This work follows the concept put forward by Egieya et al. [28] while considering the following additions and extensions:
The model is formulated on an hourly basis (previously on monthly basis in Egieya et al. [28]), where the year is divided into monthly (mp), daily (dp) and hourly (hp) time periods. Consequently, all the equations which were based on monthly periods, are now delineated to monthly, daily and hourly periods. To implement this, certain model reduction techniques are therefore introduced to reduce computational time.
Instead of subsidized prices of electricity (fixed), hourly-based auction trading prices of electricity are considered based on 2017 prices, ranging from − 42.93 to 199.00 €/MWh (between − 57.1 and 264.67 $/MWh) [33]. The highest electricity price was in August, while the lowest price was in December 2017. The hourly-based electricity price variations are illustrated in Figs. 2 and 3 for the months of August and December 2017. All the data related to electricity prices (in €/MWh) as obtained from BSP South Pool Energy Exchange [33] are presented in Additional file 1: Tables S1-S12). Furthermore, the average electricity prices (in $/MWh) for each of the considered period based on model reduction techniques and implemented in the model are also given in the Additional file 1: Tables S13-S24).
Biogas storage is incorporated to account for possible variations in electricity production, i.e. to enable storing biogas instead of electricity production at low electricity prices. However, in such situations, heat is also not produced, and thus a backup is required to generate heat from other sources.
Instead of considering only one agricultural biogas production plant as the optimal plant and with the capacity of up to 999 kW, a maximum of three biogas plants could be selected. Despite the variations in electricity production, biogas production should be constant with slight variations allowed. Thus two scenarios are performed based on the demand for methane, i) between 1.95∙106 and 2.38∙106 m3/y (average 0.9–1.1 MW of electricity produced) and ii) between 9.76∙106 and 11.93∙106 m3/y (average 4.8–5.2 MW of electricity produced).
Two additional objectives are considered besides an economic one in the form of maximizing sustainability profit [26] and the simultaneous maximization of profit with the costs and the benefits attributed to GHG burdening and unburdening. Hence, the model is upgraded to include environmental (GHG emissions) and sustainability (eco-cost and benefit and social cost and benefit) objectives.
Description of biogas supply network
The biogas supply network utilized (see Fig. 4) consists of four layers:
- i)
First layer (L1): harvesting and collection. This layer consists of a set pb of biomass feedstocks (corn, wheat and triticale grains, straw, silage, and grass silage) and different manure types (cattle, pig and poultry manure, poultry bedding and poultry slurry). For the feedstocks, characteristics such as dry matter and methane contents and biogas yields [34] are considered in the study.
- ii)
Second layer (L2): primary processing technology which is anaerobic digestion. In L2, the primary conversion product pi (a combination of biomass and waste feedstocks pb, recycled products poutpim and purchased products pbuy) is generated. These are later converted to intermediate products pm (biogas and wet digestate) or final products pd using given conversion factors.
- iii)
Third layer (L3): secondary conversion technologies involve cogeneration (CHP) combining heat and power production and physical dewatering as in [28]. It should be noted that there are other possible conversion technologies, such as biogas upgrading to biomethane [35], ammonium sulfate recovery from digestate [36] and several other, however they have not been considered in this study. The products pz (a sum of intermediate products pm, recycled product poutpin, and purchased products pbuy) are converted (using conversion factors) to the desired products pp (electricity, heat and dry digestate).
- iv)
Fourth layer (L4): demand locations.
The model considers three optional distribution modes between the layers to convey feedstocks, intermediate and final products, in the form of road, pipeline transport, and transmission lines. Besides, the model allows heat and electricity generated from the CHP and water from the dewatering plants to be reused within the supply network. For sustainable supply of all materials within the supply network, four storage facilities are also modelled at the locations of biomass and waste collection centres and primary and secondary conversion facilities, where all feedstocks and products could be stored. Additionally, it is assumed that water, electricity, and heat are excluded from storage and that the purchased materials should not be stored. Note that from the previous work of Egieya et al. [28] biogas could additionally be stored.
Similarly, as in Egieya et al. [28], certain characteristics of biomass and waste feedstocks are considered, such as different dry matter contents, methane contents and biogas yields [34]. Also, other parameters as presented in Egieya et al. [28] are considered, except instead of guaranteed purchase prices which are fixed, auction trading prices which vary hourly are considered.
For more details on the biogas production supply network methodology, the reader is referred to the paper by Egieya et al. [28].
Description of mathematical model
The mathematical model includes material and energy balances, primary and secondary conversion constraints and cost correlations. However, as the model now considers hourly production, all the variables, and equations which were based on monthly periods, are now based on monthly, daily and hourly periods.
As the hourly-based model is computationally expensive, certain model reduction techniques have been implemented based on the work by Lam et al. [37] to reduce computational time. Hence, instead of 24 h a day, three “hourly periods” or shift periods (morning, afternoon and night) are considered and are thereby defined as H1 (7 am – 2 pm), H2 (3 pm – 10 pm) and H3 (11 pm – 6 am). Furthermore, instead of 28–31 days a month, seven “daily periods” are applied based on the days of the week (Monday – Sunday) and are defined as D1: {d1, d8, d15, d22, d29}, D2: {d2, d9, d16, d23, d30}, D3: {d3, d10, d17, d24, d31}, D4: {d4, d11, d18, d25}, D5: {d5, d12, d19, d26}, D6: {d6, d13, d20, d27} and D7: {d7, d14, d21, d28}, see also Egieya et al. [20]. This is due to different electricity consumption patterns of the weekdays and weekends. All 12 months of a calendar year are on the other hand fully considered in order to preserve the variability of the model as much as practicable. Merging of time periods is done by defining the sets MPOM, DPOD and HPOH which convert the maximal number of time periods (mpo, dpo and hpo) to merged time periods (mp, dp and hp).
All the prices except electricity prices are considered at merged hourly basis as shown in Eq. (1):
$$ {\displaystyle \begin{array}{l}{P}_{p, mp, dp, hp}=\frac{\sum \limits_{mpo\in MP}\underset{\left( mpo, mp\right)\in MP OM}{\wedge }{P}_{p, mp o}}{\sum \limits_{mpo\in MP}\underset{\left( mpo, mp\right)\in MP OM}{\wedge}\mid mpo\mid },\\ {}\kern2.5em \forall p\in P\wedge p\notin \left\{ electricity\right\}, mp\subseteq MP, dp\subseteq DP, hp\subseteq HP,\left( dp, mp\right)\in DP M\end{array}} $$
(1)
where ∧ stands for logical condition (dollar operator in GAMS [38]).
As electricity prices are provided on hourly basis, they are averaged in order to more properly account for their variations. Averaging electricity prices is illustrated in Eq. (2):
$$ {\displaystyle \begin{array}{l}{P}_{electricity, mp, dp, hp}=\\ {}\kern2.5em \frac{\sum \limits_{mpo\in MPO}\sum \limits_{dpo\in DPO}\sum \limits_{hpo\in HPO}\underset{\left( mpo, mp\right)\in MPO M,\left( hpo, hp\right)\in HPO H,\left( dpo, dp\right)\in DPO D,\left( dpo, mp o\right)\in DPM}{\wedge }{P}_{electricity, mp o, dp o, hp o}}{\sum \limits_{mpo\in MPO}\underset{\left( mpo, mp\right)\in MPO M}{\wedge}\mid mp o\mid \cdot \sum \limits_{dpo\in DPO}\sum \limits_{\left( dpo, mp\right)\in DPM}\underset{\left( dpo, dp\right)\in DPO D}{\wedge}\mid dpo\mid \cdot \sum \limits_{hpo\in HPO}\underset{\left( hpo, hp\right)\in HPO H}{\wedge}\mid hpo\mid },\\ {}\kern3em \forall mp\subseteq MPO, dp\subseteq DPO, hp\subseteq HPO\end{array}} $$
(2)
The hourly-based variations in the model have been introduced with the production rate of feedstocks pb at the harvesting zone i, which is now defined based on merged hourly periods hp, merged daily periods dp and monthly periods mp (PRi, pb, mp, dp, hp in kt/period), see also Eq. (7) in Egieya et al. [28]:
$$ \sum \limits_{dp\subseteq DP}\sum \limits_{hp\subseteq HP}\underset{\left( dp, mp\right)\in DP M}{\wedge }P{R}_{i, pb, mp, dp, hp}=H{Y}_{i, pb, mp}\cdot {A}_{i, pb, mp},\kern1.62em \forall i\in I, pb\in PB, mp\in MP $$
(3)
where HYi, pb, mp is the yield of feedstocks pb in month period mp at harvesting zone i (in kt/(km2∙month)) and Ai, pb, mp is the available area for growing biomass pb at harvesting zone i in month period mp (in km2).
The equations for storages additionally consider “circular operations”. The equation for storage at the inlet of primary conversion facilities is for example defined as shown in Eq. (4), see also Eq. (9) in Egieya et al. [28].
$$ {\displaystyle \begin{array}{l} Ai{n}_{m, pi, mp, dp, hp}^{\mathrm{L}2}=\underset{{\left(m{p}_k\right)}_{k\in K},k=1\wedge {\left(d{p}_k\right)}_{k\in K},k=1\wedge {\left(h{p}_k\right)}_{k\in K},k=1}{\cup } Ai{n}_{m, pi, mp--1, dp--1, hp--1}^{\mathrm{L}2}+\\ {}\kern1.25em \underset{{\left(h{p}_k\right)}_{k\in K},k>1}{\cup } Ai{n}_{m, pi, mp, dp, hp-1}^{\mathrm{L}2}+\underset{{\left(d{p}_k\right)}_{k\in K},k>1\wedge {\left(h{p}_k\right)}_{k\in K},k=1}{\cup } Ai{n}_{m, pi, mp, dp-1, hp--1}^{\mathrm{L}2}+\\ {}\kern1.25em \underset{{\left(m{p}_k\right)}_{k\in K},k>1\wedge {\left(d{p}_k\right)}_{k\in K},k=1\wedge {\left(h{p}_k\right)}_{k\in K},k=1}{\cup } Ai{n}_{m, pi, mp-1, dp--1, hp--1}^{\mathrm{L}2}+\\ {}\kern1.25em \sum \limits_{i\in I}\sum \limits_{pb\subseteq PI}{F}_{i,m, pb, mp, dp, hp}^{\mathrm{L}1,\mathrm{L}2, net}+\sum \limits_{n\in N}\sum \limits_{poutpim\subseteq PI}{F}_{n,m, poutpim, mp, dp, hp}^{\mathrm{L}3,\mathrm{L}2, net}+\\ {}\kern1.25em \sum \limits_{pb uy\subseteq PI}{F}_{m, pb uy, mp, dp, hp}^{\mathrm{buy},\mathrm{L}2}-\sum \limits_{\left( pi,t\right)\in PI T\wedge {t}_2\in T}{F}_{m, pi,t, mp, dp, hp}^{\mathrm{L}2,\mathrm{T}}-\sum \limits_{j\in J}\sum \limits_{pn\subseteq PI}{F}_{m,j, pn, mp, dp, hp}^{\mathrm{L}1,\mathrm{L}4,\mathrm{net}}+\\ {}\kern1.25em \Big( Ai{n}_{m, pi, mp, dp, hp}^{\mathrm{L}2}+\underset{{\left(m{p}_k\right)}_{k\in K},k=1\wedge {\left(d{p}_k\right)}_{k\in K},k=1\wedge {\left(h{p}_k\right)}_{k\in K},k=1}{\cup } Ai{n}_{m, pi, mp--1, dp--1, hp--1}^{\mathrm{L}2}+\\ {}\kern1.25em \underset{{\left(h{p}_k\right)}_{k\in K},k>1}{\cup } Ai{n}_{m, pi, mp, dp, hp-1}^{\mathrm{L}2}+\underset{{\left(d{p}_k\right)}_{k\in K},k>1\wedge {\left(h{p}_k\right)}_{k\in K},k=1}{\cup } Ai{n}_{m, pi, mp, dp-1, hp--1}^{\mathrm{L}2}+\\ {}\kern1.25em \underset{{\left(m{p}_k\right)}_{k\in K},k>1\wedge {\left(d{p}_k\right)}_{k\in K},k=1\wedge {\left(h{p}_k\right)}_{k\in K},k=1}{\cup } Ai{n}_{m, pi, mp-1, dp--1, hp--1}^{\mathrm{L}2}+\\ {}\kern1.25em \underset{{\left(m{p}_k\right)}_{k\in K},k>1\wedge {\left(d{p}_k\right)}_{k\in K},k=1\wedge {\left(h{p}_k\right)}_{k\in K},k=1}{\cup } Ai{n}_{m, pi, mp-1, dp--1, hp--1}^{\mathrm{L}2}\Big)/2\cdot {\psi}_{pi, mp, dp, hp}\\ {}\kern3em \forall m\in M, pi\in PI\wedge pi\notin NOSTOR, mp\in MP, dp\in DP, hp\in HP,\left( dp, mp\right)\in DP M\ \end{array}} $$
(4)
In Eq. (4) \( Ai{n}_{m, pi, mp, dp, hp}^{\mathrm{L}2} \) represents the storage quantity of material pi in each monthly mp, daily dp and hourly time period hp at the location of primary conversion facility m, \( Ai{n}_{m, pi, mp--1, dp--1, hp--1}^{\mathrm{L}2} \) refers to the quantity of material pi in the storage tank at the beginning of January (first hour, first day and first month) which equals the quantity of material pi in the storage tank at the last hour of December (last hour, last day, last month) of the previous year. Similarly, \( Ai{n}_{m, pi, mp, dp, hp-1}^{\mathrm{L}2} \) refers to quantity of material pi in the storage tank for each month and day where the hour should not be the first hour of the day, \( Ai{n}_{m, pi, mp, dp-1, hp--1}^{\mathrm{L}2} \) refers to the quantity of material pi in the storage tank for each first hour of the day and if the day is not the first day of the month and \( Ai{n}_{m, pi, mp-1, dp--1, hp--1}^{\mathrm{L}2} \) refers to quantity of material pi in the storage tank for each first hour in a day and for each first day in a month of any given month except January (first month).
Additional terms in Eq. (4) are: \( {F}_{i,m, pb, mp, dp, hp}^{\mathrm{L}1,\mathrm{L}2, net} \) represents the net quantity of biomass and waste feedstocks pb shipped to the primary conversion location m from the harvesting location i in each considered time period (mp, dp, hp), \( {F}_{n,m, poutpim, mp, dp, hp}^{\mathrm{L}3,\mathrm{L}2, net} \) is the net flow of “recycled” material in the supply network poutpim between the secondary n and primary conversion location m, also for each considered time period. Such products are electricity, heat and water, as shown in Fig. 4. \( {F}_{m, pbuy, mp, dp, hp}^{\mathrm{buy},\mathrm{L}2} \) stands for the quantity of purchased resources pbuy to be used at L2 (primary conversion) on the location of m in each of the considered time period. \( {F}_{m, pi,t, mp, dp, hp}^{\mathrm{L}2,\mathrm{T}} \) is the flow of intermediate product pi ∈ (pb, poutpim, pbuy) from storage to technology t2 ∈ T at primary conversion location m in each time period and \( {F}_{m,j, pn, mp, dp, hp}^{\mathrm{L}1,\mathrm{L}4} \) quantifies the flow of unprocessed feedstocks pn ⊆ PI to the demand location j. The last term of Eq. (4) represents the losses of stored intermediate materials pi during the storage. Similarly as in Egieya et al. [28], it is assumed that the amount of stored intermediate products (pi ∈ PI ∧ pi ∉ NOSTOR) available in any considered time period mp, dp, hp is the average of two consecutive time periods. Parameter ψpi, mp, dp, hp represents the deterioration rate in storage which is defined on monthly basis ψpi, mp, and is then divided by the length of the daily and hourly period (cardinality of sets DP and HP), as shown in Eq. (5):
$$ {\psi}_{pi, mp, dp, hp}=\frac{\psi_{pi, mp}}{\mid dp\Big\Vert hp\mid },\kern2.25em \forall p\in P, mp\in MP, dp\in DP, hp\in HP,\left( dp, mp\right)\in DP M $$
(5)
As it was stated above, all the potential biogas plants could be selected. Since only slight variations in capacity of anaerobic digesters are allowed, two scenarios are performed based on the demand for methane, i) between 1.95∙106 and 2.38∙106 m3/y (average 0.9–1.1 MW of electricity produced) and ii) between 9.76∙106 and 11.93∙106 m3/y (average 4.8–5.2 MW of electricity produced). The capacities of methane between their upper and lower bounds are shown in Eq. (6) for lower bound and in Eq. (7) for upper bound.
$$ {\displaystyle \begin{array}{l}\sum \limits_{pi\in PI}\underset{\left( pi, methane\right)\in PI PM,\left( pi, AD\right)\in PI T}{\wedge }{F}_{m, pi, methane, AD, mp, dp, hp}^{\mathrm{L}2,P}\ge 0.9\cdot \frac{De{m}_{electricity, mp, dp, hp}}{f_{methane, electricity, CHP}^{conv,T,L3}}\cdot {y}_{m, AD}^{\mathrm{L}2,T},\\ {}\kern3.25em \forall m\in M, mp\in MP, dp\in DP, hp\in HP,\left( dp, mp\right)\in DP M\ \end{array}} $$
(6)
$$ {\displaystyle \begin{array}{l}\sum \limits_{pi\in PI}\underset{\left( pi, methane\right)\in PI PM,\left( pi, AD\right)\in PI T}{\wedge }{F}_{m, pi, methane, AD, mp, dp, hp}^{\mathrm{L}2,P}\le 1.1\cdot \frac{De{m}_{electricity, mp, dp, hp}}{f_{methane, electricity, CHP}^{conv,T,L3}}\cdot {y}_{m, AD}^{\mathrm{L}2,T},\\ {}\kern3.25em \forall m\in M, mp\in MP, dp\in DP, hp\in HP,\left( dp, mp\right)\in DP M\ \end{array}} $$
(7)
In Eq. (6) and Eq. (7) \( {F}_{m, pi, methane, AD, mp, dp, hp}^{\mathrm{L}2,P} \) represents the flowrate of methane produced from material pi using anaerobic digestion AD technology at L2 within time periods mp, dp, hp. Demelectricity, mp, dp, hp stands for the demand for electricity in each considered time period (see Eq. (8)). \( {f}_{methane, electricity, CHP}^{conv,T,L3} \) is conversion factor of methane to electricity using technology CHP, and the binary variable \( {y}_{m, AD}^{\mathrm{L}2,T} \) represents the selection of technology AD at location m. If the binary variable equals 1, AD is selected at mth location while the AD is not selected at that location when the binary variable equals to 0.
$$ {\displaystyle \begin{array}{l} De{m}_{electricity, mp, dp, hp}= cap\cdot {f}_{time}\cdot \frac{\mid mp o\mid }{\mid mp\mid}\cdot \frac{\mid dp o\mid }{\mid dp\mid}\cdot \frac{\mid hp o\mid }{\mid hp\mid },\\ {}\kern3.25em \forall mp\in MP, dp\in DP, hp\in HP,\left( dp, mp\right)\in DP M\ \end{array}} $$
(8)
where cap is capacity of electricity production (1 or 5 MW) and ftime is the fraction of time when biogas production is operating and is defined as the number of operating hours in a year divided by the total number of hours in a calendar year. In this study, value of 0.935 (8192 h/y) is assumed for ftime. The part \( \frac{\mid mp o\mid }{\mid mp\mid}\cdot \frac{\mid dp o\mid }{\mid dp\mid}\cdot \frac{\mid hp o\mid }{\mid hp\mid } \) relate to the number of all periods divided by the total number of considered periods.
Moreover, various new equations, data and variables have been included in the model to account for the two additional objectives included in this study compared to the work of Egieya et al. [28]. These equations and variables related to the additional objectives are hereby presented in the next section and the data assumed are presented in Additional file 1. The data related to GHG emissions are shown in Additional file 1: Tables S25 – S27 and the data related to sustainability profit maximization are given in Additional file 1: Tables S28 – S31.
Objectives in the study
The goal is to synthesize an optimal biogas supply network under different objective functions: (i) economic objective defined with maximizing economic profit (similar to Egieya et al. [20, 28], while excluding the tax on the profit accrued); (ii) economic and environmental objectives by maximizing economic profit while including costs and benefits due to GHG emissions (price for GHG emissions), and iii) economic, environmental and social objectives by maximizing sustainability profit [26] which includes all three sustainability objectives, economic, environmental and social (similar as in Bogataj et al. [39]). Further, four additional scenarios are performed to improve the profitability of the biogas supply network: i) price for GHG emissions is increased from the price of carbon allowances in the European Union Emissions Trading System (EU ETS) [40] up to the value of eco-costs / benefits of global warming [32], ii) the auction trading prices are increased by multiplying them with various factors, as explained above iii) the length of time period is decreased and the relation is explored between the price of biogas storage and capacity of biogas storage and electricity production and (iv) increasing the biogas plant capacity from 1 MW to 5 MW and observing its effects on economic profit.
Economic objective
The economic objective is defined with maximizing economic profit (PEconomic) from the generation of electricity, heat and digestate within the biogas supply chain network:
$$ {P}^{Economic}={R}^{Total}-{C}^{Total} $$
(9)
where RTotal is total revenue accrued ($/y) and CTotal is total cost incurred in the supply chain ($/y).
The total revenue (RTotal) is calculated as shown in Eq. (10):
$$ {\displaystyle \begin{array}{l}{R}^{Total}=\sum \limits_{m\in M}\sum \limits_{j\in J}\sum \limits_{pd\in PD}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}\underset{\left( dp, mp\right)\in DP M}{\wedge }{F}_{m,j, pd, mp, dp, hp}^{\mathrm{L}2,\mathrm{L}4, net}\cdot {P}_{pd, mp. dp, hp}+\\ {}\kern3em \sum \limits_{n\in N}\sum \limits_{j\in J}\sum \limits_{pp\in PP}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}\underset{\left( dp, mp\right)\in DP M}{\wedge }{F}_{n,j, pp, mp, dp, hp}^{\mathrm{L}3,\mathrm{L}4, net}\cdot {P}_{pp, mp, dp, hp}+\\ {}\kern2.75em \sum \limits_{m\in M}\sum \limits_{j\in J}\sum \limits_{pn\in PN}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}\underset{\left( dp, mp\right)\in DP M}{\wedge }{F}_{m,j, pn, mp, dp, hp}^{\mathrm{L}1,\mathrm{L}4, net}\cdot {P}_{pn, mp, dp, hp}\end{array}} $$
(10)
where \( {F}_{m,j, pd, mp, dp, hp}^{\mathrm{L}2,\mathrm{L}4, net} \) represents the net flowrate of direct product pd (wet digestate) produced from anaerobic digestion at site m and sold as a fertilizer in site j to farmers at each considered time period, \( {F}_{n,j, pp, mp, dp, hp}^{\mathrm{L}3,\mathrm{L}4, net} \) stands for the net flow of produced products pp (electricity, heat and dewatered digestate) from the plant n to demand j. \( {F}_{m,j, pn, mp, dp, hp}^{\mathrm{L}1,\mathrm{L}4, net} \) represents materials that do not undergo any treatment (pn) shipped directly to the demand zone in site j. Ppd, mp. dp, hp, Ppp, mp, dp, hp and Ppn, mp, dp, hp are prices of direct products (pd), produced products (pp) and products that do not undergo treatment (pn).
Total costs accrued (CTotal) in the biogas supply chain network are a sum of costs for feedstocks, purchase of additional materials needed in L2 and L3, shipment (\( T{C}_p^{Total} \)), storage (SCp), labour (LC), depreciation (DCC), maintenance (MC), and miscellaneous cost (MSC) as displayed in Eq. (11):
$$ {\displaystyle \begin{array}{l}{C}^{Total}=\sum \limits_{i\in I}\sum \limits_{pb\in PB}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}\underset{\left( dp, mp\right)\in DP M}{\wedge }P{R}_{i, pb, mp, dp, hp}\cdot {c}_{pb, mp}+\\ {}\kern2.75em \sum \limits_{m\in M}\sum \limits_{pb uy\in PB UY}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}\underset{\left( dp, mp\right)\in DP M}{\wedge }{F}_{m, pb uy, mp, dp, hp}^{\mathrm{buy},\mathrm{L}2}\cdot {c}_{pb uy, mp}+\\ {}\kern2.75em \sum \limits_{n\in N}\sum \limits_{pb uy\in PB UY}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}\underset{\left( dp, mp\right)\in DP M}{\wedge }{F}_{n, pb uy, mp, dp, hp}^{\mathrm{buy},\mathrm{L}3}\cdot {c}_{pb uy, mp}+\\ {}\kern2.75em \sum \limits_{p\in P}T{C}_p^{Total}+\sum \limits_{p\in P}S{C}_p+ LC+ DCC+ MC+ MSC\end{array}} $$
(11)
where, cpb, mp and cpbuy, mp are cost for feedstocks acquired (pb) and purchased materials (pbuy). PRi, pb, mp, dp, hp is total quantity of feedstocks harvested at site i and shipped to storage at primary conversion location, while \( {F}_{m, pbuy, mp, dp, hp}^{\mathrm{buy},\mathrm{L}2} \) and \( {F}_{n, pbuy, mp, dp, hp}^{\mathrm{buy},\mathrm{L}3} \) are quantities of additional raw materials purchased in L2 and L3 within a given monthly, daily and hourly period.
Economic and environmental objectives (economic+GHG profit)
The second objective includes economic objective and price for GHG emissions. The economic objective and equations describing it in detail are shown above (Eqs. (9)–(11)). The environmental objective is defined as a maximization of GHG unburdening and it is based and extended from the work of Bogataj et al. [39]. Avoided and released GHG emissions (unburdening and burdening) are multiplied by the price of GHG emissions (also called carbon price [41]) and included in the economic objective. Both burdening and unburdening are considered, whereby burdening is related to the negative impacts on the environment due to resource use, production and use of products, while unburdening is due to the direct utilization of harmful (waste) materials and due to substitution of environmentally more harmful products with less harmful ones [42].
The environmental objective follows the same principle of evaluation as the eco-profit calculation [30]. First, avoided and released GHG emissions are calculated with units based on t CO2 equivalent emitted per t of raw material or product except for electricity and heat which are in t CO2 equivalent emitted per MWh. The Life Cycle Assessment (LCA) principle is applied to the biogas supply network from the harvesting and collection zones to the demand zones. GHG emissions include those emissions that originate from the whole life cycle of product, from extraction of raw materials, through pre-processing and processing to disposal of harmful products, including emissions due to transportation and distribution within the supply network (similarly as in [30]).
Hence, the amounts of GHG emitted or preserved (see Eq. (12)) is a measure of the difference between GHG unburdening (GHGUB) as shown in Eq. (13) and GHG burdening (GHGB) in the supply network, presented by Eq. (14).
$$ GHG= GH{G}^{UB}- GH{G}^B $$
(12)
$$ {\displaystyle \begin{array}{l} GH{G}_p^{UB}=\sum \limits_{m\in M}\sum \limits_{t\in T}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}{F}_{m,p,t, mp, dp, hp}^{\mathrm{L}1,\mathrm{L}2, net}\cdot {c}_p^{GHG, UB}+\\ {}\kern3.25em \sum \limits_{m\in M}\sum \limits_{j\in J}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}{F}_{m,j,p, mp, dp, hp}^{\mathrm{L}2,\mathrm{L}4, net}\cdot {c}_p^{GHG, UB}\cdot {f}_p^S+\\ {}\kern3.25em \sum \limits_{n\in N}\sum \limits_{j\in J}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}{F}_{n,j,p, mp, dp, hp}^{\mathrm{L}3,\mathrm{L}4, net}\cdot {c}_p^{GHG, UB}\cdot {f}_p^S,\kern2em \forall p\in \left\{ PB, PD, PP\right\}\end{array}} $$
(13)
where, \( {c}_p^{GHG, UB} \) is the GHG emission coefficient related to unburdening or avoided GHG emissions (see Additional file 1: Table S25) for material p and \( {f}_p^S \) is substitution factor defined as the amount of produced product divided by the amount of substituted product [30]. GHG emission coefficients have been obtained from the website of Delft University of Technology, The Model of the Eco-costs / Value Ratio (EVR) [32] and checked with OpenLCA software [43] using ecoinvent 3.1 database [44] and the ecoinvent 3.1 Life Cycle Impact Assessment (LCIA) method [45] IPCC 2007.
The following substitution factors are assumed in the study: 0.9 for electricity, 0.04 for dry digestate and 0.029 for wet digestate [46].
$$ {\displaystyle \begin{array}{l} GH{G}_p^B=\sum \limits_{m\in M}\sum \limits_{t\in T}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}{F}_{m,p,t, mp, dp, hp}^{\mathrm{L}1,\mathrm{L}2, net}\cdot {c}_p^{GHG,B}+\\ {}\kern3.25em \sum \limits_{m\in M}\sum \limits_{j\in J}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}{F}_{m,j,p, mp, dp, hp}^{\mathrm{L}2,\mathrm{L}4,\mathrm{net}}\cdot {c}_p^{GHG,B}+\\ {}\kern3.25em \sum \limits_{n\in N}\sum \limits_{j\in J}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}{F}_{n,j,p, mp, dp, hp}^{\mathrm{L}3,\mathrm{L}4,\mathrm{net}}\cdot {c}_p^{GHG,B}+\\ {}\kern3.25em \sum \limits_{m\in M}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}{F}_{m,p, mp, dp, hp}^{\mathrm{buy},\mathrm{L}2}\cdot {c}_p^{GHG,B}+\sum \limits_{n\in N}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}{F}_{n,p, mp, dp, hp}^{\mathrm{buy},\mathrm{L}3}\cdot {c}_p^{GHG,B}+\\ {}\kern3.25em 2\cdot \left(\sum \limits_{x\in \left\{I,M,N\right\}}\sum \limits_{y\in \left\{M,N,J\right\}}\sum \limits_{tropt\in \left\{ road\right\}}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}{F}_{x,y,p, mp, dp, hp, tropt}^{\mathrm{L}\mathrm{a},\mathrm{L}\mathrm{b},\mathrm{tropt}}\cdot {D}_{x,y}^{\mathrm{L}\mathrm{a},\mathrm{L}\mathrm{b}}\cdot {c}_{p, tropt}^{GHG,B}\right)+\\ {}\kern2.75em \left( Ai{n}_{m, pi, mp, dp, hp}^{\mathrm{L}2}+ Aou{t}_{m, pm, mp, dp, hp}^{\mathrm{L}2}+ Ai{n}_{n, pz, mp, dp, hp}^{\mathrm{L}3}+ Aou{t}_{n, pp, mp, dp, hp}^{\mathrm{L}3}\right)\cdot {c}_p^{GHG,B}\cdot 0.05,\\ {}\kern2.75em \forall p\in \left\{ PB, PN, PD, PP, POUTPIM, POUTPIN\right\}\end{array}} $$
(14)
where \( {c}_p^{GHG,B} \) refers to the GHG emission coefficient of material p related to the released GHG emissions (burdening) (see Additional file 1: Table S26), \( {c}_{p, tropt}^{GHG,B} \) is GHG emission coefficient related to transport (see Additional file 1: Table S27). In addition, it is worth stating that the last section of the Eq. (14) illustrates the GHG emissions occurring during storage of material p over the considered time periods. Hence, in the storage, it is assumed that the burdening equals 5% of the burden of product stored.
Additional terms in Eq. (14) are: \( {F}_{x,y,p, mp, dp, hp, tropt}^{\mathrm{La},\mathrm{Lb},\mathrm{tropt}} \) shows the quantity of materials p transported from the location x in layer La to location y in layer Lb with transportation mode tropt at the considered time period mp, dp, hp, \( {D}_{x,y}^{\mathrm{La},\mathrm{Lb}} \) is the distance between object x in layer La and object y in layer Lb, and \( Ai{n}_{m, pi, mp, dp, hp}^{\mathrm{L}2} \), \( Aou{t}_{m, pm, mp, dp, hp}^{\mathrm{L}2} \), \( Ai{n}_{n, pz, mp, dp, hp}^{\mathrm{L}3} \) and \( Aou{t}_{n, pp, mp, dp, hp}^{\mathrm{L}3} \) stand for the quantity of pi stored in the inlet of L2, the quantity of pm stored in the outlet of L2, the quantity of pz stored in the inlet of L3 and for the quantity of pp stored in the outlet of L3 in the assessed time periods.
It should be noted that for simplicity of this study, the emissions given off during constructing the biogas production plants and pipelines are omitted and as such, the equipment are assumed to be used over the entire lifetime of the plant. This assumption therefore suggests relatively small contribution of GHG emissions during construction over the plant’s lifetime.
The objective which considers economic and environmental parts is defined as maximizing the economic profit while including the multiplication of the released and avoided GHG emissions with the prices of GHG emissions, \( {P}^{Economi{c}^{+ GHG}} \):
$$ {P}^{Economi{c}^{+ GHG}}={P}^{Economi c}+ GHG\cdot {p}^{GHG} $$
(15)
where pGHG stands for the prices of GHG emissions.
Economic, environmental and social objectives (sustainability profit)
The third objective considers economic, environmental and social parts which implements the concept of Sustainability profit (PSustainability), first proposed in Zore et al. [26]. PSustainability combines economic, environmental and social indicators into monetary values ($/y).
PSustainability is stipulated mathematically (see Eq. (16)) as the sum of Economic profit (PEconomic, see Eq. (9)), Eco-profit (PEco, see Eq. (17)) and Social Profit (PSocial, see Eq. (20)):
$$ {P}^{Sustainability}={P}^{Eco nomic}+{P}^{Eco}+{P}^{Social} $$
(16)
The eco-profit (PEco) [30] derives from the difference between the sum of all the eco-benefits (EB) and the eco-costs (EC) within the biogas supply network:
$$ {P}^{Eco}= EB- EC $$
(17)
Eco-benefit (EB) (see Eq. (18)) is described in monetary terms ($/y) as the sum of all positive impacts of activities/materials which unburden the environment while the eco-cost (EC) (see Eq. (19)) shows the sum of all negative impacts of activities/materials which burden the environment [30].
$$ {\displaystyle \begin{array}{l}E{B}_p=\sum \limits_{m\in M}\sum \limits_{t\in T}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}{F}_{m,p,t, mp, dp, tp}^{\mathrm{L}1,\mathrm{L}2, net}\cdot {c}_p^{EB}+\\ {}\kern3.25em \sum \limits_{m\in M}\sum \limits_{j\in J}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}{F}_{m,j,p, mp, dp, tp}^{\mathrm{L}2,\mathrm{L}4, net}\cdot {c}_p^{EB}\cdot {f}_p^S+\\ {}\kern3.25em \sum \limits_{n\in N}\sum \limits_{j\in J}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}{F}_{n,j,p, mp, dp, tp}^{\mathrm{L}3,\mathrm{L}4, net}\cdot {c}_p^{EB}\cdot {f}_p^S,\kern2em \forall p\in \left\{ PB, PD, PP\right\}\end{array}} $$
(18)
where \( {c}_p^{EB} \) is the eco-benefit coefficient (see Additional file 1: Table S28) of material or energy p ($/kg, $/kWh). It is worth stating that Eq. (18) is formulated in a similar way as Eq. (13) but considers eco-benefit coefficients instead of GHG emission coefficients related to avoided GHG emissions. The same substitution factors are assumed as previously mentioned in the case of avoided GHG emissions due to substitution of products.
$$ {\displaystyle \begin{array}{l}E{C}_p=\sum \limits_{m\in M}\sum \limits_{t\in T}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}{F}_{m,p,t, mp, dp, tp}^{\mathrm{L}1,\mathrm{L}2, net}\cdot {c}_p^{EC}+\\ {}\kern2.25em \sum \limits_{m\in M}\sum \limits_{j\in J}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}{F}_{m,j,p, mp, dp, tp}^{\mathrm{L}2,\mathrm{L}4,\mathrm{net}}\cdot {c}_p^{EC}+\\ {}\kern2.25em \sum \limits_{n\in N}\sum \limits_{j\in J}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}{F}_{n,j,p, mp, dp, tp}^{\mathrm{L}3,\mathrm{L}4,\mathrm{net}}\cdot {c}_p^{EC}+\\ {}\kern2.25em \sum \limits_{m\in M}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}{F}_{m,p, mp, dp, tp}^{\mathrm{buy},\mathrm{L}2}\cdot {c}_p^{EC}+\sum \limits_{n\in N}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}{F}_{n,p, mp, dp, tp}^{\mathrm{buy},\mathrm{L}3}\cdot {c}_p^{EC}+\\ {}\kern2.25em 2\cdot \left(\sum \limits_{x\in \left\{I,M,N\right\}}\sum \limits_{y\in \left\{M,N,J\right\}}\sum \limits_{tropt\in \left\{ road\right\}}\sum \limits_{mp\in MP}\sum \limits_{dp\in DP}\sum \limits_{hp\in HP}{F}_{x,y,p, mp, dp, tp}^{\mathrm{L}\mathrm{a},\mathrm{L}\mathrm{b},\mathrm{tropt}}\cdot {D}_{x,y}^{\mathrm{L}\mathrm{a},\mathrm{L}\mathrm{b}}\cdot {c}_{p, tropt}^{EC}\right)+\\ {}\kern2.75em \left( Ai{n}_{m, pi, mp, dp, hp}^{\mathrm{L}2}+ Aou{t}_{m, pm, mp, dp, hp}^{\mathrm{L}2}+ Ai{n}_{n, pz, mp, dp, hp}^{\mathrm{L}3}+ Aou{t}_{n, pp, mp, dp, hp}^{\mathrm{L}3}\right)\cdot {c}_p^{EC}\cdot 0.05,\\ {}\kern2.75em \forall p\in \left\{ PB, PN, PD, PP, POUTPIM, POUTPIN\right\}\end{array}} $$
(19)
whereby \( {c}_p^{EC} \) is the eco-cost coefficient for p ($/kg) (see Additional file 1: Table S29) and \( {c}_{p, tropt}^{EC} \) is eco-cost coefficient related to transport of material or energy p ($/(kg·km), $/(kWh·km)) (see Additional file 1: Table S30). More details on calculation of eco-profit could be found in Čuček et al. [30].
Eco-cost and eco-benefit coefficients have also been obtained from the website <ecocostvalue.com> [32] and checked with OpenLCA software [43], similarly as for GHG emission coefficients. Eco-costs express the amount of environmental burden a product causes based on prevention of that burden which tends to reduce the environmental pollution and materials depletion to a level which is in line with the Earth’s carrying capacity [47]. Eco-costs consider environmental burden of global warming, acidification, eutrophication, summer smog, fine dust, eco-toxicity, and the use of metals, rare earth, fossil fuels, water and land [32]. Eco-benefits on the other hand represent the avoided cost due to avoided pollution [48] and thus more sustainable solutions are obtained as it is current practice, i.e. the solutions which represent the progress toward sustainable development [48].
The concept of Social Profit (PSocial) first propounded in Zore et al. [26] is defined as the summation of paid Social Security (SS) contributions and the benefits related to creation of new jobs (BJobs) subtracting the social cost (cSocial) (see Eq. (20)). The Social Security (SS) contributions paid is a difference between average gross (\( {S}_{t, mp}^{Gross} \)) and net salaries (\( {S}_{t, mp}^{Net} \)) in a given production sector per month using a given technology t multiplied by the number of new jobs (\( {N}_{t, mp}^{Jobs} \)) created. In addition, the benefits of the new jobs created (BJobs) are the product of the average state/country social transfer for unemployed people (\( {c}_{mp}^{s, UNE} \)) and \( {N}_{t, mp}^{Jobs} \). Social costs (cSocial) are described as the level of social support made by the state/country and organization to the employee(s), and in this respect is the product of number of new jobs created (\( {N}_{t, mp}^{Jobs} \)) and sum of average state/country social transfer \( \left({c}_{mp}^{s, Country}\right) \) and organization social charge (\( {c}_{mp}^{s, Organisation} \)) per employee. Additionally, the social costs within an organization refer to activities set aside to improve the social status of employees and the community as an extension. Such activities may include team building exercises, paid vacation, free accommodation within the organization’s premises and others [26]. State/country social assistance on the other hand refers to such activities as improved health insurance, child allowance, scholarships and others. Hence, the general relation for the social profit is as given in Eq. (20):
$$ {\displaystyle \begin{array}{l}{P}^{Social}= SS+{B}^{Jobs}-{c}^{Social}=\\ {}\kern2.5em \sum \limits_{mp\in MP}\sum \limits_{t\in T}{N}_{t, mp}^{Jobs}\cdot \left[\left({S}_{t, mp}^{Gross}-{S}_{t, mp}^{Net}\right)+{c}_{mp}^{s, UNE}-\left({c}_{mp}^{s, Country}+{c}_{mp}^{s, Organisation}\right)\right]\end{array}} $$
(20)
The number of new jobs created is given by Eq. (21) where, LC is labour cost.
$$ {N}_{t, mp}^{Jobs}=\frac{LC}{\mid mp\mid \cdot {S}_{t, mp}^{Gross}},\kern1.75em \forall {t}_2\in T, mp\in MP $$
(21)
The parameters used to calculate social profit are taken from Zore et al. [26, 48] and are shown in Additional file 1: Table S31. For better understanding of mathematical model, a nomenclature of the notation used in Eqs. (1)–(21) can be found in Part B in Additional file 1.