mic and governance conditions.
The two coefficients t and c are introduced to Equations (2)-(8) as follows:
In block matrix form:
where is the diagonal matrix with the elements of the t vector along the main diagonal.
Analyzing scenarios of changing the technology matrix (A) by considering “High Efficient” and “Best Practice” technology matrix alternatives could be performed using the developed equations. Assuming that the WEF intersectoral allocation policies (as represented in C matrix) do not change, the quantitative variations of resources in any WEF nexus system due to the change of the intersectoral use intensities could be evaluated.
If the superscript “1” is used to represent the values of variables after the change in intersectoral use and the superscript “0” is used to represent the initial situation for values of variables. Assuming that WEF intersectoral allocation policies is unchanged (C0 = C1 = C), the new technology matrix A1 caused by new WEF intersectoral use intensities (t1) is then found as in equation 34.
the total outputs (x1) caused by new technology matrix A1 are then calculated as in Equation 10.
with this result for x1, it is easy to examine the changes in all elements in the intersectoral WEF use quantities (Z1 matrix) caused by A1. From the definition of intensity coefficients, we find Z1 = A11 along with t1, where is the diagonal matrix with the elements of the vector along the main diagonal.
Moreover, based on the origination of (t) and (c), it became possible interpreting the A matrix to better understand the WEF nexus system and to compare different nexus systems with each other, therefore, the A matrix of the Q-Nexus Model totally befits to be a Nexus Matrix Code or fingerprinting matrix of a nexus system.
4. Application and Analysis of Results
In order to put the introduced method in practice, the data of the WEF nexus case study of Lebanon as presented in  will be analyzed. The considered WEF nexus inflows are as follows:
Water inflows (including extraction, treatment, conveyance & distribution) (Mm3/year): 1) surface water (W1), 2) groundwater (W2), 3) desalination (W3), 4) wastewater reuse (W4), 5) recycled water and agricultural drainage water reuse (W5).
Energy inflows (evaluated in terms of primary energy equivalent in ktoe/ year on a net calorific value basis): 1) imported petroleum (E1), 2) electricity (petroleum) (E2), 3) electricity (hydro) (E3), 4) imported electricity (E4), v) electricity (wind/solar) (E5), 5) biofuels (E6).
Food inflows (including agriculture, food processing & transportation) (kt/year): 1) irrigated cereals (F1), 2) irrigated roots and tubers (F1), 3) irrigated vegetables (F2), 4) irrigated fruits (F3), 5) other Irrigated agriculture (F4), 6) livestock-meat (F5), 7) livestock-milk (F6), 8) livestock-eggs (F7), 9) fishing and aquaculture production (F8), 10) rainfed agriculture (F9), 11) imported agricultural products (F10), 12) imported livestock products-meat, milk, eggs & fish (F11).
Table 1 presents the WEF inflows intersectoral use and the corresponding end use for year 2012. There was no biofuel production in 2012 in Lebanon, the use of biomass was limited to final demand for some economic activities or household use. Therefore, the food for energy indicators is not considered in this application.
Water use intensities in energy and food inflows are calculated using equations (12) and (13), the results are presented in Table 2. The results shown in Table 3 are calculated using equations (14) and (16) and account for energy use intensities in water and food inflows. Table 4 shows the results of the food use intensities in energy and food inflows calculated using Equations (17) and (18).
The WEF intersectoral allocation coefficients are calculated using Equations (19)-(25), the results are presented in Table 5.
A scenario of replacing of gravity surface irrigation with pressurized irrigation systems for some irrigated crops is analyzed. This scenario will achieve water savings (reduction of water use intensities for some food inflows) and will require more energy. The projected water and energy intersectoral use intensities are shown in Table 6 and Table 7. Using the proposed method, it becomes easy to evaluate the new intersectoral and total outputs from the water, energy and food sectors based on this new intersectoral use intensities. In this scenario we assume that the WEF intersectoral allocation matrix (as represented in the C matrix) and the end use (as represented in the y vector), are unchanged. Table 8 presents the resulted nexus technology matrix for the considered scenario calculated using Equation (35).
The resulting changes in intersectoral quantities () caused by the change of the intersectoral use intensities are then evaluated. The results could be summarized as follows:
Table 1.Intersectoral use of WEF inflows and the corresponding end for year 2012.
Table 2. Intensities of water use in energy and food inflows.
Table 3. Intensities of energy use in water and food inflows.
Table 4. Intensities of food use in energy and food inflows.
Table 5.WEF intersectoral allocation matrix.
Table 6. Projected water use intensities in food inflows (m3/t).
Table 7. Projected energy use intensities in food inflows (toe/kt).
The output of food-related water consumption will decrease by = 96.087 Mm3 which accounts for 8.11 % decrease in water of the total water usage. This decrease in water consumption will lead to decrease in the output of the water-related energy consumption by 11.281 ktoe. Similarly, food-related energy consumption will have to increase its output by = 56.576 ktoe which accounts for 1.02% increase in energy of the total energy usage. This increase in energy consumption will lead to increase in the output of the energy- related water consumption by 0.0015 Mm3. So, the evaluation of the considered scenario shows the amount of achieved water savings and the increase in energy usage, therefore, decision makers will be informed about the quantitative impacts when adopting this policy option.
5. Conclusions and Further Developments
The WEF nexus model presented in  performs simulation of scenarios that respond to quantitative variations of the end use of WEF resources. The simulation of policy options by considering the use of high efficient WEF intersectoral use technologies necessitates a methodological addition to the model. Therefore, a methodological approach to analyzing scenarios of adopting high efficient intersectoral use technologies is introduced. The proposed approach permits the
Table 8.Resulted technology matrix for the considered scenario.
evaluation of the effects of a large number of possible WEF policy scenarios by taking advantages of best practice nexus technology matrices that are technologically most efficient. Two modeling variables are inserted into the technology matrix of the Q-Nexus Model: 1) the WEF intersectoral use intensities (t) and 2) the WEF intersectoral allocation coefficients (c).
The methodological amendment made on the Q-Nexus Model make it well suited to analyzing various technological, structural and quantitative changes scenarios:
1) It allows to analyzing scenarios of changing the technology matrix (A) by considering “High Efficient” and “Best Practice” technology matrix alternatives. It permits also to evaluate the effects of the technology shifts for future development.
2) The construction of the WEF intersectoral allocation matrix (C) allows considering “Optimized” intersectoral allocation matrix options which leads to considering “Optimized” technology matrix alternatives and calculating the resulted WEF quantities.
3) The proposed method permits interpreting the A matrix to better understand the nexus forming and to compare different nexus systems with each other. Based on the origination of (t) and (c), the A matrix of the Q-Nexus Model totally befits to be a Nexus Matrix Code or fingerprinting matrix of a nexus system.
4) It allows analyzing and comparing when effects of additional demands are offset by effects of technology change.
5) Lastly, this approach will enhance the understanding of the complex and dynamic interrelationships between water, energy and food, and support the sustainable planning and management of these finite resources to ensure WEF security and access for all.
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