Unit Commitment Scheduling Research Articles

Mixed Thermal and Renewable Energy Generation Optimization in Non-Interconnected Regions via Boolean Mapping

Global efforts aiming to shift towards renewable energy and smart grid configurations require accurate unit commitment schedules to guarantee power balance and ensure feasible operation under different complex constraints. Intelligent systems utilizing hybrid and high-level techniques have arisen as promising solutions to provide optimum exploration–exploitation trade-offs at the expense of computational complexity. To ameliorate this requirement, which is extremely expensive in non-interconnected renewable systems, radically different approaches based on enhanced priority schemes and Boolean encoding/decoding have to take place. This compilation encompasses various mappings that convert multi-valued clausal forms into Boolean expressions with equivalent satisfiability. Avoiding any need to introduce prior parameter settings, the solution utilizes state-of-the-art advancements in the field of artificial intelligence models, namely Boolean mapping. It allows for the efficient identification of the optimal configuration of a non-convex system with binary and discontinuous dynamics in the fewest possible trials, providing impressive performance. In this way, Boolean mapping becomes capable of providing global optimum solutions to unit commitment utilizing fully tractable procedures without deteriorating the computational time. The results, considering a non-interconnected power system, show that the proposed model based on artificial intelligence presents advantageous performance in terms of generating cost and complexity. This is particularly important in isolated networks, where even a-not-so great deviation between production and consumption may reflect as a major disturbance in terms of frequency and voltage.

An integrated binary metaheuristic approach in dynamic unit commitment and economic emission dispatch for hybrid energy systems

The current generation portfolio is obligated to incorporate zero-emissions energy sources, predominantly wind and solar, due to the depletion of fossil fuels and the alarming rate of global warming. In the current scenario, power engineers must devise a compromised solution that not only advocates for the adoption of renewable energy sources (RES) but also efficiently schedules all conventional power generation units to balance the increasing load demand while simultaneously minimizing fuel costs and harmful emissions that are currently addressed by Unit Commitment (UC) and Combined Economic Emission Dispatch (CEED) problem solutions. However, the integration of renewable energy resources (RES) further complicates the UC-CEED problem due to their intermittent nature. Recently, metaheuristic algorithms are acquiring momentum in resolving constrained UC-CEED problems due to their improved global solution ability, adaptability, and derivative-free construction. In this research, a computationally efficient binary hybrid version of crow search algorithm and improvised grey wolf optimization is proposed, namely Crow Search Improved Binary Grey Wolf Optimization Algorithm (CS-BIGWO) by inclusion of nonlinear control parameter, weight-based position updating, and mutation approach. Statistical results on standard mathematical functions prove the supremacy of the proposed algorithm over conventional algorithms. Further, a novel optimization strategy is devised by integrating enhanced lambda iteration with the CS-BIGWO algorithm (CS-BIGWO-λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda$$\\end{document}) to solve a day-ahead UC-CEED problem of the hybrid energy system incorporating cost functions of RES. For the model, a day-ahead forecast of wind power and solar photovoltaic power is obtained by using the Levy-Flight Chaotic Whale Optimization Algorithm optimized Extreme Learning Machines(LCWOA-ELM). The proposed algorithm is tested for the UC-CEED solution of an IEEE-39 bus system with two distinct cases: (1) without RES integration and (2) with RES integration. Several independent trial runs are executed, and the performance of the algorithms is assessed based on optimal UC schedules, fuel cost, emission quantization, convergence curve, and computational time. For case 1, the proposed algorithm resulted in a percentage reduction of 0.1021% in fuel cost and 0.7995% in emission. In contrast, for test case 2, it resulted in a percentage reduction of 0.12896% in fuel cost and 0.772% in emission with the proposed algorithm. The results validate the dominance of the proposed methodology over existing methods in terms of lower fuel costs and emissions.

Week-ahead dispatching of active distribution networks using hybrid energy storage systems

This paper presents a week-long scheduling approach to address the issues associated with uncertain stochastic generation. Specifically, the method is designed for active distribution networks (ADNs) hosting hybrid energy storages, composed by a hydrogen energy storage system (HESS) and a battery energy storage system (BESS). The inclusion of a pressurized HESS allows to balance energy over longer time periods, as opposed to methods considering only BESSs. To this end, this paper combines linearized models for the electricity grid with linearized models of the HESS to solve a tractable scheduling problem. The proposed optimal schedule consists of an active power trajectory at the grid connection point (GCP), called the dispatch plan, and the unit commitment schedule of a PEM fuel cell and electrolyzer system interfacing the electricity network with the HESS. Additionally, a bilevel model predictive control strategy is proposed, where the upper layer MPC computes a storage target accounting for the full horizon, while the lower layer computes the controllable resource setpoints to minimize the dispatch tracking error in each period. A numerical experiment shows the effectiveness of the proposed scheduling and control to accurately compute and track a dispatch plan over a full week. The results clearly show the benefits of combining a HESS with a BESS especially in periods where the prosumption is highly uncertain. Finally, we discuss the computational challenges associated with the weekly horizon and the use of a HESS that exhibits different dynamics than a BESS and propose an approach to mitigate the computational cost.

Random Furrowing for a Stochastic Unit Commitment Solution

The Unit Commitment Problem involves the inherent difficulty of obtaining optimal combinatorial power generation schedules over a future short term period. The formulation of the generalized Unit Commitment Schedule formulation involves the specific combination of generation units at several de-rated capacities during each hour of the planning horizon, the load demand profile, load indeterminateness and several other operating constraints. This largely deterministic schedule continues to find favor with several plant operators, keeping in mind the close operating time-periods involved. However, the deterministic nature of the load profile is sought to be phased out by a stochastic pattern that is realistic and mirrors real-life situations, owing to modern trends in Demand side management. This shift is in tune with the ongoing power restructuring activities of electricity power reforms. The stochastic profile is obtained by a suitably tuned 2-parameter Weibull distribution that uses appropriate shape and scale parameters. The resulting band of generated load profiles are used to evaluate net power and penal costs associated with a set of pervasive randomized probability indices. The exact UCS comprises of a specific unit absolute state corresponding to a certain time period within the planning horizon. Subsequently, regression analysis is applied to establish the correlation between the absolute states and the cumulative randomized load demand against the intervals within the planning horizon. This method is analogous to random furrowing of probabilistic demand profile.

Reinforcement learning and mixed-integer programming for power plant scheduling in low carbon systems: Comparison and hybridisation

Decarbonisation is driving dramatic growth in renewable power generation. This increases uncertainty in the load to be served by power plants and makes their efficient scheduling, known as the unit commitment (UC) problem, more difficult. UC is solved in practice by mixed-integer programming (MIP) methods; however, there is growing interest in emerging data-driven methods including reinforcement learning (RL). In this paper, we extensively test two MIP (deterministic and stochastic) and two RL (model-free and with lookahead) scheduling methods over a large set of test days and problem sizes, for the first time comparing the state-of-the-art of these two approaches on a level playing field. We find that deterministic and stochastic MIP consistently produce lower-cost UC schedules than RL, exhibiting better reliability and scalability with problem size. Average operating costs of RL are more than 2 times larger than stochastic MIP for a 50-generator test case, while the cost is 13 times larger in the worst instance. However, the key strength of RL is the ability to produce solutions practically instantly, irrespective of problem size. We leverage this advantage to produce various initial solutions for warm starting concurrent stochastic MIP solves. By producing several near-optimal solutions simultaneously and then evaluating them using Monte Carlo methods, the differences between the true cost function and the discrete approximation required to formulate the MIP are exploited. The resulting hybrid technique outperforms both the RL and MIP methods individually, reducing total operating costs by 0.3% on average.

Optimal Allocation of Curtailment Level of PV Power Output in Different Regions in Consideration of the Reduction in Aggregated PV Power

Due to the extremely high penetration of photovoltaic power generation system (PV), the power output curtailment (Cr) of PV is necessary not only to maintain supply-demand balance but also to preserve enough capacity for the frequency control. Although the Cr level for the aggregated PV in the power system service area is determined in a day-ahead unit commitment (UC) scheduling, the Cr for individual PV with different weather conditions should be allocated so that the fluctuation of aggregated PV power output is minimized. Assuming that the short-term forecasting of individual PV power output fluctuations characteristics is available, the objective of this study is to propose methods for optimal short-term allocation of Cr level among each region. The proposed methods have proven their effectiveness in meeting the predetermined average PV power ouput as well as reducing the fluctuations of the overall system.

Deep Reinforcement Learning Based Unit Commitment Scheduling under Load and Wind Power Uncertainty

The intermittent nature of renewable energy sources and fluctuating electricity demand induce significant uncertainty that needs to be tackled with computationally efficient solution techniques to provide reliable and cost-effective generation schedules of power systems. In this work, we present a deep reinforcement learning (DRL) based approach for the day-ahead scheduling of generation resources under demand and wind power uncertainties. The proposed approach yields a causal policy relying only on historical uncertainty realizations and forecast data that is trained with an actor-critic-based reinforcement learning algorithm. Through safe exploration, the DRL-based approach guarantees a feasible commitment schedule without any operational constraint violations. We conduct computational experiments on the IEEE 39-bus and 118-bus test cases to demonstrate the effectiveness of the proposed solution strategy and improvement over existing approaches, including a deterministic approach with point forecasts and the stochastic dual dynamic integer programming method. The results show that the proposed approach enjoys superior performance in terms of computational efficiency and incurred operational costs, with significant reduction in penalty costs caused by insufficient net load supply than the deterministic approach.

Real‐time supply‐demand schedule update and operation for generators and battery energy storage system based on forecasted and actual photovoltaic power outputs

AbstractThe photovoltaic (PV) power output might be frequently curtailed to maintain electricity supply‐demand balance in future power systems. In our previous study, we proposed a new method for updating the battery energy storage system (BESS) charge/discharge and the generator unit commitment (UC) schedules based on the forecasted and actual PV power outputs. The forecast dataset was updated every 3 h (eight times a day). Although the simulation results showed that the proposed method could reduce the supply‐demand imbalances, it was not clear whether the forecasted or actual values made contributions. Therefore, in this study, we propose and evaluate a real‐time scheduling and operation method using the forecasted and actual PV power outputs assuming that the forecasted dataset is updated only once a day. Numerical simulations of supply‐demand operations are conducted on the power system model of the Kanto area of Japan for one year. The results show that the previous study method has a slight advantage over proposed method in terms of curtailed PV energy and operational cost of thermal generators reduction, but the difference is very small, indicating that the contribution of the actual PV power outputs is greater than that of the forecasted PV power outputs.

Day-ahead Network-constrained Unit Commitment Considering Distributional Robustness and Intraday Discreteness: A Sparse Solution Approach

Quick-start generation units are critical devices and flexible resources to ensure a high penetration level of renewable energy in power systems. By considering the wind uncertainty and both binary and continuous decisions of quick-start generation units within the intraday dispatch, we develop a Wasserstein-metric-based distributionally robust optimization model for the day-ahead network-constrained unit commitment (NCUC) problem with mixed-integer recourse. We propose two feasible frameworks for solving the optimization problem. One approximates the continuous support of random wind power with a finite number of events, and the other leverages the extremal distributions instead. Both solution frameworks rely on the classic nested column-and-constraint generation (C&CG) method. It is shown that due to the sparsity of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{1}$</tex> -norm Wasserstein metric, the continuous support of wind power generation could be represented by a discrete one with a small number of events, and the rendered extremal distributions are sparse as well. With this reduction, the distributionally robust NCUC model with complicated mixed-integer recourse problems can be efficiently handled by both solution frameworks. Numerical studies are carried out, demonstrating that the model considering quick-start generation units ensures unit commitment (UC) schedules to be more robust and cost-effective, and the distributionally robust optimization method captures the wind uncertainty well in terms of out-of-sample tests.

Modeling the AC power flow equations with optimally compact neural networks: Application to unit commitment

Nonlinear power flow constraints render a variety of power system optimization problems computationally intractable. Emerging research shows, however, that the nonlinear AC power flow equations can be successfully modeled using neural networks. These neural networks can be exactly transformed into mixed integer linear programs and embedded inside challenging optimization problems, thus replacing nonlinearities that are intractable for many applications with tractable piecewise linear approximations. Such approaches, though, suffer from an explosion of the number of binary variables needed to represent the neural network. Accordingly, this paper develops a technique for training an “optimally compact” neural network, i.e., one that can represent the power flow equations with a sufficiently high degree of accuracy while still maintaining a tractable number of binary variables. We demonstrate the use of this neural network as an approximator of the nonlinear power flow equations by embedding it in the AC unit commitment problem, transforming the problem from a mixed integer nonlinear program into a more manageable mixed integer linear program. We use the 14-, 57-, and 89-bus networks as test cases and compare the AC-feasibility of commitment decisions resulting from the neural network, DC, and linearized power flow approximations. Our results show that the neural network model outperforms both the DC and linearized power flow approximations when embedded in the unit commitment problem. The neural network formulation most often selects a feasible unit commitment schedule, and furthermore, it only selects an infeasible schedule if both the linear and DC methods are infeasible as well. • Neural networks can be used to approximate the nonlinear AC power flow equations. • Neural networks can be transformed into MILPs and embedded in optimization problems. • We embed a compact neural network inside the unit commitment problem. • The neural network model outperforms DC and linearized power flow approximations.

IGDT-Based Realistic Scheduling of Thermal Power Generators Under Integration of Wind Turbine Generators

When Unit Commitment (UC) is considered with the integration of renewable energy, it becomes a big issue to be emphasized because of its uncertain natural impacts. The awareness on the lack of sufficient and accurate data about uncertainty and approaching towards the realistic deployment of thermal power generators is a challenge for researchers and planners in optimization works. This work proposes a framework for UC scheduling of thermal power generators using renewable energy sources with a high degree of penetration that closely matches real-time operation despite the lack of accurate data. Applying the Information Gap Decision Theory (IGDT) to deal with uncertainty, power-based UC model in which generation trajectories act as piecewise linear models was extended to consider the ramping capability and energy output during starting up and shutting down of generating units. Simulations were performed on a standard IEEE 10 unit system. Comparing with the proposed framework, the conventional energy-based UC method cannot obtain wind uncertainty values (allowable horizon of WTG forecast error) in realistic operation as much as the expected results from its scheduling and, it can also approach more serious infeasibility conditions in the realistic operation than its expected results. The load shedding and energy curtailment amounts that occur with respective wind uncertainty values can exist as maximum or minimum levels when unit scheduling is conducted with these wind uncertainty values.

Evolutionary Priority-Based Dynamic Programming for the Adaptive Integration of Intermittent Distributed Energy Resources in Low-Inertia Power Systems

The variability and uncertainty caused by the increased penetrations of renewable energy sources must be properly considered in day-ahead unit commitment, optimal power flow, and even real-time economic dispatch problems. Besides achieving minimum cost, modern generation schedules must satisfy a larger set of different complex constraints. These account for the generation constraints in the presence of renewable generation, network constraints affected by the distributed energy resources, bilateral contracts enclosing independent capacity provision, ancillary power auctions, net-metering and feed-in-tariff prosumers, and corrective security actions in sudden load variations or outage circumstances. In this work, a new method is presented to appropriately enhance the integration of distributed energy resources in low-inertia power grids. Based on optimal unit commitment schedules derived from priority-based dynamic programming, the potential of increasing the renewable capacity was examined, performing simulations for different scenarios. To ameliorate the expensive requirement of computational complexity, this approach aimed at eliminating the increased exploration-exploitation efforts. On the contrary, its promising solution relies on the evolutionary commitment of the next optimum configuration based on priority-list schemes to accommodate the intermittent generation progressively. This is achieved via the collection of mappings that transform many-valued clausal forms into satisfiability equivalent Boolean expressions.

A hybrid robust-stochastic approach for unit commitment scheduling in integrated thermal electrical systems considering high penetration of solar power

With increasing environmental concerns, limited oil reserves and governmental incentives, installation of photovoltaic (PV) systems have gained substantial momentum in smart electric distribution systems (EDS), which is considered to be a great success. That said, widespread integration of these PV units, comes with uncertainty issues that puts the flexibility of the system at stake. In this regard, local district heating systems (DHS) are considered to be an effective tool in enhancing the flexibility of the power system on account of their high thermal inertia. In this study, we propose hybrid robust (RO)- stochastic programming (SP) optimization framework for commitment scheduling of distributed generation units and local DHS considering high penetration of PV units. Furthermore, the impact of flexible thermal and electrical loads on total operational cost and overall flexibility of the system is evaluated. The SP is utilized to deal obscurity of solar radiation, and RO is utilized to handle price uncertainties. The uncertainty management via the electrical storage system is also scrutinized. The results are evaluated by different cases studies implemented on an integrated IEEE 33-bus EDS and an 8-node DHS. Eventually, the mixed integer linear problem is solved with a standard CPLEX solver.

Integrating learning and explicit model predictive control for unit commitment in microgrids

In this paper, we apply flexibility-based operational planning method to microgrid (MG) unit commitment (UC). The problem is formulated based on model predictive control (MPC) paradigm. Conventionally, such paradigm requires the online solution of a mixed-integer optimization problem, which faces difficulties in practical field implementations in a low-power computing microgrid controller. The explicit model predictive control (EMPC) can address this problem of MPC by computing the control laws of MPC in an explicit form offline to enable fast online computation. However, the complexity of the explicit control laws usually grows exponentially with the dimension of the problem, which hinders the application of EMPC in larger systems. In this paper, we integrate learning and EMPC to develop a computationally efficient and rigorous approach for implementing flexibility-based unit commitment paradigms in a microgrid controller with limited computational power. The computational complexity of the proposed approach can be adjusted to meet the hardware limitation of any given microgrid controller, while preserving as much as possible the optimality of the full-fidelity EMPC. This overcomes the drawbacks of traditional EMPC. Moreover, compared to the existing learning-based methods for accelerating optimization algorithms, the proposed approach is able to handle the variables and constraints of the original unit commitment problem systematically, which guarantees the feasibility of its output unit commitment schedules. We conduct case studies to demonstrate the effectiveness of the proposed approach.

Investigating the market effects of increased RES penetration with BSAM: A wholesale electricity market simulator

Currently electricity markets worldwide encounter a transition phase into cleaner energy. This is specially the case in Greece, where not only the market structure changes to a harmonized EU target model, but also ‘green electricity’ is projected to reach 50% share by 2030 according to the National Energy and Climate Plan of Greece. This paper presents the extension of the Business Strategy Assessment Model (BSAM) into a computationally efficient central-dispatch wholesale market model. BSAM integrates agent-based profit-maximizing policy making with unit commitment scheduling, into one novel tool that can be used to explore the effects of mid-term market design policies in wholesale electricity markets. BSAM is subsequently utilized to model the Greek market’s transitional phase and explore the effects of even higher RES penetration scenarios. Results indicate that the planned short-to-mid-term “de-lignification” and transition of the Greek power system is feasible and will not lead to an excessive RES curtailment situation, but would yield significant CO2 emissions’ reductions and small reductions in the system marginal price, with even more ambitious RES shares also being possible.

Gaussian process-based Bayesian optimization for data-driven unit commitment

Global efforts aiming to shift towards de-carbonization give rise to remarkable challenges for power systems and their operators. Modern power systems need to deal with the uncertain and volatile behavior of their components (especially, renewable energy generation); this necessitates the use of increased operating reserves. To ameliorate this expensive requirement, intelligent systems for determining appropriate unit commitment schedules have risen as a promising solution. This is especially the case for weak power systems with low dispatching flexibility and high dependency on imported fossil fuels. In this work, we introduce a radically new paradigm for addressing the optimal unit commitment problem, that is capable of accounting for the largely unaddressed challenge of the uncertain and volatile behavior of modern power systems. Our solution leverages widely adopted developments in the field of uncertainty-aware machine learning models, namely Bayesian optimization. This framework enables the effective discovery of the best possible configuration of a volatile system with uncertain and unknown dynamics, without the need of introducing restrictive prior assumptions. Based on appropriately selected acquisition function and Gaussian process regression, it constitutes a radically different from existing approaches, which heavily rely on heuristic approximations and do not allow to account for volatile behavioral patterns. On the contrary, it guarantees global optimum solutions in non-convex optimization tasks in the least possible number of trials. The demonstrated results show better performance in terms of total production cost and number of function evaluations, inspiring system operators to better schedule their power networks in the forthcoming, de-carbonized grids.

Network-constrained unit commitment with piecewise linear AC power flow constraints

This paper presents a new network-constrained unit commitment (UC) model, which is formulated as a mixed-integer linear programming (MILP) optimization problem with embedded ac power flow (AC-PF) constraints. Piecewise linear AC-PF equations are formulated, which constrain the UC dispatch schedule within permissible transmission capacity and voltage deviation limits, solely by executing the main optimization problem. The proposed model is tested on a six-bus and the IEEE RTS-79 test systems, using GAMS software, whereas the obtained AC-PF results are validated against DIgSILENT/Powerfactory package. The UC schedules are comparatively assessed with detailed results presented in the relevant literature for the same test systems, employing different UC+AC-PF models.

Stochastic day-ahead unit commitment scheduling of integrated electricity and gas networks with hydrogen energy storage (HES), plug-in electric vehicles (PEVs) and renewable energies

This study investigates stochastic day-ahead unit commitment scheduling of integrated natural gas and power systems that consist of a non-gas-fired unit (NGFU), gas-fired unit (GFU), combined heat and power (CHP) unit, and renewable production units considering AC power flow. Natural gas storage is incorporated into the problem to support the gas network during peak natural gas demand hours by reducing pipeline congestion. Moreover, integrating hydrogen energy storage (HES) along with new flexible technologies, such as power to gas (P2G) system and demand response program (DRP) is proven to reduce total operation cost by facilitating renewable energy dispatch and shifting peak load demand to off-peak hours. A sensitivity analysis is carried out to evaluate the impact of thermal load increment on the operation of the CHP unit as its thermal and electrical output are interrelated variables. A residential plug-in electric vehicle (PEV) charging station is integrated into the problem to observe its impact on the operation of the networks, as PEVs introduce a new unplanned and uncertain load to the system. Furthermore, the sensitivity of the problem for PEV penetration is investigated. The uncertain parameters of the problem are electrical load demand, PEVs’ load demand, and wind power. Eventually, the proposed mixed integer non-linear problem (MINLP) is applied to several case studies involving an integrated 6-bus power system and 6-node gas network.

Power System Resilience Enhancement in Typhoons Using a Three-Stage Day-Ahead Unit Commitment

We propose a three-stage resilient unit commitment model which considers uncertain typhoon paths and line outages to improve the power system resilience against typhoon events. The proposed solution coordinates resources in response to the worst-case scenario for each possible typhoon path. The optimal decision is based on the characterization of the power system schedule into three stages of preventive control, emergency control, and restoration. Preventive control is performed before the typhoon occurs by quickly adjusting the three-stage resilient unit commitment schedule; emergency control is conducted during the typhoon by shedding local loads to meet the power balance, while other control strategies are assumed to be unavailable due to possible interruptions in the communication system; restoration is realized after the typhoon, when resources are optimally dispatched to repair the outages of critical devices and recover the normal operation state of the power system quickly. Considering the typhoon path uncertainty, we have introduced a stochastic model for possible typhoon paths where all possible affected lines along each typhoon path are assumed to be on outage during the typhoon. Accordingly, we explore the strategy for co-optimizing the three stages in unit commitment. The proposed model is tested on the IEEE 118-bus system and the real-world provincial system to verify its effectiveness.

Increasing the Photovoltaic Hosting Capacity in Autonomous Grids and Microgrids via Enhanced Priority-List Schemes and Storage

European Union has seen a rapid increase in renewable energy sources during the last decade. The variability and uncertainty caused by the increased penetrations of renewable generation must be properly considered in day-ahead unit commitment to retain the stable operation of conventional power plants. In this work, we present an enhanced method to determine the hosting capacity of photovoltaic energy in an autonomous grid. Based on optimal unit commitment schedules derived from priority list-schemes, we examine the potential of increasing the hosting capacity performing annual simulations for different scenarios in the presence of electricity storage. According to the obtained results, the application of storage eliminates the reliability expenses of load shedding and spinning reserve deficits. Hence, the actual hosting capacity is appropriately retrieved based on the renewable generation curtailment during each case study. However, sustainable solutions are achieved at higher penetration levels, reaching a near of 20% with respect to photovoltaic systems. The proposed solution could be efficiently utilized to determine the photovoltaic hosting capacity of microgrids in islanded or interconnected mode.