Voltage Limits Research Articles

An improved system design method for cell-based energy storage systems: A combination of a constraint satisfaction problem with an extended Ragone plot

Over the past three decades, advancements in electrical energy storage technologies have substantially enhanced performance and reduced costs, rendering them viable for diverse applications. Cell-based energy storage systems, such as batteries and supercapacitors, are particularly notable for their adaptability and scalable interconnection capabilities. However, the design process for such systems lacks uniform standards, presenting a gap in research for the conceptual system design. This paper introduces an improved system design method (SDM), addressing critical limitations of existing approaches: (a) Shifting from current-based to power-based requirements; (b) Enabling flexible adaptation of operational design points beyond fixed datasheet specifications; (c) Ensuring harmonization of energy storage design with other system components, notably power electronics. Central to our SDM is the novel “extended Ragone plot” (ERP), which maps cell-specific energy–power relations and allows for flexible adjustment of operational limits. Since it can be determined experimentally, the subsequent design method is model-free and can efficiently be executed for different cell types. In the SDM, this ERP is coupled with a constraint satisfaction problem (CSP) to compute a comprehensive design solution space that accommodates all application-specific requirements and component interrelations. It is implemented in a holistic, interactive tool, facilitating mathematical optimization and enabling visual analysis of energy storage system designs. As a proof-of-concept, two case studies on the design of lithium-ion battery systems for stationary grid supply demonstrate the method’s superiority over conventional approaches. While in the first case, an optimal design can be determined immediately, in the second case, there are no valid design solutions based on the cell’s initial operating point. Utilizing the ERP, a practical design solution is achieved by reducing the maximum cell power by approximately 9%, thereby modifying the cell’s available voltage and current limits. This approach provides a possible solution for scenarios where a system design based on data sheets alone is not feasible. A comparative analysis and graphical synthesis of both case studies validate this hypothesis, illustrating the novel system design method’s ability to precalculate and evaluate diverse design scenarios in advance. The source code and data used in this study are openly accessible, offering the SDM as an interactive tool for analyzing and designing energy storage systems.

Optimal position collaborative control of dual permanent magnet synchronous motor system

In order to improve the control accuracy and synchronous performance of the multi-motor control system, this paper proposes an optimal position collaborative control strategy for dual permanent magnet synchronous motor system. Firstly, based on the mathematical model of permanent magnet synchronous motor, an incremental predictive model is established to suppress constant and slow time-varying disturbances caused by changes in rotational inertia and load torque. Afterwards, the predicted compensation amount after multi-step prediction is obtained by minimizing the cost function which contains the synchronous error and tracking error simultaneously, and compensated to the given current loop in advance without changing the original dual motor system structure. Then the controllability and observability of the system are analyzed, and it can be proved that the system is asymptotically stable based on the optimal control theory. Secondly, considering the constraints such as maximum voltage and current limits, the optimal solution under the constraints is output to obtain the reference currents i q1 and i q2. Finally, it is demonstrated through the experiments that the proposed control strategy can achieve advance prediction compensation control of synchronous error, and the dynamic response speed is improved, while ensuring its steady-state performance remains unchanged.

Effective Charging of Commercial Lithium Cell by Triboelectric Nanogenerator with Ultrahigh Voltage Energy Management.

It is an increasingly mature application solution that triboelectric nanogenerator (TENG) supplies power to electronic devices through its power management system (PMS). However, the previous PMS is able to manage a limited voltage magnitude and the energy storage elements are limited to capacitors. This work proposes an ultrahigh voltage PMS (UV-PMS) to realize the charging of commercial lithium cells (LCs) by TENG. The design of UV-PMS enables energy management of TENGs with ultrahigh open-circuit voltages up to 3500V and boosts the peak charging current from 30.9µA to 2.77mA, an increase of 89.64 times. With the introduction of UV-PMS, the effective charging capacity of LC charged by a TENG at a working frequency of 1.5Hz for 1h comes to 429.7µAh, making a 75.3 times enhancement compared to charging by TENG directly. The maximum charging power comes to 1.56mW. The energy storage efficiency is above 97% and the overall charge efficiency can be maintained at 81.2%. This work provides a reliable strategy for TENG to store energy in LC, and has promising applications in energy storage, LC's life, and self-powered systems.

Conduction thermography for non-destructive assessment of fatigue cracks in metallic materials

Metallic materials are widely used in many applications of mechanical and aerospace engineering. Due to severe environmental and loading conditions, components and structures can experience cracking phenomena. In this regard, different non-destructive techniques were developed in the last years to detect and monitor cracks that can propagate during the actual loading conditions.This work deals with the application of the conduction thermography as a non-destructive technique for the offline control and inspection of different notched metallic specimens characterized by short fatigue cracks. The potentials of the technique have been demonstrated considering a low-cost set-up consisting of a power supply of low current and limited voltage, and a microbolometer sensor. Moreover, to further simplify the inspection, the specimens were not black coated. Quantitative results, in terms of crack lengths, have been obtained using a new procedure of data analysis and then compared to those provided by thermoelastic stress analysis, a well-established technique adopted for the detection and monitoring of cracks.

L-Index-Based Technique for Voltage Collapse Prediction and Voltage Stability Enhancement in Electrical Power Systems

Recent years have witnessed a notable increase in the occurrence of blackouts, especially in developing nations, attributed to the continuously growing demand on modern power networks. Given that the demand shows no signs of abating and is projected to increase further in the coming years, additional research on power system stability is imperative. This study, therefore, investigates voltage stability assessment in power systems using the L-index methodology, focusing on the Nigerian 28-bus system and the IEEE system. The L-index offers a practical means of identifying weak buses and evaluating voltage stability margins. Calculating L-index values for load buses under diverse conditions identifies critical points, with higher values indicating vulnerability. The research investigates injecting reactive power at load buses to prevent collapse, comparing outcomes with and without compensation. Analyzing the L-index's performance across varied loading scenarios confirms its precision in predicting breakdown points and identifying critical buses. Load flow analysis of the Nigerian 28-Bus system reveals that only bus 16 exceeds voltage limits, while line analysis shows total power losses. Increasing loadability exposes bus 16 as the weakest, supported by its low voltage magnitude. The research confirms bus 16 as the system's weakest point, guiding corrective measures to enhance stability and prevent collapse. Utilizing Matlab for implementation, this study contributes valuable insights into system vulnerability and provides a framework for improving voltage stability in power systems.

Analysis of voltage limit‐induced barrier for connecting inverter‐based distributed generators to medium voltage networks: Australian case studies

AbstractInverter‐based distributed generators (IBDGs), mainly solar photovoltaic, connected in medium‐voltage (MV) networks cause challenges, such as voltage limit violations, for distribution network service providers (DNSPs), and require advanced network management strategies to mitigate these challenges. A theoretical analysis of the voltage limit‐induced barrier to IBDG connection and their export limits due to the change in network characteristics is imperative for developing new strategies. The authors formulated a relationship between the network equivalent impedance and the IBDG's connection point in the network and further explored the link between the network equivalent impedance and voltage magnitude due to the IBDG connection point. The authors also assessed the voltage limit‐induced barrier to IBDG connections in MV networks and proposed solutions to overcome issues with the dynamic export limit of IBDGs. Four representative Australian MV networks are analysed in DIgSILENT PowerFactory under different scenarios, such as variation in IBDG location and static and dynamic export limits. The authors found that an IBDG connected at the end of the network can achieve better performance in supporting the network voltage. An IBDG with a dynamic export limit can export three times more energy than the static export limit, which benefits both the DNSPs and IBDG owners.

Performance evaluation of novel 9-level RSMLI topology for grid-tied solar-PV system

The shortage of traditional fossil fuels like coal, petrol and natural-gas are increased day-by-day, fulfills the most of energy demand. Most of engineers are trying to maximize the energy demand by employing renewable energy with existing micro-grid system. Owing to merits, the solar-PV system plays a significant alternative among all other renewable energy sources due to abundant and virtuous nature. For grid-tied solar-PV system, the cascaded H-bridge multilevel inverter is the most significant over the classical 2-level inverter due to provision of isolated input DC sources. But the cascaded H-bridge topology is designed for limited voltage levels due to its larger number of switches for higher voltage levels, high cost, large-size, and more weight. To alleviate these demerits, a reduced-switch multilevel inverter has been generally preferable for higher voltage levels. In this work, a novel 9-level reduced-switch multilevel inverter (RSMLI) topology has been proposed by utilizing low number of switching devices. The performance of proposed novel 9-level RSMLI topology has been verified in grid-tied solar-PV system by using MATLAB/Simulink tool, simulation results are presented with attractive comparisons.

Molybdenum Sulfide Nanoflowers as Electrodes for Efficient and Scalable Lithium-Ion Capacitors.

Hybrid supercapacitors (HSCs) bridge the unique advantages of batteries and capacitors and are considered promising energy storage devices for hybrid vehicles and other electronic gadgets. Lithium-ion capacitors (LICs) have attained particular interest due to their higher energy and power density than traditional supercapacitor devices. The limited voltage window and the deterioration of anode materials upsurged the demand for efficient and stable electrode materials. Two-dimensional (2D) molybdenum sulfide (MoS2) is a promising candidate for developing efficient and durable LICs due to its wide lithiation potential and unique layer structure, enhancing charge storage efficiency. Modifying the extrinsic features, such as the dimensions and shape at the nanoscale, serves as a potential path to overcome the sluggish kinetics observed in the LICs. Herein, the MoS2 nanoflowers have been synthesized through a hydrothermal route. The developed LIC exhibited a specific capacitance of 202.4 F g-1 at 0.25 A g-1 and capacitance retention of >90 % over 5,000 cycles. Using an ether electrolyte improved the voltage window (2.0 V) and enhanced the stability performance. The ex-situ material characterization after the stability test reveals that the storage mechanism in MoS2-LICs is not diffusion-controlled. Instead, the fast surface redox reactions, especially intercalation/deintercalation of ions, are more prominent for charge storage.

Dynamic voltage mapping of the ventricular tachycardia substrate

Abstract Background Substrate mapping techniques that predict ventricular tachycardia (VT) isthmus components in sinus or paced rhythm are necessary as mapping during VT is often not possible. Voltage maps lack accuracy in differentiating scar, traditionally displayed as tissue <0.50mV, from bordering arrhythmogenic substrate. We developed a practical technique aiming to improve the utility of a voltage map during VT ablation, termed ‘Dynamic Voltage Mapping’ (DVM). Methods Consecutive patients with post-infarct VT at our centre underwent left ventricle (LV) voltage mapping (mean 4762 points) with Carto3 v.7® and a PentaRay®, or Ensite X® and a HD Grid mapping catheter. Both mapping systems offer algorithms that allow substrate activation maps to be superimposed and analysed on the surface of a voltage display (i.e. Carto Ripple Maps, or Ensite X activation vectors). DVM is a process that sequentially adjusts the voltage display to highlight scar tissue in areas devoid of Ripple bars, or in areas where activation vectors point in multiple unfathomable directions (‘vector disarray’). This approach is summarised in Figures 1 & 2 and was retrospectively performed in all cases. This alternative scar voltage limit was termed the ‘DVM scar threshold’ (mV). Tissue below this threshold was termed ‘DVM-scar’, and tissue immediately surrounding (up to 0.5mV) was termed "DVM-Borderzone (BZ)". Results Twenty-five patients (mean age 64.4±11.0 years, 88% male, LV ejection fraction 31.8±6.9%, 16.4±8.5 years since infarct) were studied (n=15 Carto; n=10 EnSite). Across all cases, the mean shell area with voltage <0.50 mV was 36.8±20.3 cm2. The mean DVM scar threshold was 0.24±0.07 mV (Carto: 0.22±0.07 mV [range 0.12–0.35 mV]; EnSite: 0.26±0.08 [range 0.18–0.50 mV]). The mean DVM-scar area was 18.4±14.3 cm2, significantly smaller than the traditional 0.50mV value (p=0.0002). In the 8 patients where stable VT was mapped, the VT isthmus sites co-located within the DVM-BZ in all cases, exemplified in the figures. Pace-mapping at the DVM-BZ matched the remaining unmappable VT morphologies. Ablation lesions were retrospectively superimposed onto the DVM map and generally co-located within the DVM-BZ. No ablation was delivered in tissues >0.5mV. At median follow-up of 8.8 months, VT burden significantly reduced (median 3 episodes in 3-months pre vs. 0 post; p<0.001), with 76% of patients VT-free. Conclusion DVM appears to improve the delineation of the post-infarct VT substrate, and is feasible using two commercially-available mapping systems. The mean voltage threshold that differentiated DVM-Scar from BZ tissue was significantly lower than the traditional 0.50 mV scar definition, at 0.24mV, though ranged widely, suggesting that scar thresholds require individualised substrate definition. Targeted ablation of DVM-BZ may be effective at reducing future VT, though requires prospective validation.Figure 1Figure 2

Massive, long-lived electrostatic potentials in a rotating mirror plasma

Hot plasma is highly conductive in the direction parallel to a magnetic field. This often means that the electrical potential will be nearly constant along any given field line. When this is the case, the cross-field voltage drops in open-field-line magnetic confinement devices are limited by the tolerances of the solid materials wherever the field lines impinge on the plasma-facing components. To circumvent this voltage limitation, it is proposed to arrange large voltage drops in the interior of a device, but coexist with much smaller drops on the boundaries. To avoid prohibitively large dissipation requires both preventing substantial drift-flow shear within flux surfaces and preventing large parallel electric fields from driving large parallel currents. It is demonstrated here that both requirements can be met simultaneously, which opens up the possibility for magnetized plasma tolerating steady-state voltage drops far larger than what might be tolerated in material media.

Optimal sizing and siting of single tunned passive filter based on jellyfish algorithm

<p class="MsoNormal" style="margin-top: 10pt; text-align: justify;"><span lang="EN-US" style="font-family: 'times new roman', times, serif; font-size: 14pt;">The presence of electrical harmonics in low voltage networks causes many problems for customers, the most important of which is the decrease in the power quality beyond the permissible limits, which leads to a decrease in voltage for long periods, a decrease in power factor and a large losses in the electrical network due to the deformation of the electric waveform of voltage and current. The aim of this paper is to design an optimal passive filter and its optimal location to suppress the harmonic distortion in the distribution system that results from nonlinear loads. This filter will use single objective functions and multi objective functions that will minimize the proposed filter cost, the real power losses, total harmonic distortion "THD" and individual harmonic distortion "IHD". This will be done under the equality constraint of power balance and some inequality constraints such as filter parameters limits, quality factor limits, voltage limits, harmonic distortion limits. The optimal passive filter parameters and location will be calculated using the modern optimization techniques such jellyfish optimization technique. IEEE 33 bus radial system will be used for demonstrating results obtained by the proposed method.</span></p>

Investigation of the effect of load weight on the discharge capacity of lithium–O2 batteries using carbon-paper–based electrodes

We prepared carbon electrodes with nearly uniform thickness and varying loading weights using carbon paper (CP) as a support material and current collector for single-walled carbon nanotubes (SWCNTs). These electrodes were used as the positive electrode in a lithium–O2 battery, tested at a lower voltage limit of 2.4 V with a fixed current of 0.26 mA/cm2 (0.5 mA). Results demonstrated that the discharge gravimetric capacity (capacity normalized per weight of SWCNTs) did not change significantly for loading weights ≥0.42 mg/cm2. However, for loading weights <0.42 mg/cm2, the discharge gravimetric capacity exhibited an inverse correlation with weight. Analysis of the prepared electrodes revealed that at very low SWCNT loading weights, the sheets of SWCNTs became extremely thin, leading to occasional tears. Nonetheless, these structural changes had no significant impact on the specific surface area or contact resistance. Further analysis of the post-test electrodes using scanning electron microscopy and X-ray diffraction, along with considerations of the carbon paper's surface area as a percentage of the total electrode surface, led to the conclusion that at low loading weights, CP actively contributes to the discharge process and serves as a deposition site for Li2O2. Consequently, when the weight of the active material falls below a certain threshold, there is a risk of misperception, potentially leading to an overestimation of the discharge capacity.

Research on harmonic suppression characteristics and control methods for low-frequency resonance suppressor

The problem of low-frequency harmonics and their resonance seriously threatens the safety of the fusion power supply system, and is one of the key issues that urgently need to be solved during high-power operation of fusion devices. The low-frequency resonance suppressor (LFRS) can compensate for the low frequency harmonic suppression effect of passive filter and active filter and the limitation of voltage and capacity, which is a new approach to solve the low frequency resonance problem of fusion power system. However, the performance of the LFRS is affected by the power grid parameters and the fundamental current circulation between the active filter and passive filter branches. The low frequency harmonic current and its resonance suppression characteristics are analyzed based on the multi-reference vector control strategy in this paper. On this foundation, a current double closed loop control method is proposed, which can ensure the safe and reliable operation of the LFRS system and the suppression effect of low frequency harmonic current. The correctness and effectiveness of the theoretical analysis are verified by RT-LAB Hardware-in-the-loop simulation.

On the structural evolution of pristine P2-type Na0.67Ni0.33Mn0.67O2 for symmetric full cell application

P2 type Na0.67Ni0.33Mn0.67O2 (NNMO) is a widely studied layered oxide cathode material for sodium-ion batteries (SIBs) because of its high capacity (160 mAhg-1 practically achieved) and broad voltage window (2–4.5 V). The redox activity in NNMO is shown by both Ni2+ and Mn4+ ions, making it a prospective symmetric SIB electrode material. The average working potential of the symmetric full cell is less due to the limited average working voltages of both the redoxes in a half cell between 2 V and 4.5 V, resulting in poor energy density. Herein, we report the feasibility of the NNMO symmetric full cell by widening the operational voltage window of NNMO. The electrochemical performance of NNMO half cells between 0.01 V and 4 V and the structural evolution during cycling are mapped by ex-situ XRD. Na0.44MnO2-like tunnel-type structure (Pbam) with high Na+ content (>0.90) exists at potentials close to the Mn3+/4+ redox reaction during cycling. The phase change was relatively reversible, implying that the NNMO anode works at such low voltages without the hazard of severe capacity losses. The symmetric full cell made using NNMO cycled between 0.01 V and 3 V and, at C/10 rate, delivered a capacity of ∼68 mAhg−1.

Understanding and Weakening Photon Recycling in Solar Cells to Approach the Radiative Limit.

Photon recycling plays an important role in the light outcoupling of state-of-the-art solar cells and is considered a necessary condition to achieve the radiative limit of open-circuit voltage (VOC) and efficiency. However, due to the impact of photon recycling on bulk and surface radiation of solar cells being overlooked, experimental demonstrations on the accuracy of existing photon recycling models are scarce and some contrary theoretical results also emerge. Here, the relationship between photon recycling and radiation processes, as well as the corresponding VOC losses of solar cells based on the principle of detailed balance is clarified. It is shown that weakening photon recycling is more effective to boost the device performance than enhancing it, promoting the theoretical efficiencies of solar cells, such as perovskite, Si, and GaAs, to 98.5%, 94.9%, and almost 100% of their radiative limit, respectively. Moreover, weakening photon recycling also helps to maintain higher efficiency when the internal radiative efficiency decreases, which benefits higher device stability. This work provides an in-depth understanding of the role of photon recycling in solar cells and helps to push efficiency to a new limit.

Photoelectrochemical CO2 reduction in tandem photoelectrode cells: Interpretation of apparent photocurrents

Photoelectrochemical reactors harness solar energy to accelerate heterogeneous chemical reactions and reduce power consumption for a given reaction rate. The tandem photoelectrochemical cell design presented here employs a light penetrable TiO2 photoanode||Cu2O photocathode configuration that enables carbon dioxide turnover at limited cell voltages. During polarization tests, the tandem configuration demonstrated a photocurrent density of 0.2 mA/cm2 at a cell voltage of -2.6 V. That is 2-fold higher than the photocurrent density obtained with a solo photoanode (Cu||TiO2) and five orders of magnitude higher compared to the solo photocathode (Cu2O||Ni) design. Moreover, electrolysis under illumination (photoelectrolysis) showed a current density increase of 3.9 mA/cm2 with respect to electrolysis in the dark. However, this photocurrent density was merely apparent and was caused by light-triggered changes in the surface morphology and phase of Cu2O, which in turn affected the catalytic response. Moreover, the apparent partial ethylene photocurrent density and total partial current density during photoelectrolysis was 44 %, the highest among the detected products. That selectivity is attributed to the combined effect of Cu2O photoreduction and the associated local pH rise at the electrode-electrolyte interface.

A Redox-Electrodialysis Model with Zero Fitting Parameters: Insights into Process Limitations, Design, and Material Interventions

Redox-Electrodialysis (r-ED) is an electrochemical desalination cell architecture that has recently received considerable interest, due to its low energy demand relative to electrochemical desalination technologies that rely on electrode-based ion removal. To further improve the energy efficiency of r-ED, we developed a lumped mathematical model with no adjustable parameters to investigate the various sources of overpotential within the cell. Existing models of electrodialysis and r-ED cells either do not accurately incorporate all phenomena contributing to the overpotential or utilize empirical fitting parameters. The model developed here indicates that ohmic overpotentials, especially in the diluate chamber, are the most significant contributors to energy losses. Based on this insight, we hypothesized that adding an ion exchange resin wafer in the diluate compartment would increase the ionic conductivity and decrease the energy demand. Experimental results showed an 18% reduction in specific energy use while achieving the same degree of salt removal (20 mM to 12 mM). Furthermore, the resin wafer enabled complete desalination to potable drinking levels at a current density previously unachievable within practical operating voltage limits (4.93 mA cm−2). We also expanded the model to explore differences in r-ED energy use between configurations using multiple cells and a single cell with increased area.

Particle Swarm Optimization (PSO) And Evolutionary Programming (EP) Technique for Optimal Placement and Sizing DG for Minimizing Loss in IEEE-30 Bus System

In recent years, most growing nations worldwide have experienced an unavoidably rising demand for electricity, which may result in a dropping of the voltage profile and a reduction in system stability, both of which have the potential to overload the power generation system significantly. One of the ways to reduce loss system network in this study is by distributed generation (DG) allocation, which the DG optimal location and sizing are the two most significant considerations for integration of DG. Improper placement and DG sizing in the power system not only result in increased total power losses but also affect the electrical system’s capability to be function properly. This paper provides a comparative analysis of two meta- heuristic optimization techniques, that is, Particle Swarm Optimization (PSO) and Evolutionary Programming (EP), to examine the optimal placement and sizing of distributed-generation (DG) for photovoltaic systems with the aim of minimizing the losses in interconnected distributed generation, IEEE-30 bus system. The objective function of this study is minimization the power loss with considered some constraints which are voltage limits and DG sizing limits are being covers in this study. The optimization issue has been solved by working with MATLAB/m-files software. The outcomes derived from the simulation results in this study demonstrated that PSO performance of two units of DG is better compared to EP algorithm. The voltage profile was improved drastically, and the power loss was reduced as lowest as 17.4486MW in the first case, 17.4368MW in the second case, and 17.4179MW in the third case.

Side chain-engineered pseudocapacitive semiconducting polymer electrodes for symmetrical supercapacitors operable at wide potentials

Pseudocapacitive electrodes based on inorganic metal oxides exhibit superior capacitance. However, they suffer from limited operational voltage, resulting in low energy and power densities. To address these issues, redox semiconducting polymer (SP)-based pseudocapacitive electrodes, integrating electron donor and acceptor functionalities, offer a promising solution. In this study, we introduce two novel pseudocapacitive SP electrodes composed of highly planar electron-donating building blocks 4,8-bis ((5-octylthiophen-2-yl)ethynyl)benzo[1,2-b:4,5-b'] dithiophene (TBDT) with an electron-accepting building blocks benzo[c][1,2,5]thiadiazole (BT) and its side chain manipulated counterpart, i.e., OBT, denoted as SP1 and SP2, respectively. Among them, SP2 demonstrates a maximum specific capacitance of 117 F/g at 1 A/g. Furthermore, a symmetric supercapacitor fabricated for full-cell applications exhibits a significant energy density, reaching 29.1 Wh kg−1, and a maximum power density of 12.4 kW kg−1 within the wide operating potential window of 2.5 V. The exceptional performance of the pseudocapacitive electrode can be attributed to its highly planar backbone, nanoporous morphology, low crystallinity, and low lowest unoccupied molecular orbital levels. Overall, this new class of pseudocapacitive polymer electrode holds great potential for diverse sustainable energy storage technologies, addressing the limitations of traditional inorganic metal oxide-based electrodes by offering wide potentials window.

Influence of Load Models and DG Operational Power Factor on Optimization of Shunt Capacitor Placement in Distribution Grids Considering Voltage Stability

The paper proposes a mathematical model of mixed-integer nonlinear programming (MINLP) to optimize the location and size of shunt capacitors in power distribution grids with distributed generation (DG), considering voltage stability constraints. At the same time, the paper also analyzes the influence of the load models, such as voltage-dependence load (ZIP load) and constant power load, together with DG’s operating power factor on the optimal solution. The objective function of the proposed optimization formulation is minimizing the total expenses, including the capital investment of capacitors and the expense incurred by network energy loss. This MINLP model includes constraints in the current and critical loading conditions, such as the system of power flow equations, nodal voltage limits, branch thermal limits, capacitor-related constraints, and restrictions of minimum security level. The proposed optimization model was evaluated on a modified IEEE 33-node distribution grid using the KNITRO commercial solver with the GAMS programming language. The calculation results show that the voltage stability constraints, load models, and operational power factor of the DG units have an essential influence on the optimal position and capacity of the shunt capacitors.