Model Description¶

Sets¶

Name	Symbol	Description
Years	n	Year of the project
Periods	t	Period that are divided the years
Scenarios	s	Scenarios analalized

Parameters¶

Data analysis parameters¶

Name	Unit	Description
Startdate	Day	Start date of the analysis
PlotTime	Days	Number of days for the plot of energy dispatch
PlotDay	Day	Start date for the dispatch plot
PlotScenario		Scenario to be plot

PV parameters¶

Name	Unit	Description
PVNominalCapacity	W/unit	Nominal capacity of one PV unit
InverterEfficiency	%	efficiency of the inverter to transform DC energy to AC
PVinvesmentCost	USD/W	Investment Cost to install PV panels
PVEnergyProduction	Wh	The yield of energy of one PV unit in the period (i,t)

Battery bank parameters¶

Name	Unit	Description
ChargeBatteryEfficiency	%	The efficiency of the battery to charge energy
DischargeBatteryEfficiency	%	The efficiency of the battery to discharge energy
DeepOfDischarge	%	Minimum percentage of energy of the nominal capacity of the battery
MaximunBatteryChargeTime	hour	Maximum time to charge from 0 % to a 100 % of energy in the battery
MaximunBatteryDischargeTime	hour	Maximum time to discharge from 100 % to a 0 % of energy in the battery
BatteryInvesmentCost	USD/Wh	Investment cost to install a Wh of batteries
BatteryRepostionTime	Years	Time for the remplacement of the battery

Diesel generator parameters¶

Name	Unit	Description
GeneratorEfficiency	%	Generator efficiency to transform heat into electricity
LowHeatingValue	W/L	Low heating value of the diesel
DieselCost	USD/L	Diesel cost
GeneratorInvesmentCost	USD/W	Investment cost to install a diesel generator

Energy balance parameters¶

Name	Unit	Description
EnergyDemand (s,t)	W	The total energy demand of the system for each scenario.
LostLoadProbability	%	The percentage of the demand that the micro-grid has to provide
ValueOfLostLoad	USD/W	The price of the load that is not supply to the system

Project parameters¶

Name	Unit	Description
Periods	Hours	Number of periods of the year
Years	Years	Number of years in the project
DeltaTime	Hours	Time step of the analysis of the energy flow
PorcentageFunded	%	Percentage of the total investment that is Funded by a bank or another entity
MaintenanceOperationCostPV	%	Percentage of the total investment spend in operation and management of PV
MaintenanceOperationCostBattery	%	Percentage of the total investment spend in operation and management of the battery
MaintenanceOperationCostGenerator	%	Percentage of the total investment spend in operation and management of the genset
DiscountRate	%	Discount rate of the project
InterestRate	%	Interest rate of the loan
ProbalityOccurrence (s)	%	Probability of occurrence of each scenario
N	Years	Years of duration of the project

Variables¶

PV variables¶

Name	Unit	Description
PVUnits	unit	Number of installed PV
TotalEnergyPV (s,t)	Wh	Energy generated for all the PVs in the system in each scenario
OyMCostPV	USD	Cost of the OyM of the PV during the life time of the proyect

Battery variables¶

Name	Unit	Description
BatteryNominalCapacity	Wh	Nominal capacity of the battery bank
EnergyBatteryDischarge (s,t)	Wh	Energy that flows out of the battery in each scenario
EnergyBatteryCharge (s,t)	Wh	Energy that flows in to the battery in each scenario
StateOfChargeBattery (s,t)	Wh	Energy inside the battery in each scenario
MaximunChargePower	W	Maximum charge power
MaximunDischargePower	W	Maximum discharge power

Diesel generator variables¶

Name	Unit	Description
GeneratorNominalCapacity	W	Nominal capacity of the diesel generator
DieselConsumed (s,t)	L	Diesel consumed to produce energy
GeneratorEnergy (s,t)	Wh	Energy produced by the diesel generator
DieselCostTotal (s)	USD	Cost of the diesel during the life time of the project

Energy balance variables¶

Name	Unit	Description
LostLoad (s,t)	Wh	Energy not supply by the system in each scenario
EnergyCurtailment (s,t)	Wh	Curtailment of solar energy in each scenario
LostLoadCostTotal (s)	USD	Cost of the Lost load during the life time of the project

Project variables¶

Name	Unit	Description
FinancialCost	USD	Annual constant payment for the loan adquire to finance the project
ScenarioNetPresentCost	USD	Net present cost of each scenario
InitialInversion	USD	Value of the inital inversion of the project
OyMCost	USD	Total cost of the Operation and maintenence during the life time of the project
FinancialCostTotal	USD	Total cost of the payment for the loan during the life time of the project
BatteryRepositionCost	USD	Cost for the reposition of the battery

Modeling of the system¶

Objective function¶

The objective function will minimize the sum of the multiplication of the net present cost of each scenario and their probability of occurrence.

$Objective Funtion = \sum _s\mathit{ScenarioNetPresentCost}_s \cdot \mathit{ProbalityOccurrence}_s$

The net present cost of each scenario is computed with the following equation:

$\mathit{ScenarioNetPresentCost}_s = InitialInversion + OyMCost + FinancialCostTotal + BatteryRepositionCost + \mathit{DieselCostTotal}_s + \mathit{LostLoadCostTotal}_s$

The total investment equation is:

$InitialInversion = (PVinvestmentCost \cdot PVNominalCapacity \cdot PVUnits +BatteryNominalCapacity \cdot BatteryInvestmentCost + GeneratorInvestmentCost \cdot GeneratorNominalCapacity ) \cdot (1 - PorcentageFunded)$

The OyMCost is calculated by the following equation:

$OyMCostPV = PVinvesmentCost \cdot PVNominalCapacity \cdot PVUnits \cdot MaintenanceOperationCostPV$

$OyMCostBattery = BatteryNominalCapacity \cdot BatteryInvesmentCost \cdot MaintenanceOperationCostBattery$

$OyMCostGenerator = GeneratorInvesmentCost \cdot GeneratorNominalCapacity \cdot MaintenanceOperationCostGenerator$

$OyMCost = \sum _n\frac{ OyMCostPV + OyMCostBattery + OyMCostGenerator} {(1 + DiscountRate)^{n}}$

The financial cost is a fix amount, that is payed each period to pay the loan acquire to finance a percentage of the initial investment and is calculated with the following equation:

$FinancialCost = \frac{INV \cdot PorcentageFunded \cdot InterestRate} {1 - (1 +InterestRate)^{-N}}$

The total cost incurred in the lifetime of the project for the financial cost is calculated with equation:

$FinancialCostTotal = \sum _n\frac{FinancialCost} {(1+ DiscountRate)^{n}}$

The replacement cost is given by the fallowing equation:

$\mathit{ReplacementCost}_{10} = \frac{BatteryNominalCapacity \cdot BatteryInvesmentCost} {(1+ DiscountRate)^{N}}$

The Diesel cost is calculated by:

$\mathit{DieselCostTotal}_s = \sum _n\frac{\sum _t\mathit{DieselConsumed}_{s,t} \cdot DieselCost} {(1+ DiscountRate)^{n}}$

Finally the cost for the unmment load is calculated with the following equation:

$\mathit{LostLoadCostTotal}_s = \sum _n\frac{\sum _t\mathit{LostLoad}_{s,t} \cdot ValueOfLostLoad} {(1+ DiscountRate)^{n}}$

PV model¶

The equation that model the PV array energy yield is given by:

$\mathit{TotalEnergyPV}_{s,t} = \mathit{PVEnergyProduction}_{s,t} \cdot \mathit{InverterEfficiency} \cdot \mathit{PVUnits}$

Diesel generator¶

The fuel consumption is modeled by:

$\mathit{DieselConsumed}_{s,t} = \mathit{GeneratorEnergy}_{s,t} / (\mathit{GeneratorEfficiency} \cdot \mathit{LowHeatingValue})$

In order to ensure that the generator will not exceed his capacity the fallowing constraint is added to the model:

$\mathit{GeneratorNominalCapacity} \cdot \mathit{DeltaTime} \geq \mathit{GeneratorEnergy}_{s,t}$

Battery bank¶

The state of charge of the battery is modeled by:

$t=1: \mathit{StateOfChargeBattery}_{s,1} = BatteryNominalCapacity - \mathit{EnergyBatteryCharge}_{s,1} \cdot \mathit{ChargeBatteryEfficiency} - \mathit{EnergyBatteryDischarge}_{s,1} \cdot \mathit{DischargeBatteryEfficiency}$

$t>1: \mathit{StateOfChargeBattery}_{s,t} = BatteryNominalCapacity - \mathit{EnergyBatteryCharge}_{s,t} \cdot \mathit{ChargeBatteryEfficiency} - \mathit{EnergyBatteryDischarge}_{s,t} \cdot \mathit{DischargeBatteryEfficiency}$

In this equations is important to highlight that in the period 1 the stated of charge of the batterie is equal to the total capacity of the battery.

In order to ensure the durability of the battery a minimum depth of discharge (%) and maximum charge are establish as a constraint:

$BatteryNominalCapacity \cdot DeepOfDischarge \leq \mathit{StateOfChargeBattery}_{s,t} \leq BatteryNominalCapacity$

The maximum power of charge and discharge are modeled as follow:

$MaximunChargePower = BatteryNominalCapacity/MaximunBatteryChargeTime MaximunDischargePower = BatteryNominalCapacity/MaximunBatteryDischargeTime$

The flow of energy is into and out of the battery is restricted by:

$\mathit{EnergyBatteryCharge}_{s,t} \geq - MaximunChargePower \cdot DeltaTime \mathit{EnergyBatteryDischarge}_{s,t} \leq MaximunDischargePower \cdot DeltaTime$

Energy constraints¶

In order to ensure a perfect match between generation and demand, an energy balance is created as a constraint.

$\mathit{EnergyDemand}_{s,t} = \mathit{TotalEnergyPV}_{s,t} + \mathit{DieselConsumed}_{s,t} + \mathit{EnergyBatteryCharge}_{s,t} + \mathit{EnergyBatteryDischarge}_{s,t} + \mathit{EnergyCurtailment}_{s,t} + \mathit{LostLoad}_{s,t}$

This constraint is used to ensure that a percentage of the demand will always be supply and is express as follow:

$LostLoadProbability = \frac{\sum _t\mathit{LostLoad}_{s,t}} {\sum _t\mathit{EnergyDemand}_{s,t}}$