**1. Introduction**

Water scarcity is serious in China, and per capita water resources are only 25% of the average world level [1]. The spatial and temporal distribution of water resources is extremely unbalanced [2]. Over 80% of water resources are reserved in southern China, while northern China sustains 47% of people with less than 20% of water resources. Water scarcity has become a major bottleneck restricting the sustainable development of the economy and society [3]. The Chinese governmen<sup>t</sup> has implemented the strictest regulation to mitigate water scarcity, e.g., constraining the total use of water resources within 700 billion cubic meters by 2030 [4]. Industry sectors consumed 126 billion cubic meter water accounting for 20% of the total water consumption in 2018 [5]. Among all industrial sectors, the power system is the largest water consumer contributing to 70% of the total industrial water consumption [6]. Therefore, investigating electricity-related water consumption is a key aspect to mitigate water scarcity in China.

To investigate water stress posed by the power system, previous studies have quantified direct water use or consumption of power generation in China [7,8], focusing on different power generation technologies [9,10], environmental impacts [11,12] and future scenarios analysis [13,14]. To further quantify the life cycle water consumption of functional electricity (kWh), virtual water (VW) was introduced in the investigation for electricity-water nexus. The concepts of virtual water and water footprint were proposed to quantify water consumption through the life cycle of a commodity [15,16]. The life cycle water use of power generation was estimated through different methods [17,18] or at different spatial scales [19–21]. For instance, Feng et al. [17] used the hydro-life-cycle-analysis method to calculate water coefficients of eight power generation types, and they found the water coefficient of hydropower is higher than that of thermal power. Besides, electricity is transferred from arid western provinces to eastern provinces. The mismatch between energy resources and water resources in China has raised many concerns regarding potential water-related impacts from power transmission.

According to the virtual water strategy [16], VW flows are transferred from the source to the sink and thus reallocates the water resources on both sides. In the power system, VW is delivered from power generation provinces to load hubs through electricity transmission, which may induce water scarcity. Multiple studies have explored the virtual water network through electricity transmission in recent years. For instance, Zhu et al. [20] and Guo et al. [21] developed a node-flow model to calculate the virtual water transmission (VWT) among six sub-national grids in the year 2010 and short-time span (2007–2012), respectively. Zhang et al. [22] quantified the interprovincial VWT embodied in thermal power transmission and adjusted VW into virtual scarce water (VSW) in 2011, by improving the spatial resolution to the provincial level and combining the water stress index (WSI). The aforementioned studies proved that VWT extremely exacerbated the local water scarcity in power generation provinces, especially in northwestern provinces. In USA, Chini et al. [23] explored the blue water and grey water transfers embodied in power grids and provided clear grey water and blue water network. In addition, some studies identified whether the virtual water flows mitigate or exacerbate the water scarcity by comparing the WSI of source and sink [24]. However, there is a lack of study for the evolution of VWT embodied in electricity transmission.

Electricity transmission has increased by 4.2 times in the last decade [25], resulting in a significant increase of VWT. To explore the evolution of VWT, the decomposition analysis is introduced to reveal the driving forces of the evolution of the virtual water network. The decomposition analysis was proposed to quantify different contributions of factors to the change of an indicator between two periods. Structural decomposition analysis (SDA) and index decomposition analysis (IDA) are two widely used approaches to investigate electricity consumption related indicators, such as carbon emissions [26,27] and virtual water consumption [28,29]. Logarithmic Mean Divisia Index (LMDI) is the most popular IDA method with low-resolution data requirements and simplicity. It could also be used to investigate electricity-related carbon emission [30] and water consumption [31]. However, unlike SDA, LMDI cannot distinguish final demand and intermediate demand, thus, the indirect impacts of change in final demand cannot be estimated. To estimate the contribution of indirect impacts to the overall changes of energy-related indicators, the SDA method was widely used. Nevertheless, SDA is restricted by its high-quality data requirements based on the input-output (IO) table. Research [32] proposed a modified SDA based on the electricity transmission table to analyze the contribution of factors (especially for the change of power transmission) to the increase of carbon emission in the power sector. Electricity-related water footprint differs from carbon footprint because water resources vary from region to region when carbon emissions have the same effects on climate change regardless of location. Paper [33] investigated the driving forces for the evolution of VWT embodied in thermal power generation, and the results showed that water efficiency improvements were the main driver to the decrease of VWT at the national level. However, the analysis in paper [33] is not comprehensive, because only thermal power was considered, and the differences of driving forces in various provinces were not included. During the last decades, China's power generation and transmission have expanded significantly, which was largely driven by renewable energy technologies. Hydropower has a higher water coe fficient than thermal power because of evaporation in reservoir. A deep investigation of the evolution of virtual water transfers is necessary for the understanding of water-electricity nexus.

This study investigated the long-time series evolution of virtual (scarce) water network embodied in interprovincial electricity trade, and further identified the driving factors to the change of the virtual (scarce) water transmission by using a modified SDA model. Compared to previous studies, the novelty and significance of this work are as follows: (1) water consumption for all power generation technologies (including thermal, hydropower, wind, solar, and nuclear) are included; (2) VW network and VSW network are both investigated to represent the imbalance spatial distribution of water resources in di fferent provinces; and (3) by using a modified SDA model, a high-resolution decomposition results are analyzed to capture the contribution of di fferent factors at both national level and provincial level.

The structure of this study is into two parts. In part 1, Sections 2.1 and 2.2 constructed the node-flow model of the virtual water network, and the SDA model for investigating the driving factors, respectively. Sections 2.3 and 2.4 introduced the spatial and temporal area of the paper and the data sources. In part 2, Section 3 shows the results for the VW network evolution and its driving factors. Sections 4.1 and 4.2 discussed the impacts of di fferent policies between 2005 to 2014 and advice for Chinese policymaker, while Section 4.3 discussed the limitation and advantages in this study.

## **2. Materials and Methods**

#### *2.1. Modeling Virtual Water Network Embodied in Electricity Trade*

Based on the node-flow model proposed in previous studies [20,21], we modelled inter-provincial electricity transmission as a network with nodes and flows. Each province and each electricity trade is assumed as a node and a flow, respectively. For each node, the total power demand is supplied by local power generation and power imported from other nodes; local plants can only outflow power to satisfy the local demand and other nodes. The transmission loss is ignored in this study, and the balance between power generation and power demand for each node can be expressed below:

$$\sum\_{i=m}^{i=1} ED\_i + \sum\_{i=m, i \neq j}^{i=1} ET\_{ij} = \sum\_{i=m}^{i=1} EG\_i + \sum\_{j=m, j \neq i}^{j=1} ET\_{ji} \tag{1}$$

where *EGi* is the electric power generation of node *i*, *EDi* is the power demand of node *i*, and *ETij* is the electricity flow from node *i* to node *j*.

The balance between power demand and power generation can be adapted to the virtual water transfers by Equation (2):

$$\sum\_{i=m}^{i=1} E G\_{\text{i}} \cdot dw\_{i} + \sum\_{i=m, i \neq j}^{i=1} E T\_{\text{i}j} \cdot tw\_{j} = \left( \sum\_{i=m}^{i=1} E D\_{\text{i}} + \sum\_{j=m, j \neq i}^{j=1} E T\_{\text{j}i} \right) \cdot tw\_{i} \tag{2}$$

where *dwi* represents the direct water consumption factor for functional unit power generation in province *i*, and *twj* represents the embodied water footprint (total water consumption factor) of functional unit power supplied by province *j*.

We assume that all electricity feeding into province *i* is totally mixed, i.e., from di fferent sources. The embodied water footprint of electricity supplied to local users is thus the same as the electricity exported to other nodes for provinces *i*. To calculate *twi*, Equation (2) can be rearranged into Equations (3) and (4):

$$\sum\_{i=-m}^{i=1} EG\_{i^\cdot} dw\_i = \left(\sum\_{i=m}^{i=1} ED\_i + \sum\_{j=m, j\neq i}^{j=1} ET\_{ji}\right) tw\_i - \sum\_{i=m, i\neq j}^{i=1} ET\_{i^\cdot\_j} tw\_j \tag{3}$$

ˆ

$$
\triangle \text{G} \cdot \text{DW} = \text{M} \cdot \text{TW} \tag{4}
$$

where *EG* represents the diagonal matrix of electricity generation *EGi*, *DW* represents the column of *dwi*, *TW* represents the column of *twi*, and *M* is rearranged as electricity trade matrix given by Equation (5):

$$M = \begin{bmatrix} \sum\_{j \neq 1} ET\_{1j} + E\_{d,1} & -ET\_{21} & \cdots & -ET\_{m1} \\ & -ET\_{12} & \sum\_{j \neq 2} ET\_{2j} + E\_{d,2} & \cdots & -ET\_{m2} \\ & \vdots & \vdots & \ddots & \vdots \\ & -ET\_{1m} & -ET\_{2m} & \cdots & \sum\_{j \neq m} ET\_{2j} + E\_{d,2} \end{bmatrix} \tag{5}$$

The vector of total water coefficient can be calculated in the following:

$$T\mathcal{W} = \mathcal{M}^{-1} \cdot \triangle D \cdot DW = H \cdot DW \tag{6}$$

By using Equation (6), we create a linear map between the direct water coefficient and the virtual water coefficient, thus link the water footprint from the electricity consumption side to the production side. Each element in matrix *H*, *hij*, represents the complete electricity transmission, which is different from the direct electricity transmission, because it includes higher-order electricity transmission. The role of *H* is the same as the role of Leonfief matrix in the input–output analysis which transforms the direct consumption to the total consumption.

#### *2.2. Virtual Water Transmission and Decomposition Analysis Model*

SDA can estimate the contribution of different factors to the overall evolution of an indicator, and it has been widely used to investigate the energy-related emissions. However, SDA is highly depended on the IO table, which restricts the application of itself. According to a previous study [32], a modified SDA model applying the electricity transmission table (not IO table) is used in our study. Combing the power generation and demand of each province, we develop the power transmission table as follows:

$$AT = \left\{ AT\_{\text{ij}} \right\}\_{m \times m'} AT\_{\text{ij}} = ET\_{\text{ij}} / ED\_{\text{j}} \tag{7}$$

$$BT = \left\{ BT\_{\text{ij}} \right\}\_{m \times m'} BT\_{\text{ij}} = ET\_{\text{ij}} / EG\_{\text{i}} \tag{8}$$

where *ATji* in matrix *AT* is the power generation in *ith* province feed into *jth* province's functional unit consumption, thus *AT* can be expressed as the power demand structure. *BTij* is the number of power outflows to *jth* provinces in *ith* provinces' power generation, which can be regarded as the power generation structure.

Therefore, matrix *AT* can be regarded as a power demand table and matrix *BT* should be considered as the power generation structure table. In this case, the total power outflow of a province can be represented as:

$$E\_{\rm out} = AT \cdot ED \tag{9}$$

where *ED* refers to the column for power demand in all provinces and electricity inflow of each province can be represented as:

$$E\_{\rm in} = BT^T \cdot EG \tag{10}$$

Therefore, the electricity generation can be represented as:

$$EG = E\_{\rm out} - E\_{\rm in} + ED = AT \cdot ED - BT^T \cdot EG + ED \tag{11}$$

$$
\dot{\mathbf{E}}\cdot\mathbf{E}\mathbf{G} = \left(\mathbf{I} + BT^T\right)^{-1} \cdot \left(\mathbf{I} + AT\right) \cdot ED \tag{12}
$$

The *VWT* embodied in electricity transmission can be calculated as:

$$VWT = T\mathcal{W}^T \cdot E\_{\text{in}} = T\mathcal{W}^T \cdot BT^T \cdot EG = D\mathcal{W}^T \cdot H^T \cdot BT^T \cdot \left(I + BT^T\right)^{-1} \cdot \left(I + AT\right) \cdot ED \tag{13}$$

where *TW<sup>T</sup>* is the transpose of *TW*, and *I* refers to the identity matrix. To classify the driving factor of *VWT*, Equation (13) could be further transformed into Equation (14):

$$VWT = DW^T \cdot H^T \cdot \text{PG} \cdot \text{PD} \cdot ED \tag{14}$$

where *DW* is the direct water coefficient, which is dominated by the power generation mix of each province, and *H* is the revised power transmission structure.

Matrix *DW* can be regarded as the power generation mix, while *H* can be regarded as a power transmission structure, and matrix *ED* can be regarded as the power demand factor. As is described in the previous study [32], at the national level, factor *PG* and *PD* can be expressed as power generation structure and power demand structure. At the province level, it should be considered as the external power generation structure and the external power demand structure.

Compared to the SDA model using the input–output Table, this modified model uses electricity transmission data, and could introduce power generation structure and power demand structure to the VWT change. The complete additive decomposition method can eliminate the residual of decomposition [32,34], Equation (15) is thus expressed as an example to decompose the VWT between two periods.

$$\begin{aligned} \Delta VWT\_t &= VWT\_{t+1} - VWT\_t\\ &= E(\Delta DW) + E(\Delta H) + E(\Delta PD) + E(\Delta PG) + E(\Delta ED) \end{aligned} \tag{15}$$

where *E*(Δ*DW*) refers to the contribution of the power generation mix, *E*(Δ*H*) refers to the contribution of the electricity transmission structure, *E*(Δ*PD*) refers to the contribution of the (external) power demand structure, *E*(Δ*PG*) refers to the contribution of the (external) power generation structure, and *E*(Δ*ED*) refers to the contribution of the power demand.

In an SDA model, to obtain a complete decomposition form, each factor should be weighted by Laspeyres or Paasche weights [27]. However, SDA would produce N! different decomposition forms when it has N factors. The N decomposition forms represent N ways to eliminate residuals. In previous studies [28,32], to reduce the computational complexity, residuals are divided equally among factors by using the average residuals for different decomposition forms. This study also uses the averages of different forms to eliminate residuals.

The decomposition formulation of modified SDA model between the two periods can be expressed as follows:

*E*Δ*DW<sup>T</sup>* = 15 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ <sup>Δ</sup>*DWT*·*Ht*·*PGt*·*PDt*·*EG* + <sup>Δ</sup>*DWT*·*Ht*+1·*PGt*+1·*PDt*+1·*EGt*+<sup>1</sup> + 120 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ <sup>Δ</sup>*DWT*·*Ht*+1·*PGt*·*PDt*·*EGt* + <sup>Δ</sup>*DWT*·*Ht*·*PGt*+1·*PDt*·*EGt*<sup>+</sup> <sup>Δ</sup>*DWT*·*Ht*·*PGt*·*PDt*+<sup>1</sup> ·*EGt* + <sup>Δ</sup>*DWT*·*Ht*·*PGt*·*PDt* ·*EGt*+1+ <sup>Δ</sup>*DWT*·*Ht*·*PGt*+1·*PDt*+<sup>1</sup> ·*EGt*+<sup>1</sup> + <sup>Δ</sup>*DWT*·*Ht*+1·*PGt*·*PDt*+<sup>1</sup> ·*EGt*+1+ <sup>Δ</sup>*DWT*·*Ht*+1·*PGt*+1·*PDt*·*EGt*+<sup>1</sup> + <sup>Δ</sup>*DWT*·*Ht*+1·*PGt*+1·*PDt*+1·*EGt* ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ + 130 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ <sup>Δ</sup>*DWT*·*Ht*+1·*PGt*+1·*PDt*·*EGt* + <sup>Δ</sup>*DWT*·*Ht*+1·*PGt*·*PDt*+1·*EGt*<sup>+</sup> <sup>Δ</sup>*DWT*·*Ht*+1·*PGt*·*PDt*·*EGt*+<sup>1</sup> + <sup>Δ</sup>*DWT*·*Ht*·*PGt*+1·*PDt*+<sup>1</sup> ·*EGt*+ <sup>Δ</sup>*DWT*·*Ht*·*PGt*+1·*PDt* ·*EGt*+<sup>1</sup> + <sup>Δ</sup>*DWT*·*Ht*·*PGt*·*PDt*+<sup>1</sup> ·*EGt*+<sup>1</sup> ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (16)

*E*Δ*H<sup>T</sup>* = 15 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *DWt<sup>T</sup>*·Δ*HT*·*PGt*·*PDt*·*EGt* + *DWt*+1*T*·Δ*HT*·*PGt*+1·*PDt*+1·*EGt*+<sup>1</sup> + 120 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *DWt*+1*T*·Δ*HT*·*PGt*·*PDt*·*EGt* + *DWt<sup>T</sup>*·Δ*HT*·*PGt*+1·*PDt*·*EGt*<sup>+</sup> *DWt<sup>T</sup>*·Δ*HT*·*PGt*·*PDt*+<sup>1</sup> ·*EGt* + *DWt<sup>T</sup>*·Δ*HT*·*PGt*·*PDt* ·*EGt*+1+ *DWt<sup>T</sup>*·Δ*HT*·*PGt*+1·*PDt*+<sup>1</sup> ·*EGt*+<sup>1</sup> + *DWt*+1*T*·Δ*HT*·*PGt*·*PDt*+<sup>1</sup> ·*EGt*+1+ *DWt*+1*T*·Δ*HT*·*PGt*+1·*PDt*·*EGt*+<sup>1</sup> + *DWt*+1*T*·Δ*HT*·*PGt*+1·*PDt*+1·*EGt* ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ + 130 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ *DWt*+1*T*·Δ*HT*·*PGt*+1·*PDt*·*EGt* + *DWt*+1*T*·Δ*HT*·*PGt*·*PDt*+1·*EGt*<sup>+</sup> *DWt*+1*T*·Δ*HT*·*PGt*·*PDt*·*EGt*+<sup>1</sup> + *DWt<sup>T</sup>*·Δ*HT*·*PGt*+1·*PDt*+<sup>1</sup> ·*EGt*+ *DWt<sup>T</sup>*·Δ*HT*·*PGt*+1·*PDt* ·*EGt*+<sup>1</sup> + *DWt<sup>T</sup>*·Δ*HT*·*PGt*·*PDt*+<sup>1</sup> ·*EGt*+<sup>1</sup> ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (17) *E*(Δ*PG*) = 15 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *DWt<sup>T</sup>*·*Ht<sup>T</sup>*·Δ*PG*·*PDt*·*EGt* + *DWt*+1*T*·*Ht*+1*T*·Δ*PG*·*PDt*+1·*EGt*+<sup>1</sup> + 120 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *DWt*+1*T*·*Ht<sup>T</sup>*·Δ*PG*·*PDt*·*EGt* + *DWt<sup>T</sup>*·*Ht*+1*T*·Δ*PG*·*PDt*·*EGt*<sup>+</sup> *DWt<sup>T</sup>*·*Ht<sup>T</sup>*·Δ*PG*·*PDt*+<sup>1</sup> ·*EGt* + *DWt<sup>T</sup>*·*Ht<sup>T</sup>*·Δ*PG*·*PDt* ·*EGt*+1+ *DWt<sup>T</sup>*·*Ht*+1*T*·Δ*PG*·*PDt*+<sup>1</sup> ·*EGt*+<sup>1</sup> + *DWt*+1*T*·*Ht<sup>T</sup>*·Δ*PG*·*PDt*+<sup>1</sup> ·*EGt*+1+ *DWt*+1*T*·*Ht*+1*T*·Δ*PG*·*PDt*·*EGt*+<sup>1</sup> + *DWt*+1*T*·*Ht*+1*T*·Δ*PG*·*PDt*+1·*EGt* ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ + 130 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ *DWt*+1*T*·*Ht*+1*T*·Δ*PG*·*PDt*·*EGt* + *DWt*+1*T*·*Ht<sup>T</sup>*·Δ*PG*·*PDt*+1·*EGt*<sup>+</sup> *DWt*+1*T*·*Ht<sup>T</sup>*·Δ*PG*·*PDt*·*EGt*+<sup>1</sup> + *DWt<sup>T</sup>*·*Ht*+1*T*·Δ*PG*·*PDt*+<sup>1</sup> ·*EGt*+ *DWt<sup>T</sup>*·*Ht*+1*T*·Δ*PG*·*PDt* ·*EGt*+<sup>1</sup> + *DWt<sup>T</sup>*·*Ht<sup>T</sup>*·Δ*PG*·*PDt*+<sup>1</sup> ·*EGt*+<sup>1</sup> ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (18) *E*(Δ*PD*) = 15 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *DWt<sup>T</sup>*·*Ht<sup>T</sup>*·*PGt*·Δ*PD*·*EGt* + *DWt*+1*T*·*Ht*+1*T*·*PGt*+1·Δ*PD*·*EGt*+<sup>1</sup> + 120 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *DWt*+1*T*·*Ht<sup>T</sup>*·*PGt*·Δ*PD*·*EGt* + *DWt<sup>T</sup>*·*Ht*+1*T*·*PGt*·Δ*PD*·*EGt*<sup>+</sup> *DWt<sup>T</sup>*·*Ht<sup>T</sup>*·*PGt*+1·Δ*PD*·*EGt* + *DWt<sup>T</sup>*·*Ht<sup>T</sup>*·*PGt*·Δ*PD*·*EGt*+1<sup>+</sup> *DWt<sup>T</sup>*·*Ht*+1*T*·*PGt*+1·Δ*PD*·*EGt*+<sup>1</sup> + *DWt*+1*T*·*Ht<sup>T</sup>*·*PGt*+1·Δ*PD*·*EGt*+1<sup>+</sup> *DWt*+1*T*·*Ht*+1*T*·*PGt*·Δ*PD*·*EGt*+<sup>1</sup> + *DWt*+1*T*·*Ht*+1*T*·*PGt*+1·Δ*PD*·*EGt* ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ + 130 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ *DWt*+1*T*·*Ht*+1*T*·*PGt*·Δ*PD*·*EGt* + *DWt*+1*T*·*Ht<sup>T</sup>*·*PGt*+1·Δ*PD*·*EGt*<sup>+</sup> *DWt*+1*T*·*Ht<sup>T</sup>*·*PGt*·Δ*PD*·*EGt*+<sup>1</sup> + *DWt<sup>T</sup>*·*Ht*+1*T*·*PGt*+1·Δ*PD*·*EGt*<sup>+</sup> *DWt<sup>T</sup>*·*Ht*+1*T*·*PGt*·Δ*PD*·*EGt*+<sup>1</sup> + *DWt<sup>T</sup>*·*Ht<sup>T</sup>*·*PGt*+1·Δ*PD*·*EGt*+<sup>1</sup> ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (19) *E*(Δ*EG*) = 15 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *DWt<sup>T</sup>*·*Ht<sup>T</sup>*·*PGt*·*PDt*·Δ*EG* + *DWt*+1*T*·*Ht*+1*T*·*PGt*+1·*PDt*+1·Δ*EG* + 120 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *DWt*+1*T*·*Ht<sup>T</sup>*·*PGt*·*PDt*·Δ*EG* + *DWt<sup>T</sup>*·*Ht*+1*T*·*PGt*·*PDt*·Δ*EG*<sup>+</sup> *DWt<sup>T</sup>*·*Ht<sup>T</sup>*·*PGt*+1·*PDt*·Δ*EG* + *DWt<sup>T</sup>*·*Ht<sup>T</sup>*·*PGt*·*PDt*+1·Δ*EG*<sup>+</sup> *DWt<sup>T</sup>*·*Ht*+1*T*·*PGt*+1·*PDt*+1·Δ*EG* + *DWt*+1*T*·*Ht<sup>T</sup>*·*PGt*+1·*PDt*+1·Δ*EG*<sup>+</sup> *DWt*+1*T*·*Ht*+1*T*·*PGt*·*PDt*+1·Δ*EG* + *DWt*+1*T*·*Ht*+1*T*·*PGt*+1·*PDt*·Δ*EG* ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ + 130 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ *DWt*+1*T*·*Ht*+1*T*·*PGt*·*PDt*·Δ*EG* + *DWt*+1*T*·*Ht<sup>T</sup>*·*PGt*+1·*PDt*·Δ*EG*<sup>+</sup> *DWt*+1*T*·*Ht<sup>T</sup>*·*PGt*·*PDt*+1·Δ*EG* + *DWt<sup>T</sup>*·*Ht*+1*T*·*PGt*+1·*PDt*·Δ*EG*<sup>+</sup> *DWt<sup>T</sup>*·*Ht*+1*T*·*PGt*·*PDt*+1·Δ*EG* + *DWt<sup>T</sup>*·*Ht<sup>T</sup>*·*PGt*+1·*PDt*+1·Δ*EG* ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (20)

As we can see, even though the average has been used in this model, the structure of SDA is still very complex.
