*4.2. Deducing Analytical Surrogate Model for IPM*

As shown in Equation (26), the gradient of functions *F*,*G* and *H* are needed in the process of IPM. The gradient of *F*, *G* and *H<sup>c</sup>* can be obtained directly. However, since the surrogate model is a "black-box", the gradient of *H<sup>s</sup>* cannot be simply calculated. Based on Equations (22) and (23), the gradient of *H<sup>s</sup>* can be transformed into:

$$\begin{array}{l}\bigtriangledown\!\Gamma = \bigtriangledown^{2}\!\Gamma\_{\mathfrak{c}} = \bigtriangledown^{2}\!\Psi\_{\mathfrak{c}}(\mathbf{X})\,,\,\mathfrak{c}\in\mathcal{S}\_{\mathfrak{c}}\\\bigtriangledown^{2}\!\mathbf{H}\_{\mathfrak{s}} = \bigtriangledown^{2}\!\Gamma\_{\mathfrak{c}} = \bigtriangledown^{2}\!\Psi\_{\mathfrak{c}}(\mathbf{X})\end{array}\tag{31}$$

Next, by DBN backwards process, Equation (31) is deduced to get the Jacobin and Hessian matrix, which are also known as sensitivities of transient stability against optimization variables. See Appendix A for the detailed DBN backwards process. The algorithm flow and implementation of the proposed method are shown in Figure 1.

**Figure 1.** The flow chart of the proposed method.
