## 3 Problem Formulation  
This paper introduces the distributed optimization method for solving EDP for the technological asset group, where control policy focuses on minimizing operation costs from the owner's perspective. The main task of EDP is to minimize the total operational costs. At the same time, the overall consumption and limitation of individual technological assets (e.g., generators (generally, controllable units such as generators, controllable loads, etc.)) are considered [8; 43; 52; 53]. The objective function can be formulated as:  
\[\min_{P_{i}}\sum_{i=1}^{N}C_{i}(P_{i}), \tag{1}\]  
where \(N\) represents the number of technological assets, \(P_{i}\) is the active power injection related to the \(i\)-th asset and \(C_{i}\) denotes the cost function of the \(i\)-th asset. Each agent in \(N\), i.e., technological assets, is part of the considered network graph denoted as \(\mathcal{G}\). The graph \(\mathcal{G}\) indicates the overall shape of the network topology, and on its basis, the communication matrix is constructed later in Section 4.4. In the next step, the objective function (1) can be detailed to include considered assets such as conventional generating units (i.e., generators), renewable generating units, battery systems, and power grid connection [32; 54]. In a distributed optimization approach, each term of an objective function is optimized by an individual agent, which interacts with other ones to achieve optimal system-wide consensus. Consequently, the agents representing these generating resources have their cost functions, which may differ from each other. The distributed formulation of the objective function minimizing operational costs across all network agents can be expressed as [10; 26; 30; 55; 56]:  
\[\min\Bigl{[}\underbrace{\sum_{i=1}^{N_{c}}C_{i}(P_{c_{i}})}_{\text{Conventional}}+ \underbrace{\sum_{i=1}^{N_{c}}C_{s_{i}}(P_{s_{i}})+\sum_{i=1}^{N_{w_{b}}}C_{w _{i}}(P_{w_{i}})}_{\text{Renewables}}+\underbrace{\sum_{i=1}^{N_{b}}C_{b_{i}}( P_{b_{i}})}_{\text{Energy storage}}+\underbrace{C_{\mathcal{G}}(P_{\mathcal{G}})}_{\text{Grid}}, \tag{2}\]  
where variable \(N_{c}\) stands for conventional generating assets (microturbines, diesel generators), \(N_{s}\) for photovoltaic plants, \(N_{w}\) for wind power plants, and \(N_{b}\) for battery systemswithin the network. Additionally, \(P_{c_{i}},P_{s_{i}},P_{w_{i}}\), and \(P_{b_{i}}\) represent the power output of generators by type and index \(i\). These types include conventional, solar, wind, and battery systems. Furthermore, \(C_{c_{i}},C_{s_{i}},C_{w_{i}}\), and \(C_{b_{i}}\) denote price functions for generators of the respective type and index \(i\). Variable \(C_{g}\) denotes the price function of energy provided or taken from an external large network. Therefore, the variable \(P_{g}\) indicates the power provided or received by this network [37; 38]. Renewable resources, such as solar and wind power plants, are uncontrollable [54; 57], and their generating performance depends on current weather conditions [55]. The power balance equation can be defined as follows [23; 26]:  
\[\sum_{i=1}^{N_{c}}P_{c_{i}}+\sum_{i=1}^{N_{c}}P_{s_{i}}+\sum_{i=1}^{N_{m}}P_{ w_{1}}+\sum_{i=1}^{N_{b}}P_{b_{i}}+P_{g}=P_{D}, \tag{3}\]  
where \(P_{D}\) is overall demand power and \(P_{(\cdot)}\) stands for the above-mentioned power injections of the assets. Some technological assets such as energy storage [56], or power grid connections can vary their power flow direction. In the case of discharge or power grid supply, the outputs of these assets are considered positive, otherwise negative. The need to meet the total power demand while respecting the generator limitations, particularly the minimum and maximum power outputs, aligns with the previously defined formulation [10; 26; 58]. These constraints can be expressed as:  
\[