Electrical distribution networks are the power system component responsible for providing electrical service to end-users in medium- and low-voltage levels in urban and rural areas [1,2]. These grids are typically constructed with a radial structure using AC technologies to reduce investment costs and simplify the coordination of the protective devices [3]. However, due to the recent advances in power electronic converters, renewable energy resources, and energy storage devices, is possible to operate electrical distribution networks using DC technologies [4-6], with the following advantages:

Low power losses in comparison with AC distribution networks [7].

In the specialized literature, the analyses of DC networks are made in the context of providing electric distribution applications in urban and rural areas [1,10]. The main approaches correspond to: a) optimal location and sizing of distributed generators in DC grids [11], b) optimal location and operation of battery energy storage systems [12-14], c) optimal reconfiguration of DC feeders [15], d) and planning of DC networks [1], among others. Based on the importance of these studies, we focus on the problem of the optimal location and sizing of distributed generators in DC networks considering daily load and PV curves to minimize the number of daily energy losses and the harmful greenhouse gas emissions to the atmosphere. Next, some recent works regarding DC distribution networks are presented below.

The authors in [16] have presented optimization models to locate and size distributed generators in DC grids using mixed-integer nonlinear programming model, which is solved with optimization tools available in the general algebraic modeling system, i.e., GAMS, and heuristic optimization methods such as black hole optimizer, and population-based incremental learning. References [17] and [18] have proposed convex optimization models to place and size distributed generators in DC grids considering conic and semidefinite approximations considering the peak load conditions. For their part, authors in [12] and [19] have presented MINLP models to operate battery energy storage systems in DC grids, and their solutions are reached with heuristic algorithms and convex reformulations. As another contribution, the authors of [9] have proposed a conic reformulation of the power flow problem for DC grids considering conic constraints by transforming the hyperbolic relation between voltages and currents in the power balance equations into conic constraints [20]. Additional studies regarding DC distribution networks are listed in Table 1.

Table 1

Source: Authors

Considering the previous state-of-the-art, there is a gap in which this research contributes through a novel mixed-integer convex programming model to minimize greenhouse gas emissions in rural distribution networks with the optimal installation and sizing of PV sources. The main advantage of the proposed reformulation from the exact MINLP model is to reach global optimum, which can ensure in the mixed-integer convex model. This model uses a combination of the Branch & Bound and the interior point methods with zero duality gap compared to the exact formulation [34]. The proposed convex model's effectiveness and robustness are tested in two radial DC distribution tests, which are composed of 33 and 69 node test feeders. Besides, this paper makes comparisons with MINLP solvers available in the GAMS optimization tool.

The remainder of this research is organized as follows: Section 2 presents the exact MINLP formulation for the optimal placement and sizing of PV sources in DC distribution grids for rural application to minimize the amount of greenhouse gas emissions the atmosphere produced by diesel generators. Section 3 presents the conic transformation of the power balance equations using the product's hyperbolic equivalent between continuous variables, becoming the exact MINLP model into a mixed-integer convex one. Section 4 describes the main aspects of the mixed-integer convex programming using the Branch & Bound method. Section 5 describes the main characteristics of the IEEE 33- and IEEE 69-node test feeders as well as the daily load and PV generation curves, respectively. Section 6 presents the numerical results in both test feeders using the CVX tool and the MOSEK solver in the MATLAB programming environment and their comparisons with the GAMS MINLP solvers [11]. Section 7 shows the main conclusions derived from this study and further works.

A mixed-integer nonlinear programming model allows describing the problem of the optimal location and sizing of PV sources in DC distribution networks to minimize the daily energy losses considering the calculation of the amount of greenhouse gas emissions to the atmosphere by diesel generators. In such a model the continuous part is related to the power flow variables, i.e., currents, voltages, and powers; and the integer part is related with the possibility of locating a PV source or no in a particular node of the grid [35].

The objective functions for the MINLP model to locate and size PV sources in DC grids can be the represented as follows.

where E_losses represents the objective function value associated with the daily energy losses, 𝑅_𝑗K is the resistive parameter of the line between nodes j and k; 𝑖_𝑗K,_t is the current that flows in the branch that connects nodes j and k at time t. G _emissions represents the objective function value regarding the amount of greenhouse gas emissions to the atmosphere produced by diesel generators during a typical day of operation CGG^k _emissions is the coefficient of greenhouse gas emissions in the diesel generator connected at node 0 p_0k,t is the amount of power generation sent from the slack node for the first line, i.e., the line that connects nodes 0 to k Δt is the length of the fraction of time where the power generation is assumed as a constant. This period is typically 30 or 60 minutes. Note that T and E are the sets that contains all the periods in which is divided the typical operative day and the set that contains all the branches of the network, respectively. The total nodes of the network are denoted by N.

Remark 1: In this research we select as objective function the minimization of the energy losses in all the branches of the network, i.e.,Equation (1)for a typical day of operation, and the amount of greenhouse emissions calculated as a function of the amount of power injected by the slack source, i.e., applyingEquation (2).

The problem of the optimal location and sizing of PV sources in distribution grids for rural areas must accomplish the power balance equations and the capacities of the devices, among other constraints. The set of constraints is presented below.

where 𝑝_jk,t (𝑝_km,t) is the power flow in the branch that connects nodes j(k) and k(m) in the period t; p_k,t is the power demand at node k in the period t; y_k ^pv is the size of the PV source connected at node k; p_k,t ^pv,nom represents the generation value of the PV source at node k in the period t, note that this curve is normalizes in percentage; v_j,t (v_k,t) is the voltage value at node j(k)at time t; i_jk ^maxis the thermal bound associated with the caliber of the conductor in the line that connects nodes j and k; 𝑣^min and 𝑣_max are the minimum and maximum voltage limit for all the nodes of the grid at any period; p_k ^pv,max is the maximum size allowed for a PV source connected a node k; x_k ^pv is the binary variable associated with the location (x_k ^pv=1) or not (x_k ^pv=0) of a PV source at node k; and N_max ^pv is the number of PV sources available for installation in the DC network.

Remark 2: The MINLP model defined from (1) to (9) is a mixed-integer non-convex optimization model due to the square variables present inEquations (2)and (3) as well as the product of these in (4). However, this model can be transformed into a mixed-integer convex one using the conic representation of the power balance equations presented in [36].

The optimization model (1)-(9) can be interpreted as follows: Equation (1) formulates the objective function of the optimization problem which is related with the minimization of the amount of the daily energy losses; Equation (2) determines the quantity of greenhouse gas emissions emitted by diesel generators that feeds rural distribution grids using DC technologies; Equation (3) is known in the specialized literature as the branch power flow constraint that guarantees the power balance at each node of the grid at each period of time; Equation (4) defines the voltage drop at each branch of the network as a function of the power and current flows at each period of time; Equation (5) shows the definition of the power in an electrical element as function of its voltage and current (i.e., Tellegen’s theorem); Inequality constraints (6) and (7) determine the thermal limits associated with the calibers of the conductors in all the branches and the voltage regulation bounds in the nodes of the grid, respectively. Inequality constraint (8) determines if a PV source is installed or no at node k, while inequality constraint (9) limits the number of PV sources that can be connected to the DC grid.

Remark 3: The optimization model presented from (1) to (9) is only applicable to radial DC distribution networks since it was developed based on the concept of branch power flow proposed in [37] where only exists the possibility of having one path between each node and the slack source.

The transformation of the exact MINLP model (1)-(9) into a mixed-integer convex model using conic constraints will be presented in the next section.

The transformation of the MINLP model (1)-(9) into a convex one with binary and continuous variables is made through the usage of auxiliary variables that allow changing voltages and currents to rewrite Equations (3) to (4) as affine expressions. Let us define u_k,t = v² _k,t and l_jk,_t=l² _jk,t; using them in Expressions (3) and (4) it is possible to build affine planes; however, the main complication of the model is the power definition in (5). To transform this equation into a convex one, let us use the hyperbolic equivalent of the product between two variables as follows (note that sets notation was eliminated for easy comprehension of the mathematical procedure):

Equation (10) is a conic equality constraint that is still non-convex due to the presence of the equality symbol [17]. However, as described in [38], this symbol can be replaced by a low equal symbol, which allows transforming it into a convex constraint, as represented below.

Now, with expression (11) and the auxiliary variables previously defined, the optimization model (1)-(9) is transformed from a MINLP structure to a mixed-integer convex one as presented below:

Remark 3: The mixed-integer convex model can be solved using a hybrid optimization algorithm based on the Branch & Bound method combined with a modification of the interior point method for convex models with the main advantage that the global optimum finding is guarantee [34].

The main aspects of the solution methodology will be presented in the next section.

The mixed-integer convex reformulation proposed in this research is a convex reformulation problem with integrality constraints on some variables [17]. Therefore, we take advantage of the fact that the mixed-integer convex reformulation proposed is a convex problem, which can be solved efficiently with some integer programming solvers such as the Branch & Bound (B&B) algorithm [34].

The B&B algorithm in each bifurcation ( takes the value of “0” or “1”) generates a convex problem, which is solved with an interior-point method. In a child bifurcation (B1, B2, …, BN) conforming to a B&B tree (which is a convex problem), the problem must be solved from its main fork (B0). This produces a series of secondary branch problems, which are solved for entire partitions where their primary branch is a lower bound for the convex problem. This methodology is efficient despite having many variables and continues until reaching the best binary solution, which is the global optimum of the problem. This methodology is illustrated in Fig. 1.

Figure 1 Source: Authors

Finally, Fig. 2 shows the flow chart of the proposed optimization approach for exact minimization of the energy losses and the CO2 emissions in isolated DC distribution networks using PV sources.

Figure 2 Source: Authors

To validate the proposed mixed-integer convex optimization model's effectiveness and robustness for locating and sizing PV sources in DC grids, two radial test feeders composed of IEEE 33- and 69-node are considered. The electrical configuration between nodes in both test feeders is depicted in Fig. 3.

Figure 3 Source: Authors

The parametric information of these DC distribution grids can be consulted in [22]. In addition, the daily load and the normalized PV curves are listed in Table 2.

Table 2

Source: Authors

Regarding the greenhouse gas emissions, the information about diesel generators presented in [39]. Here, the most relevant gas emissions for medium size diesel generators (those with capabilities lower than 10 MW) is the carbon dioxide, i.e., CO2, with an average emissions rate of 612.35 kg/MWh.

The evaluation of the exact MINLP model and the proposed mixed-integer convex model are made in the GAMS optimization software and the CVX with the MOSEK solver in MATLAB [17], respectively. We implement both optimization models in a personal computer AMD Ryzen 7 3700U, 2.3 GHz, 16 GB RAM with 64-bits Windows 10 Home Single Language. All algorithms developed in this paper are available at File Exchange.

The optimal location and sizing PV sources in electric distribution networks operated using DC technologies have been addressed in this research from exact mathematical optimization. The exact MINLP model was transformed into a mixed-integer convex model with the main advantage that the global optimum can be ensured by applying the Branch & Bound method combined with an interior point approach for conic programming.

Numerical results in the IEEE 33-node test feeder show that the proposed MIC approach found a better reduction of daily energy losses than BONMIN and COUEENE solvers in the GAMS optimization package that were stuck in local optimal solutions.

For the IEEE 69-node test feeder, it was observed that depending on the number of the PV sources the amount of CO2 emissions and the daily energy losses presents a saturation regarding their possible reductions. Besides, it was noted that for two PV sources, the reduction of daily energy losses was about 33.90 %; and for three PV sources, the total reduction was about 34.25 %, i.e., the additional gain when an additional PV source installed was only 0.35 %.

As future works, the following research can be conducted: (i) to extend the proposed MIC approach to AC to the location of renewable energy resources considering the presence of battery energy storage systems align the AC grid; and (ii) the application of the proposed MIC to the problem of the dynamic reactive power compensation in AC grids considering FACTS.