With the growing level of uncertainties in today's power systems, the vulnerability analysis of a power system with uncertain parameters becomes a must. This paper proposes a two-stage adaptive robust optimization (ARO) model for the vulnerability analysis of power systems. The main goal is to immunize the solutions against all possible realizations of the modeled uncertainty. In doing so, the uncertainties are defined by some predetermined intervals defined around the expected values of uncertain parameters. In our model, there are a set of first-stage decisions made before the uncertainty is revealed (attacker decision) and a set of second-stage decisions made after the realization of uncertainties (defender decision). This setup is formulated as a mixedinteger trilevel nonlinear program (MITNLP). Then, we recast the proposed trilevel program to a single-level mixed-integer linear program (MILP), applying the strong duality theorem (SDT) and appropriate linearization approaches. The efficient off-the-shelf solvers can guarantee the global optimum of our final MILP model. We also prove a lemma which makes our model much easier to solve. The results carried out on the IEEE RTS and modified Iran's power system show the performance of our model to assess the power system vulnerability under uncertainty.
This paper examines the effects of reactive power dispatch, losses, and voltage profile on the results of the interdiction model to analyze the vulnerability of the power system. First, an attacker-defender Stackelberg game is introduced. The introduced game is modeled as a bilevel optimization problem where the attacker is modeled in the upper level and the defender is modeled in the lower level. The AC optimal power flow (ACOPF) is proposed as the defender's tool in the lower-level problem to mitigate the attack consequences. Our proposed ACOPF-based mathematical framework is inherently a mixed-integer bilevel nonlinear program (MIBNLP) that is NP-hard and computationally challenging. This paper linearizes and then transforms it into a one-level mixed-integer linear program (MILP) using the duality theory and some proposed linearization techniques. The proposed MILP model can be solved to the global optimum using state-of-the-art solvers such as Cplex. Numerical results on two IEEE systems and Iran's 400-kV transmission network demonstrate the performance of the proposed MILP for vulnerability assessment. We have also compared our MILP model with the DCOPF-based approach proposed in the relevant literature. The comparative results show that the reported damage measured in terms of load shedding for the DCOPF-based approach is always lower than or equal to that for the ACOPF-based approach and these models report a different set of critical lines, especially in more stressed and larger power systems. Also, the effectiveness and feasibility of the proposed MILP model for power-system vulnerability analysis are discussed and highlighted.
This paper deals with general bilinear feasibility problems. A nonlinear transformation is introduced that reformulates a general bilinear feasibility problem as a Linear Matrix Inequality (LMI) problem augmented with a single non-convex quadratic constraint. The single non-convex quadratic constraint has a regular concave constraint function. Due to the LMI part of this formulation, it is easier to analyze, and we prove that the solution space of this formulation is located inside several ellipsoids and outside a sphere. This leads to our proposed Inside-Ellipsoid and Outside-Sphere (IEOS) model for general bilinear feasibility problems. Then, the feasibility analysis of our proposed IEOS model is performed. The related necessary feasibility conditions and sufficient feasibility conditions are theoretically developed. Moreover, an iterative algorithm for solving our IEOS model is also proposed.Two applications including matrix-factorization problem in control systems and power-flow prob-lem in power systems are considered to evaluate the practicality of our proposed approach. Both problems are formulated as IEOS models. It is shown that our proposed model can provide more accurate solutions to these problems as compared to previous competing approaches in the relevant literature.
In this paper, we propose a new mathematical model for power flow problem based on the linear and nonlinear matrix inequality theory. We start with rectangular model of power flow (PF) problem and then reformulate it as a Bilinear Matrix Inequality (BMI) model. A Theorem is proved which is able to convert this BMI model to a Linear Matrix Inequality (LMI) model along with One Nonconvex Quadratic Constraint (ONQC). Our proposed LMI-ONQC model for PF problem has only one single nonconvex quadratic constraint irrespective of the network size, while in the rectangular and BMI models the number of nonconvex constraints grows as the network size grows. This interesting property leads to reduced complexity level in our LMI-ONQC model which in turn makes it easier to solve for finding a PF solution. The non-conservativeness, iterative LMI solvability, well-defined and easy-to-understand geometry and pathwise connectivity of feasibility region are other important properties of proposed LMI-ONQC model which are discussed in this paper. An illustrative two-bus example is carefully studied to show different properties of our LMI-ONQC model. We have also tested our LMI-ONQC model on 30 different power-system cases including four ill-conditioned systems and compared it with a group of existing approaches. The numerical results show the promising performance of our LMI-ONQC model and its solution algorithm to find a PF solution.
This paper presents a methodology for frequency regulation in a microgrid involving renewable energy sources (RES) using a dynamic controller, which is an output feedback controller (OFC). The parameters of OFC are tuned by searching the design space of the controller. Since the RES model is not exactly known, the uncertain model is derived and the OFC is considered for it. The goal of controller tuning is to find appropriate parameters of the controller such that the norm of frequency deviations, even in presence of uncertainties in the RES parameters is minimized. An algorithm based on searching the controller design space is suggested to find the suitable controller gains. The algorithm assumes the controller parameters lie in a convex space and searches the space systematically such that an appropriate solution is found. The method is proved mathematically and two theorems are mentioned, accordingly. Finally, a simulated model of a RES is utilized for algorithm evaluation and the results demonstrate the algorithm capability in optimal frequency regulation.
With the rise of distributed energy resources and the increasing activation of flexibility resources by Distribution Systems Operators (DSOs), the Transmission System Operators (TSOs) need to co-ordinate their actions with those of the DSOs. This research uses a look-ahead multi-interval (LA-MI) framework for analyzing this coordination and explores two formulations. Firstly in the exogenous DSO model, a mixed-integer linear program is developed to reflect the pragmatic approach in many real situations whereby the TSO can only anticipate statistically the actions of the DSO. In the embedded DSO model, as a comparator, we propose a new organizational setup for the TSO-DSO operational coordination mechanism. In the resulting bilevel decomposition, a new method to calculate Benders cuts is developed and tested on a modified IEEE 118-bus test system as a transmission network and two modified IEEE 33-bus test systems as distribution networks. The benefits of the LA-MI coordination framework are substantial in comparison with the current Look-Ahead Single-Interval (LA-SI) coordination framework widely used in Europe.
The use of the ground as the current return path often presents planning and operational challenges in power distribution networks. This study presents optimization-based models for the optimal selection of conductor sizes in Single Wire Earth Return (SWER) power distribution networks. By using mixed integer non-linear programming (MINLP), models are developed for both branch-wise and primary-lateral feeder selections from a discrete set of overhead conductor sizes. The models are based on a mathematical formulation of the SWER line, where the objective function is to minimize fixed and variable costs subject to constraints specific to SWER power flow. Load growth over different time periods is considered. The practical application is tested using a case study extracted from an existing SWER distribution line in Namibia. The results were consistent for different network operating scenarios.
Power flow in earth return distribution systems typically depends on geographical location and specific earth properties. The planning of such systems has to take into account different operational and safety constraints from conventional distribution systems. This work presents the mathematical modeling and planning of Single Wire Earth Return (SWER) power distribution networks. The SWER load flow is modeled and formulated as an optimization problem. Then by using a heuristic iterative procedure, a planning algorithm is developed for the SWER system. The developed procedure includes optimal feeder routing and overhead conductor selection for both primary and lateral feeders with load growth over several time periods. A 30 node test network extracted from a rural area in Uganda is used to test the algorithm's practical application to give reasonable and consistent results. The model presented can be used in planning SWER networks for areas which have previously not been electrified as well as determining suitable upgrades for existing SWER distribution feeders. The algorithm's mathematical modeling and simulations were done using the General Algebraic Modeling System (GAMS).
The planning of distribution networks with earth return is highly dependent on the ground's electrical properties. This study incorporates a load flow algorithm for Single Wire Earth Return (SWER) networks into the planning of such systems. The earth's variable conductive properties are modelled into the load flow algorithm and the model considers load growth over different time periods. It includes optimal conductor selection for the SWER system and can also be used to forecast when an initially selected conductor will need to be upgraded. The planning procedure is based on indices derived through an iterative heuristic process that aims to minimise losses and investment costs subject to load flow constraints. A case study in Uganda is used to test the model's practical application.
The HVDC systems built based on the voltage source converters (VSC) can bring several benefits to the AC power systems. Better voltage profile, increasing power flow controllability, lower ohmic loss, and higher transfer capability are some major benefits of such systems. This paper investigate the impact of VSC-type DC grids installed in the AC power systems on the ohmic network losses. This is done by formulating a convex optimization problem which minimises the ohmic losses (both in AC and DC grids) subject to the technical constraints of both AC and DC system. The formulated optimisation problem is a conic optimisation problem which can be solved using the commercially available optimisation softwares. The conic AC-DC optimal power flow, CAD-OPF, is coded in GAMS platform and solved using the MOSEK solver. The IEEE 30-bus example system is modeled and studied.
Recent research shows that non-convex AC OPF problem can be recast as a convex Semidefinite (SD) problem or Second Order Cone Programming (SOCP) problem. This paper presents a stochastic SOCP OPF (SSOCP-OPF) model for power systems connected to the wind farms. This is performed by reformulating the original non-convex OPF problem using more practical parameters of the power system. Finally, we obtain a convex optimization problem through some wellknown approximations and an exact relaxation incorporating the stochastic nature of wind power. One of the advantage of convex SOCP problems, which are a general form of linear problems, is that they can be efficiently solved through Interior PointMethods (IPMs). The proposed OPF model takes advantage of both DC-OPF models (solution efficiency) and full AC-OPF models (solution accuracy). As an application of SSOCP-OPF model, we study the impact of wind power uncertainty on the transmission loss in the power systems. To evaluate the proposed stochastic model modified IEEE 30-bus test system is used. The optimization problem is coded in GAMS platform and solved using its embedded interior point optimizer MOSEK.
Recent research shows that non-convex OPF problem can be recast as a convex Semidefinite Programming (SDP) problem or Second Order Cone Programming (SOCP) problem. However, in the most SOCP OPF problems, there are some cases that conic relaxation results in a miscalculation of negative Local Marginal Prices (LMPs). This paper reviews the SOCP formulation of the optimal power flow problem proposed in [1] and then proposes one way of generating negative Locational Marginal Prices, LMPs, using this SOCP formulation. The proposed model is coded in GAMS and its built MOSEK solver and tested on a modified version of IEEE-30 test system.
This paper presents a second order cone programming (SOCP) formulation of the optimal power flow problem for AC-DC systems with voltage source converter (VSC) technology. Approximation techniques have been used to derive the SOCP formulation of the AC-DC OPF problem. Later, the SOCP formulation can be solved using the interior point method (IPM) by considering the limits on AC-DC grid. The accuracy of SOCP formulation of AC OPF has been proven with numerical examples using IEEE 14-bus, IEEE 30-bus, and IEEE 57-bus example systems. The results of the SOCP formulation are compared with available commercial software. Then a DC system with VSC technology is modeled in the IEEE 30-bus example system. The SOCP formulation of AC-DC OPF is applied to the modified IEEE 30-bus example system and the results are discussed. The limitations of derived SOCP formulation are also discussed.
This letter presents a second-order cone formulation for AC power flow problem. The power flow equations are first derived as functions of more practical variables of power systems and then placed in a second-order cone programming (SOCP) problem. The proposed conic power flow (CPF) model can be solved efficiently through IPMs, and at the same time, it has a very good accuracy as compared to the full AC power flow model. Also, the proposed CPF can efficiently handle the ill-conditioned networks. The numerical efficiency and good accuracy of the model are shown by simulating various case studies.
The multi-terminal HVDC systems and their embedded DC networks are considered as smart grids technology which improve economic efficiency of the power system. This technology allows better voltage profile in the power system by better allocation of the generation sources. Also, it can help in improving the economic efficiency of the system by substituting the high-cost generation with low-cost generation. In order to assess the technical benefit of this smart grids technology, this paper presents an optimal power flow formulation for AC grids with embedded DC networks built from multi-terminal HVDC systems. The objective function of this AC-DC OPF formulation is the total active dispatch costs. The constraints consist of (a) AC grid constraints, (b) constraints from multi-terminal HVDC systems, and (c) DC grid constraints. The formulated AC-DC OPF is a mixed-integer nonlinear optimisation problem. The formulation is coded in GAMS platform and tested on IEEE 30 Bus system.
In a liberalised wholesale electricity market risk-averse market participants need some form of financial instrument to offset the risks of spot price variation across locations and across time. Some liberalised wholesale electricity markets seek to facilitate transactions across separately-priced nodes by making available an instrument known as a Financial Transmission Right (or FTR). But FTRs are flawed as a hedging instrument. They do not necessarily make available the full set of funds required to allow market participants to hedge locational price differences. Furthermore, conventional FTRs, which are associated with a volume which is fixed in advance, are not useful for hedging transactions where the volume depends on market conditions at the time. This paper proposes introducing a new form of transmission right which mimics the operation of a 'cap' hedge contract. This transmission right can be combined into a portfolio which provides the natural backing for the price-dependent volume-varying hedge that most market participants require. We consider that this new design of transmission rights offers promise as an approach for facilitating hedging and improving market outcomes in wholesale electricity markets.
The incentive on an electricity generating firm to exercise market power depends strongly on the volume the firm has pre-sold in the forward or hedge markets. Therefore, in order to forecast the effect of mergers and other market developments on market power outcomes, it is essential to model the hedging decisions of dominant generating firms. This paper shows that a dominant firm's profit-maximizing choice of hedge level depends on the extent to which the hedge price varies with the firm's hedging decision. In the case in which the hedge price is independent of the firm's hedge level, the optimal choice of hedging is an "all or nothing" decision. In this case, there is no equilibrium level of hedging in pure strategies. This outcome may explain the observed lack of hedge market liquidity in wholesale electricity markets with substantial market power. We also model the equilibrium hedging outcome in a two-stage Cournot oligopoly and show that, even if the hedge price is independent of the hedging decisions of the firms, a rational expectations equilibrium can exist with high levels of hedging if there are enough firms in the market.
Around the world, the electricity industry is in the process of undergoing a fundamental transition. Twenty years ago, electricity was primarily generated at large, industrial-scale generating plants, and transported in one direction to consumers via the transmission and distribution networks. The large generators were typically closely integrated into the operation of the transmission and distribution networks. Electricity consumers, on the other hand, were treated as essentially passive. This paradigm has changed and will change further. Around the world, a number of regions have chosen to introduce competition and competitive markets into the generation of electricity. In most of these regions, the operation of generation and transmission is coordinated through market mechanisms. This required a substantial change in the way the electricity industry is organised and operated.
An energy community is defined as a group of individually metered electricity customers with generation, storage, and load assets, who may virtually exchange electricity within the group. This definition captures virtual net metering, virtual storage, and peer-to-peer trading. We show that such an energy community may both reduce the retail payments, through the arbitrage of retail price differences across customers, and improve the efficiency of dispatch of generation and load within the community. The creation of an energy community exacerbates the impact of inefficient retail tariffs, and prevents pricing of local network congestion. Tariff reform is a prerequisite for energy communities.
The incentive on an electricity generatingfirm to exercise market power depends strongly on thevolume the firm has pre-sold in the forward or hedge markets.Therefore, in order to forecast the effect of mergersand other market developments on market power outcomes,it is essential to model the hedging decisions ofdominant generating firms. This paper shows that a dominantfirm’s profit-maximizing choice of hedge level dependson the extent to which the hedge price varies withthe firm’s hedging decision. In the case in which the hedgeprice is independent of the firm’s hedge level, the optimalchoice of hedging is an “all or nothing” decision. In thiscase, there is no equilibrium level of hedging in purestrategies. This outcome may explain the observed lack ofhedge market liquidity in wholesale electricity marketswith substantial market power. We also model the equilibriumhedging outcome in a two-stage Cournot oligopolyand show that, even if the hedge price is independent of thehedging decisions of the firms, a rational expectationsequilibrium can exist with high levels of hedging if thereare enough firms in the market.
In a typical liberalized wholesale electricity market, the output of controllable units is determined at regular intervals through a dispatch process. However, since the physical limits of power systems must be respected down to time scales shorter than the dispatch interval, in practice system operators must also be able adjust the output of controllable units over very short time frames - typically through the procurement and dispatch of reserves, ancillary services or balancing services. To date, the procurement and dispatch of balancing services has been guided by heuristics and rules of thumb. Yet the approach to the dispatch of balancing services can have a significant impact on the pre-contingent or system normal dispatch of the power system. In principle, improvements in the efficiency of the dispatch of balancing services could significantly improve the efficiency of the utilization of power system assets. This paper observes that the dispatch of balancing services should correspond to optimal dispatch in a dispatch process with a very short dispatch interval. We also identify a set of conditions which the procurement and dispatch of balancing services should satisfy and compare those conditions to the current arrangements for ancillary services in the Australian National Electricity Market.
The literature on optimal dispatch of wholesale power systems implicitly assumes that market participants are risk-neutral. But, in practice, most wholesale electricity market participants behave as though they are risk averse, seeking to insulate themselves from the market risks they face. In this context, achieving the overall social-welfare maximum requires simultaneously finding both the optimal dispatch and optimal hedging arrangements. Assuming that market participants have mean-variance preferences, we show that the dispatch task can be separated from the hedging task. We show how market participants can achieve a perfect hedge by forming a portfolio of inter-temporal hedge contracts. Departing from the previous literature, we assume the system operator is risk averse. We show how the system operator can achieve a perfect hedge using a portfolio of inter-nodal hedging instruments which we refer to as generalised Financial Transmission Rights. The total risk experienced by market participants when optimally hedged is equal to the variation in the total surplus or total economic welfare. This approach therefore leads naturally to a form of merchant transmission investment where network upgrade decisions are carried out by a coalition of risk-bearers in the market. In addition, we propose a natural extension in which transmission network operators provide a form of insurance against network outages, and face the correct social incentive for avoiding network outages. This approach resolves a number of outstanding issues in the economic analysis of power markets.
Many market-based power systems have implemented a form of 'look-ahead dispatch' which simultaneously solves for the optimal dispatch and prices over several intervals into the future. A few papers have pointed out that the dispatch outcomes which emerge from look-ahead dispatch may not be time consistent. We emphasise that this time inconsistency is not inherent in look-ahead dispatch but is a consequence of the assumption of linear cost and utility functions, which is arguably a special case. Various augmentations to the dispatch process to resolve the time inconsistency problem have been proposed, but these augmentations suffer from the drawback that they do not allow the power system to efficiently adjust to new information. We query whether it is necessary to implement multi-interval real-time markets. We show how under certain assumptions, a sequence of oneshot dispatch processes will achieve the efficient outcome.
For many years, researchers have hoped to find a mechanism for improving incentives for operation, and investment in network assets, based on the value of financial transmission rights. However, attempts to do so using conventional fixed-volume financial transmission rights have failed. In this chapter, we introduce a new concept of financial transmission rights, referred to as generalized FTRs. We demonstrate that, when generators, loads, and the system operator are risk averse they can perfectly hedge by trading in a portfolio of conventional hedge contracts, and, in the case of the system operator, the generalized FTRs. A risk-neutral market participant, referred to here as the trader, takes on the remaining risk in the market and collectively has a payoff equal to the total economic welfare created in the market. The trader therefore has an incentive to augment the network if and only if it is socially beneficial to do so. We illustrate how this gives risk to efficient merchant investment decisions using simple network examples.
With increasing scarcity in water resources around the world, policy-makers are increasingly looking to water trading arrangements to ensure that available surface water is used as efficiently as possible. However, traded water must be able to be transported from its source to its point of use. In many real-world applications this transportation occurs over a gravity-fed river network. The physical characteristics of this network (such as the shape of weirs) determines how quickly water flows from point to point, as well as setting upper and lower limits on the rate of flow. The choice of the path of injections and extractions which maximises overall welfare must take into account these physical characteristics. This paper characterises the profile of water injections and extractions, and the corresponding path of prices, that maximises overall economic welfare, subject to the hydrology of a stylised river network. We illustrate the outcomes of the model in simple water networks, and show how even relatively simple changes in supply and demand conditions can lead to dynamic variation in the profile of prices and injection/extraction across the river system. We show how this model can act as the foundation for a smart market for water trading arrangements.
Meeting peak demand is a major concern for grid operators, sometimes amplified by the liberalization of the electricity industry. Peak resources operate during a short period, at a high operating cost, and face difficulties to recover their investments. Sending the proper market signals to trigger long term generation investments in these types of generating units is a critical issue. Different incentivizing approaches have been adopted worldwide to cope with this issue, from a simple administratively-determined capacity payment, to complex but field-proven market designs complying with grid constraints. In France, the so-called law "NOME" (New Organization of the Electricity Market) represents a new step in establishing a fair, transparent and efficient energy market. It also enforces market participants and system operator to design a mechanism for long-term capacity procurement. This paper will introduce the motivation for developing capacity markets and describe an advanced mechanism that has been derived from the Alstom Grid clearing engine currently in operation in North America. An adaptation of this centralized market place to the situation in Europe will be discussed and simulated with realistic dataset representing the market structure of one particular European country.
In this paper, we will consider the problems of Revenue Adequacy (RA) and Hedging to Risk (H2R), faced by the Independent System Operators (ISOs) and holders of Financial Transmission Rights (FTRs) (or, Congestion Revenue Rights, or CRRs as they are variously known), respectively. It is well known, that the main driver for these two problems is the difference in the topology of the network that is used while solving the FTR auction and allocation process, to that used in the Day Ahead (DA) or Real Time (RT) market dispatch calculations. As we will see in this paper, that the problems of RA and H2R form a set of conflicting requirements, especially when situations corresponding to changing network topologies are considered. We will, therefore, in the present work, propose a newer type of FTR such that both the above-mentioned problems are averted. We will also present the revised auction mechanism of this new FTR, in order to incentivize both the ISOs and the potential holders to sell and purchase them, respectively.
The integration of wind power (WP) into power systems has led to concerns on the adequacy of primary frequency control. Such concerns have currently arisen in interconnections with high penetration of WP and may appear in other systems where high penetration of WP is planned. This paper proposes a method for predicting the level of WP penetration that may lead to primary frequency control inadequacy, considering a empirically-validated model of primary frequency control, worst-case scenario conditions and no frequency response from WP machines. The NORDEL interconnection is considered as a case study.
The optimal controlled islanding of power systems is a practical solution to prevent system blackouts if the boundary of islands and corrective control actions in each island are carefully specified. This paper establishes a model for the controlled islanding problem by a proposed mixed integer linear program (MILP). The frequency-arresting (FA) and frequency-stabilising (FS) constraints are linearised and incorporated in our FA-FS-constrained MILP model to prevent triggering of load shedding (LS) relays. This achievement makes our model capable of handling low-inertia networks. Intentional LS and stepwise generation curtailment are corrective actions accommodated for frequency control and power mismatch considerations. These corrective actions increase the degree of freedom and broaden the feasible space of the MILP model under the envisaged tough operating conditions. Our model's computational efficiency is improved using a proposed graph theory-based network reduction technique. The basic groups of coherent generators, determined in the offline mode, are aggregated as equivalent buses using an extended Steiner tree method. A graph-path determination technique is also proposed to generate disconnection constraints (between equivalent buses of incoherent areas) and bus-allocation constraints. Simulation results on the IEEE 39-bus test system and a 76-bus case study verify the proposed network reduction technique's effectiveness and the MILP model.
Frequency-arresting (FA) adequacy has received an increasing attention in operation of power systems with intermittent, electronically connected wind and sfolar generation. Presence of high penetration of renewable energy leads to reduced amount of inertia in the power system and the need for more sustainable market design. This paper addresses the FA adequacy by integrating Nadir and rate of change of frequency constraints in unit commitment formulation. Here, a mixed-integer linear program (MILP) is proposed for addressing the FA adequacy. The proposed MILP model of FA adequacy is non-convex and accordingly, FA adequacy pricing requires non-convex pricing techniques. The explicit and implicit uplift payment techniques are proposed and implemented for FA adequacy pricing using the authors' proposed MILP model. The primal–dual formulation is employed for implicit uplift technique. The proposed MILP model of FA adequacy along with its pricing techniques is carefully studied through an illustrative example system. For further discussions, the Nordic 44-node test network is studied.
This paper presents a Hidden Markov Model (HMM) based method to predict the prices and trading volumes in the electricity balancing markets. The HMM are quite powerful in modelling stochastic processes where the underlying dynamics are not apparent. The proposed method provides both one hour and 12-36 hour ahead forecasts. The first is mostly useful to wind/solar producers in order to compensate their production imbalances while the second is important when submitting the offers to the day ahead markets. The results are compared to the ones from Markov-autoregressive model.
In this paper, a method for Reactive Power Planning (RPP) is proposed which ensures voltage stability in a system with cross-border electricity flows. One or more cross-border tie-lines connect the Source-area, containing cheap generation, to the Sink-area which is willing to import this cheap power. A mathematical method is used to maximize the economic benefit of the Sink-area with respect to the cost of the installed reactive compensators which maximize the Net Transfer Capacity (NTC) of the tie-lines. An analytic expression for the NTC does not exist and two methods are henceforth proposed to approximate the NTC: (1) approximation by a piecewise linear function and (2) by a polynomial obtained with statistical regression. The method is programmed in GAMS and formulated as a Mixed-Integer Non-Linear Programming problem (MINLP).
Increasing the Net Transfer Capacity (NTC) of tie-lines between different grid areas may lead to a decrease in the cost to generate electricity in one area if this cost is more expensive than the cost of importing power from neighbouring areas over those tie-lines. An effective means to increase the cross-border transmission capacity is by installing reactive power compensation devices. In this paper, a multi-objective optimization is devised that optimally locates and sizes reactive compensation devices in a grid, so that the net benefit of increasing the NTC value of a tie-line is maximized and the maximum voltage stability indicator (L-index) in the grid area is minimized, indicating there is enough voltage stability margin. The genetic algorithm NSGA-II is implemented to perform the optimization. The IEEE 30-bus and 14-bus test grids are used as two interconnected areas.
In this paper, the voltage stability of the Swedish grid is studied by a twofold approach. Firstly, the loads within Sweden are increased and the effect on the voltage of the tie-line buses is examined. Secondly, the exchanging of power between the two neighboring countries, Denmark and Finland, is taken into account. The main objective of this paper is to show that different reactive power compensation scenarios within the Swedish network and also in the vicinity of the tie-lines will improve the voltage stability of the whole system and also will increase the maximum power transfer on the tie-lines. With the method of this paper, we will be able to suggest the best places where the reactive power compensation can improve the voltage stability. Finally, an optimization problem for installing reactive power compensation from economic point of view is formulated. The CIGRE Nordic 32-A model has been used as test system and all the simulations have been done in PSS/E and Matpower.
In order to have a reliable and secure power system, frequency stability should be guaranteed. Primary Frequency Response (PFR) is used to arrest the frequency deviation upon a contingency. The significant penetration of wind generation in power grids has negative effect on the PFR, as the contribution of wind machines to the total system inertia is low and classical wind turbines do not have PFR capabilities. This study uses two metrics of frequency control, ROCOF and frequency Nadir to analyze the cost increase for maintaining adequacy in the presence of wind power. The Nordic 32-A test grid, representing the Swedish grid is used as case study on which the method is applied.
Problem definition: With the rise of renewables and the decline of fossil fuels, electricity markets are shifting toward a capacity mix in which low-cost generators (LCGs) are dominant. Within this transition, policymakers have been considering whether current market designs are still fundamentally fit for purpose. This research analyses a key aspect: the design of real-time imbalance pricing mechanisms. Currently, markets mostly use either single pricing or dual pricing as their imbalance pricing mechanisms. Single-pricing mechanisms apply identical prices for buying and selling, whereas dual-pricing mechanisms use different prices. The recent harmonization initiative in Europe sets single pricing as the default and dual pricing as the exception. This leaves open the question of when dual pricing is advantageous. We compare the economic efficiency of two dual-pricing mechanisms in current practice with that of a single-pricing design and identify conditions under which dual pricing can be beneficial. We also prove the existence of an optimal pricing mechanism. Methodology/results: We first analytically compare the economic efficiency of single-pricing and dual-pricing mechanisms. Furthermore, we formulate an optimal pricing mechanism that can deter the potential exercise of market power by LCGs. Our analytical results characterize the conditions under which a dual pricing is advantageous over a single pricing. We further compare the economic efficiency of these mechanisms with respect to our proposed optimal mechanism through simulations. We show that the proposed pricing mechanism would be the most efficient in comparison with others and discuss its practicability. Managerial implications: Our analytical comparison reveals market conditions under which each pricing mechanism is a better fit and whether there is a need for a redesign. In particular, our results suggest that existing pricing mechanisms are adequate at low/moderate market shares of LCGs but not for the high levels currently envisaged by policymakers in the transition to decarbonization, where the optimal pricing mechanism will become more attractive.
Growing population and increasing air pollution in different countries around the world have incentivized the need to expand the renewable power plant capacities. Small-scale renewable power plants, especially solar and wind power plants, are usually not able to compete in the wholesale electricity markets partly due to their small size. This problem has been resolved by the concept of the Virtual Power Plants (VPP), where a collection of small-scale power plants form a VPP and participates in the wholesale electricity markets. In this paper, the VPP capacity investment problem is formulated and solved. A multi-objective function is considered for VPP investment problem. To test, the formulated optimization problem, three different investment scenarios are considered and optimization problems are solved centrally. The results confirm the utility of our developed optimization model.
The scheduling of virtual power plants (VPPs) has received much attention in the last few years. VPP refers to the integration of several power plant units together, which is considered as a single power plant. In this paper, three VPPs are considered, each of which includes different power plant units and must supply the desired load. Besides supplying the desired load, they should maximize their profit. Decentralized optimization method has been used to optimize these three VPPs. The reason for using decentralized method is to increase network security and also to not need a central computer. On the other hand, using the decentralized optimization method increases the speed of problem solving. Finally, the obtained results have been compared with the centralized method. The simulations show that almost the same result has been obtained by using different optimization methods. These results increase the tendency to use decentralized methods in VPPs.
The energy system in the Nordic countries faces changes driven by increasing integration with the rest of Europe and changes to the generation mix. These developments pose challenges with respect to future network development and operation. We focus on three major aspects: market integration; generation and network adequacy; the need for more flexibility and frequency control. We describe factors behind these problems and present possible solutions within the Nordic context. One conclusion is that supranational cooperation should be further improved.
This paper presents a simultaneous active and reactive power/voltage dispatch methodology using a mixture of continuous and discrete control variables. The continuous control variables are active power and reactive power/voltage magnitudes of generator buses, while the discrete ones are transformer-tap settings. The Lambda iteration based economic dispatch is used to minimize the fuel cost and active power dispatch among the generators. During this process, the voltage magnitude of load buses may move away from the desired value. The genetic algorithm is applied for optimal reactive power/voltage dispatch among generators and determination of tap settings of under load tap changers (ULTCs). This results in globally improvement of voltage regulation of load buses. Considering the fact that the reactive power/voltage of generators are continuous variables while the transformer tap settings are discrete variables, a method is proposed to combine these variables into one suitable chromosome. The IEEE 30-bus test system is used to illustrate the effectiveness and feasibility of proposed approach.
Karyotypes of three taxa (seven populations) of the genus Thymus from different geographic origins is presented. The secondary basic numbers in all populations was x= 15 that probably originate from a basic number x=7.The ploidy levels were different among populations belongs to T. daenensis and T. kotschyanus species (2x and 4x). Detailed karyotype analysis allows us to group the different populations and to postulate relationships among them.
Power systems are experiencing a decrease of synchronous generation along with increased penetration of inverter based renewable generation leading to reduced system inertia and a need for flexible resources. Non-generating resources such as thermostatically controlled loads (TCLs) are flexible due to their thermal energy storage capacity. When aggregated, TCLs can arbitrage energy prices and provide reserves to the power system. We approach the operational flexibility of the TCLs by modeling a risk-averse aggregator that controls decentralized TCLs and aims to maximize its own profit. The high number and low power rating of residential TCLs makes it difficult to model and assess their flexibility potential on national level. Thus, we make use of a high-level thermal energy storage model for aggregations of TCLs to quantify their flexibility potential. We present a method to aggregate temperature, TCL parameters, and building stock data into a thermal battery equivalent. We propose a multi-period multi-market multi-zonal two-stage chance constrained rolling horizon optimization problem formulation for the risk-averse day-ahead self-scheduling problem of a price-taker TCL aggregator bidding in energy and reserve markets under uncertainty and recast the problem as a linear program. We perform several case studies in the Swedish power system based on a survey of single -and two-family dwellings with electric heating and assess the flexibility potential. Additionally, a sensitivity analysis provides insights regarding market design and policy implications.
This paper defines and discusses three types of benefits for economic analysis of transmission investment policies. These benefits are termed (1) The Efficiency Benefit, (2) The Static Competition Benefit, and (3) The Dynamic Competition Benefit. To provide more insights to these benefits, an example system is studied. The results show that a transmission network service provider can and should use the transmission investment policies to improve the competition (static and dynamic) in the supply electricity industry. Further research is necessary to model the static and dynamic competition benefit in the process of transmission expansion planning.
A new steady state modeling of unified power flow controller (UPFC) is proposed in this paper. Using this model, factors that affect the objective function of electricity market as a result of UPFC installation in power grid has been decomposed into four components, including line series impedance increase, shunt reactive power compensation, inphase component of series voltage and quadrature component of series voltage. A UPFC has been placed in different points of a test system and impact of each component on objective function of electricity market has been measured by simulation and compared with results from analytical method. Both active and reactive power spot prices are calculated and their relation with settings of UPFC series part has been studied. Also, numerical results shows that the necessary cost to improve security of electricity market decreases by UPFC installation.
This paper derives a mathematical structure for investment decisions of a profit-maximising and strategic producer in liberalised electricity markets. The paper assumes a Cournot producer in an energy market with nodal pricing regime. The Cournot producer is assumed to have revenue from selling energy to the pool. The investment problem of the strategic producer is modelled through a leader-follower game in applied mathematics. The leader is the strategic producer seeking the optimal mix of its investment technologies and the follower is a stochastic estimator. The stochastic estimator forecasts the reactions of other producers in the market in response to the investment decisions of the producer in question. The stochastic estimator takes the investment decisions of the producer and it calculates the stochastic prices. The mathematical structure is a stochastic linear bilevel programming problem. This problem is reformulated as a stochastic MILP problem which can be solved using the commercially available software packages. Finally, the developed mathematical structure is applied to a six-node example system to highlight the strengths of the whole approach.
Market power analysis is one of the major issues facing regulators of wholesale electricity markets. The exercise of market power both distorts wholesale price signals and reduces the efficiency of the operation of and investment in the wholesale electricity market. This paper deals with a systematic way for quantifying and visualising market power. The paper first proposes three indicators termed the System Market Power Indicator, SMPI, the Producer Market Power Indicator, PMPI, and the Nodal Market Power Indicator, NMPI. The game theory in applied mathematics and the concept of social welfare in microeconomics are used in formulating of these indicators. The SMPI finds the total cost of exercising market power by generating companies. The contribution of a specific generating company in system market power is calculated using the PMPI. The NMPI finds the contribution of each power system node in the total market power cost. Then after, a colour contour map is used to visualise the exercise of market power and its associated cost. The proposed market power indicators are applied to the modified Garver’s example system to show the promising performance of these indicators.
This paper develops a mathematical tool for modelling market power cost in transmission expansion planning decisions. The mathematical modelling is based on the game theory in applied mathematics and the concept of social welfare in microeconomics. We assume the generating companies as Cournot players and the Transmission System Operators as a regulated social transmission planner. To tackle the multiple Nash equilibria problem, the concept of worst-Nash equilibrium is defined and mathematically formulated. The developed mathematical structure is a mixed-integer linear programming problem. This closed form mathematical structure can be solved efficiently using the available computational packages.
In this paper, we derive and evaluate a new mathematical structure for market-based augmentation of the transmission system. The closed-form mathematical structure can capture both the efficiency benefit and competition benefit of the transmission capacity. The Nash solution concept is employed to model the price-quantity game among GenCos. The multiple Nash equilibria of the game are located through a characterisation of the problem in terms of minima of the function. The worst Nash equilibrium is used in the mechanism of transmission augmentation. The worst Nash equilibrium is defined as the one which maximises the social cost, total generation cost + total value of lost load. Thorough analysis of a simple three-node network is presented to clearly highlight the mechanism of the derived mathematical structure from different perspectives.