Stabilization of MT-HVDC grids via passivity-Based control and convex optimization

Stabilization of DC resistive grids with CPLs via the PBC theory and convex optimization Oscar Danilo Montoyaa,b, Walter Gil-Gonz´ alezc, Alejandro Garcesd, Federico Serrae, Jesus C. Hern´ andezf,∗ aFacultad de Ingenier´ıa, Universidad Distrital Francisco Jos´e de Caldas. Carrera 7 No. 40B - 53, Bogot´a D.C 11021, Colombia. bLaboratorio Inteligente de Energ´ıa, Universidad Tecnol´ogica de Bol´ıvar, Cartagena 131001, Colombia. cGrupo GIIEN, Facultad de Ingenier´ıa, Instituci´on Universitaria Pascual Bravo, Campus Robledo, Medell´ın 050036, Colombia. dPrograma de Ingenier´ıa El´ectrica, Universidad Tecnol´ogica de Pereira. AA: 97 - Postcode: 660003 - Pereira, Colombia. eAutomatic Control Laboratory, Facultad de Ingenier´ıa y Ciencias Agropecuarias, Universidad Nacional de San Luis, Villa Mercedes, San Luis 5730, Argentina. fDepartment of Electrical Engineering, University of Ja´en, Campus Lagunillas s/n, Edificio A3, 23071 Ja´en, Spain. Abstract This paper tackles the problem of stabilizing of direct-current (DC) networks with constant power loads (CPLs) interfaced with power electronic converters. A primary-secondary control is proposed to guarantee the stable operation of the grid under load variations. In the primary control, a port-Hamiltonian formulation is developed to represent the DC network dynamics. A passivity-based control (PBC) is proposed in order to guarantee stability according to Lyapunov’s theory. In the secondary control, a convex optimal power flow formulation based on a semidefinite programming (SPD) model is used to define the primary controller’s set point. The proposed stabilization scheme can be applied over grid-connected or isolated DC microgrids indistinctly and allows the stable operation of the grid considering CPLs and renewable generation. Simulation results were carried out in MATLAB by using CVX and ODE packages. These prove the applicability and efficiency of the proposed primary-secondary control approach. Keywords: Convex optimization; direct-current networks; passivity-based control; primary-secondary control; port-Hamiltonian formulation; stabilization of electrical networks. 1. Introduction The presence of direct-current (DC) networks in conventional power systems has increased due to speed advances and cost reduction in power electronic converter technologies [1]. These situations have transformed electrical grids from a vertical structure given by generation, transmission, and distribution to a horizontal structure with a high penetration of distributed energy resources (DERs) that are mainly grouped in renewable generation [2] and energy storage devices [3, 4]. The integration of these devices has generated the concept of microgrids [5], which allow the interconnection of multiple DERs under the grid-connected [6] or grid-isolated modes [7]. Note that the microgrid concept applies to both AC or DC networks under the same operation philosophy [8]. When a DC microgrid operates under the grid-connected mode, the main objective for all DERs is to operate as the power control to maximize the power extraction of all the renewable energy resources and storing capabilities in the energy storage devices [5, 9]. Nevertheless, in the case of the grid-isolated mode, DERs need to be operated coordinately to maintain the voltage profile on the network by preserving the maximum and minimum operative limits of the energy storage and renewable ∗Corresponding author Email addresses: [email protected] (Oscar Danilo Montoya), [email protected] (Walter Gil-Gonz´ alez), [email protected] (Alejandro Garces), [email protected] (Federico Serra), [email protected] (Jesus C. Hern´ andez) generation capabilities as well as supporting the constant power consumption of the entire microgrid [7, 10]. The islanded mode can occur when an external event (short-circuit or programmed maintenance) generates grid outages or when the microgrid operates in a non-interconnected area (rural or remotes zones) [2, 5]. The operation of microgrids under islanded mode requires the integration of advanced control strategies to operate all power electronic converters associated to each DER [9, 11] and optimization techniques to define the best operative point as a function of renewable energy availability, energy storage conditions, constant power load consumption, and grid topology [6, 12]. In the control layer, the primary control strategy stabilizes the DC network at a particular operating point defined by a set of voltage profiles v0[13–15]. In contrast, in the optimization layer, a secondary control defines the best operative point v0as a function of the grid operative conditions [6, 16]. In the specialized literature on DC stand-alone grids, these grids are proposed as multiple linear and nonlinear controllers for primary control purposes. For example, in [17], a port-Hamiltonian model in conjunction with the interconnection and damping assignment passivity-based control (PBC) is presented for the primary control at stand-alone microgrids. In [16], an optimal power flow from a dynamic analysis based on port-Hamiltonian models for controlling power flows in high-voltage direct-current (HVDC) systems was proposed. This same approach was adopted in [18] via port-Hamiltonian models for resistive networks. Preprint submitted to Electric Power Systems Research December 29, 2020 The authors in [19, 20] proposed consensus strategies for power-sharing in DC networks with multiple constant power loads (CPLs) and DERs. While in [13], the authors combined the consensus algorithm to a plug-and-play control approach for power control in DC networks. The authors in [21] presented a detailed revision of different control architectures for the voltage and power regulation of DC networks via a hierarchical control. Some distributed control approaches were also discussed in [22, 23] considering grid topology variations or communication delays. The authors in [11] proposed a general nonlinear controller with feedback based on the Lyapunov control theory to stabilize DC networks with multiple CPLs and actuators for islanded grid applications by modeling each power electronic converter. In [24] a predictive fuzzy control model for the dynamic stabilization of DC microgrids is proposed, and in [14] a nonlinear stabilization method for DC networks using a curvature Kalman filter was recently proposed. In the secondary control, most of the publications reported in the specialized literature focus on an optimal power flow (OPF) solution via metaheuristic or exact mathematical techniques. The authors in [12] presented a hybrid Gauss-Seidel genetic algorithm for solving the OPF problem in DC networks with multiple CPLs and voltage-controlled nodes. The authors in [7] provided a convex optimization model based on second-order cone programming for solving the OPF problem in islanded DC networks with CPLs, while [25] proposed a semidefinite programming (SDP) model for determining the voltage stability index on DC networks via OPF formulation. The authors in [3] presented a multi-period SDP model for the economic dispatch of renewable generation and batteries in DC networks. The authors in [26] proposed an SDP model for an OPF solution in high-voltage DC networks by including a quadratic formulation of the power losses in the DC-DC converters; whereas [6] proposed a convex model for solving the OPF problem via Taylor’s series approximation of the power balance equations. Based on the review mentioned in the above state-of-the-art, we propose a combination of the PBC theory for primary (stabilization) control purposes on DC networks. This guarantees globally asymptotic stability properties by employing Lyapunov’s theory by representing the DC network as a port-Hamiltonian system. The main advantage of this approach, in comparison with the recent contribution presented in [17], is that our proposed PBC controller does not require communication channels in the control step and avoids parametric dependence on the capacitive filter values at each power electronic converter. The approach in [17] required this information to develop its PBC control strategy. Additionally, to define the desired operative point for the proposed PBC approach, we proposed as a secondary control approach an OPF methodology based on an SDP model. This optimization strategy guarantees uniqueness in the solution as presented in [3]. This primary-secondary control exploits the most important advantages of nonlinear control and convex optimization to propose a robust DC grid operation strategy that was not previously found in the specialized literature. The remainder of this paper is organized as follows: Section 2 presents the mathematical modeling of the DC network by highlighting its dynamic representation for primary control purposes and its static formulation for secondary control interests. Section 3 presents the mathematical foundation of the PBC theory for nonlinear port-Hamiltonian systems and the different controller designs, including the proportional and PI PBC proposed approaches. Section 4 shows the selection of the desired operative point via convex optimization by showing the transformation of the original non-convex OPF model into a convex representation via an SDP relaxation. In section 5, a pseudo-code compacts the proposed primary-secondary control methodology, while Section 6 shows a numerical example composed of four nodes, two CPLs, a battery energy storage system, and a photovoltaic (PV) system, which are used for validating the proposed primary–secondary control approaches. Finally, in Section 7, the main conclusions derived from this work are presented. 2. DC network modeling The mathematical representation of an electrical DC network depends on the control or optimization objectives [6]. In the case of the control, a dynamic model is required to design the controller, which can be linear or nonlinear [16]. In the optimization case, static models under steady-state conditions are usually proposed [7]. Nevertheless, for the primary-secondary stabilization (control) approaches, both models are required [17]. Fig. 1 shows a typical structure of a DC network shaped by CPLs, distribution (transmission) lines, renewable generation, energy storage systems, and linear loads [27]. 4 53 12 6 DC DC PV DC DC CPL DC DC - + BESS Linear load Resistive network DC DC Converter Fig. 1. Schematic representation of a DC network with distributed energy resources and linear loads. To obtain a dynamic model from the DC configuration presented in Fig.1, it is necessary to focus on the power electronic converters [17]. In contrast, an optimization model can be achieved by focusing on the resistive network 2 configuration by assuming an ideal operation of those converters [1, 28]. 2.1. Dynamic DC grid modeling Let us consider that any CPL node interconnected to a power electronic converter can be modeled from the grid point of view as presented in Fig. 2 [17]. By applying Kirchhoff’s first law at node k, the next nonlinear first order differential equation is achieved: ck˙vk=pkv−1 k−gkvk−ik(1) where ckcorresponds to the capacitive shunt filter associated whit the power electronic converter connected at node k,gk represents a linear resistive load connected at node kmodeled as a constant conductance component, pkcorresponds to the constant power value managed by the power electronic converter, which is positive for generators and negative for CPLs, ikis the total net injected current at node k, and vk represents its voltage dynamics. ck Node k gk pk vk ik Fig. 2. A detailed model of a constant power load controlled by a power electronic converter. For compactness purposes on the dynamic modeling given by Eq. (1), all power electronic converter nodes were grouped in the set Swith cardinality s, which implies that Eq. (1) can be rewritten as: CS˙vS=diag−1(vS)pS−GSvS−iS(2) where CS∈Rs×sis a positive definite matrix that contains all the capacitive effects of the power electronic converter nodes. GS∈Rs×sis a positive semidefinite matrix that contains all the conductive effects at the nodes contained at set S;diag−1(vS)∈ Rs×scorresponds to a positive definite matrix with the inverse of each contained profile at its diagonal; iS∈Rs×1represents the vector of injected currents while vS∈Rs×1corresponds to the voltage profiles at the power electronic converter nodes. Now, let us focus on the resistive network that interconnects the power electronic converter nodes to the other step of linear load nodes (see Fig. 1) . This grid can be easily modeled by an admittance nodal relation between net injected currents and voltage nodes by applying the well–known voltage nodal method as presented below: iS iR!="GSS GSR GRS GRR # vS vR!(3) where GSS ∈Rs×smodels the conductance relation between power electronic converter nodes, GSR ∈Rs×rand GRS ∈ Rr×sdefine the admittance relation between power electronic converter nodes and linear nodes; whereas GRR ∈Rr×r determines the conductance relation between linear nodes. Notice that rcorresponds to the cardinality of the set R associated with the linear nodes; finally iR∈Rr×1and vR∈ Rr×1correspond to the net injected current and voltage profiles at linear nodes. Notice that the net injected current at linear nodes is identically to zero [4], i.e., iR, since not constant power loads, distributed generators, energy storage systems or grid interconnections appears in these nodes (see Fig. 4), which implies that by applying the Kron’s reduction over Eq. (3), the following result is reached: iS=GSS −GSRG−1 RRGRSvS=YSSvS(4) where YSS ∈Rs×srepresents the equivalent admittance relation between power-controlled nodes after applying Kron’s reduction method. Finally, to achieve the final dynamic modeling of the DC networks for control purposes the expression in Eq. (4) is substituted in Eq. (2) which produces the following: CS˙vS=diag−1(vS)pS−KSvS(5) where KS=GS+YSS. 2.2. Static DC grid modeling Conventionally, a DC grid can be modeled by a set of nonlinear non–convex equations for power flow and optimal power flow problems [7, 25]. These models assume that all the voltage profiles of the DC network have been stabilized for a primary control strategy, which implies that their dynamics are completely null [3]. Considering this assumption, ˙vS=0 implies that Eq. (5) under steady state conditions can be rewritten as the following: pS=diag (vS)KSvS(6) Power system readers widely know expression in Eq. (6) as the set of power flow equations for DC networks with CPLs. Besides, some recent investigations have also demonstrated that these equations can be solved by Newton–Raphson methods [10], Gauss-Seidel approximations [4, 12], or linear methods [1]. Nevertheless, in this paper, those equations are essential input for formulating a convex optimization problem with a unique solution to determine the proposed passivity-based controller’s set point, as will be discussed in the next section [6]. 3. Stabilization by passivity–based control The PBC approach corresponds to a well-studied and strong mathematically founded nonlinear control theory for designing control methods under the Lagrangian or Hamiltonian representation by guaranteeing stable operating according Lyapunov’s theory in a closed-loop operation [29, 30]. The PBC method exploits the strong relation between 3 the energy storage in the system and its dynamics to generate a control strategy by preserving their passive and dissipative properties [31, 32]. Based on the well-known passive properties of electrical networks, we proposed a controller’s design by using a port-Hamiltonian formulation [33]. Proposition 1. The reduced dynamic system in Eq. (5) can be represented as a port-Hamiltonian system with an affine structure as follows: D˙x=(M−N)x+G(x)u(7) where D ∈ Rs×sis a positive define matrix known as the inertia matrix by its similarities with mechanical systems; M ∈ Rs×s and N ∈ Rs×scorresponds to the interconnection and damping matrix, which are skew–symmetric and positive semidefinite, respectively. x ∈Rs×1represents the vector of the state variables, G(x)∈Rs×scorresponds to the positive define input matrix and u ∈Rs×1is the input vector. Proof. The proof of this proposition is straightforward by comparing Eq. (5) to Eq. (7) and making the following definitions: x=vS,u=pS,D=CS(8) M=0,N=KS,G(x)=diag−1(vS) (9) To define a control law that allows moving the current operative point xto the desired operative point x?, let us define the dynamics of the error in the state variables as ˜x=x−x?; besides, it is clear that for DC grids x?is constant, which implies that ˙ ˜x=˙x. Proposition 2. A control input u exists such that the port-Hamiltonian system in Eq. (7) takes a gradient form for a closed-loop operation as presented below: D˙x=−N?˜x(10) where N?is a positive definite matrix that guarantees that the dynamic behavior described by Eq. (10) be exponentially stable according to Lyapunov’s theory. Proof. To determine the control input law that allows transforming Eq. (7) into Eq. (10), both equations are equaled by assuming that G(x)is a full–rank square invertible matrix. This is a reasonable assumption since the voltage profiles xmust be guaranteed for DC grid operation and contained between xmin and xmax, such that xmin >0. Considering this assumption, then the control input takes the following form, and the proof is completed: u=G−1(x) (Nx− N?˜x)(11) where N?∈Rs×scorresponds to a positive definite matrix that contains all control gains. The proposed control law defined by Eq. (11) was initially proposed by [17]. Nevertheless, the control law’s main disadvantage is its dependence on the measurements of all the voltage profiles at the power electronic converter nodes, since Ncorresponds to a full rank matrix. For providing an alternative control law that avoids depending on voltage measurements at each power electronic converter, we propose a modification to Eq. (11) as follows: u=G−1(x)Nx?− N?˜x(12) Proposition 3. The control law in Eq. (12) guarantees a global asymptotic stable response over the dynamic system in Eq. (7). Proof. To obtain a closed–loop dynamic behavior of the system in Eq. (7), let us substitute Eq. (12) with Eq. (7) and rearrange some terms as follows: D˙x=−(N+N?)˜x(13) To proof the exponential stability of the dynamic system given by Eq. (3), let us define the following candidate Lyapunov function [30]: V(˜x)=1 2˜xTD˜x(14) Applying the temporal derivative of V( ˜x) and considering that D=DT, the following result is achieved: ˙ V(˜x)=−˜xT(N+N?)˜x(15) The result given by Eq. (15) shows that the dynamic system in Eq. (13) is asymptotically stable according to Lyapunov’s theory [33]. Besides, it is also exponentially stable according to Lyapunov’s theory, that is, ˙ V(˜x)≤ −βV(˜x)if we select β≤λmin(D−1(N+N?)). Remark 1. An integral action can be added to the controller for reducing steady state errors without affecting the exponential stability of the closed-loop dynamic system as presented below: u=G−1(x)Nx?− N?˜x− Kiz ˙z=˜x(16) where Ki=KT icorresponds to a symmetric matrix that contains all integral gains. For proving the asymptotically stability of these PI controller refer to [29]. 4. Convex OPF for references’ selection The setpoints’ determination for primary controllers corresponds to a second control layer, which is typically associated with optimization control purposes. Hence, the system dynamics is usually negligible since this optimization layer focuses on obtaining the desired operative point under steady-state conditions. To reach this objective, the static DC model presented in Eq. (6) is involved inside optimal power flow problems as a balanced power set of constraints [6]. 4 This paper focuses on a convex transformation of the classical nonlinear non-convex OPF problem since the convex model guarantees a global optimum solution with low computational effort [26, 34]. The nonlinear non-convex conventional OPF problem has the following form: Model 1 (Non convex OPF-DC). Minimize ploss =xTYSS x(17) subject to pS=pg S−pd S=diag (x)KSx(18) pg,min S≤pg S≤pmax g(19) xmin ≤x≤xmax (20) where ploss ∈R+corresponds to the objective function value associated to the active power losses, pg S∈Rs×sand pd S∈ Rs×srepresent the power generation and load consumption at distributed energy resource nodes and CPLs, respectively. Notice that the solution of the optimization problem given from Eq. (17) to Eq. (20) corresponds to the setpoint x?for the control problem treated previously. Nevertheless, the solution of Model 1 is not easy since the power flow balance in Eq. (18) is nonlinear and non-convex. For this reason, we use the convex reformulation of the OPF problem as proposed in [3] to obtain its approximated solution to guarantee uniqueness in the solution [35]. For reformulating Model 1 as a convex approximation via SDP formulation, let us define X=xxTas a matrix of variables, which produces in the model of Eqs. (17) to (20) the following transformation: Model 2 (SDP convex OPF-DC). Minimize ploss =trace (X)(21) subject to pg S−pd S=diag (KSX)(22) pg,min S≤pg S≤pmax g(23) 1Sx2 min ≤X≤1Sx2 max (24) X=XT0 (25) where trace(·) calculates the sum of all components in the diagonal of a square matrix, 1S∈Rs×scorresponds to a matrix filled by ones. Notice that Model 2 is convex since the rank matrix constrain rank(X)=1 was relaxed as recommend in [25]. It can be observed that the result of the SDP is a rank s matrix, which represents a problem. Therefore, it is necessary to employ a general decomposition to recover the vector xfrom matrix X. To achieve this, it uses the decomposition method by means of eigenvalues and eigenvectors [3, 26] as follows: X= s X k=1 λkWkWT k(26) where λkand Wkrepresent eigenvalues and their corresponding eigenvectors in each time period, respectively. If the representation for the problem as an SDP is good enough, s−1 eigenvalues close to zero are expected. Therefore, the rank of Xis an approximation to one and can be achieved as follows: X≈λmWmWT m(27) where λmrepresents the maximum eigenvalue (i.e other eigenvalues are close to zero) in each time period. According to this approximation, it is possible to recover the vector xas follows: x≈pλmWT m(28) The main advantage of this methodology is its efficiency and precision [36]. 5. Proposed primary–secondary control scheme The pseudo-code presented in Algorithm 1 shows the coordinate control strategy for stabilizing DC networks by using PBC theory in conjunction with convex optimization. Data: Define DC grid topology and read the available power generation and consumptions. k=1; for t≥t0do if k== 1then Solve the convex model in Eqs. (21) to (25); Apply at each converter the control law in Eq. (11), (12), or (16); k=0; else if Any power generation has changed? then k=1; end if Any CPL has changed? then k=1; end end end Algorithm 1. Proposed primary–secondary control strategy for stabilizing DC networks. This primary-secondary control scheme solves recursively the proposed convex OPF model each time the DC grid has any change to update the control input so that the DC network remains stable. 5 6. Numerical example Let us suppose the existence of a DC network composed by six nodes with four power electronic converters that manage two CPLs, a PV array and a battery energy storage system. After applying Kron’s reduction, the following matrix is reached considering per unit representation: YSS = 116.96 −28.07 −88.889 0 −28.07 191.81 −70.175 −93.567 −88.889 −70.175 159.06 0 0−93.567 0 93.567  Notice that in this grid non–resistive effects are associated with power electronic converters, i.e., GS=0. For this test system, simulation parameters are listed in Table 1. Table 1 Power consumption at each CPL node and power available at the PV system. Node T1 [p.u] T2 [p.u] T3 [p.u] T4 [p.u] Capacitance [µF] 1 (Slack) — — — — 1500 2 (CPL) 1.5 2.0 2.0 1.5 1200 3 (CPL) 1.0 3.0 4.0 2.5 750 4 (PV) 2.0 2.5 3.0 1.0 1000 6.1. Main numerical results The OPF model’s solution by using the convex proposed reformulation estimates a total power loss of 0.4377 p.u. with a tracking error lower than 1 ×10−3% in comparison to the exact nonlinear non-convex OPF model. This validates its application, as reported in [3, 25]. Fig. 3 reports the voltage profile of the controller proposed by Eq. (12) in comparison to the reported solution in Eq. (11) and the proposed PI control defined in Eq. (16). For simulation purposes, the slack node is assumed to maintain the voltage grid profile at 1.0 p.u. Fig. 3 presents all the dynamic behaviors of the voltage profile in the four-node test system. Each demand and generation conditions are reported in Table 1. All the PBC controllers enhance their objectives with minimal errors; nevertheless, it is important to observe that when voltage profiles pass form higher values to lower values, all controllers present first-order dynamic responses (see Fig 3(a)). Notwithstanding, when voltage profiles pass from lower values to higher values (see Figs. 3(b) and 3(c)), the PBC controller reported in [17] conserves the first-order dynamic behavior; whereas the proposed P-PBC and PI-PBC methods exhibit higher slopes with multiple overpasses. This situation is attributable to the fact that our approaches do not require crossing information between converters, which implies that their responses are only influenced by the set point given by the OPF solution and their local voltage measurements. However, the PBC controller proposed in [17] is soft since it uses crossed information between converters, which softens the dynamic voltage behavior. It is a considerable disadvantage in terms of physical implementation due to the communication requirements. 0.00 0.05 0.10 0.15 0.20 0.970 0.975 0.980 0.985 0.990 (a) v2[p.u] Ref. P–PBC P–PBC [17] PI–PBC 0.00 0.05 0.10 0.15 0.20 0.960 0.970 0.980 0.990 (b) v3[p.u] 0.00 0.05 0.10 0.15 0.20 0.980 0.990 1.000 1.010 Time [s] (c) v4[p.u] Fig. 3. Dynamic performance at each voltage profile in the CPL and PV nodes: (a) voltage behavior in the CPL connected at node 2, (b) voltage behavior in the CPL connected at node 3, and (c) voltage behavior at the PV node. Fig. 4 depicts the dynamic performance of the control outputs (power generated or demanded) for each PBC technique implemented; notice that those graphics the proposed PI–PBC and P–PBC have speeder performance when compared to the P–PBC controller presented in [17]. Additionally, when power consumptions in Table 1 (second and third rows) are observed and compared to results in Figs. 4(b) and 4(c), it is clear that after the transient periods both loads operated as CPLs, which implies that the solution of the static model given by Eq. (6) is always satisfied under steady-state conditions as proposed in [1, 4, 10]. On the other hand, when the fourth row of Table 1 is compared to the dynamic power performance at the PV node in Fig. 4(d), we can conclude that not all power available in this renewable energy resource is injected into the network (see the period of time comprehended between 0 s to 0.150 s). Those limits are not required for minimizing the total power losses by using the proposed convex model. Furthermore, in the last simulation period, the maximum available power generation at the PV system (1.0 p.u) is used, as can be confirmed in Fig. 4(d). Finally, the active power performance at slack node 1 presented in Fig. 4(a) proves that this battery system (or also equivalent grid node) supports the remaining active power to guarantee the power balance at the grid as voltage profile holds constant at 1.0 p.u. 6 0.00 0.05 0.10 0.15 0.20 0.0 1.0 2.0 3.0 4.0 (a) pg 1[p.u] Ref. P–PBC P–PBC [17] PI–PBC 0.00 0.05 0.10 0.15 0.20 0.0 1.0 2.0 (b) pd 2[p.u] 0.00 0.05 0.10 0.15 0.20 0.0 1.0 2.0 3.0 4.0 (c) pd 3[p.u] 0.00 0.05 0.10 0.15 0.20 0.0 1.0 2.0 3.0 Time [s] (d) ppv 4[p.u] Fig. 4. Dynamic behavior of the active power at each power-controlled node in the system: (a) power generated by the slack node, (b) power consumption by the CPL connected at node 2, power consumption by the CPL connected at node 3, and (d) power generated by the PV node. 6.2. Additional comments We compared the effectiveness of the SDP approximation to solve the OPF problem with the nonlinear non-convex formulation (Model 1) in the general algebraic modeling system (GAMS), and the results are contrasted to the solution of Model 2 provided by the CVX package from MATLAB. Table 2 presents the solution of Model 1 using the Artelys Knitro 10.3.0 solver as well as the solution of Model 2 provided by CVX. The SDP model’s voltage profiles present estimation errors lower than 5.695 ×10−3% compared to the nonlinear exact model. These results confirm the convex model’s accuracy and efficiency in terms of voltage calculation for the OPF problem in DC networks, as confirmed by different authors in [3, 25, 26]. On the other hand, Table 3 shows the behavior of the eigenvalues of the matrix of variables Xused in the SDP model. This behaviour also confirms that relaxing the rank of this matrix for obtaining a convex model is an adequate approximation for solving OPF problems since the values in Table 3 confirm that there exists a unique eigenvalue λ4that is different from zero. In contrast, the other eigenvalues are closer Table 2 Voltage profiles obtained by solving Model 1 and Model 2 with GAMS and CVX. GAMS-KNITRO Period [s] v1[p.u] v2[p.u] v3[p.u] v4[p.u] T1 1.00000000 0.98805336 0.98839370 1.00041814 T2 1.00000000 0.97964434 0.97163307 1.00134435 T3 1.00000000 0.97702266 0.96379547 1.00190671 T4 1.00000000 0.97542262 0.97302896 0.98625905 CVX-MATLAB Period [s] v1[p.u] v2[p.u] v3[p.u] v4[p.u] T1 1.00000000 0.98799709 0.98834360 1.00036228 T2 1.00000000 0.97958790 0.97158273 1.00128854 T3 1.00000000 0.97696623 0.96374498 1.00185099 T4 1.00000000 0.97536681 0.97297903 0.98620382 to zero, as supposed in (28). Table 3 Eigenvalues’ behavior for the SDP convex approximation. Period [s] λ1λ2λ3λ4 T1 -7.338×10−13 2.052×10−14 3.0228×10−12 3.9537 T2 -3.264×10−13 -1.4948×10−14 2.8167×10−12 3.9061 T3 -6.7974×10−12 -5.4389×10−12 -2.0424×10−12 3.887 T4 -8.7551×10−14 1.9794×10−13 1.4432×10−12 3.8706 7. Conclusion This paper proposed a hybridization of a PBC and convex optimization via SPD to address DC networks’ primary-secondary control with multiple DERs and CPLs. The primary control strategy was based on the PBC methods that guarantee global asymptotic stability using Lyapunov’s theory in a closed-loop operation. Furthermore, the proposed PBC controllers were based on proportional and PI actions that allow reaching the stabilization control objective by using only local measurements. This showed a clear advantage compared PBC methods reported in literature. Simultaneously, the set point of the PBC methods (the secondary control stage) was defined with convex optimization, which relaxes the nonlinear non-convex model to guarantee uniqueness in the OPF solution. Different simulation results showed that the proposed primary-secondary control strategy’s was efficienct and effective for stabilizing the DC network varying time load conditions. CRediT author statement O.D. Montoya: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data Curation, Writing-Original Draft, Writing-Review Editing, Visualization, Supervision, Project administration, Funding acquisition. W. Gil-Gonzalez: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data Curation, Writing-Original Draft, Writing-Review Editing, Visualization. A. Garces: Conceptualization, 7 Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data Curation, Writing-Original Draft, Writing-Review Editing, Visualization, Supervision, Project administration, Funding acquisition. F. Serra: Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data Curation, Writing-Original Draft, Writing-Review Editing, Visualization. J.C. Hernandez: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data Curation, Writing-Original Draft, Writing-Review Editing, Visualization. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was partially supported by the Administrative Department of Science, Technology, and Innovation of Colombia (COLCIENCIAS) through the National Scholarship Program (calling contest 727-2015). References [1] O. D. Montoya, L. Grisales-Nore˜ na, D. Gonz´ alez-Montoya, C. Ramos-Paja, A. Garces, Linear power flow formulation for low-voltage DC power grids, Electr. Power Syst. Res. 163 (2018) 375 – 381. doi:https://doi.org/10.1016/j.epsr.2018.07.003. [2] R. Farooq, L. Mateen, M. Ahmad, S. Q. Akbar, H. A. Khan, N. A. 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