Intertemporal Constraints Research Articles

An adaptive demand response framework using price elasticity model in distribution networks

Price elasticity model (PEM) is an appealing and modest model for assessing the potential of flexible demand in demand response (DR). It measures the customer’s demand sensitivity through elasticity in relation to price variation. However, application of PEM is partially apprehensible on attributing the adaptability and adjustability along with intertemporal constraints in DR. Thus, this article presents an adaptive economic DR framework with its attributes via a dynamic elasticity approach to model customer’s demand sensitivity. This dynamic elasticity is modeled through the deterministic and stochastic approaches. Both approaches envision the notion of load recovery for shiftable/flexible loads to make the proposed framework adaptive and adjustable relative to price variation. In stochastic approach, a geometric Brownian motion is employed to emulate load recovery in addition to intertemporal constraint of load flexibility. The proposed mathematical model shows what should be the customers elasticity value to achieve the factual DR. The numerical study is carried out on standard IEEE 33 distribution system bus load data to assess its technical and socio-economic impact on customers and is also compared with the existing model.

Enhancing the Flexibility of Storage Integrated Power System by Multi-Stage Robust Dispatch

The ever-increasing integration of variable wind energy requests for a power system with high flexibility. In this paper, we formulate the real-time economic dispatch problem as a multi-stage robust program to leverage flexible resources, especially energy storage systems, in a broader timescale, and also to avoid the risk of high costs in critical scenarios. The multi-stage robust program is decomposed to the dynamic programming form and the fast robust dual dynamic programming (FRDDP) algorithm is developed to solve it efficiently. To tackle the nonconvexities caused by the inter-temporal constraints of ESSs, an estimation-correction method is developed, and measures of the approximation accuracy are presented. Comprehensive case studies comprising units up to 1371 are carried out to verify the effectiveness of the proposed model. Numerical results show that compared with the existing ones, the proposed multi-stage robust model achieves lower cost under the worst-case scenario and shows better scalability in large-scale systems.

Missing incentives for flexibility in wholesale electricity markets

In most liberalized electricity markets, flaws in short-term price formation have led to a “missing money” problem wherein average energy prices are too low to support an efficient level of capacity. Recent growth of wind and solar generation has exposed another category of flaws related to intertemporal constraints, such that energy prices are not volatile enough to support an efficient level of flexibility. In theory, electricity markets convey the value of flexibility through time-varying prices, encouraging market participants to invest in resources able to match production and consumption profiles to grid needs. Real-world market clearing procedures, however, make several simplifications that suppress volatility relative to the theoretical ideal. This paper describes five mechanisms by which current wholesale electricity market price formation may fail to provide full-strength incentives for flexible resources. Instead of restoring price volatility, market designers may choose to define flexibility products that act as a proxy for full-strength prices. However, these products cannot replicate theoretically ideal incentives precisely, with potentially important consequences for small-scale and distributed resources. The analysis could help guide ongoing efforts to ensure that systems have the capability to respond to growing variability and uncertainty in an efficient way.

Reserve constrained dynamic economic dispatch in multi-area power systems: An improved fireworks algorithm

The multi-area economic dispatch (MAED) plays a major role in an interconnected power system to determine the generation dispatch strategy and satisfy the load demands as well as the technical constraints. Tie-line capacity between different areas, upper/lower generation limits, generator ramp rates, transmission losses, valve point effect, disjoint prohibited zones as well as spinning reserve are considered as the MAED problem constraints. Due to hourly load variations, the MAED problem should be solved dynamically to coincide the realistic conditions which build up the multi-area dynamic economic dispatch (MADED) problem. Considering these constraints besides the intertemporal constraints which depends to the optimal conditions of each hour with respect to other time steps may increase the problem complexity. In this regard, an improved version of fireworks algorithm which is equipped with two effective cross-generation mutation strategies is deployed to handle the non-linear, non-smooth, non-convex, multi-dimensional and multi-constraints MADED problem. Furthermore, a new constraint handling scheme is developed to correct the potential solutions in feasible search space. Finally, two small and large systems are used to investigate the capability of the developed algorithm in finding the global or near-global optima.

An Efficient Hybrid Approach to Solve Bi-objective Multi-area Dynamic Economic Emission Dispatch Problem

Single period economic dispatch cannot handle the intertemporal constraints in multi-period environment. To cope with this issue, the extension of economic dispatch over multiple time intervals (i.e., dynamic economic dispatch) has been introduced that considers the intertemporal constraints between different time intervals. Another issue is determining the most economical generation dispatch that could supply the area demand without violating the tie-line capacity, which cannot be solved by conventional economic dispatch problems. However, this study shows that the most economic schedule of power generation cannot satisfy echo-system expectation; therefore, making a compromise between fuel cost and environmental issues, a hot-button subject in industrialized nations, seems to be crucial. To reach the goals a bi-objective multi-area dynamic economic dispatch approach, which can handle intertemporal and multi-area constraints concurrently, is proposed to assist power system operators more and more. Finally, a hybrid algorithm, namely gray wolf optimizer-particle swarm optimization is introduced to solve the proposed problem and also a set of benchmark problems. By implementing the proposed approach on two small (10-unit, three areas) and large (40-unit, four areas) scale test systems, about 3.1% and 3.3% improvement in generation cost is obtained, respectively compare to the best reported results in the literature.

Missing Incentives for Flexibility in Wholesale Electricity Markets

In most liberalized electricity markets, flaws in short-term price formation have led to a missing money problem wherein average energy prices are too low to support an efficient level of capacity. Recent growth of wind and solar generation has exposed another category of flaws related to intertemporal constraints, such that energy prices are not volatile enough to support an efficient level of flexibility. In theory, electricity markets convey the value of flexibility through time-varying prices, encouraging market participants to invest in resources able to match production and consumption profiles to grid needs. Real-world market clearing procedures, however, make several simplifications that suppress volatility relative to the theoretical ideal. This paper describes five mechanisms by which current wholesale electricity market price formation may fail to provide full-strength incentives for flexible resources. Instead of restoring price volatility, market designers may choose to define flexibility products that act as a proxy for full-strength prices. However, these products cannot replicate theoretically ideal incentives precisely, with potentially important consequences for small-scale and distributed resources. The analysis could help guide ongoing efforts to ensure that systems have the capability to respond to growing variability and uncertainty in an efficient way.

A parallel multi-period optimal scheduling algorithm in microgrids with energy storage systems using decomposed inter-temporal constraints

Because microgrids have relatively high share of renewable energy sources and energy storage systems (ESSs) compared with existing large-scale power systems, the inter-temporal constraints such as the generators’ ramp-rates and the state-of-charge of the ESSs have a much greater impact on system operation. Therefore, in this paper, the optimization of the microgrid operation, the commitment of generators and the charging/discharging of ESSs, is formulated as a mixed-integer nonlinear programming (MINLP) problem with inter-temporal constraints. In order to find the optimal solution to the problem effectively, we propose a parallel computation method based on the generalized Bender’s decomposition and the optimality condition decomposition. The method has the structure which is suitable for parallel computation and the convergence to the optimal solution is greatly improved compared with conventional sequential optimization methods. The proposed method is applied to the CIGRE medium-voltage microgrid benchmark system and the simulation results show that the proposed method has a potential for facilitating full-scale parallel computation ability and the application to the real-time operation of microgrid system.

A Multi-Period Market Design for Markets With Intertemporal Constraints

The participation of renewable, energy storage, and resources with limited fuel inventory in electricity markets has created the need for optimal scheduling and pricing across multiple market intervals for resources with intertemporal constraints. In this paper, a new multi-period market model is proposed to enhance the efficiency of markets with such type of resources. It is also the first market design that links a forward market and a spot market through the coordination of schedule and price under the multi-period paradigm, achieving reliability, economic efficiency and dispatch-following incentives simultaneously. The forward market solves a multi-period model with a long look-ahead time horizon whereas the spot market solves a series of multi-period dispatch and pricing problems with a shorter look-ahead time horizon on a rolling basis. By using the forward schedules and opportunity costs of intertemporal constraints as a guideline, the spot market model is able to produce economically efficient dispatch solutions as well as prices that incentivize dispatch following under the perfect forecast condition. The proposed scheme is applied to the dispatch and pricing of energy storage resources. Numerical experiments show that the proposed scheme outperforms the traditional myopic method in terms of economic efficiency, dispatch following and reliability.

Time decomposition strategy for security‐constrained economic dispatch

A horizontal time decomposition strategy to reduce the computation time of security-constrained economic dispatch (SCED) is presented in this study. The proposed decomposition strategy is fundamentally novel and is developed in this paper for the first time. The considered scheduling horizon is decomposed into multiple smaller sub-horizons. The concept of overlapping time intervals is introduced to model ramp constraints for the transition from one sub-horizon to another sub-horizon. A sub-horizon includes several internal intervals and one or two overlapping time intervals that interconnect consecutive sub-horizons. A local SCED is formulated for each sub-horizon with respect to internal and overlapping intervals’ variables/constraints. The overlapping intervals allow modelling intertemporal constraints between the consecutive sub-horizons in a distributed fashion. To coordinate the subproblems and find the optimal solution for the whole operation horizon distributedly, accelerated auxiliary problem principle is developed. Furthermore, the authors present an initialisation strategy to enhance the convergence performance of the coordination strategy. The proposed algorithm is applied to three large systems, and promising results are obtained.

Cross-product manipulation with intertemporal constraints: An equilibrium model

The use of uneconomic virtual transactions in day-ahead electricity markets with the intent to benefit related financial positions constitutes cross-product manipulation, and has emerged as a policy concern in recent years. Developing analytical frameworks and models to explain the means for achieving sustained day-ahead price manipulation is a challenge. This paper presents a two-stage equilibrium model of day-ahead price manipulation to enhance the value of financial transmission rights (FTRs). We cast the problem as a Stackelberg game between manipulating traders in the day-ahead market (leaders) and generating firms, grid operator and traders without FTRs in the day-ahead and real-time markets (followers). The model accounts for features specific to electricity systems, like intertemporal constraints of power generating units and real-time uncertainty, and considers imperfect competition as a condition allowing manipulation in equilibrium. We simulate hourly financial trading and operations decisions in a small test system for 24 hours. Results suggest that cross-product manipulation is sustained in equilibrium only when both physical and financial participants engage in Cournot competition. Further, as a result of loop flows, price separation between FTR source and sink may be induced by virtual transactions at network locations that are not on the FTR path.

Multi-Time Step Service Restoration for Advanced Distribution Systems and Microgrids

Modern power systems are facing increased risk of disasters that can cause extended outages. The presence of remote control switches, distributed generators (DGs), and energy storage systems (ESS) provides both challenges and opportunities for developing post-fault service restoration methodologies. Inter-temporal constraints of DGs, ESS, and loads under cold load pickup conditions impose extra complexity on problem formulation and solution. In this paper, a multi-time step service restoration methodology is proposed to optimally generate a sequence of control actions for controllable switches, ESSs, and dispatchable DGs to assist the system operator with decision making. The restoration sequence is determined to minimize the unserved customers by energizing the system step by step without violating operational constraints at each time step. The proposed methodology is formulated as a mixed-integer linear programming model and can adapt to various operation conditions. The proposed method is validated through several case studies that are performed on modified IEEE 13-node and IEEE 123-node test feeders.

Electricity pricing theory based on continuous time commodity model

The theory of spot pricing forms the basis of power market design in several countries. However, this theory has two major drawbacks. First, it is based on the traditional hourly scheduling/dispatch model, it ignores the critical influence of time continuity in the electric power production/consumption on cost and power system operation, and it fails to address intertemporal constraints. The second drawback is that it assumes that electric products are homogeneous in the same dispatch period and cannot distinguish among the base, intermediate, and peak load powers with different technical and economic characteristics. This study aims to present a continuous time commodity model of electricity, which comprise spot pricing and load duration pricing methods, to overcome the abovementioned shortcomings of the spot pricing theory. Market optimization models in traditional electricity market theories were then changed accordingly from a static optimization problem to a functional optimization problem. The feasibility of load duration pricing was validated by strict mathematical derivation. The proposed theory and methods will provide new concepts and theoretical foundation for the development of electric power markets globally and in China.

Assessing the impact of sampling and clustering techniques on offshore grid expansion planning

Due to the ongoing large-scale connection of non-dispatchable renewable energy sources to the power systems, short- to long-term planning models are challenged by an increasing level of variability and uncertainty. A key contribution of this article is to explore and assess the implications of different dimension reduction approaches for long-term Transmission Expansion Planning (TEP) models. For the purpose of this study, a selection of sampling and clustering techniques are introduced to compare the resulting sample errors with a variety of sampling sizes and two different scaling options of the original data set. Based on the generated samples, a range of TEP model runs are carried out to investigate their impacts on investment strategies and market operation in a case study reflecting offshore grid expansion in the North Sea region for a 2030 scenario. The evaluations show that dimension reduction techniques performing well in the sampling and clustering process do not necessarily produce reliable results in the large-scale TEP model. Future work should include ways of incorporating inter-temporal constraints to better capture medium-term dynamics and the operational flexibility in power system models.

Modeling marginal CO2 emissions in hydrothermal systems: Efficient carbon signals for renewables

Reducing local and global emissions is one of the drivers behind the promotion of renewable energy technologies (RETs). Some RET projects (e.g. solar, marine power, and some biomass and wind) usually require supplemental income from carbon credits in order to make them economically viable. This paper presents the mathematical development of nodal marginal CO2 emission metrics for hydrothermal systems. These metrics are obtained from optimal power flow and hydrothermal dispatch models considering hydro uncertainty. The proposed model allows a more accurate assessment of the emission reduction effect of RET projects and the provision of efficient carbon signals to the system. Modeling of hydro reservoirs and intertemporal constraints also allows incorporating the emission mitigation potential of water storage. The concepts of marginal CO2 emissions for hydrothermal systems are first exemplified using a small test system, and then simulations on a real hydrothermal system illustrate how these metrics can inform investment and operational decisions. Theoretical and simulation results show that transmission and hydro storage constraints neglect may provide wrong signals for RET investors, missing the incentives to install in areas where the potential for emissions reduction is higher (where renewable energy displaces a larger amount of coal or diesel). Policy makers can use the proposed model to assess the cost-effectiveness of emission reduction policies.

Resilience Enhancement With Sequentially Proactive Operation Strategies

Extreme weather events, many of which are climate change related, are occurring with increasing frequency and intensity and causing catastrophic outages, reminding the need to enhance the resilience of power systems. This paper proposes a proactive operation strategy to enhance system resilience during an unfolding extreme event. The uncertain sequential transition of system states driven by the evolution of extreme events is modeled as a Markov process. At each decision epoch, the system topology is used to construct a Markov state. Transition probabilities are evaluated according to failure rates caused by extreme events. For each state, a recursive value function, including a current cost and a future cost, is established with operation constraints and intertemporal constraints. An optimal strategy is established by optimizing the recursive model, which is transformed into a mixed integer linear programming by using the linear scalarization method, with the probability of each state as the weight of each objective. The IEEE 30-bus system, the IEEE 118-bus system, and a realistic provincial power grid are used to validate the proposed method. The results demonstrate that the proposed proactive operation strategies can reduce the loss of load due to the development of extreme events.

Aggregators’ Optimal Bidding Strategy in Sequential Day-Ahead and Intraday Electricity Spot Markets

This paper proposes a probabilistic optimization method that produces optimal bidding curves to be submitted by an aggregator to the day-ahead electricity market and the intraday market, considering the flexible demand of his customers (based in time dependent resources such as batteries and shiftable demand) and taking into account the possible imbalance costs as well as the uncertainty of forecasts (market prices, demand, and renewable energy sources (RES) generation). The optimization strategy aims to minimize the total cost of the traded energy over a whole day, taking into account the intertemporal constraints. The proposed formulation leads to the solution of different linear optimization problems, following the natural temporal sequence of electricity spot markets. Intertemporal constraints regarding time dependent resources are fulfilled through a scheduling process performed after the day-ahead market clearing. Each of the different problems is of moderate dimension and requires short computation times. The benefits of the proposed strategy are assessed comparing the payments done by an aggregator over a sample period of one year following different deterministic and probabilistic strategies. Results show that probabilistic strategy reports better benefits for aggregators participating in power markets.

Sensitivity‐based relaxation and decomposition method to dynamic reactive power optimisation considering DGs in active distribution networks

With the development of active distribution networks, new challenges such as overvoltage and power loss become critical. The reactive power optimisation serves as a voltage control measure to minimise the total transmission loss by coordinating the continuous and discrete reactive power compensators while guaranteeing the specific physical and operating constraints. To address the daily operating times of discrete control variables, the dynamic reactive power optimisation (DRPO) is set up to minimise total energy loss over several time periods when considering the inter-temporal constraints. However, DRPO is in fact a large-scale mixed integer non-linear non-convex programming that is difficult to solve. Therefore, second-order cones are employed to relax the non-convex power flow equations to obtain a mixed integer second order cone programming model. Furthermore, a sensitivity-based relaxation and decomposition method is proposed to further improve the computational performance. Solution quality and computational performance are compared with traditional methods on IEEE-33, 123 and 615-bus systems as well as two real-world distribution networks in China. The Results demonstrate that the fast performance and effectiveness of the proposed technique.

On the Solution of Revenue- and Network-Constrained Day-Ahead Market Clearing Under Marginal Pricing—Part I: An Exact Bilevel Programming Approach

The first of this two-paper series addresses a practical day-ahead auction model, where generation revenue constraints are explicitly incorporated in the problem formulation, as routinely done in several national electricity markets across Europe. The revenue-constrained market-clearing procedure includes the effect of the transmission network, inter-temporal constraints associated with generation scheduling, demand-side bidding, and marginal pricing. This auction design is an instance of price-based market clearing which features two major complicating factors. First, locational marginal prices become decision variables of the optimization process. In addition, producer revenues are formulated as bilinear and highly nonconvex products of power outputs and market-clearing prices. The resulting problem is formulated as a mixed-integer nonlinear bilevel program with bilinear terms for which available solution techniques rely on heuristics, approximations, or modeling simplifications. This paper presents a novel and exact methodology whereby the original problem is recast as an equivalent single-level mixed-integer linear program. As a consequence, finite convergence to optimality is guaranteed and the use of standard commercial software is allowed. The proposed transformation is based on duality theory of linear programming, Karush-Kuhn-Tucker optimality conditions, and integer algebra results. In the second part of this two-paper series, numerical results from several case studies illustrate the effective performance of the proposed solution approach.

Technology adoption under time-differentiated market-based instruments for pollution control

Peak concentrations of ground-level ozone pose health risks to millions of U.S. citizens across the U.S. In order to reduce peak ozone concentrations, nitrogen oxide (NOx) emissions from the power sector, among others, have been regulated with technology-based standards or, more commonly in recent years, market-based instruments such as cap-and-trade programs. However, the lack of temporal flexibility in current designs of these market-based instruments limits their cost-effectiveness on days forecasted to have the highest levels of pollution, including ozone precursors such as NOx, and as further emission reductions are sought, the marginal cost of these approaches increases dramatically. In this paper, we compare three regulatory schemes for reducing NOx emissions on high-ozone days: time-differentiated pricing, which prices NOx emissions only on days with high-ozone concentrations; undifferentiated pricing, which represents current NOx emission regulations; and technology-based standards. We develop a novel model that captures for the first time both the short- and long-term response of generators, through redispatching and control technology adoption, to a dynamic pricing scheme such as time-differentiated pricing. Unlike prior studies on time-differentiated pricing, the heart of our model, a unit commitment model, accounts for inter-temporal constraints on power generation that may be crucial to accurately capturing the response of generators to a transient price signal. We apply this model to the Texas power system and find that while control technology adoption (specifically selective catalytic reduction) does occur at very high time-differentiated prices, time-differentiated pricing mainly affects emissions and costs through redispatching of gas- for coal-fired generation. Furthermore, we show that time-differentiated pricing, due to its targeted pricing mechanism, provides a more cost-effective approach than undifferentiated pricing or technology-based standards for reducing NOx emissions on high-ozone days, but is not cost-effective at reducing summer-wide NOx emissions. Our results illustrate the trade-offs between these regulatory approaches and suggest that states should consider dynamic pricing schemes such as time-differentiated pricing for achieving further reductions in peak ozone concentrations.

Integrated Electrical and Gas Network Flexibility Assessment in Low-Carbon Multi-Energy Systems

In power systems with more and more variable renewable sources, gas generation is playing an increasingly prominent role in providing short-term flexibility to meet net-load requirements. The flexibility provided by the gas turbines in turn relies on the flexibility of the gas network. While there are several discussions on the ability of the gas network in providing this operational flexibility, this has not been clearly modeled or quantified. In addition, the gas network may also be responsible for supplying heating technologies, and low-carbon scenarios see a tighter interaction between the electricity, heating and gas sectors, which calls for a holistic multi-energy system assessment. On these premises, this paper presents an original methodology to quantify the flexibility the gas network can provide to the power system, as well as the constraints it may impose on it, with also consideration of different heating scenarios. This is achieved by a novel multi-stage integrated gas and electrical transmission network model, which uses electrical DC OPF and both steady-state and transient gas analyses. A novel metric that makes use of the concept of zonal linepack is also introduced to assess the integrated gas and electrical flexibility, which is then used to impose gas-related inter-network inter-temporal constraints on the electrical OPF. Case studies are performed for the Great Britain transmission system for different renewables and heating scenarios to demonstrate the proposed integrated flexibility assessment methodology, provide insights into the effects of changes to the heating sector on the multi-energy system's combined flexibility requirements and capability, and assess how the electrical network can experience local generation and reserve constraints related to the gas network's lack of flexibility.