*2.2. Structural Decomposition Analysis (SDA)*

As the primary fossil energy consumptions formulated in Equation (3) include both domestically produced and imported energy, the SDA technique can be performed, focusing on either of these two sources. From Equation (1), the change in the consumption of domestically produced energy in China between 2007 and 2012 can be calculated by the following:

$$
\Delta \mathbf{d} = \mathbf{d}\_{12} - \mathbf{d}\omega = \hat{\mathbf{e}} \mathbf{L}\_{12} \mathbf{f}\_{12} - \hat{\mathbf{e}} \mathbf{L}\omega \mathbf{f}\omega \tag{4}
$$

The driving factors can be found as the final demand effect ( Δ**d***F*) and the Leontief inverse effect (Δ**d***L*), respectively. The final demand effect refers to the energy-use change induced by the shifts of the final demand while holding the Leontief inverse constant. The Leontief inverse effect represents the energy-use change generated from the Leontief inverse shift, with a given final demand. We take

the average of all of the possible decompositions in order to qualify each effect. Dietzenbacher and Los (1998), and Hoekstra and Van Den Bergh (2002) provide the theoretical details in their studies [20,21].

$$
\Delta \mathbf{d}\_F = \frac{1}{2} \left( \hat{\mathbf{e}} \mathbf{L}\_{\mathcal{O}} \Delta \mathbf{f} + \hat{\mathbf{e}} \mathbf{L}\_{12} \Delta \mathbf{f} \right) \tag{5}
$$

$$
\Delta \mathbf{d}\_L = \frac{1}{2} \left( \hat{\mathbf{e}} \Delta \mathbf{L} \mathbf{f}\_{0\mathcal{T}} + \hat{\mathbf{e}} \Delta \mathbf{L} \mathbf{f}\_{12} \right) \tag{6}
$$

where Δ represents the change of a factor.

We further decompose the Leontief inverse effect ( Δ**d***L*) into trade effect ( Δ**d***T*) and technology effect ( Δ**d***B*). Moreover, the technology effect is identified along three dimensions, as the energy-use change associated with shifts in the non-energy input, energy composition, and energy input level. Trade effect refers to the energy change induced by the shifts of the input factors' sourcing locations. Technology effect refers to the combined effect of the non-energy input, energy composition, and energy input level effects. Firstly, the non-energy input effect ( Δ**d***G*) is the energy-use change generated from the change in non-energy inputs in production, when holding the other factors constant. Secondly, energy composition effect ( Δ**d***C*) refers to the energy-use change induced by the energy mix shifts in the production process. Thirdly, the energy input level effect ( Δ**d***E*) is the energy-use change brought about by the sum of all kinds of energy consumption variation per unit of output (i.e., energy efficiency).

Furthermore, we have the following matrix decomposition of the Leontief inverse matrix [22].

$$
\Delta \mathbf{L} = \mathbf{L}\_{12} - \mathbf{L}\_{07} = \frac{1}{2} (\mathbf{L}\_{12} \Delta \mathbf{A} \mathbf{L}\_{07} + \mathbf{L}\_{07} \Delta \mathbf{A} \mathbf{L}\_{12}) \tag{7}
$$

In this study, we re-defined the input coefficient matrices of 2007 and 2012, as follows:

$$\mathbf{A}\_{\mathcal{O}\mathcal{T}} = \mathbf{T}\_{\mathcal{O}\mathcal{T}} \diamond \mathbf{B}\_{\mathcal{O}\mathcal{T}} \tag{8}$$

$$\mathbf{A}\_{12} = \mathbf{T}\_{12} \odot \mathbf{B}\_{12} \tag{9}$$

Here, **T** is the inter-provincial trade coefficient matrix of China, showing the proportion supplied from province *r* of the total intermediate delivery from sector *i*, required for producing one unit of sector *j* of province *s*, and it can be defined as follows:

$$\mathbf{T} = (t\_{ij}^{rs}) = \left(\frac{a\_{ij}^{rs}}{\sum\_{r=1}^{R} a\_{ij}^{rs}}\right) = \begin{bmatrix} \mathbf{T}^{11} & \cdots & \mathbf{T}^{1R} \\ \vdots & \ddots & \vdots \\ \mathbf{T}^{R1} & \cdots & \mathbf{T}^{RR} \end{bmatrix} \tag{10}$$

where **<sup>T</sup>***rs*(*<sup>r</sup>*, *s* = 1, . . . , *R*) is the inter-provincial trade coefficient submatrix for the goods and services flowing from province *r* to *s*. **B** in the right-hand sides of Equations (8) and (9) is the technical coefficient matrix of the Chinese provinces, and it can be written as follows:

$$\mathbf{B} = \begin{bmatrix} \mathbf{B}^1 & \cdots & \mathbf{B}^R \\ \vdots & \ddots & \vdots \\ \mathbf{B}^1 & \cdots & \mathbf{B}^R \end{bmatrix} \tag{11}$$

where **B***s* = (*bsij*)=( *R* ∑ *<sup>r</sup>*=1 *arsij* ) is the technical coefficient submatrix of a specific province (*s*), showing the total intermediate delivery from sector *i* required for producing one unit of sector *j* of province *s*. It should be noted that - denotes the Hadamard product.

We further propose decomposing the technical coefficient submatrix of a specific province (*s*), **B***s* (*s* = 1, . . . , *<sup>R</sup>*), as follows:

$$\begin{array}{c} \text{Energy sector} \\ \begin{array}{c} \text{Energy sector} \\ \vdots \\ \text{Energy sector} \end{array} \\ \begin{array}{c} \text{Energy sector} \\ \begin{array}{c} \\ \text{Non-energy sector} \end{array} \end{array} \left[ \begin{array}{c} 0 & \dots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \dots & 0 \\ 0 & \dots & 0 \\ \vdots & \ddots & \vdots \\ \text{Non-energy sector} \end{array} \right] \\ \begin{array}{c} \text{Non-energy sector} \\ \begin{bmatrix} \frac{p\_{11}^{e}}{\sum\_{i=1}^{e}p\_{i1}^{e}} & \dots & \frac{p\_{1N}^{e}}{\sum\_{i=1}^{e}p\_{N}} \\ \vdots & \ddots & \vdots \\ \vdots & \ddots & \vdots \\ \frac{p\_{11}^{e}}{\sum\_{i=1}^{e}p\_{11}} & \dots & \frac{p\_{1N}^{e}}{\sum\_{i=1}^{e}p\_{N}} \\ 0 & \dots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \dots & 0 \\ \end{array} \right] \\ \begin{array}{c} \text{S}^{e}\_{1} \\ \vdots \\ \text{S}^{e}\_{1} \end{array} \right] \end{array} \tag{12}$$

where **G***<sup>s</sup>* is the technical coefficient submatrix for the non-energy sectors, **C***<sup>s</sup>* is the energy composition matrix showing the energy mix information of each sector, and **E***s* is the diagonal matrix with the overall energy input level (energy intensity in this study) of each sector. *K* is the total number of energy sectors.

Using the refined decomposition factors of Equations (10), (11), and (12), the input coefficient matrices of 2007 and 2012 can be formulated, respectively, as follows:

$$\mathbf{A}\_{07} = \mathbf{T}\_{07} \odot \mathbf{B}\_{07} = \mathbf{T}\_{07} \odot \left(\mathbf{G}\_{07} + \mathbf{C}\_{07}\mathbf{E}\_{07}\right) \tag{13}$$

$$\mathbf{A}\_{12} = \mathbf{T}\_{12} \circ \mathbf{B}\_{12} = \mathbf{T}\_{12} \circ (\mathbf{G}\_{12} + \mathbf{C}\_{12}\mathbf{E}\_{12}) \tag{14}$$

Accordingly, Δ**L** in Equation (6) can be further decomposed into the changes in the structural factors of **T**, **G**, **C**, and **E**, as follows:

$$
\Delta \mathbf{L}\_{T} = \frac{1}{4} \left[ \mathbf{L}\_{12} \left\{ \Delta \mathbf{T} \odot \left( \mathbf{B}\_{07} + \mathbf{B}\_{12} \right) \right\} \mathbf{L}\_{07} + \mathbf{L}\_{07} \left\{ \Delta \mathbf{T} \odot \left( \mathbf{B}\_{07} + \mathbf{B}\_{12} \right) \right\} \mathbf{L}\_{12} \right] \tag{15}
$$

$$
\Delta\mathbf{L}\_{\rm G} = \frac{1}{4} [\mathbf{L}\_{12} \{ (\mathbf{T}\_{0\mathcal{T}} + \mathbf{T}\_{12}) \diamond \Delta\mathbf{G} \} \mathbf{L}\_{0\mathcal{T}} + \mathbf{L}\_{0\mathcal{T}} \{ (\mathbf{T}\_{0\mathcal{T}} + \mathbf{T}\_{12}) \diamond \Delta\mathbf{G} \} \mathbf{L}\_{12}] \tag{16}
$$

$$\begin{array}{ll} \Delta \mathbf{L}\_{\mathsf{C}} = & \frac{1}{8} [\mathbf{L}\_{12} \{ (\mathbf{T}\_{07} + \mathbf{T}\_{12}) \diamond \Delta \mathbf{C} (\mathbf{E}\_{07} + \mathbf{E}\_{12}) \} \mathbf{L}\_{07} \\ & + \mathbf{L}\_{\mathsf{O}7} \{ (\mathbf{T}\_{07} + \mathbf{T}\_{12}) \diamond \Delta \mathbf{C} (\mathbf{E}\_{07} + \mathbf{E}\_{12}) \} \mathbf{L}\_{12} \end{array} \tag{17}$$

$$\begin{array}{ll} \Delta \mathbf{L}\_E = & \frac{1}{8} [\mathbf{L}\_{12} \{ (\mathbf{T}\_{07} + \mathbf{T}\_{12}) \diamond (\mathbf{C}\_{07} + \mathbf{C}\_{12}) \Delta \mathbf{E} \} \mathbf{L}\_{07} \\ & + \mathbf{L} \boldsymbol{\sigma} \{ (\mathbf{T}\_{07} + \mathbf{T}\_{12}) \diamond (\mathbf{C}\_{07} + \mathbf{C}\_{12}) \Delta \mathbf{E} \} \mathbf{L}\_{12} \end{array} \tag{18}$$

*Energies* **2019**, *12*, 699

We can finally quantify the Leontief inverse effect-driven energy-use change as the following four factors.

$$\begin{array}{l} \mathsf{\bf \Delta \mathbf{f}\_{L}} = \frac{1}{2} \left( \hat{\mathsf{e}} \texttt{\bf \Delta \mathbf{f}\_{0T}} + \hat{\mathsf{e}} \texttt{\bf \Delta \mathbf{f}\_{12}} \right) = \frac{1}{2} \left( \hat{\mathsf{e}} \texttt{\bf \Delta \mathbf{L}\_{T} \mathbf{f}\_{0T}} + \hat{\mathsf{e}} \texttt{\bf \Delta \mathbf{L}\_{T} \mathbf{f}\_{12}} \right) \left( \texttt{\bf \Delta \mathbf{e}} \texttt{\bf \text{eff} \, \mathrm{erf}} \right) \\ + \frac{1}{2} \left( \hat{\mathsf{e}} \texttt{\bf \Delta \mathbf{L}\_{C} \mathbf{f}\_{0T}} + \hat{\mathsf{e}} \texttt{\bf \Delta \mathbf{L}\_{C} \mathbf{f}\_{12}} \right) \left( \texttt{\bf \text{Non} \, \mathrm{er} \, \mathrm{arg} \, \mathrm{input} \, \mathrm{eff \, \mathrm{erf}}} \right) \\ + \frac{1}{2} \left( \hat{\mathsf{e}} \texttt{\bf \Delta \mathbf{L}\_{C} \mathbf{f}\_{0T}} + \hat{\mathsf{e}} \texttt{\bf \Delta \mathbf{L}\_{C} \mathbf{f}\_{12}} \right) \left( \texttt{\bf \text{Energy composition effect}} \right) \\ + \frac{1}{2} \left( \hat{\mathsf{e}} \texttt{\bf \Delta \mathbf{L}\_{E} \mathbf{f}\_{0T}} + \hat{\mathsf{e}} \texttt{\bf \Delta \mathbf{L}\_{E} \mathbf{f}\_{12}} \right) \left( \texttt{\bf \text{Energy input level} } \texttt{\bf$$

Similarly, we decomposed the changes in the imported energy of China between 2007 and 2012 (see Supporting Information).
