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Average values of optimal power flow objective for real, forecasted, optimal voltage and phase angles based on Equation (18).
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Distribution networks are typically unbalanced due to loads being unevenly distributed over the three phases and untransposed lines. Additionally, unbalance is further increased with high penetration of single-phased distributed generators. Load and optimal power flows, when applied to distribution networks, use models developed for transmission gr...
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... and reactive power for all the phases can be observed in Figures 3 and 4. Using the three schedules shown in Figures 3 and 4, load flows are performed using THELM described in Section 2. Loads flows are performed for all intervals and are represented using box-plots. Figure 5 describes the averaged objective function values based on Equation (18). It can be observed that the three-phase unbalance has been reduced from 0.879 for real and forecasted profiles to 0.529 for optimal profiles which accounts for 39% unbalance minimization based on the defined objective function (see Section 4). ...
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... schedules for these buses were generated and used to produce voltages using THELM and the results were described in Figure 6. It can be observed that the three-phase voltage unbalance has reduced up to 39% and the optimal average objective function values can be observed in Figure 5. ...
Citations
... Compared with the steady-state problem, time-domain simulation is much more time-consuming and also suffers divergence issues especially in DAEs. To overcome those challenges induced by traditional numerical computational methods, the holomorphic embedding method (HEM) is proposed initially to solve steady-state problems such as power flow computations [6−9] , voltage security assessments [10][11][12] , transfer capacity analysis [13] , contingency analysis [14,15] , network reductions [16,17] , control strategy design [18][19][20] and optimal power flow [21,22] due to satisfying convergence rate, efficient and robust properties, and non-iterative feature, and later is extended to accelerate dynamic simulations [23,24] . To further investigate the value of the HEM in power system analysis, it is very important to summarize present related research and then give a direction for future research. ...
The holomorphic embedding method (HEM) stands as a mathematical technique renowned for its favorable convergence properties when resolving algebraic systems involving complex variables. The key idea behind the HEM is to convert the task of solving complex algebraic equations into a series expansion involving one or multiple embedded complex variables. This transformation empowers the utilization of complex analysis tools to tackle the original problem effectively. Since the 2010s, the HEM has been applied to steady-state and dynamic problems in power systems and has shown superior convergence and robustness compared to traditional numerical methods. This paper provides a comprehensive review on the diverse applications of the HEM and its variants reported by the literature in the past decade. The paper discusses both the strengths and limitations of these HEMs and provides guidelines for practical applications. It also outlines the challenges and potential directions for future research in this field.
... • Holomorphic embedding power flow for three-phase networks [8]. ...
... x ∈ X ⊂ C transformers (6) (i, p) ∈ T bt ⊆ I × P bus-terminal topology (7) (c, i) ∈ T bus ⊆ C × I component-bus topology (8) (c, i, p) ∈ T term ⊆ T bus × P component-bus-terminal topology (9) ...
This report serves as a technology description of a Julia-based re-implementation of the fixed-point current injection algorithm, available in PowerModelsDistribution.jl [1]. This report does not describe a novel method for solving unbalanced power flow problems. It merely provides a description of the fixed point iteration variant of the current injection method, inspired by the existing open-source implementation in OpenDSS1 [2]. The current injection method is commonly conceived as a system of nonlinear equalities solved by Newton s method [3, 4]. However, as Roger Dugan points out in the OpenDSS documentation, the fixed point iteration variant commonly outperforms most methods, while supporting meshed topologies from the ground up. We note that the unbalanced power flow algorithm in turn relies on matrix solvers for sparse systems of equations. In the context of circuits and factorizing nodal admittance matrices, the sparsity-exploiting KLU solver [5] has proven to be both reliable and scalable. OpenDSS uses KLU. This report documents work-in-progress, and the authors aim to update it when learnings are obtained or more features are added to the implementation in PowerModelsDistribution.jl. The authors invite collaborators to contribute through pull requests on the repository.
... In [17,18], the authors interestingly proposed a modified holomorphic embedding method for hybrid meshed AC-DC microgrid and for remote voltage control, respectively; however, the time intensive PA is still applied. e authors in [19,20] showed the benefits from utilizing the HELM approach for three-phase unbalanced optimal power flow and for probabilistic power flow for small distribution test networks; however, the long time needed to solve the power flow equations impeded extending the application to larger distribution test networks. In [21,22], the authors notably used static approximate Newton-Raphson-Sidel (ANRS) and the developed down-hill (DDH) algorithm methods to derive a reliable and efficient holomorphic approach to evaluate dynamic available transfer capability and proposed distributed slack bus model formulation for the holomorphic embedding load flow method, respectively; however, Padé Approximation is still used. ...
... Equation (17) is then rewritten in the matrix forms given in (18)- (20) to solve for bus voltages. ...
... Equation (20) can be solved for V either iteratively or recursively. To solve the equation recursively, equations (16) and (20) are embedded with parameter, α as shown in (21) and (22). is embedding implies that α is a parameter for constant power injections. is embedding is simpler than the embedding method used in [10,11,24] as discussed in the previous section and results in faster computation as demonstrated in the next section. e voltage functions V 1 (α), V(α), and W * (α * ) satisfy Cauchy-Riemann equations and have continuous first partial derivatives; thus, they are holomorphic. ...
This paper proposes a novel holomorphic embedding approach for solving the nonlinear power flow equation for meshed electric distribution networks with ZIP load model. In the proposed approach, bus voltages are modelled as holomorphic functions in the constant power injections and then expanded using Maclurin series. The Z-bus matrix is implicitly used to calculate Maclurin series coefficients for the expanded voltage functions in a recursive manner. The necessary and sufficient conditions for the convergence of expanded voltage functions are found. Performance evaluations show that the proposed approach solves the nonlinear power flow equations faster than the existing approaches when applied to 18-, 33-, 69-, 141-, 3239-, 5701-, and 6921-bus distribution network test cases.
... The solution of this MINLP model was achieved using the General Algebraic Modeling System (GAMS) and the large-scale solver BONMIN [31,32]. In addition, we propose two recursive validations using the successive approximation power flow method in the MATLAB programming environment [33] and the Newton-Raphson power flow method in DigSILENT software [34], which allow the final value of the λ−coefficient to be verified via an iterative procedure by fixing the sizes and locations of the dispersed generators based on the solution provided by GAMS software. The main contributions of this study include: (i) the validation of the optimal value of the λ-coefficient using two recursive power flow methods programmed in MATLAB and DigSILENT programming environments based on the successive approximation power flow method and the Newton-Raphson approach, respectively; and (ii) the evaluation of multiple simulation scenarios with different numbers of available dispersed generators with limitations of 40% and 60% in the total injection of active power by the distributed sources. ...
This study addresses the problem of the maximization of the voltage stability index (λ-coefficient) in medium-voltage distribution networks considering the optimal placement and sizing of dispersed generators. The problem is formulated through a mixed-integer nonlinear programming model (MINLP), which is solved using General Algebraic Modeling System (GAMS) software. A numerical example with a 7-bus radial distribution network is employed to introduce the usage of GAMS software to solve the proposed MINLP model. A new validation methodology to verify the numerical results provided for the λ-coefficient is proposed by using recursive power flow evaluations in MATLAB and DigSILENT software. The recursive evaluations allow the determination of the λ-coefficient through the implementation of the successive approximation power flow method and the Newton–Raphson approach, respectively. It is effected by fixing the sizes and locations of the dispersed sources using the optimal solution obtained with GAMS software. Numerical simulations in the IEEE 33- and 69-bus systems with different generation penetration levels and the possibility of installing one to three dispersed generators demonstrate that the GAMS and the recursive approaches determine the same loadability index. Moreover, the numerical results indicate that, depending on the number of dispersed generators allocated, it is possible to improve the λ-coefficient between 20.96% and 37.43% for the IEEE 33-bus system, and between 18.41% and 41.98% for the IEEE 69-bus system.
... One additional consideration we have not investigated is to consider the possibility of having an unbalanced electrical grid, whereas in our case we make the implicit assumption that the grid is balanced between the three electrical phases. Techniques for OPFs considering unbalanced networks can be found in [247] or in [85] where a low-voltage grid is considered (LV networks are natural use-cases for unbalanced OPF). Another important consideration in an ideal OPF for the DSO should be the ability to take integer variables into account, as several levers available to the DSO naturally require discrete variables to be modelled (as line switches for instance). ...
The increasing integration of renewable energy sources has a long-lasting impact on the electrical grid, and the liberalisation of the energy sector has significantly changed its regulatory environment. In particular, the distribution network has become an area of interactions of competitive actors, while being managed by a single actor: the distribution system operator (DSO). Among the DSO’ challenges is the short-term operational planning: the selection and activation of levers to ensure the safe exploitation of the grid, taking into account the forecasts of grid users’ activities. Decisions in this context are based on the mathematical model of the Optimal Power Flow (OPF). Sources of uncertainties on these latter forecasts are growing due to the increasing number of actors on the grid.The focus of this thesis is on the integration of uncertainties on power production and consumption in the OPF, using chance-constraints. The resulting probabilistic OPF model is a non-convex non-smooth optimization problem with a Difference-of-Convex (DoC) structure. The class of DoC functions is large enough to include convex, concave, and approximations of arbitrarily precision of every continuous function, while offering strong regularity properties that one can leverage to derive a generic optimisation algorithm.A first contribution of this work is the development of an original bundle algorithm for the class of DoC constrained DoC problems. Chance-constraints are proved to be DoC, and a DoC approximation of chance-constraints is proposed before being applied to the probabilist OPF. A characterization of the first-order information of probabilist functions is presented, based on a variational study of these latter functions. This characterization highlights the variety of choices when it comes to solving chance-constrained programs. Four explicit formulations of probabilist OPFs are then proposed, and their DoC structure is proved. The algorithm’s performance, the impact of parametrisation on its behaviour and the interest of each model are numerically validated on a 33 nodes network. Besides the reasonable computing times, this methodology isparticularly relevant as, differently to other works in literature, the electrical viability and validity of a solution are directly accessible.
... Because of the rapid growth of large-scale wind farms, wind energy is playing an increasingly important role in domestic and international power markets as a sustainable and cost-effective renewable energy source. Wind's highly unpredictable capacity, on the other hand, can trigger nonlinear characteristics in the wind power, which can have a number of negative consequences for the wind power system's reliability [19][20][21][22][23][24][25]. As a result, developing an accurate and efficient power prediction model is needed to preserve the grid's reliability while also improving the equal planning, dispatching, control, and risk assessment capabilities. ...
The operation complexity of the distribution system increases as a large number of distributed generators (DG) and electric vehicles were introduced, resulting in higher demands for fast online reactive power optimization. In a power system, the characteristic selection criteria for power quality disturbance classification are not universal. The classification effect and efficiency needs to be improved, as does the generalization potential. In order to categorize the quality in the power signal disturbance, this paper proposes a multi-layer severe learning computer auto-encoder to optimize the input weights and extract the characteristics of electric power quality disturbances. Then, a multi-label classification algorithm based on rating is proposed to understand the relationship between the labels and identify the various power quality disturbances. The two algorithms are combined to construct a multi-label classification model based on a multi-level extreme learning machine, and the optimal network structure of themulti-level extreme learning machine as well as the optimal multi-label classification threshold are developed. The proposed method can be used to classify the single and compound power quality disturbances with improved classification effect, reliability, robustness, and anti-noise performance, according to the experimental results. The hamming loss obtained by the proposed algorithm is about 0.17 whereas ML-RBF, SVM and ML-KNN schemes have 0.28, 0.23 and 0.22 respectively at a noise intensity of 20 dB. The average precision obtained by the proposed algorithm 0.85 whereas theML-RBF, SVM andML-KNN schemes indicates 0.7, 0.77 and 0.78 respectively.
... Steady-state transformer models are frequently employed in power system load flow simulations and other applications in which component modeling can be carried out in quasi-stationary regime [1][2][3][4][5][6][7][8][9]. Different transformer models can be selected according to frequency range [10][11][12] and the required trade-off between precision and complexity. ...
Measurements obtained in transformer tests are routinely used for computing associated steady-state model parameters, which can then be used for load flow simulation and other modeling applications. The short circuit and open circuit tests are most commonly performed with this purpose, allowing estimation of series and parallel branch transformer parameters. In this study, an extended model is proposed which does not employ the usually assumed cantilever circuit approximation and explicitly accounts for transformer connection resistance. An estimation of the proposed model parameters is enabled by usage of additional measurements yielded by the direct current (DC) resistance test. The proposed approach is validated by means of an experiment carried out on a real distribution power transformer, whose results demonstrate that the proposed model and parameter computation approach effectively decompose total transformer resistance into winding and contact components. Furthermore, the numerical results show that contact resistance is not negligible especially for low voltage windings, which reinforces the usefulness of the proposed model in providing detailed modeling of transformer resistances.
... The methodology developed in this paper includes a novel heuristic for three-phase LV network congestion management which estimates network headroom based on both thermal and voltage limits. This heuristic provides a more scalable approach than centralised threephase optimal power flow (OPF) which is the established approach to LV network congestion management in the literature [8][9][10]. Novel aspects of the proposed methodology are that the network headroom https://doi. ...
... In academic literature, a common solution to managing flexible assets to solve congestion in electricity distribution networks is the use of OPF. There are numerous examples of OPF being applied to distribution networks including [36] and [37], and more recently threephase OPF has been developed [8,9], including open source software that has the capability to model LV network constraints as part of multi-period market optimisation [10]. These methods can provide the optimal solution in terms of maximising the use of flexibility at LV, however they can be limited by their tractability in terms of the required computational power and time required to solve non-linear AC OPF formulations. ...
... The zonal power flow limit, Vlim , for each network is calculated from the minimum power flow that resulted in a minimum voltage of 0.94 p.u. 9 An example of the estimation of Vlim for network 1, zone 11 is shown below in Fig. 24, where Vlim 11 =26.6 kVA. The points on Fig. 24 correspond to the 12,673 timesteps of each HP penetration case combined (from 0 to 100% HP penetration in increments of 25%). ...
The decarbonisation of heat and transport using heat pumps (HPs) and electric vehicles (EVs) will require significant investment in low voltage (LV) networks both in terms of network reinforcement and in the provision of flexibility to avoid network upgrades where appropriate. In this paper, a heuristic methodology is presented to estimate headroom available for domestic EV charging optimisation in LV networks at different penetrations of HPs and a novel zonal approach is applied to EV optimisation. It was found that optimised charging of EVs can allow for a significantly higher penetration of EVs for a given HP penetration within the network, without the need for reinforcement. Significant improvements in terms of network hosting capacity were realised: for example, an increase from 34% EV and 50% HP penetration for dumb charging to 72% EV and 57% HP penetration for optimised charging was available for one particular case study. The level of improvement in hosting capacity was found to be strongly dependent on particular network topology and pre-existing demand; this reinforces the need for further study in unlocking the potential synergies of EV and HP uptake.
... where EQ u (·) corresponds to the nonlinear functions associated with the active and reactive power balances at each node [15]; P d,h k, f and Q d,h k, f are the active and reactive power demand values associated with node k at phase f in the period of time h; and V h k, f is the RMS value of the voltage at phase f at node k in the period of time h, where the voltage angle is defined as θ h k, f . The voltage regulation bounds in all the nodes of the network can be constrained as follows: ...
The optimal expansion of AC medium-voltage distribution grids for rural applications is addressed in this study from a heuristic perspective. The optimal routes of a distribution feeder are selected by applying the concept of a minimum spanning tree by limiting the number of branches that are connected to a substation (mixed-integer linear programming formulation). In order to choose the caliber of the conductors for the selected feeder routes, the maximum expected current that is absorbed by the loads is calculated, thereby defining the minimum thermal bound of the conductor caliber. With the topology and the initial selection of the conductors, a tabu search algorithm (TSA) is implemented to refine the solution with the help of a three-phase power flow simulation in MATLAB for three different load conditions, i.e., maximum, medium, and minimum consumption with values of 100%, 60%, and 30%, respectively. This helps in calculating the annual costs of the energy losses that will be summed with the investment cost in conductors for determining the final costs of the planning project. Numerical simulations in two test feeders comprising 9 and 25 nodes with one substation show the effectiveness of the proposed methodology regarding the final grid planning cost; in addition, the heuristic selection of the calibers using the minimum expected current absorbed by the loads provides at least 70% of the calibers that are contained in the final solution of the problem. This demonstrates the importance of using adequate starting points to potentiate metaheuristic optimizers such as the TSA.
... It uses optimality conditions using Lagrangian functions with objective and constraint derivatives. In [9], a novel class C algorithm is presented in which a reliable load flow is coupled with a heuristic optimization method. This method helps to overcome the challenges presented by classes A and B, which are used in this paper. ...
... In this paper, a solution to OPF using the non-convex optimization method is chosen. This is based on the method developed in [9]. It uses a three-phase unbalanced holomorphic embedding load flow method (HELM) with a genetic algorithm to generate optimal set-points, a HELM-OPF method. ...
... It uses a three-phase unbalanced holomorphic embedding load flow method (HELM) with a genetic algorithm to generate optimal set-points, a HELM-OPF method. The reason for using HELM is due to its robustness and ability to converge to a high voltage operable solution irrespective of its initial conditions (very high or low loading conditions) [9]. Using HELM, OPF is given access to the entire search space. ...
This paper presents control relationships between the low voltage distribution grid and flexibilities in a peer-to-peer local energy community using a stratified control strategy. With the increase in a diverse set of distributed energy resources and the next generation of loads such as electric storage, vehicles and heat pumps, it is paramount to maintain them optimally to guarantee grid security and supply continuity. Local energy communities are being introduced and gaining traction in recent years to drive the local production, distribution, consumption and trading of energy. The control scheme presented in this paper involves a stratified controller with grid and flexibility layers. The grid controller consists of a three-phase unbalanced optimal power flow using the holomorphic embedding load flow method wrapped around a genetic algorithm and various flexibility controllers, using three-phase unbalanced model predictive control. The control scheme generates active and reactive power set-points at points of common couplings where flexibilities are connected. The grid controller’s optimal power flow can introduce additional grid support functionalities to further increase grid stability. Flexibility controllers are recommended to actively track the obtained set-points from the grid controller, to ensure system-level optimization. Blockchain enables this control scheme by providing appropriate data exchange between the layers. This scheme is applied to a real low voltage rural grid in Austria, and the result analysis is presented.
