Optimal design of total integrated residential complexes involving water-energy-waste nexus

Abstract

This paper presents a multi-objective optimization formulation for enhancing the sustainable development of a residential complex. The approach accounts for the water-energy-waste nexus of the complex and enables various pathways for system integration. For conserving the fresh water demands, the proposed model includes the synthesis of water networks while accounting for wastewater reclamation and recycle and rainwater harvesting. The proposed model also incorporates the optimal design of a residential cogeneration unit to satisfy the demands for electric power and hot water. An absorption refrigeration system is considered to utilize waste heat and provide the needed refrigeration. The emitted carbon dioxide is fed to an algae growth system, which is integrated with the use of reclaimed water. A solid-waste gasification system is considered to provide electric power and heat to the residential complex. The optimization approach accounts for all the interactions of the involved units and for the seasonal variabilities of the system. A case study for a residential complex of Mexico is solved.

Appendices

Appendix: Mathematical model formulation for solving the addressed problem

This section presents the proposed mathematical model for solving the addressed problem, which is described as follows.

Rainwater collecting system

To determine the amount of harvested rainwater in a time period t ($F_{t}^{{\mathrm{RW}}}$), the conditioned collecting area is multiplied ($A^{{\mathrm{RW}}}$) times the precipitation in that period (${\text{Precip}}_{t}$):

$$F_{t}^{\rm RW} = A^{\rm RW} {\text{Precip}}_{t} { , }\,\forall t \in T$$

(1)

Rainwater storage system

The capacity for the rainwater storage system (${\text{Cap}}^{{\mathrm{RWSS}}}$) should be greater than the water stored over the considered periods ($F_{t}^{{\mathrm{RW}}}$):

$${\text{Cap}}^{{\mathrm{RWSS}}} \ge F_{t}^{{\mathrm{RW}}} , \;\forall t \in T$$

(2)

$${\text{Cap}}^{{\mathrm{RWSS}}} \ge F_{t}^{{\mathrm{SRW}}} , \,\forall t \in T$$

(3)

Balance in the rainwater storage system

The stored water in the rainwater storage system over the period of time t ($F_{t}^{{\mathrm{SRW}}}$) is equal to the stored water at the end of previous time period ($F_{t-1}^{{\mathrm{SRW}}}$), plus the inlet water to the storage system ($F_{t}^{{\mathrm{RW}}}$), minus the outlet water from the storage system that is sent to the residential complex ($f_{t}^{\text{RW-residential}}$), to the algae system ($f_{t}^{\text{RW-algae}}$), to the boiler ($f_{t}^{\text{RW-boiler}}$), the prime mover (in this case an internal combustion engine, ICE) ($f_{t}^{\rm RW-ICE}$), to the gardens ($f_{t}^{\text{RW-garden}}$), and to the cold water sale ($f_{t}^{\text{RW-sale}}$):

$$\begin{aligned} F_{t}^{{\mathrm{SRW}}} = F_{t-1}^{{\mathrm{SRW}}} + F_{t}^{{\mathrm{RW}}}-f_{t}^{\text{RW-residential}}-f_{t}^{\text{RW-algae}}-f_{t}^{\text{RW-boiler}}-f_{t}^{\rm RW-ICE}-f_{t}^{\text{RW-garden}} \hfill \\-f_{t}^{\text{RW-sale}} { , }\,\forall t \in T \hfill \\ \end{aligned}$$

(4)

Fresh water needed

The total fresh water consumed in the time period t ($F_{t}^{{\mathrm{FW}}}$) is equal to the sum of the inlet water from the residential complex ($f_{t}^{\text{FW-residential}}$), the algae system ($f_{t}^{\text{FW-algae}}$), the boiler ($f_{t}^{\text{FW-boiler}}$), the internal combustion engine ($f_{t}^{\text{FW-ICE}}$), and the garden ($f_{f}^{\text{FW-garden}}$), which is stated as follows:

$$F_{t}^{{\mathrm{FW}}} = f_{t}^{\mathrm{FW-residential}} + f_{t}^{\mathrm{FW-algae}} + f_{t}^{\mathrm{FW-boiler}} + f_{t}^{\mathrm{FW-ICE}} + f_{t}^{\mathrm{FW-garden}},\,\forall t \in T$$

(5)

Capacity for the fresh water storage tank

The capacity for the fresh water storage tank (${\text{Cap}}^{{\mathrm{FWSS}}}$) should be greater than the water stored over the considered periods ($F_{t}^{{\mathrm{FW}}}$):

$${\text{Cap}}^{{\mathrm{FWSS}}} \ge F_{t}^{{\mathrm{FW}}} , \;\forall t \in T$$

(6)

Internal combustion engine

An internal combustion engine (ICE) is considered to produce in site the electricity needed in the residential complex. This prime mover uses heat at high temperature, which comes from the combustion of natural gas, or resides from the residential complex or biomass from the algae system. Furthermore, this internal combustion engine discharges significant amounts of waste heat, and this can be used to heat cold water and to satisfy the residential needs of hot water for domestic uses, or this waste heat can be used to run the absorption refrigeration system and to produce cooling needed in the residential complex. Then, the following relationships are needed to model the interactions of these units and to design the internal combustion engine.

Electricity generated in site by the internal combustion engine

The electricity generated for the ICE ($E_{t}^{\rm ICE}$) is calculated as follows:

$$E_{t}^{\rm ICE} = \alpha^{\text{ICE-E}} F_{t}^{\text{NG-ICE}} ,\;\forall t \in T$$

(7)

where $\alpha^{\text{ICE-E}}$ is an electricity conversion factor for the ICE and $F_{t}^{\text{NG-ICE}}$ is the natural gas required for the ICE to generate electricity.

Needed cold water for the ICE

The cold water required for the cogeneration system ($F_{t}^{\text{CW-ICE}}$) is equal to the cold water conversion factor for the ICE ($\alpha^{\text{ICE-CW}}$) multiplied by the inlet natural gas ($F_{t}^{\text{NG-ICE}}$) to the system:

$$F_{t}^{\text{CW-ICE}} = \alpha^{\text{ICE-CW}} F_{t}^{\text{NG-ICE}} ,\;\forall t \in T$$

(8)

Balance for the cold water in the ICE

The total cold water sent to the ICE ($F_{t}^{\text{CW-ICE}}$) is obtained from the rainwater storage system ($f_{t}^{\rm RW-ICE}$) and from the fresh water ($f_{t}^{\text{FW-ICE}}$):

$$F_{t}^{\text{CW-ICE}} = f_{t}^{\rm RW-ICE} + f_{t}^{\text{FW-ICE}} ,\;\forall t \in T$$

(9)

Distributed electricity from the ICE

The electricity distributed by the ICE ($E_{t}^{\rm ICE}$) is equal to the sum for the electricity sent to the residential complex ($e_{t}^{\text{ICE-residential}}$), to the algae system ($e_{t}^{\text{ICE-algae}}$), to the greywater treatment ($e_{t}^{\rm ICE-GWT}$), to the wastewater treatment ($e_{t}^{\text{ICE-WWT}}$), and to the sale ($e_{t}^{\text{ICE-sale}}$):

$$E_{t}^{\rm ICE} = e_{t}^{\text{ICE-residential}} + e_{t}^{\text{ICE-algae}} + e_{t}^{\rm ICE-GWT} + e_{t}^{\text{ICE-WWT}} + e_{t}^{\text{ICE-sale}} ,\;\forall t \in T$$

(10)

Generated hot water in the ICE

To calculate the hot water generated by the ICE ($H_{t}^{\rm ICE}$), the hot water factor for the ICE ($\beta^{\text{ICE-HW}}$) is multiplied by the inlet natural gas into the ICE ($F_{t}^{\text{NG-ICE}}$):

$$H_{t}^{\rm ICE} = \beta^{\text{ICE-HW}} F_{t}^{\text{NG-ICE}} ,\;\forall t \in T$$

(11)

Needed natural gas in the ICE

The natural gas needed to operate the ICE ($F_{t}^{\text{NG-ICE}}$) can be obtained for purchase ($f_{t}^{\text{ NG-purchased-ICE}}$), for the gas treatment unit ($f_{t}^{\text{NG-gastreatment-ICE}}$), and for the gasification system ($f_{t}^{\text{NG-gasification-ICE}}$):

$$F_{t}^{\text{NG-ICE}} = f_{t}^{\text{ NG-purchased-ICE}} + f_{t}^{\text{NG-gastreatment-ICE}} + f_{t}^{\text{NG-gasification-ICE}} ,\;\forall t$$

(12)

Distributed hot water from the ICE

The hot water generated by the ICE ($H_{t}^{\rm ICE}$) can be sent to the residential complex ($h_{t}^{\text{ICE-residential}}$), to the absorption refrigeration system ($h_{t}^{\rm ICE-ARC}$) and for sale ($h_{t}^{\text{ICE-sale}}$):

$$H_{t}^{\rm ICE} = h_{t}^{\text{ICE-residential}} + h_{t}^{\text{ICE-sale}} + h_{t}^{\rm ICE-ARC} ,\;\forall t$$

(13)

Generated flue gases in the ICE

The flue gases generated by the cogeneration system ($G_{t}^{\rm ICE}$) is equal to the emission conversion factor by the ICE ($\gamma^{\rm ICE-g}$) multiplied by the inlet natural gas into the system ($F_{t}^{\text{NG-ICE}}$):

$$G_{t}^{\rm ICE} = \gamma^{\rm ICE-g} F_{t}^{\text{NG-ICE}} , \, \,\forall {\text{t}}$$

(14)

Distributed flue gases from the ICE

The flue gases for the ICE ($G_{t}^{\rm ICE}$) can be distributed to the algae system ($Fg_{t}^{{^{\text{ICE-algae}} }}$) and to the discharge ($Fg_{t}^{{^{\text{ICE-discharge}} }}$):

$$G_{t}^{\rm ICE} = Fg_{t}^{\text{ICE-algae}} + Fg_{t}^{{^{\text{ICE-discharge}} }} , \;\forall t$$

(15)

Balance in the boiler

There are needed some energy and water balances in the boiler to design this unit and to determine the operating conditions through the different time periods, which are stated as follows.

Produced hot water in the boiler

The hot water produced by the boiler ($H_{t}^{\text{HW-boiler}}$) is equal to the boiler conversion factor ($\beta^{\rm boiler}$) multiplied by the inlet natural gas to the boiler ($F_{t}^{\text{NG-boiler}}$):

$$H_{t}^{\text{HW-boiler}} = \beta^{\rm boiler} F_{t}^{\text{NG-boiler}} , \;\forall t$$

(16)

Needed natural gas in the boiler

The natural gas needed in the boiler ($F_{t}^{\text{NG-boiler}}$) is equal to the natural gas coming from the gasification system ($f_{t}^{\text{NG-gasification-boiler}}$), plus the natural gas from the gas treatment unit ($f_{t}^{\text{NG-gastreatment-boiler}}$) and from purchase ($f_{t}^{\text{NG-boiler-purchased}}$):

$$F_{t}^{\text{NG-boiler}} = f_{t}^{\text{NG-boiler-purchased}} + f_{t}^{\text{NG-gasification-boiler}} + f_{t}^{\text{NG-gastreatment-boiler}} , \;\forall t$$

(17)

Distributed hot water from the boiler

The hot water produced in the boiler ($H_{t}^{\text{HW-boiler}}$) can be sent to the residential complex ($h_{t}^{\text{boiler-residential}}$), to the absorption refrigeration system ($h_{t}^{\text{boiler-ARC}}$) and for sale ($h_{t}^{\text{boiler-sale}}$):

$$H_{t}^{\text{HW-boiler}} = h_{t}^{\text{boiler-residential}} + h_{t}^{\text{boiler-sale}} + h_{t}^{\text{boiler-ARC}} { , }\,\forall t$$

(18)

Needed cold water in the boiler

The inlet cold water to the boiler ($F_{t}^{\text{CW-boiler}}$) is equal to the outlet hot water for the unit ($H_{t}^{\text{HW-boiler}}$):

$$F_{t}^{\text{CW-boiler}} = H_{t}^{\text{HW-boiler}} , \,\forall t$$

(19)

Balance for the cold water in the boiler

The inlet cold water to the boiler ($F_{t}^{\text{CW-boiler}}$) is the sum of the cold water from the rainwater storage system ($f_{t}^{\text{RW-boiler}}$) plus the cold water from the fresh water storage system ($f_{t}^{\text{FW-boiler}}$):

$$F_{t}^{\text{CW-boiler}} = f_{t}^{\text{RW-boiler}} + f_{t}^{\text{FW-boiler}} { , }\,\forall t$$

(20)

Flue gases produced by the boiler

The flue gases produced by the boiler ($G_{t}^{\rm boiler}$) are calculated through the efficiency conversion factor from the boiler and the hot water produced in the boiler:

$$G_{t}^{\rm boiler} = h^{\rm boiler} H_{t}^{\text{HW-boiler}} , \,\forall t$$

(21)

Distributed flue gases from the boiler

The flue gases produced by the boiler ($G_{t}^{\rm boiler}$) can be distributed as follows:

$$G_{t}^{\rm boiler} = Fg_{t}^{\text{ boiler-algae}}+ Fg_{t}^{\text{ boiler-discharge}} , \,\forall t$$

(22)

where $Fg_{t}^{\text{ boiler-algae}}$ corresponds to the gases sent to the algae system and $Fg_{t}^{\text{ boiler-discharge}}$ to the gases sent to the discharge.

Balances for the absorption refrigeration cycle

An absorption refrigeration cycle is considered for waste heat recovery system from the internal combustion engine as well as to provide the needed cooling in the housing complex. The proposed relationships to model and design this unit are the following.

Produced refrigeration in the absorption refrigeration cycle

The produced refrigeration in the absorption refrigeration cycle (ARC) ($R_{t}^{\rm ARC}$) is calculated by the refrigeration factor ($\beta^{\text{R-ARC}}$) and the inlet hot water into the refrigeration system ($H_{t}^{\text{HW-ARC}}$):

$$R_{t}^{\rm ARC} = \beta^{\text{R-ARC}} H_{t}^{\text{HW-ARC}} , \,\forall t$$

(23)

Distribution for the refrigeration from the ARC

The refrigeration generated by the ARC ($R_{t}^{\rm ARC}$) can be distributed to satisfy the demand in the residential complex ($r_{t}^{\text{ARC-residential}}$) and for sale ($r_{t}^{\text{ARC-sale}}$):

$$R_{t}^{\rm ARC} = r_{t}^{\text{ARC-residential}} + r_{t}^{\text{ARC-sale}} { , }\,\forall t$$

(24)

Needed hot water in the ARC

The hot water needed for the ARC ($H_{t}^{\text{HW-ARC}}$) can be obtained from purchase ($h_{t}^{\text{purchase-ARC}}$), from the boiler ($h_{t}^{\text{boiler-ARC}}$) and/or from the ICE ($h_{t}^{\rm ICE-ARC}$):

$$H_{t}^{\text{HW-ARC}} = h_{t}^{\text{purchase-ARC}}+ h_{t}^{\text{boiler-ARC}} + h_{t}^{\rm ICE-ARC} { , }\,\forall t$$

(25)

Algae system

An algae system is considered for trapping the generated CO₂ emissions and to produce energy from biofuels and biomass. This unit can be useful for improving the sustainability of the system and this must account for the available resources in the residential complex. The needed relationships for modeling this system are the following.

Captured CO₂

The captured CO₂ by the algae system ($G_{t}^{\rm algae}$) is equal to the CO₂ from the boiler ($Fg_{t}^{\text{ boiler-algae}}$), plus the CO₂ from the ICE ($Fg_{t}^{\text{ICE-algae}}$), and the CO₂ from the gasification system ($Fg_{t}^{\text{ gasification-algae}}$):

$$G_{t}^{\rm algae} = Fg_{t}^{\text{ boiler-algae}}+ Fg_{t}^{\text{ICE-algae}} + Fg_{t}^{\text{ gasification-algae}} { , }\,\forall t$$

(26)

Needed water for the algae system

The water needed for the algae system ($F_{t}^{\text{W-inlet-algae}}$) is obtained from harvested rainwater ($f_{t}^{\text{RW-algae}}$), from freshwater ($f_{t}^{\text{FW-algae}}$) and/or from reclaimed water produced by the greywater treatment unit ($f_{t}^{\text{reclaim-algae}}$):

$$F_{t}^{\text{W-inlet-algae}} = f_{t}^{\text{RW-algae}} + f_{t}^{\text{FW-algae}} + f_{t}^{\text{reclaim-algae}} { , }\,\forall t$$

(27)

The water needed for the algae system ($F_{t}^{\mathrm{W-inlet-algae}}$) is equal to the algae water factor ($\alpha^{{\mathrm{algaeW}}}$) multiplied for the inlet CO₂ ($G_{t}^{\rm algae}$) to the system:

$$F_{t}^{\mathrm{W-inlet-algae}} = \alpha^{{\rm algaeW}} G_{t}^{\rm algae} , \,\forall t$$

(28)

Needed electricity in the algae system

The electricity needed for the algae system ($E_{t}^{\rm algae}$) is equal to the algae electricity factor ($\alpha^{\text{algae-E}}$) multiplied by the gases inlet to the algae system ($G_{t}^{\rm algae}$):

$$E_{t}^{\rm algae} = \alpha^{\text{algae-E}} G_{t}^{\rm algae} { , }\,\forall t$$

(29)

Satisfied electricity in the algae system

The electricity required by the algae system ($E_{t}^{\rm algae}$) can be obtained from the ICE ($e_{t}^{\text{ICE-algae}}$) and from purchase ($e_{t}^{\text{purchased-algae}}$):

$$E_{t}^{\rm algae} = e_{t}^{\mathrm{ICE-algae}} + e_{t}^{\mathrm{purchased-algae}} , \;\forall t$$

(30)

Produced biodiesel in the algae system

The produced biodiesel by the algae system ($F_{t}^{\mathrm{Biodisel}}$) is equal to the biodiesel conversion factor ($\alpha^{\mathrm{Biodisel}}$) times the inlet gases to the algae system ($G_{t}^{\rm algae}$):

$$F_{t}^{Biodisel} = \alpha^{\mathrm{Biodisel}} G_{t}^{\rm algae} , \,\forall t$$

(31)

Produced wastewater in the algae system

The algae system also produces wastewater, which is calculated as follows:

$$F_{t}^{\text{WW-algae}} = \alpha^{\text{WW-algae}} F_{t}^{\text{W-inlet-algae}} { , }\,\forall t$$

(32)

where $F_{t}^{\text{WW-algae}}$ is the wastewater produced by the algae system, $\alpha^{\text{WW-algae}}$ is the conversion factor for the wastewater in the algae system, and $F_{t}^{\text{W-inlet-algae}}$ is the inlet water to the algae system.

Needed water for gardening

One of the main water consumers in the housing complex corresponds to gardening ($F_{t}^{\text{W-garden}}$), and the proposed approach considers that this can be obtained from harvested rainwater ($f_{t}^{\text{RW-garden}}$), freshwater ($f_{t}^{\text{FW-garden}}$), and reclaimed water by the greywater treatment unit ($f_{t}^{\text{reclaim-garden}}$):

$$F_{t}^{\text{W-garden}}= f_{t}^{\text{RW-garden}} + f_{t}^{\text{FW-garden}}+ f_{t}^{\text{reclaim-garden}} , \,\forall t$$

(33)

Demands of the residential complex

The electricity, hot water, cold water, and cooling demands from the residential complex must be satisfy through the integrated system, and for modeling these aspects the following relationships are proposed.

Cold water demand in the residential complex

The cold water demand in the residential complex is modeled as follows:

$$F_{t}^{\text{CW-residential}} = f_{t}^{\text{RW-residential}} + f_{t}^{\text{FW-residential}} { , }\,\forall t$$

(34)

where $F_{t}^{\text{CW-residential}}$ is the cold water needed in the residential complex for different domestic uses, $f_{t}^{\text{RW-residential}}$ is the water used in the residential complex from the harvested rainwater, and $f_{t}^{\text{FW-residential}}$ is the freshwater required for the residential complex.

Hot water demand in the residential complex

The hot water demand in the residential complex is modeled as follows:

$$H_{t}^{{\mathrm{residential}}} = h_{t}^{\text{ICE-residential}} + h_{t}^{\text{boiler-residential}} , \,\forall t$$

(35)

where $H_{t}^{{\mathrm{residential}}}$ is the warm water needed in the residential complex, $h_{t}^{\text{ICE-residential}}$ is the warm water from the ICE, and $h_{t}^{\text{boiler-residential}}$ is the one from the boiler.

Needed electricity

The electricity needed by the residential complex ($E_{t}^{{\mathrm{residential}}}$) is obtained from the ICE ($e_{t}^{\text{ICE-residential}}$) and from purchase ($e_{t}^{{\mathrm{purchased}}}$):

$$E_{t}^{{\mathrm{residential}}} = e_{t}^{\text{ICE-residential}} + e_{t}^{{\mathrm{purchased}}} { , }\,\forall t$$

(36)

Needed cooling

The cooling needed in the residential complex ($R_{t}^{{\mathrm{residential}}}$) is generated and sent from the ARC ($r_{t}^{\text{ARC-residential}}$) and from purchase ($r_{t}^{\text{purchased-residential}}$):

$$R_{t}^{{\mathrm{residential}}} = r_{t}^{\text{ARC-residential}} + r_{t}^{\text{purchased-residential}} ,\, \,\forall t$$

(37)

Greywater generated in the residential complex

One of the most important problems in the residential complexes corresponds to the significant amount of greywater produced, and if this is discharged without any treatment it represents a tremendous environmental problem. This way, in this paper, the proposed superstructure contemplates the treatment and reuse of this greywater, and the implemented relationships to model this issue are the following.

Generated greywater in the residential complex

The generated greywater in the residential complex ($F_{t}^{\text{GW-residential}}$) is calculated as follows:

$$F_{t}^{\text{GW-residential}} = \alpha^{\text{GW-residential}} (F_{t}^{\text{CW-residential}} + H_{t}^{{\mathrm{residential}}} ){ , }\,\forall t$$

(38)

where $\alpha^{\text{GW-residential}}$ is a conversion factor for the greywater generated in the residential complex, $F_{t}^{\text{CW-residential}}$ is the inlet cold water to the residential complex, and $H_{t}^{{\mathrm{residential}}}$ is the inlet warm water to the residential complex.

Treating greywater

The inlet water into the greywater treatment unit ($F_{t}^{\text{Inlet-GW}}$) is calculated by the sum from the residential water ($F_{t}^{\text{GW-residential}}$) plus the water from the algae system ($F_{t}^{\text{WW-algae}}$):

$$F_{t}^{\text{Inlet-GW}} = F_{t}^{\text{GW-residential}} + F_{t}^{\text{WW-algae}} { , }\,\forall t$$

(39)

Generated reclaimed water

The treated water by the greywater treatment unit ($F_{t}^{\text{reclaimed-GW}}$) is equal to the reclaimed water conversion factor ($\alpha^{\text{reclaimed-GW}}$) multiplied by the inlet greywater to the treatment unit ($F_{t}^{\text{Inlet-GW}}$):

$$F_{t}^{\text{reclaimed-GW}} = \alpha^{\text{reclaimed-GW}} F_{t}^{\text{Inlet-GW}} \, ,\quad \, \,\forall t$$

(40)

Needed electricity in the greywater treatment unit

The needed electricity for the greywater treatment unit ($E_{t}^{\rm GW}$) is calculated as follows:

$$E_{t}^{\rm GW} = \alpha^{\mathrm{GW-E}} F_{t}^{\text{Inlet-GW}} , \,\forall t$$

(41)

where $\alpha^{\mathrm{GW-E}}$ is the electricity conversion factor for the greywater treatment unit and $F_{t}^{\text{Inlet-GW}}$ is the inlet water into the unit.

Satisfied electricity in the greywater treatment

The electricity needed for the greywater treatment unit ($E_{t}^{\rm GW}$) can be satisfied by the ICE ($e_{t}^{\rm ICE-GWT}$) and from purchase ($e_{t}^{\text{ purchased-GWT}}$):

$$E_{t}^{\rm GW} = e_{t}^{\rm ICE-GWT} + e_{t}^{\text{ purchased-GWT}} , \,\forall t$$

(42)

Distribution of reclaimed greywater

The treated water from the greywater treatment unit ($F_{t}^{\text{reclaimed-GW}}$) can be sent to the algae system ($f_{t}^{\text{reclaimed-algae}}$), to the garden ($f_{t}^{\mathrm{reclaimed-garden}}$) and/or to the drainage ($f_{t}^{\text{reclaimed-drainage}}$):

$$F_{t}^{\text{reclaimed-GW}} = f_{t}^{\mathrm{reclaimed-algae}} + f_{t}^{\mathrm{reclaimed-garden}} + f_{t}^{\text{reclaimed-drainage}} { , }\,\forall t$$

(43)

Generated gases in the greywater treatment unit

In the greywater treatment unit, some emissions are produced ($F_{t}^{\text{NG-GWT}}$), which are calculated multiplying the natural gas conversion factor times the greywater treatment ($\alpha^{\text{NG-GWT}}$) by the inlet water to the system ($F_{t}^{\text{Inlet-GW}}$):

$$F_{t}^{\text{NG-GWT}} = \alpha^{\text{NG-GWT}} F_{t}^{\text{Inlet-GW}} , \;\forall t$$

(44)

Black wastewater generated in the residential complex

The black water generated in the residential complex is modeled as follows.

Generated black wastewater

The generated black wastewater in the residential complex ($F_{t}^{\rm WW}$) is calculated by the sum of the water from the residential complex ($F_{t}^{\text{CW-residential}}$) plus the warm water for the residential complex ($H_{t}^{{\mathrm{residential}}}$), multiplied by the wastewater conversion factor for the residential complex ($\alpha^{\text{WW-residential}}$):

$$F_{t}^{\rm WW} = \alpha^{\text{WW-residential}} (F_{t}^{\text{CW-residential}} + H_{t}^{{\mathrm{residential}}} ){ , }\,\forall t$$

(45)

Treated black wastewater

The treated black wastewater by this unit ($F_{t}^{\text{treated-WW}}$) is equal to the inlet water into the black wastewater unit ($F_{t}^{\rm WW}$) multiplied by the factor for the wastewater treatment unit ($\alpha^{\text{treating-WW}}$):

$$F_{t}^{\text{treated-WW}} = \alpha^{\text{treating-WW}} F_{t}^{\rm WW} , \;\forall t$$

(46)

Needed electricity in the black wastewater treatment unit

The black wastewater treatment unit needs electricity for functioning; this electricity ($E_{t}^{\rm WW}$) is calculated as follows:

$$E_{t}^{\rm WW} = \alpha^{WW-E} F_{t}^{\rm WW} { , }\,\forall t$$

(47)

where $\alpha^{\mathrm{WW-E}}$ is the electricity factor for the wastewater and $F_{t}^{\rm WW}$ is the inlet water to the black wastewater treatment unit.

Satisfied electricity in the wastewater treatment

To satisfy the electricity for the unit ($E_{t}^{\rm WW}$), this can be obtained from the ICE ($e_{t}^{\text{ICE-WWT}}$) and/or for purchase ($e_{t}^{\text{purchased-WWT}}$):

$$E_{t}^{\rm WW} = e_{t}^{\text{ICE-WWT}} + e_{t}^{\text{purchased-WWT}} { , }\,\forall t$$

(48)

Distribution of treated wastewater

The treated black wastewater ($F_{t}^{\text{treated-WW}}$) only can be sent to the drainage ($F_{t}^{\text{WW-Drainage}}$), which is modeled as follows:

$$F_{t}^{\text{treated-WW}} = F_{t}^{\text{WW-Drainage}} { , }\,\forall t$$

(49)

Natural gas generated in the black wastewater treatment unit

The black wastewater treatment unit generates natural gas ($F_{t}^{\text{NG-WW}}$), which is calculated with the natural gas factor for the wastewater treatment unit ($\alpha^{\text{NG-WW}}$) multiplied by the inlet flowrate to the wastewater treatment unit ($F_{t}^{\rm WW}$):

$$F_{t}^{\text{NG-WW}} = \alpha^{\text{NG-WW}} F_{t}^{\rm WW} { , }\,\forall t$$

(50)

Natural gas treatment unit

It is needed a treatment unit to treat the generated raw gases to be useful to use in different applications, and to model this unit the following relationships are needed.

Inlet natural gas to the treatment unit

The natural gas inlet to the treatment unit ($F_{t}^{\text{NGT-Inlet}}$) is composed by the gases from the greywater treatment unit ($F_{t}^{\text{NG-GWT}}$) and gases from the wastewater treatment ($F_{t}^{\text{NG-WW}}$):

$$F_{t}^{\text{NGT-Inlet}} = F_{t}^{\text{NG-GWT}} + F_{t}^{\text{NG-WW}} { , }\,\forall t$$

(51)

Treated natural gas

The outlet gases from the natural gas treatment system ($F_{t}^{\text{ NG-Outlet}}$) are calculated as follows:

$$F_{t}^{\text{ NG-Outlet}} = \alpha^{\text{NG-NGT}} F_{t}^{\text{NGT-Inlet}} { , }\,\forall t$$

(52)

where $\alpha^{\text{NG-NGT}}$ is the natural gas conversion factor for the natural gas treatment unit and $F_{t}^{\text{NGT-Inlet}}$ is the one inlet to the system.

Distribution of treated natural gas

The outlet gases from the natural gas treatment unit ($F_{t}^{\text{ NG-Outlet}}$) can be utilized in the ICE ($f_{t}^{\text{NG-gastreatment-ICE}}$), the boiler ($f_{t}^{\text{NG-gastreatment-boiler}}$), the gasification system ($f_{t}^{\text{NG-gastreatment-gasification}}$) and can be sold to an external company ($f_{t}^{\text{NG-gastreatment-sale}}$):

$$\begin{aligned} F_{t}^{\text{ NG-Outlet}} = f_{t}^{\text{NG-gastreatment-ICE}} + f_{t}^{\text{NG-gastreatment-boiler}} + f_{t}^{\text{NG-gastreatment-gasification}} \hfill \\ + f_{t}^{\mathrm{NG-gastreatment-sale}} , \,\forall t \hfill \\ \end{aligned}$$

(53)

Generated solid waste

In the residential complexes, there is a huge production of solid waste. This solid waste generates a lot of environmental problems. Therefore, in the proposed superstructure there is considered the gasification for the solid waste, and this way to avoid the production of this waste and simultaneously to produce electricity, which is needed as input in several units of the integrated residential complex. The proposed relationships for modeling the gasification systems are stated as follows.

Produced NG in the gasification process

The natural gas produced by the gasification process ($F_{t}^{\text{NG-gasification}}$) is equal to the gasification conversion factor for the unit ($\alpha^{\rm gasification}$) multiplied by the produced solid waste in the residential complex ($F_{t}^{{\mathrm{solidwaste}}}$):

$$F_{t}^{\text{NG-gasification}} = \alpha^{\rm gasification} F_{t}^{{\mathrm{solidwaste}}} { , }\,\forall t$$

(54)

Distribution of the generated NG in the gasification process

The natural gas produced in the gasification system ($F_{t}^{\text{NG-gasification}}$) can be sent to the ICE ($f_{t}^{\text{NG-gasification-ICE}}$), to the boiler ($f_{t}^{\text{NG-gasification-boiler}}$) and/or can be sale to an external company ($f_{t}^{\text{NG-gasification-sale}}$):

$$F_{t}^{\text{NG-gasification}} = f_{t}^{\text{NG-gasification-ICE}} + f_{t}^{\text{NG-gasification-boiler}} + f_{t}^{\text{NG-gasification-sale}}{ , }\,\forall t$$

(55)

Generated flue gases

In the gasification process, flue gases are generated, which are calculated as follows:

$$G_{t}^{\rm gasification} = \alpha^{\text{fluegases-gasification}} F_{t}^{{\mathrm{solidwaste}}} { , }\,\forall t$$

(56)

where $G_{t}^{\rm gasification}$ is the flow of gases generated by the gasification process, $\alpha^{\text{fluegases-gasification}}$ is the flue gas conversion factor for the gasification process, and $F_{t}^{{\mathrm{solidwaste}}}$ is the inlet solid waste to the gasification process.

Distributed flue gases from the gasification process

The gases generated in the gasification process ($G_{t}^{\rm gasification}$) can be sent to the algae system ($g_{t}^{\text{ gasification-algae}}$) and/or to the discharge ($g_{t}^{\text{gasification-discharge}}$):

$$G_{t}^{\rm gasification} = g_{t}^{\text{ gasification-algae}} + g_{t}^{\mathrm{gasification-discharge}} , \;\forall t$$

(57)

Needed fuel in the gasifier

The gasifier needs fuel to operate, which is calculated as follows:

$$F_{t}^{\text{NG-needed-gasification}} = \alpha^{\text{NG-needed-gasification}} F_{t}^{{\mathrm{solidwaste}}} { , }\,\forall t$$

(58)

where $F_{t}^{\text{NG-needed-gasification}}$ is the needed fuel in the gasifier, $\alpha^{\text{NG-needed-gasification}}$ is the natural gas factor for the gasifier, and $F_{t}^{{\mathrm{solidwaste}}}$ is the inlet solid waste inlet to the gasifier.

Balance for the needed NG in the gasifier

The gases needed in the gasifier ($F_{t}^{\text{NG-needed-gasification}}$) are obtained from the NG treatment unit ($f_{t}^{\text{NG-gastreatment-gasification}}$) and from purchase ($f_{t}^{\text{NG-gasification-purchased}}$):

$$F_{t}^{\text{NG-needed-gasification}} = f_{t}^{\text{NG-gastreatment-gasification}} + f_{t}^{\text{NG-gasification-purchased}} { , }\,\forall t$$

(59)

Operating costs

There are several operating costs associated to the system, and these are calculated as follows.

Fresh water cost

The cost for the total fresh water (${\text{Cost}}^{{\mathrm{FW}}}$) required in the process is equal to the unitary cost for the fresh water (${\text{UC}}^{\text{FW}}$) times the total fresh water required in all the time periods:

$${\text{Cost}}^{{\mathrm{FW}}} = {\text{UC}}^{\text{FW}} \sum\limits_{t} {F_{t}^{{\mathrm{FW}}} }$$

(60)

NG cost

The cost for the natural gas (${\text{Cost}}^{{\mathrm{NG}}}$) utilized in the process is calculated by the natural gas purchased for the ICE, plus the natural gas purchased for the boiler, plus the natural gas purchased for the gasification process, multiplying by the unit cost for the natural gas (${\text{UC}}^{\text{NG}}$):

$${\text{Cost}}^{{\mathrm{NG}}} = {\text{UC}}^{{\text{NG}}} \left(\sum\limits_{t} f_{t}^{{\text{NG-purchased-ICE}}} + \sum\limits_{t} f_{t}^{{\text{NG-boiler-purchased}}} + \sum\limits_{t} {f_{t}^{{\text{NG-gasification-purchased}}}}\right)$$

(61)

Electricity cost

The total cost for the electricity needed (${\text{Cost}}^{{\mathrm{E}}}$) is equal to the electricity purchased for the residential complex, plus the electricity purchased for the algae system, plus the electricity purchased for the greywater treatment, and plus the electricity that cannot be satisfied for other system for the wastewater treatment, multiplying by the unitary cost for electricity (${\text{UC}}^{\text{E}}$):

$${\text{Cost}}^{{\mathrm{E}}} = \sum\limits_{t} {{\text{UC}}^{\text{E}} } \left(e_{t}^{\text{purchased-residential}} + e_{t}^{\text{purchased-algae}} + e_{t}^{\text{ purchased-GWT}} + e_{t}^{\text{purchased-WWT}} \right)$$

(62)

Cooling cost

The cost for the needed refrigeration (${\text{Cost}}^{{\mathrm{ref}}}$) in the residential complex is equal to the sum of the refrigeration required in each period multiplied by the unit cost for refrigeration (${\text{UC}}^{\text{ref}}$):

$${\text{Cost}}^{{\mathrm{ref}}} = {\text{UC}}^{\text{ref}} \sum\limits_{t} {r_{t}^{\text{purchased-residential}} }$$

(63)

Total operating cost

The total operating cost for the system (${\text{TotOpCost}}$) is calculated by the sum of the fresh water cost, plus the natural gas cost, plus the cost for the purchased electricity, plus the cost for the refrigeration required in the process:

$${\text{TotOpCost}} = {\text{Cost}}^{{\mathrm{FW}}} + {\text{Cost}}^{{\mathrm{NG}}} + {\text{Cost}}^{{\mathrm{E}}} + {\text{Cost}}^{{\mathrm{ref}}}$$

(64)

Sales

In the proposed integrated scheme for the residential complex, there are several products that can be sold to external users, and the sales obtained must be determined, which is stated as follows.

Cold water sale

The sale of cold water (${\text{Sale}}^{{\mathrm{CW}}}$) to an external company is an option. This consists in the total cold water sold from each period of time, multiplying by the sale price of the cold water (${\text{US}}^{\text{CW}}$) in the respective periods, which depends on the external client:

$${\text{Sale}}^{{\mathrm{CW}}} = {\text{US}}^{\text{CW}} \sum\limits_{t} {f_{t}^{\text{RW-sale}} }$$

(65)

Hot water sale

To satisfy water demands in the residential complex, hot and warm water is required, but in some cases there can be produced more hot water that the one required. Thus, this can be sold to an external company and this is calculated as follows:

$${\text{Sale}}^{{\mathrm{HW}}} = {\text{US}}^{\text{HW}} (\sum\limits_{t} {h_{t}^{\text{boiler-sale}} + \sum\limits_{t} {h_{t}^{\text{ICE-sale}} )} }$$

(66)

where ${\text{Sale}}^{{\mathrm{HW}}}$ is the total hot water sale and ${\text{US}}^{\text{HW}}$ is the unitary sale price for the hot water.

Electricity sale

The excess of produced electricity by the ICE can be sold to external clients. The electricity sale (${\text{Sale}}^{{\mathrm{E}}}$) is equal to the unitary sale price for the electricity (${\text{US}}^{\text{e}}$) multiplied by the sum of the electricity sold in each time period:

$${\text{Sale}}^{{\mathrm{E}}} = {\text{US}}^{\text{e}} \sum\limits_{t} {e_{t}^{\text{ICE-sale}} }$$

(67)

Refrigeration sale

The refrigeration sold (${\text{Sale}}^{{\mathrm{R}}}$) to external clients is calculated multiplying the unit cost of refrigeration (${\text{US}}^{\text{R}}$) times the sum of sold refrigeration in all the time periods:

$${\text{Sale}}^{{\mathrm{R}}} = {\text{US}}^{\text{R}} \sum\limits_{t} {r_{t}^{\text{ARC-sale}} }$$

(68)

NG sale

The natural gas that is produced in the gas treatment unit and in the gasification process that is not exploited in the proposed scheme can be sold. This sold NG (${\text{Sale}}^{{\mathrm{NG}}}$) is equal to the natural gas sold by the gas treatment, plus the natural gas produced in the gasifier in all the time periods, multiplying by the unit cost for natural gas (${\text{US}}^{\text{NG}}$):

$${\text{Sale}}^{{\mathrm{NG}}} = {\text{US}}^{\text{NG}} \left(\sum\limits_{t} {f_{t}^{\mathrm{NG-gastreatment-sale}} } + \sum\limits_{t} {f_{t}^{\text{NG-gasification-sale}}} \right)$$

(69)

Biofuel sale

The biofuel sold by the process (${\text{Sale}}^{{\mathrm{biofuel}}}$) is equal to the unit sale cost for biofuel (${\text{US}}^{\text{biofuel}}$) by the total biofuel flow in all the periods:

$${\text{Sale}}^{{\mathrm{biofuel}}} = {\text{US}}^{\text{biofuel}} \sum\limits_{t} {F_{t}^{{\mathrm{biofuel}}} }$$

(70)

Total sales

The total sales (${\text{TotSales}}$) for this process is equal to the sum of the sale by the cold water (${\text{Sale}}^{{\mathrm{CW}}}$), plus the hot water (${\text{Sale}}^{{\mathrm{HW}}}$), plus the electricity (${\text{Sale}}^{{\mathrm{E}}}$), plus the refrigeration (${\text{Sale}}^{{\mathrm{R}}}$), plus the natural gas (${\text{Sale}}^{{\mathrm{NG}}}$) and biofuel sale (${\text{Sale}}^{{\mathrm{biofuel}}}$):

$${\text{TotSales}} = {\text{Sale}}^{{\mathrm{CW}}} + {\text{Sale}}^{{\mathrm{HW}}} + {\text{Sale}}^{{\mathrm{E}}} + {\text{Sale}}^{{\mathrm{R}}} + {\text{Sale}}^{{\mathrm{NG}}} + {\text{Sale}}^{{\mathrm{biofuel}}}$$

(71)

Existence for units

When there is needed a unit in the integrated complex, there is required to determine the associated fixed cost. This way, it is needed to determine the existence of the proposed units in the integrated residential complex. This is modeled through binary variables, and for each unit the following relationships are required.

Rainwater collecting system

The used area for the rainwater collecting system ($A^{{\mathrm{RW}}}$) must be lower than the greatest area available for collecting rainwater (${\text{A}}^{\text{RW-MAX}}$) multiplied by the associated binary variable ($y^{{\mathrm{RW}}}$) used for determining the existence of the devise. The binary variable can be zero or one; if the device exists it is one, whereas if the device does not exist it is zero.

$$A^{{\mathrm{RW}}} \le {\text{A}}^{\text{RW-MAX}} y^{{\mathrm{RW}}}$$

(72)

Capital cost for the rainwater collecting system

The unit fixed (${\text{UFC}}^{\text{\rm RW}}$) and variable cost for the rainwater collecting system define the associated capital cost. The binary variable establishes the fixed part of the cost, when the collecting unit exists this part is activated and if this does not exist the fixed part is not activated:

$${\text{CapCost}}^{{\mathrm{RW}}} = {\text{UFC}}^{\text{\rm RW}} y^{{\mathrm{RW}}} + A^{{\mathrm{RW}}} {\text{UVC}}^{\text{\rm RW}} + {\text{Cap}}^{{\mathrm{RWSS}}} {\text{UVC}}^{\text{RWSS}}$$

(73)

where ${\text{UVC}}^{\text{\rm RW}}$ and ${\text{UVC}}^{\text{RWSS}}$ are the unit variable costs for the collecting area and for the storage tank for rainwater, respectively, and ${\text{Cap}}^{{\mathrm{RWSS}}}$ is the capacity for the rainwater storage system.

Fresh water storage unit

The capacity for the fresh water storage unit (${\text{Cap}}^{{\mathrm{FWSS}}}$) must be lower than the greatest available storage unit to be able to install in the residential complex (${\text{Cap}}^{\text{FWSS-MAX}}$) multiplied by the associated binary variable ($y^{{\mathrm{FWSS}}}$) for the existence of the storage device:

$${\text{Cap}}^{{\mathrm{FWSS}}} \le {\text{Cap}}^{\text{FWSS-MAX}} y^{{\mathrm{FWSS}}}$$

(74)

Capital cost for the fresh water storage unit

The fixed (${\text{UFC}}^{\text{\rm FWSS}}$) and variable costs for the fresh water storage system define the associated capital cost. The binary variable establishes the fixed part of the cost, when the unit exists this part is activated and if this does not exist the fixed part is not activated. ${\text{UVC}}^{\text{\rm FWSS}}$ refers to the unit variable cost for the fresh water storage unit:

$${\text{CapCost}}^{{\mathrm{FWSS}}} = {\text{UFC}}^{\text{\rm FWSS}} y^{{\mathrm{FWSS}}} + {\text{UVC}}^{\text{\rm FWSS}} {\text{Cap}}^{{\mathrm{FWSS}}}$$

(75)

Capital cost for the boiler

The capacity for the boiler (${\text{Cap}}^{\rm boiler}$) must be greater than the hot water required for this ($H_{t}^{{^{\text{HW-boiler}} }}$):

$${\text{Cap}}^{\rm boiler} \ge H_{t}^{\text{HW-boiler}} { , }\,\forall t$$

(76)

Existence for the boiler

The capacity for boiler (${\text{Cap}}^{\rm boiler}$) must be lower than the greatest hot water required in a period (${\text{Cap}}^{\text{boiler-MAX}}$) multiplied by the associated binary variable ($y^{\rm boiler}$) for the existence of the boiler:

$${\text{Cap}}^{\rm boiler} \le {\text{Cap}}^{\text{boiler-MAX}} y^{\rm boiler}$$

(77)

$${\text{CapCost}}^{\rm boiler} = {\text{UFC}}^{{\rm boiler}} y^{\rm boiler} + {\text{UVC}}^{{\rm boiler}} {\text{Cap}}^{{\rm boiler}}$$

(78)

The capital cost for the boiler (${\text{CapCost}}^{\rm boiler}$) is equal to the unit fixed cost for the boiler (${\text{UFC}}^{\text{boiler}}$) multiplied by the respective binary variable plus the unit variable cost for the boiler (${\text{UVC}}^{\text{boiler}}$) times the capacity for the boiler.

Capital cost for the ICE

The capacity for the ICE (${\text{Cap}}^{\rm ICE}$) must be greater than the maximum electricity needed in all the considered periods ($E_{t}^{\rm ICE}$):

$${\text{Cap}}^{\rm ICE} \ge E_{t}^{\rm ICE} { , }\,\forall t$$

(79)

Existence for the ICE

The capacity for the ICE (${\text{Cap}}^{\rm ICE}$) must be lower than the greatest electricity needed in a period (${\text{Cap}}^{\text{ICE-MAX}}$) multiplied by the associated binary variable ($y^{\rm ICE}$) for the existence of the ICE:

$${\text{Cap}}^{\rm ICE} \le {\text{Cap}}^{\text{ICE-MAX}} y^{\rm ICE}$$

(80)

The capital cost for the ICE (${\text{CapCost}}^{\rm ICE}$) is calculated with the unit fixed cost for the ICE (${\text{UFC}}^{\text{ICE}}$) multiplied by the respective binary variable, plus the unit variable cost for the ICE (${\text{UVC}}^{\text{ICE}}$) times the capacity for the ICE:

$${\text{CapCost}}^{{\rm ICE}} = {\text{UFC}}^{{\rm ICE}} y^{{\rm ICE}} + {\text{UVC}}^{{\rm ICE}} {\text{Cap}}^{{\rm ICE}}$$

(81)

Cost for the ARC

The capacity for the absorption refrigeration cycle (${\text{Cap}}^{\rm ARC}$) must be greater than the maximum refrigeration required for the residential complex in a given time period ($R_{t}^{\rm ARC}$):

$${\text{Cap}}^{\rm ARC} \ge R_{t}^{\rm ARC} { , }\,\forall t$$

(82)

The capacity for the ARC (${\text{Cap}}^{\rm ARC}$) must be lower than the greatest refrigeration needed in all the time periods (${\text{Cap}}^{\text{ARC-MAX}}$) multiplied by the associated binary variable ($y^{\rm ARC}$) for the existence of the ARC:

$${\text{Cap}}^{\rm ARC} \le {\text{Cap}}^{\text{ARC-MAX}} y^{\rm ARC}$$

(83)

The capital cost for the ARC (${\text{CapCost}}^{\rm ARC}$) is equal to the unit fixed cost for the ARC (${\text{UFC}}^{\text{ARC}}$) multiplied by the respective binary variable plus the unit variable cost for the ARC (${\text{UVC}}^{\text{ARC}}$) for the capacity for the absorption refrigeration cycle:

$${\text{CapCost}}^{{\rm ARC}} = {\text{UFC}}^{{\rm ARC}} y^{{\rm ARC}} + {\text{UVC}}^{{\rm ARC}} {\text{Cap}}^{{\rm ARC}}$$

(84)

Algae system

The capacity for the algae system (${\text{Cap}}^{\rm algae}$) must be greater than the one needed to produce the required gases in all the time periods ($G_{t}^{\rm algae}$):

$${\text{Cap}}^{\rm algae} \ge G_{t}^{\rm algae} { , }\,\forall t$$

(85)

The capacity for the algae system must be lower than the greatest capacity needed in all the time periods (${\text{Cap}}^{\text{algae-MAX}}$) multiplied by the associated binary variable ($y^{\rm algae}$) for the existence of the algae system.

$${\text{Cap}}^{\rm algae} \le {\text{Cap}}^{\text{algae-MAX}} y^{\rm algae}$$

(86)

The capital cost for the algae system (${\text{CapCost}}^{\rm algae}$) is equal to the unit fixed cost (${\text{UFC}}^{\text{algae}}$) multiplied by the respective binary variable plus the unit variable cost for the algae system (${\text{UVC}}^{\text{algae}}$) times the capacity for the system:

$${\text{CapCost}}^{\rm algae} = {\text{UFC}}^{{\rm algae}} y^{{\rm algae}} + {\text{UVC}}^{{\rm algae}} {\text{Cap}}^{{\rm algae}}$$

(87)

Greywater treatment unit

The capacity for the greywater treatment unit (${\text{Cap}}^{\rm GW}$) must be greater than the inlet water to the unit ($F_{t}^{\text{Inlet-GW}}$):

$${\text{Cap}}^{\rm GW} \ge F_{t}^{\text{Inlet-GW}} { , }\,\forall t$$

(88)

The capacity for the greywater treatment unit must be lower than the greatest inlet water into the system in all time periods (${\text{Cap}}^{\text{GW-MAX}}$) multiplied by the associated binary variable ($y^{\rm GW}$) times the existence of the greywater unit:

$${\text{Cap}}^{\rm GW} \le {\text{Cap}}^{\text{GW-MAX}} y^{\rm GW}$$

(89)

The unit fixed cost (${\text{UFC}}^{\text{GW}}$) and variable cost for the greywater treatment unit define the associated capital cost. The binary variable establishes the fixed part of the cost, when the treatment exists this part is activated and if this does not exist the fixed part is not activated. ${\text{UVC}}^{\text{GW}}$ refers to the unit variable cost for the greywater treatment unit:

$${\text{CapCost}}^{{\rm GW}} = {\text{UFC}}^{{\rm GW}} y^{{\rm GW}} + {\text{UVC}}^{{\rm GW}} {\text{Cap}}^{{\rm GW}}$$

(90)

Black wastewater treatment unit

The capacity for the black wastewater treatment unit (${\text{Cap}}^{\rm WW}$) must be greater than the inlet water to this unit, and it should be lower than the capacity for the maximum inlet water in all time periods (${\text{Cap}}^{\text{WW-MAX}}$) multiplied by its binary variable ($y^{\rm WW}$):

$${\text{Cap}}^{\rm WW} \ge F_{t}^{\rm WW} { , }\,\forall t$$

(91)

$${\text{Cap}}^{\rm WW} \le {\text{Cap}}^{\text{WW-MAX}} y^{\rm WW}$$

(92)

The capital cost for the black wastewater treatment unit (${\text{CapCost}}^{\text{WW}}$) is equal to the unit fixed cost (${\text{UFC}}^{\text{WW}}$) multiplied by its respective binary variable, plus the unitary variable cost (${\text{UVC}}^{\text{WW}}$) times the wastewater multiplied by the capacity of the wastewater treatment unit:

$${\text{CapCost}}^{{\rm WW}} = {\text{UFC}}^{{\rm WW}} y{}^{\rm WW} + {\text{UVC}}^{{\rm WW}} {\text{Cap}}^{{\rm WW}}$$

(93)

Gas treatment unit

The capacity for the gas treatment unit (${\text{Cap}}^{\rm {NGT}}$) must be greater than the inlet gases to this unit, and it should be lower than the capacity for the maximum inlet gases in all time periods (${\text{Cap}}^{\text{NGT-MAX}}$) multiplied by its binary variable ($y^{\rm {NGT}}$):

$${\text{Cap}}^{\rm {NGT}} \ge F_{t}^{\rm {NGT}} { , }\,\forall t$$

(94)

$${\text{Cap}}^{\rm {NGT}} \le {\text{Cap}}^{\text{{NGT-MAX}}} y^{\rm {NGT}}$$

(95)

The capital cost for the gas treatment unit (${\text{CapCost}}^{\rm NGT}$) is equal to the unit fixed cost for the gas treatment unit (${\text{UFC}}^{\text{NGT}}$) multiplied by its respective binary variable, plus the unitary variable cost (${\text{UVC}}^{\text{NGT}}$) for the gas treatment multiplied by the capacity of the same unit:

$${\text{CapCost}}^{\rm NGT} = {\text{UFC}}^{\rm NGT} y^{\rm NGT} + {\text{UVC}}^{\rm NGT} {\text{Cap}}^{\rm NGT}$$

(96)

Gasification unit

The capacity for the gasifier (${\text{Cap}}^{\rm gasification}$) must be greater than the solid waste inlet to this unit, and it should be lower than the capacity for the maximum inlet solid waste in all time periods (${\text{Cap}}^{\text{gasification-MAX}}$) multiplied by its binary variable ($y^{\rm gasification}$):

$${\text{Cap}}^{\rm gasification} \ge F_{t}^{\text{NG-gasification}} { , }\,\forall t$$

(97)

$${\text{Cap}}^{\rm gasification} \le {\text{Cap}}^{\text{gasification-MAX}} y^{\rm gasification}$$

(98)

The capital cost for the gasification process (${\text{CapCost}}^{\rm gasification}$) is equal to the unit fixed cost for the gasifier unit (${\text{UFC}}^{\text{gasification}}$) multiplied by its respective binary variable, plus the unitary variable cost (${\text{UVC}}^{\text{gasification}}$) for the gasifier unit multiplied by the capacity of this unit:

$${\text{CapCost}}^{{\rm gasification}} = {\text{UFC}}^{{\rm gasification}} y^{{\rm gasification}} + {\text{UVC}}^{{\rm gasification}} {\text{Cap}}^{{\rm gasification}}$$

(99)

Total capital cost for the integrated system for the residential complex

The total capital cost (${\text{TotCapCost}}$) for the whole process is equal to the sum of the capital cost of each process, including the capital cost for the rainwater collecting system, plus the freshwater storage system, plus the boiler, plus the internal combustion engine, plus the absorption refrigeration cycle, plus the algae system, plus the greywater treatment unit, plus the black wastewater treatment unit, plus the treatment of NG gases, and plus the gasification unit:

$$\begin{aligned} {\text{TotCapCost}} &= {\text{CapCost}}^{{\rm RW}} + {\text{CapCost}}^{{\rm FWSS}} + {\text{CapCost}}^{{\rm boiler}} \\& \quad + {\text{CapCost}}^{{\rm ICE}} + {\text{CapCost}}^{{\rm ARC}} + {\text{CapCost}}^{{\rm algae}} \\ &\quad +{\text{CapCost}}^{{\rm GW}}+ {\text{CapCost}}^{{\rm WW}} \\ & \quad + {\text{CapCost}}^{{\rm NGT}} + {\text{CapCost}}^{{\rm gasification}} \end{aligned}$$

(100)

Maximum demands for external sales

There are needed some constraints for the maximum products that can be sold to external users, and these are stated as follows.

External fresh water maximum demand

The maximum demand of fresh water for external users ($\text{F}_{t}^{\mathrm{RW-MAX-Dem}}$) must be greater than the sold water ($f_{t}^{\text{RW-sale}}$):

$$f_{t}^{\text{RW-sale}} \le {\text{F}}_{\text{t}}^{\text{RW-MAX-Dem}} { , }\,\forall t$$

(101)

External hot water maximum demand

The maximum demand of hot water for external users ($\text{H}_{t}^{{\text{HW}-MAX-Dem}}$) must be greater than the hot water sold, which is produced by the boiler ($h_{t}^{Boiler-sale}$) plus the one produced by the ICE ($h_{t}^{\text{ICE-sale}}$):

$$h_{t}^{\text{boiler-sale}} + h_{t}^{\text{ICE-sale}} \le {\text{H}}_{\text{t}}^{\text{HW-MAX-Dem}} { , }\,\forall t$$

(102)

External electricity maximum demand

The maximum demand of electricity for external users ($\text{E}_{t}^{{\text{E}-MAX-Dem}}$) must be greater than the electricity sale ($e_{t}^{\text{ICE-sale}}$) to an external company:

$$e_{t}^{\text{ICE-sale}} \le {\text{E}}_{\text{t}}^{\text{E-MAX-Dem}} { , }\,\forall t$$

(103)

External refrigeration maximum demand

The maximum demand of refrigeration for external users ($\text{R}_{t}^{ARC-MAX-Dem}$) must be greater than the refrigeration sold to external users ($r_{t}^{\text{ARC-sale}}$):

$$r_{t}^{\text{ARC-sale}} \le {\text{R}}_{\text{t}}^{\text{ARC-MAX-Dem}} { , }\,\forall t$$

(104)

External NG maximum demand

The maximum demand of natural gas for external users ($\text{F}_{t}^{NG-MAX-Dem}$) must be greater than the natural gas sold to external users and that comes from the gasification process ($f_{t}^{\text{NG-gasification-sale}}$) plus the one produced by the gas treatment unit ($f_{t}^{\mathrm{NG-gastreatment-sale}}$):

$$f_{t}^{\text{NG-gasification-sale}}+ f_{t}^{\mathrm{NG-gastreatment-sale}} \le {\text{F}}_{\text{t}}^{\text{NG-MAX-Dem}} { , }\,\forall t$$

(105)

$$F_{t}^{{\mathrm{biofuel}}} \le {\text{F}}_{\text{t}}^{\text{Biofuel-MAX-Dem}} { , }\,\forall t$$

(106)

Objective Functions

Total annual cost for the integrated system

The total annual cost (${\text{TAC }}$) is equal to the total operating cost (${\text{TotOpCost}}$) plus the total capital cost (${\text{TotCapCost}}$) minus the total sales (${\text{TotSales}}$):

$${\text{TAC }} = {\text{TotOpCost}} + {\text{TotCapCost}}-{\text{TotSales}}$$

(107)

Fresh water consumption

The total fresh water consumption ($FFwTot$) in the integrated system is equal to the summation of the fresh water used in each period of time:

$$FFwTot = \sum\limits_{t} {FFw_{t} }$$

(108)

GHGC emissions

The greenhouse gas emissions ($GDischargeTotal$) are equal to the emissions discharged to the environment by the ICE ($Fg_{t}^{\mathrm{Ice-Discharge}}$) plus the emissions by the boiler ($Fg_{t}^{\mathrm{Boiler-Discharge}}$) and the gasification process ($Fg_{t}^{\mathrm{Gasification-Discharge}}$) as follows:

$$GDischargeTotal = \sum\limits_{t} {Fg}_{t}^{\mathrm{Ice-Discharge}} + \sum\limits_{t} {Fg_{t}^{\mathrm{Boiler-Discharge}} + \sum\limits_{t} {Fg_{t}^{\mathrm{Gasification-Discharge} } }}$$

(109)