## 5 Problem Formulation
### Problem Re-Formulation  
Instead of attempting to solve the original Problem 1 directly, we present a modified convex MINLP problem that is equivalent to Problem 1. To this end, we exploit a _Log-transformation_ technique [19] on Problem 1. First, we introduce new variables for this transformation as follows:  
\[y_{i}(k)=x_{i}(k)+1,\ \forall i\in\mathcal{I},\ \forall k\in\mathcal{K}. \tag{15}\]  
Obviously, \(y_{i}(k)\in\{1,2\},\ \forall i\in\mathcal{I},\forall k\in\mathcal{K}\). Then, the additional variables for reformulation are defined as follows:  
\[\overline{y}_{i}(k) =\ln(y_{i}(k)), \tag{16}\] \[\overline{f}_{i}(k) =\ln(f_{i}(k)), \tag{17}\]  
where \(\ln(\cdot)\) denotes \(\log_{e}(\cdot)\). Therefore, \(x_{i}(k)=y_{i}(k)-1=\exp(\overline{y}_{i}(k))-1\) and \(f_{i}(k)=\exp(\overline{f}_{i}(k))\). Let \(z_{i}(k)\in\{0,1\},\ \forall i\in\mathcal{I},\ \forall k\in\mathcal{K}\) denote the new binary integer variables. We set \(\overline{Y}=(\overline{y}_{\mathcal{I}},\overline{y}_{\mathcal{I}},\cdots, \overline{y}_{\mathcal{I}})\), \(\overline{F}=(\overline{f}_{1},\overline{f}_{2},\cdots,\overline{f}_{\mathcal{ I}})\) and \(Z=(z_{1},z_{2},\cdots,z_{\mathcal{I}})\) where \(\overline{y}_{i}=(\overline{y}_{i}(1),\cdots,\overline{y}_{i}(\mathcal{K}))\), \(\overline{f}_{i}=(\overline{f}_{i}(1),\cdots,\overline{f}_{i}(\mathcal{K}))\) and \(z_{i}=(z_{i}(1),\cdots,z_{i}(\mathcal{K}))\), as the auxiliary vectors. Then, we propose the reformulated problem as follows:  
**Problem 2** (Log-transformed Problem). : \[\min_{\overline{Y},\overline{F},Z} \sum_{\forall k\in\mathcal{K}}\max[\sum_{\forall i\in\mathcal{I}}\{p_{i }^{\mathrm{STD}}+\delta_{i}\cdot b_{i}\cdot(\exp(\overline{y}_{i}(k)+\overline {f}_{i}(k))-\exp(\overline{f}_{i}(k)))\}\cdot\tau-r(k),0]\cdot c(k),\] (18) s.t. \[\exp(f_{i}^{-})\leq\exp(f_{i}(k))\leq\exp(f_{i}^{+}),\ \forall i\in \mathcal{I},\ \forall k\in\mathcal{K},\] \[a_{i}\leq k\cdot(\exp(\overline{x}_{i}(k))-1)\leq d_{i},\ \forall k\in \mathcal{K},\] (19) \[\sum_{\forall k\in[K]}\frac{(\exp(\overline{y}_{i}(k)+\overline {f}_{i}(k))-\exp(\overline{f}_{i}(k)))\cdot\tau}{\lambda_{i}\cdot b_{i}}\geq R ^{i},\ \forall i\in\mathcal{I},\] (20) \[\overline{y}_{i}(k)=\ln(2)\cdot z_{i}(k),\ \forall i\in \mathcal{I},\ \forall k\in\mathcal{K},\] (21) \[z_{i}(k)\in\{0,1\},\ \forall i\in\mathcal{I},\ \forall k\in \mathcal{K}.\] (22)  
Transforming the constraints (16) for our log-transformation is not straightforward, so we employ a modeling technique based on a special-ordered set of type-I (SOS-1), as described in [19]. This technique, combined with constraints (21) and (22), ensures that \(\overline{y}_{i}(k)\) takes values in \(\{\mathbf{ln}(1)=0,\ \mathbf{ln}(2)=1\}\) for all \(i\in\mathcal{I}\) and \(k\in\mathcal{K}\). Additionally, we have developed a modified formulation for Problem 2 that eliminates the max term from the objective function. For more details, please see the Appendix in [20]. Algorithm 1 outlines the steps of our proposed RA-FS approach. If there is no feasible solution to meet the DJR \(R_{i}\) for a specific \(i\)-th DL job within its deadline \(d_{i}\), the algorithm drops that request (as shown in line 2). Subsequently, for all incoming time slots, a GRU-based predictor forecasts the future generation of renewable energy sources (referenced in line 3). Then, the MINLP solver tackles the convex optimization Problem (2) (mentioned in line 4), determining the optimal solution (\(\mathbf{X}^{*}\), \(\mathbf{F}^{*}\)). Utilizing this solution, our approach then executes the actual frequency scaling and allocates the DL jobs accordingly.