_{1}

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The quantitative assessment framework of the water, energy and food (WEF) nexus proposed by [1] permits the analysis of the WEF as an interconnected system of resources that directly and indirectly affect one another. The model performs simulation of policy options and scenarios that respond to quantitative variations of the use of WEF resources. One of the key outcomes of the mathematical formulation of the model is the WEF nexus intersectoral technology matrix. In order to take advantages and analyzing policy options of adopting high efficient intersectoral use technologies, WEF intersectoral use intensities and intersectoral allocation coefficients are introduced to the technology matrix of the nexus model proposed in [1]. The developed method is then applied to evaluate the WEF nexus case study of Lebanon. Lastly, the conclusions and further developments are presented.

The important interactions between water energy and food are imposing new challenges to research and scientific modelling, especially that these resources are driven by many same key drivers, such as population growth, urbanization, mobility, economic development, climate change, technology modifications, governance, market, regulations and international trade [

This paper presents methodological supplement to the model presented in [

A quantitative assessment framework of the WEF nexus is proposed in [

The end use quantities (y) are those used in the socio-economic system which cover households demands, government demands, rest of the economy demands, losses, accumulation (storage) and exports.

A set of inflows were identified to represent the WEF sectors, for example, surface water, groundwater, desalination, wastewater and drainage water reuse inflows covering the water sector, imported petroleum and all types of electricity including renewable energy, covering the energy sector, and irrigated cereals, irrigated roots and other food production items covering the area of food.

It is important to mention that these inflows are particularly identified for the Lebanese case study presented in [

A set of equations are developed in [

If we denote by:

n: number of water resources inflows (i.e. surface water, groundwater, desalination…);

m: number of energy resources inflows (i.e. petroleum, natural gas, electricity, renewable energy…);

h: number of food resources inflows (i.e. irrigated crop, animal production, fisheries…);

^{th} water resource inflow in the j^{th} energy resource inflow;

^{th} water resource inflow in the j^{th} food resource inflow;

^{th} energy resource inflow in the j^{th} water resource inflow;

^{th} energy resource inflow in the j^{th} energy resource inflow;

^{th} energy resource inflow in the j^{th} food resource inflow;

^{th} food resource inflow in the j^{th} energy resource inflow;

^{th} food resource inflow in the j^{th} food resource inflow.

The expanded intersectoral intensity coefficients of the water energy food nexus system are defined as follows:

where^{th} water resource inflow, total use of the j^{th} energy resource inflow and total use of the j^{th} food resource inflow, respectively.

In block matrix form:

where A is the WEF nexus technology matrix.

The water for water and food for water relationships are considered quantitatively negligible, so they are set equal to zero.

In terms of nexus quantification, when the intersectoral and end use values are constructed, then we can start evaluating scenarios, for example, what if we produce more food, what if there will be droughts, what if we reduce losses and wastes, and so forth.

The total outputs (x) caused by end use quantities (y) are related by the following equations [

and

where I is the identity matrix.

The whole core of the WEF nexus is contained in the matrix L. It is easy to examine the changes in all elements in the intersectoral WEF use quantities (Z) caused by (y) [

The first type of simulation that could be performed using the Q-Nexus Model consists of assuming variations of the end use values represented in the model by the (y) vector. These scenarios are analyzed for any planned policy related to increasing of resources demand or reducing losses. A second type of simulation could be also performed using the model by analyzing scenarios of changing the technology matrix (A) by considering “High Efficient” and “Best Practice” technology matrix alternatives. However, this necessitates a methodological addition to the model presented in [

The WEF “High Efficient” or “Best Practice” intersectoral use technologies can be defined as those for which the ratios of the intersectoral quantities used to produce one unit of resource are relatively low.

If we denote by:

^{th} energy inflow;

^{th} food inflow;

^{th} water inflow;

^{th} energy inflow;

^{th} food inflow;

^{th} energy inflow;

^{th} food inflow.

The WEF intersectoral use intensities (t) and intersectoral allocation coefficients (c) are proposed to be integrated within the “technology matrix”, these two policy aspects are defined as follows:

1) WEF intersectoral use intensities (t):

WEF intersectoral use intensities (t) (may be referred also as footprints) were selected because they are able to refer to the efficiency of the intersectoral use technologies. The lower the value of resources use intensities (or footprints), the better the resources usage in the production of other resources inflows.

The WEF intersectoral use intensities (t) are calculated as follows:

Intensity of water use in the j^{th} energy inflow:

Intensity of water use in the j^{th} food inflow:

Intensity of energy use in the j^{th} water inflow:

Intensity of energy use in the j^{th} energy inflow:

Intensity of energy use in the j^{th} food inflow:

Intensity of food use in the j^{th} energy inflow:

Intensity of food use in the j^{th} food inflow:

The WEF intersectoral use intensities t represent the effectiveness of the nexus system and they could be used to compare the performance of different WEF nexus systems. Moreover, the intensities t will serve to identify sustainable policy options by evaluating and analyzing the effects of technology changes on quantitative assessment of WEF nexus. For instance, the effect of the replacement of gravity irrigation with pressurized irrigation system, which achieve water savings and require more energy, could be analyzed by changing the corresponding water and energy use intensities.

The construction of the technology matrix based on the best practice WEF intersectoral use intensities will be addressed in the subsequent paragraphs.

2) WEF intersectoral allocation coefficients (c):

The WEF intersectoral allocation coefficients are defined as follows:

In block matrix form:

where C is the WEF intersectoral allocation matrix. WEF nexus intersectoral allocation policies vary from one system to another. Each country is characterized by specific WEF nexus system in accordance to its natural, social, economic and governance conditions.

The two coefficients t and c are introduced to Equations (2)-(8) as follows:

In block matrix form:

where

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 (C^{0} = C^{1} = C), the new technology matrix A^{1} caused by new WEF intersectoral use intensities (t^{1}) is then found as in equation 34.

the total outputs (x^{1}) caused by new technology matrix A^{1} are then calculated as in Equation 10.

with this result for x^{1}, it is easy to examine the changes in all elements in the intersectoral WEF use quantities (Z^{1} matrix) caused by A^{1}. From the definition of intensity coefficients, we find Z^{1} = A^{11} along with t^{1}, where

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.

In order to put the introduced method in practice, the data of the WEF nexus case study of Lebanon as presented in [

Water inflows (including extraction, treatment, conveyance & distribution) (Mm^{3}/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).

Water use intensities in energy and food inflows are calculated using equations (12) and (13), the results are presented in

The WEF intersectoral allocation coefficients are calculated using Equations (19)-(25), the results are presented in

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

The resulting changes in intersectoral quantities (