Total flow (production - consumption between all processes) + deficit = demand + excess # which translates to: # total_commodity_flow + deficit - excess == constant ("static" demand) M[j, length(plants) + j*2 + 1:2 - 2] <- c(1, -1) sign[j] <- "==" constant[j] <- demandi[j] # Plant flows by activity M[j, 1:length(plants)] <- varsi[[names(demandi)[j]]] } rowi <- length(demandi) diag(M[ (rowi + 1):(rowi + length(plants)), 1:length(plants) ]) <- 1 diag(M[ (rowi + length(plants) + 1):(rowi + 2 * length(plants)), 1:length(plants) ]) <- 1 constant[rowi + 1:length(plants)] <- lower constant[rowi + length(plants) + 1:length(plants)] <- upper sign[rowi + length(plants) + 1:length(plants)] <- "<=" rowi <- rowi + 2 * length(plants) # Excess and deficit must both be positive diag(M[ rowi + 1:(2 * length(demandi)), length(plants) + #length(commodities) + 1:(2 * length(demandi)) ]) <- 1 # Operating cost M[nrows, 1:length(plants)] <- opcost * optim_time_period_adj M[nrows, nvars] <- -1 sign[nrows] <- "==" # Objective function (profits - penalty) objective <- rep(0, nvars) # Commodity prices #objective[length(plants) + (1:length(commodities))[!is.na(fuelPricej)]] <- result(fuelPricei)[fuelPricej[!is.na(fuelPricej)]] # Penalize heat deficit heavily, # electricity deficit has no impact on profit assuming company sells to customers at the purchase price # cooling deficit has no effect if ("Heat" %in% names(demandi)) { objective[length(plants) + match("Heat", names(demandi)) * 2 - 1] <- -1e6 } # Electricity excess can be sold in the markets, # heat and cooling (and other?) excess, however, cannot, hence they need to be deducted from profits objective[ length(plants) + which(names(demandi) != "Electricity") * 2 ] <- - fuelPricei[match(names(demandi)[names(demandi) != "Electricity"], names(fuelPricei))] * optim_time_period_adj # Operation cost objective[nvars] <- -1 # Unit profit objective[1:length(plants)] <- unitpi # Do actual linear programming temp <- lp( "max", objective, M, sign, constant ) # Format results outi <- varsi[rep(1, nvars + 1), include, drop = FALSE] outi[["Process_variable_type"]] <- rep( c("Activity", "Flow", "Misc"), c(length(plants), nvars - length(plants) - 1, 2) ) outi[["Process_variable_name"]] <- c( as.character(plants), paste(rep(names(demandi), each = 2), c("Deficit", "Excess"), sep = ""), "Operation cost", "Profit" ) outi[["Result"]] <- c(temp\$solution, temp\$objval) return(outi) } #aggregate(out\$Result[out\$Process_variable_name == "Profit"], out[out\$Process_variable_name == "Profit",colnames(out) %in% c("Time", "Temperature") | grepl("Policy", colnames(out))], mean) #aggregate(out\$Result[out\$Process_variable_name == "HeatDeficit"], out[out\$Process_variable_name == "HeatDeficit", colnames(out) %in% c("Time", "Temperature") | grepl("Policy", colnames(out))], mean) #aggregate(out\$Result[out\$Process_variable_name == "Hanasaari"], out[out\$Process_variable_name == "Hanasaari", colnames(out) %in% c("Time", "Temperature") | grepl("Policy", colnames(out))], mean) # Error handling optfsecure <- function(varsi) { ret <- tryCatch( optf(varsi), error = function(e) return(NULL) ) if (!is.null(ret)) { return(ret) } else { warning(paste("EnergyNetworkOptim failed optimising a permutation with error:", geterrmessage())) return(data.frame()) } } out <- ddply(vars@output, colnames(vars@output)[vars@marginal & !colnames(vars@output) %in% exclude], optfsecure) out <- Ovariable(output = out, marginal = !grepl("Result\$", colnames(out))) return(out) } ) fuelUse <- Ovariable("fuelUse", dependencies = data.frame( Name = c( "EnergyNetworkOptim", "EnergyNetworkDemand", "energyProcess", "EnergyFlowHeat", "temperdays" ) ), formula = function(...){ # Calculate flows resulting from plant activity EnergyFlow <- EnergyNetworkOptim[EnergyNetworkOptim\$Process_variable_type == "Activity", colnames(EnergyNetworkOptim@output) != "Process_variable_type"] colnames(EnergyFlow@output)[colnames(EnergyFlow@output) == "Process_variable_name"] <- "Plant" EnergyFlow <- EnergyFlow * energyProcess # Realised consumption of centrally produced commodities (demand - deficit or prod - excess) # NOTE: centralized flows which consume electricity for example should be separated # from consumer demand and consumption, so prod - excess is not accurate deficit <- EnergyNetworkOptim[ grepl("Deficit\$", EnergyNetworkOptim\$Process_variable_name), colnames(EnergyNetworkOptim@output) != "Process_variable_type" ] colnames(deficit@output)[colnames(deficit@output) == "Process_variable_name"] <- "Fuel" deficit\$Fuel <- gsub("Deficit\$", "", deficit\$Fuel) # Assume that the power company purchases any electricity deficit from the markets in order to satisfy demand result(deficit)[deficit\$Fuel == "Electricity"] <- 0 #excess <- EnergyNetworkOptim[ # grepl("Excess\$", EnergyNetworkOptim\$Process_variable_name), # colnames(EnergyNetworkOptim@output) != "Process_variable_type" #] #colnames(excess@output)[colnames(excess@output) == "Process_variable_name"] <- "Fuel" #excess\$Fuel <- gsub("Excess\$", "", excess\$Fuel) #temp <- oapply(EnergyFlow, NULL, sum, c("Plant", "Burner")) #real <- oapply(EnergyFlow[EnergyFlow\$Fuel %in% unique(excess\$Fuel)], NULL, sum, c("Plant", "Burner")) #real <- real - excess real <- EnergyNetworkDemand - deficit # Combine real\$Burner <- "None" real\$Plant <- "Domestic" heating <- EnergyFlowHeat[!EnergyFlowHeat\$Fuel %in% unique(deficit\$Fuel),] heating <- oapply(heating, NULL, sum, "Consumable") heating\$Plant <- "Domestic" heating\$Heating <- NULL EnergyFlow <- OpasnetUtils::combine(0 - EnergyFlow, real, heating) EnergyFlow <- unkeep(EnergyFlow, sources = TRUE, prevresults = TRUE) #EnergyFlowTest <- oapply(EnergyFlow, c("Time", "Temperature", "Fuel"), sum) #ggplot(EnergyFlowTest@output, aes(x = Temperature, y = Result, group = Fuel, color = Fuel)) + geom_line() + facet_wrap( ~ Time) EnergyFlow <- EnergyFlow * temperdays * 3600 * 24 EnergyFlow <- oapply(EnergyFlow, cols = c("Temperature"), FUN = sum) return(EnergyFlow) } ) EnergyNetworkCost <- Ovariable("EnergyNetworkCost", dependencies = data.frame( Name = c( "plantParameters", "EnergyNetworkOptim", "temperdays" ), Ident = c( NA, "Op_en5141/EnergyNetworkOptim", # [[Energy balance]] NA ) ), formula = function(...) { oper <- plantParameters[plantParameters@output\$Parameter == "Max" , colnames(plantParameters@output) != "Parameter"] result(oper)[result(oper) != 0] <- 1 oper <- plantParameters * oper # Take the first year when a plant is operated and put all investment cost there. investment <- oper[oper@output\$Parameter == "Investment cost" , colnames(oper@output) != "Parameter"] investment <- investment[result(investment) > 0 , ] investment <- investment[order(investment@output\$Time) , ] investment <- investment[!duplicated(investment@output[investment@marginal & colnames(investment@output) != "Time"]) , ] investment <- unkeep(investment, sources = TRUE) #investment <- oapply(investment, cols = "Plant", FUN = sum) maintenance <- oper[oper@output\$Parameter == "Management cost" , colnames(oper@output) != "Parameter"] maintenance <- unkeep(maintenance, sources = TRUE) #maintenance <- oapply(maintenance, cols = c("Plant"), FUN = sum) operation <- EnergyNetworkOptim[EnergyNetworkOptim@output\$Process_variable_name == "Operation cost" , ] operation <- operation * temperdays * 10 * 1E-6 # For 10-year periods, € -> M€ operation <- oapply(operation, cols = c("Temperature"), FUN = sum) operation <- unkeep(operation, cols = c("Process_variable_name", "Process_variable_type"), sources = TRUE, prevresults = TRUE) operation <- operation * Ovariable(output = data.frame(Plant = "Operation", Result = 1), marginal = c(TRUE, FALSE)) cost <- OpasnetUtils::combine(investment, maintenance, operation) marginals <- character() for(i in colnames(cost@output)[cost@marginal]) { if(any(is.na(cost@output[[i]]))) marginals <- c(marginals, i) } if(length(marginals) > 0) { cost@output <- fillna(cost@output, marginals) warning(paste("In combine had to fillna marginals", marginals, "\n")) } return(cost) } ) objects.store(EnergyConsumerDemandTotal, EnergyFlowHeat, EnergyFlowOther, EnergyNetworkDemand, EnergyNetworkOptim, fuelUse, EnergyNetworkCost) cat("Ovariables EnergyConsumerDemandTotal, EnergyFlowHeat, EnergyFlowOther, EnergyNetworkDemand, EnergyNetworkOptim, EnergyNetworkCost and fuelUse stored.\n") ```

### Fuel use and fuel shares in generic processes

There is an alternative way for calculating fuel use. It is based on the idea that for each heating type, there is a constant share of fuels used. For some heating types, this is generic and is shown on this page. For some others, the constant is case-specific and is determined on a case-specific page.

The table below contains connections of heating types and fuel usage in generic situations. There may be case-specific differences, which must be handled separately.

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Fuel use in different heating types(-)
ObsHeatingBurnerFuelFractionDescription
1WoodHouseholdWood1
2OilHouseholdLight oil1
3GasHouseholdGas1
4Heating oilHouseholdLight oil1
5CoalHouseholdCoal1
6Other sourcesHouseholdOther sources1
7No energy sourceHouseholdOther sources1
8GeothermalGridElectricity0.3Geothermal does not sum up to 1 because more heat is produced than electricity consumed.
9Centrifuge, hydro-extractorGridElectricity0.3Not quite clear what this is but presumably a heat pump.
10Solar heater/ collectorGridElectricity0.1Use only; life-cycle impacts omitted.
11ElectricityGridElectricity1
12DistrictUndefinedHeat1

 ``` # This is code Op_en5141/fuelSharesgeneric (only generic) on page [[Energy balance]]. library(OpasnetUtils) fuelSharesgeneric <- Ovariable("fuelSharesgeneric", ddata = "Op_en5141", subset = "Fuel use in different heating types") # [[Energy balance]] colnames(fuelSharesgeneric@data) <- gsub("[ \\.]", "_", colnames(fuelSharesgeneric@data)) #fuelSharesgeneric@data <- merge(fuelSharesgeneric@data, data.frame(Time = 1900:2080)) objects.store(fuelSharesgeneric) cat("Object fuelSharesgeneric initiated!\n") ```

 ```## This is code Op_en5141/fuelUse on page [[Energy balance]]. library(OpasnetUtils) fuelUse <- Ovariable("fuelUse", dependencies = data.frame( Name = c("energyUse", "fuelShares") ), formula = function(...) { out <- energyUse * fuelShares return(out) } ) objects.store(fuelUse) cat("Ovariable fuelUse stored.\n") ```

### Old version with a set of linear equations

• Energy balances are described as input = output on a coarse level (called classes) where the structure is the same or similar to the OECD energy balance tables. If possible, this is described on the Energy balance method level and it is shared by all cities.
• On more detailed (variable level in the matrix), the fraction of each variable of the total class are described separately. Fractions are city specific and they are described on city level in a separate table.
• Based on the fraction table, detailed equations with variables are created. The format will be fraction * class total = variable.
• The last fraction has zero degrees of freedom when the class total is given. However, it must have a variable and thus a row in the fraction table. The result for that variable is an empty cell (which results in NA).
• Unlike in the previous version, all variables are given either as values or equations, and the user interface is not used for BAU. In contrast, user interface or decision table may be used to derive values for alternative scenarios.
• To make this work, the city-specific fraction data must be defined as ovariable (so that it can be changed with a decision table), and also the energy balance method must be described asa ovariable. How are we going to make the two interplay, as we may want to have several cities?
• Define one city ovariable and evaluate energy balance with that. The ovariable has a generic name. Then, define a new city ovariable with the same name and re-evaluate the energy balance ovariable; this must be done so that the two cities are appended rather than replaced.
• city ovariables are appended first into a large fraction table, and then that is used to create the large energy balance matrix. ←--#: . This is clearly better. --Jouni 17:09, 21 February 2013 (EET) (type: truth; paradigms: science: defence)
• The city-specific ovariable may have Iter and other indices. A separate matrix is created and solved for each unique combination of indices. This makes it possible to have a very flexible approach.
• We should check if the energy balance matrix (see Matti's Excel) has city-specific equations. If possible, energy transformations are described as generic equations on the energy balance method.
• Structure of OECD Energy balance tables (data):
• Fuel (given as observation columns in OECD table)
• Activity (row in OECD table)
• Description
• Structure of the generic process table
• Equation,
• Col,
• Result,
• Description? ⇤--#: . This does not join up in a coherent way. --Jouni 17:09, 21 February 2013 (EET) (type: truth; paradigms: science: attack)
• Columns for fraction table
• Class
• Item
• Result (fraction)
• Indices as needed
Example table for making matrices from text format equations. CHPcapacity describes which of the piecewise linear equations should be used. Policy is a decision option that alters the outcome. Dummy is only for compatibility but it is not used.

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Equations(GWh /a)
ObsCHPcapacityPolicyEquationDummyDescription
1BiofuelCHP renewable = CHP peat1Biofuel policy contains half biofuels, half peat
2BAUCHP renewable = 89.241
3CHP peat + CHP renewable + CHP oil = CHP heat + CHP electricity1
4CHP peat = 90-98*CHP oil1
5CHP electricity = 0.689*CHP heat1
6CHP<1000H heat = 0.08*CHP heat1Small heat plants reflect the total heat need
7CHP>1000CHP heat + CHP electricity = 10001But production capacity of CHP may be overwhelmed, decoupling CHP heat and H heat.
8H biogas + H oil = H heat1
9H oil = 18.973*H biogas1
10Bought electricity + CHP electricity = Cons electricity1
11CHP heat + H heat = Cons heat1
12Cons electricity = 900-11001
13Cons heat = 900-10001
Example table to describe the details about nonlinear equations.

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Nonlinearity parameters(GWh /a)
ObscritVarcritIndexrescolcritLocLowcritLocHighcritValue
1Cons heatCHPcapacityResultCHP<1000CHP>10001080

This table is fetched if there are no nonlinearities. Therefore, there is no need to copy it to the case study page.

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No nonlinearities(GWh /a)
ObscritVarcritIndexrescolcritLocLowcritLocHighcritValue
1
This table is fetched if there are no modelled upstream variables that would affect the equations.

Data updated successfully!

No modelled upstream variables(-)
ObsenergybalanceVarsResult
1

Stored objects below used by Energy balance in Kuopio.

 ```library(OpasnetUtils) library(ggplot2) solveMatrix <- function(equations, directinputtemp, N. = 0, verbose = FALSE, dbug = FALSE, ...) { if (N. == 0) N. <- openv\$N N <- N. # Parse equations using regular expressions out <- gsub(" ", "", equations) out <- strsplit(out, "=") out <- lapply(out, strsplit, split = "+", fixed = TRUE) nterms <- rapply(out, length) # number of terms on each side of each equation signs <- rep( rep(c(1,-1), length(out)), # lhs - rhs = 0 nterms ) eqn <- factor(rep(rep(1:length(equations), each = 2), nterms)) # equation number out <- unlist(out) k <- rep(1, length(out)) x <- out # Equation factors out <- strsplit(out, split = "*", fixed = TRUE) temp <- sapply(out, length) > 1 for (i in (1:length(out))[temp]) { k[i] <- out[[i]][1] x[i] <- out[[i]][2] } # Constants temp <- substr(x, 1, 1) %in% 0:9 # is interpretable (probably) for (i in (1:length(out))[temp]) { k[i] <- x[i] x[i] <- "Constant" } # Direct input (substitutes for constants) if (nrow(directinputtemp) > 0) { l <- tapply(eqn, data.frame(eqn), length) di <- directinputtemp\$energybalanceVars dires <- directinputtemp\$directinputResult for (i in 1:length(di)) { temp <- eqn %in% names(l[l == 2]) & x %in% c("Constant", di[i]) temp <- tapply(temp, eqn, sum) k[eqn %in% names(temp[temp == 2]) & x == "Constant"] <- dires[i] } } # Interpret possible distribution regular expressions (also converts text to numeric) out <- data.frame(eqn, x, signs, Result = k) out <- interpret(out, N = N) marg <- colnames(out)[colnames(out) %in% c("eqn", "x", "Iter")] out\$k <- out\$Result * out\$signs # Separate constants and vars and make them into arrays temp <- out[out\$x != "Constant", ] b <- out[out\$x == "Constant", ] b <- tapply(b\$k, b[marg[-2]], I) b[is.na(b)] <- 0 temp <- tapply(temp\$k, temp[marg], I) if ("Iter" %in% marg) { temp <- temp[,dimnames(temp)\$x != "Constant",] } else { temp <- temp[,dimnames(temp)\$x != "Constant"] } temp[is.na(temp)] <- 0 if ("Iter" %in% marg) { out <- list() for (i in 1:N) { ret <- tryCatch( out[[i]] <- solve(temp[,,i], -b[,i]), error = function(...) return(NULL) ) if (is.null(ret)) { print("Faulty matrix:") oprint(temp[,,i]) stop(geterrmessage()) } } out <- data.frame( energybalanceVars = names(unlist(out)), Iter = rep(1:N, each = length(out[[1]])), Result = unlist(out) ) } else { ret <- tryCatch( out <- solve(temp, -b), error = function(...) return(NULL) ) if (is.null(ret)) { print("Faulty matrix:") oprint(temp) stop(geterrmessage()) } out <- data.frame( energybalanceVars = names(out), Result = out ) } return(out) } energy.balance <- Ovariable( # NOTE! energy.balance requires that there exists object N that defines the number of iterations. name = "energy.balance", dependencies = data.frame( Name = c("balance", "nonlinearity", "directinput") ), formula = function(...) { # Column Equation: change factor into character and demarginalise. balance@data\$Equation <- as.character(balance@data\$Equation) balance@marginal[colnames(balance@output) == "Equation"] <- FALSE # Take those indices that are not needed within solveMatrix and merge them to the balance from the directinput. directinputindex <- directinput@output[directinput@marginal] directinputindex <- directinputindex[!colnames(directinputindex) %in% c("Iter", "energybalanceVars")] indices <- unique(merge(balance@output[balance@marginal], directinputindex)) # Look at each unique combination of index locations and solve the set of equations to each. out <- data.frame() for(k in 1:nrow(indices)) { equations <- merge(balance@output, indices[k , , drop = FALSE]) directinputtemp <- merge(directinput@output[!is.na(result(directinput)),], indices[k, colnames(indices) != "Iter", drop = FALSE]) directinputtemp <- directinputtemp[ , c("energybalanceVars", "directinputResult")] temp <- solveMatrix(as.character(equations[ , "Equation"]), directinputtemp, N. = N, ...) out <- rbind(out, merge(indices[k , , drop = FALSE], temp)) } if(!(nonlinearity@output\$critVar[1] == "" | is.na(nonlinearity@output\$critVar[1]))) { # Go through the non-linear equations one at a time and find which values to use in which situation. for(i in 1:nrow(nonlinearity@output)) { critVar <- gsub(" ", "", as.character(nonlinearity@output\$critVar[i])) critIndex <- as.character(nonlinearity@output\$critIndex[i]) rescol <- as.character(nonlinearity@output\$rescol[i]) critLocLow <- as.character(nonlinearity@output\$critLocLow[i]) critLocHigh <- as.character(nonlinearity@output\$critLocHigh[i]) critValue <- result(nonlinearity)[i] marginal <- colnames(out) %in% c(colnames(balance@output)[balance@marginal], rescol, "Iter") choos <- out[out\$energybalanceVars == critVar , marginal] choos <- choos[ choos[critIndex] == critLocLow & choos[rescol] <= critValue | choos[critIndex] == critLocHigh & choos[rescol] > critValue , colnames(choos) != rescol # Remove result column ] out <- merge(out, choos) } } # Create the marginal: those from the parents and "energybalanceVars" but not nonlinear indices. outmarginal <- c( colnames(balance@output)[balance@marginal], colnames(nonlinearity@output)[nonlinearity@marginal], colnames(directinput@output)[directinput@marginal], "energybalanceVars" ) outmarginal <- outmarginal[!outmarginal %in% nonlinearity@output\$critIndex] outmarginal <- colnames(out) %in% outmarginal out <- new("ovariable", name = "energy.balance", output = out, marginal = outmarginal) return(out) } ) objects.store(solveMatrix, energy.balance) cat("Objects solveMatrix and energy.balance stored with code initiate.\n") ```

How to give uncertain parameters?

• In equations, the content is interpreted only inside solveMatrix. Therefore, the typical approach where all unique index combinations are run one at a time does not work.
• There should be an update in parameter interpretation for terms with one entry only. It can no longer be based on as.numeric, if distributions (=text) is allowed.
• If it starts with [a-z.] it is a variable name.
• If it starts with [0-9<\\-] it is a parameter value.
• Instead of params[[i]] and [[vars]] vectors, a data.frame will be created with Result as the params column.
• The data.frame is then interpreted with N = N. If parameters are probabilistic, Iter column will appear.
• When all parameters have been interpreted, check if Iter exists.
• If Iter exists, make a for loop for all values of Iter.
• Create a matrix from the parameters and solve.
• Rbind the result to a data.frame with Iter.
• Return the output.
• Old code with an input table with columns Equation, Col, Result, Description: [1]

Helsinki energy decision 2015
In English
Assessment Main page | Helsinki energy decision options 2015
Helsinki data Building stock in Helsinki | Helsinki energy production | Helsinki energy consumption | Energy use of buildings | Emission factors for burning processes | Prices of fuels in heat production | External cost
Models Building model | Energy balance | Health impact assessment | Economic impacts
Related assessments Climate change policies in Helsinki | Climate change policies and health in Kuopio | Climate change policies in Basel
In Finnish
Yhteenveto Helsingin energiapäätös 2015 | Helsingin energiapäätöksen vaihtoehdot 2015 | Helsingin energiapäätökseen liittyviä arvoja | Helsingin energiapäätös 2015.pptx
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