Load Control Research Articles

Lyapunov-based estimation and control of velocity and load in rotating machines via Luenberger globally-convergent observer

This paper presents the design of a Lyapunov-based Luenberger-like observer to simultaneously estimate the non-available angular velocity and the unknown constant load torque of rotating machines. This proposal assumes that the only measurable output is the rotor angle, which is considered to be defined in the circle, its natural topology. This provides nonlinear dynamics where the Lyapunov-based Luenberger-like observer globally converges. Moreover, a trajectory tracking controller for the motor angular velocity is also designed by feeding back the observer estimations. Convergence analysis of the observer-controller system is performed using Lyapunov-like theory. The effectiveness and performance of the system are validated through tests performed on an experimental platform.

Control algorithm for dynamic solar shadings: A simulation study for office buildings based on ISO 52016-3

In 2019, the building sector was accountable for emitting 12GtCO2, equivalent to 21 % of global GHG emissions. To achieve carbon neutrality by 2050, the building sector must change its pace. One of the ways to achieve this goal is through the installation of dynamic solar shadings in buildings. This study focuses on a single office in two locations characterised by a temperate climate: Liège (Belgium) and Milan (Italy). Two control strategies are designed for Venetian and Roller blinds, one with and one without glare evaluation. They both integrate horizontal illuminance, room occupancy, indoor operative temperature and vertical irradiance according to a multi-criteria approach based on ISO 52016-3. The control strategy aims to balance visual comfort, heating, artificial lighting and cooling energy needs and considers user satisfaction by evaluating the shading activation time. The control algorithms are applied and validated on a DesignBuilder shoebox model. Regardless of location, the control strategy that includes glare control improves the user's visual comfort in terms of light quantity and discomfort glare. However, a total annual energy needs increase is registered independently of the shading. Conversely, if glare is not included in the control strategy, control of thermal loads is observed. This work contributed to developing ISO 52016-3 shading control scenarios for offices and is intended for shading producers, solar shading associations, façade engineers, facility managers and the scientific community working on solar shading simulation and analysis.

The system of accounting and control of dose loads on the lacrimal apparatus during radiotherapy of thyroid cancer

The system of accounting and control of dose loads on the lacrimal apparatus during radiotherapy of thyroid cancer

Empowered or enchained? Exploring consumer perspectives on Direct Load Control

Demand Response has become a central focus in policy discussions, attracting heightened attention for its potential key role in fostering enhanced demand flexibility. Direct Load Control (DLC) holds particular promise, requiring reduced consumer involvement compared to strategies relying on manual responses to time-varying pricing. However, the participation rate of residential consumers in DLC programs remains low, emphasizing the need for a more profound understanding of consumers' perspectives on DLC. Drawing from 15 in-depth interviews with Swedish households participating in a program involving direct load control of heat pumps, this study aims to explore the experiences and perceptions of consumers, providing in-depth understanding on factors that may serve as motivators, barriers and enablers for participation. Key findings include: financial benefits and interest in technology were the main motivations for participation; pre-existing knowledge and awareness of energy-related matters shaped consumers’ attitudes to DLC; trust in the service provider was a key enabling factor for participation. The study further suggests that DLC does not inevitably lead to a perceived loss of control among participants, but, if implemented in alignment with their specific conditions, needs and preferences, may also foster a sense of empowerment.

ПІДТРИМКА ТОЧНОСТІ ЗНАЧЕНЬ КЕРУЮЧИХ ПАРАМЕТРІВ ЕЛЕКТРОГІДРАВЛІЧНОГО НАВАНТАЖЕННЯ ПРИ ШТАМПОВЦІ З ПРОСТОРОВО-ЧАСОВИМ КЕРУВАННЯМ

The article is devoted to ensuring the stability of the technological process of electrohydraulic stamping of thin-sheet large-sized parts. This process is realised on multi-contour EH presses equipped with the capabilities of space-time control of the workpiece load. This control ensures the production of high-quality parts and high technical and economic performance of the process.The control parameters include the geometry of the press energy release zone, the loading sequence of the workpiece sections and the value of the inter-electrode distance. The first parameters change little under specified operating conditions. The third parameter is subject to change over a specified number of discharges due to cavitation and EDM erosion. Violation of the specified value of the interelectrode distance leads to a violation of the energy release mode and its loss.As a result of the experiments, the mechanism of cavitation destruction of the high-voltage insulation of the working electrode was determined and a solution was proposed for the conditions of winding the high-voltage insulation with a glass fibre roving, as well as electrical insulation in the areas adjacent to the discharge zone.Several variants of the design of the end part of the working electrode, which is still directly in the EG discharge zone, were experimentally investigated.The ratio of the duration of the pre-discharge stage to the characteristic time of the discharge circuit was chosen as a criterion parameter of resistance to electrical erosion of the discharge edges. The duration of the pre-discharge stage was determined from the discharge voltage oscillograms.According to the given dependences of the above ratio on the number of discharges, the best design from the considered options was determined.The used value of this ratio for the selected design corresponds to ~ 2500 discharges at a voltage of 20...25 kV. This number of discharges is on average comparable to the operation during five shifts, after which it is necessary to restore the discharge edges of the electrode. To simplify the operations of changing electrodes, we propose a design in which the electrodes are fixed by clamping them with elastic washers.It is concluded that the goal has been achieved: maintaining stable operation of the EG press with high technical and economic indicators of the entire technological process at a given quality of parts manufacturing.It is concluded that the goal has been achieved: maintaining stable operation of the EG press with high technical and economic indicators of the entire technological process at a given quality of parts manufacturing.Objective of the work - is to evaluate the effectiveness of the proposed technical solutions to increase the stability of the working electrodes and simplify their repair and restoration.

Research on load prediction of low-calorific fuel fired gas turbine based on data and knowledge hybrid model

The high-precision load prediction technology plays a vital role in load control and health management for gas turbines. Low-calorific fuel fired gas turbines pose an especially significant challenge for load prediction due to the frequent and wide fluctuations of gas parameters. First, an improved hybrid-dimension physical-based model is proposed, which is corrected based on segmented operation data and integrates zero-dimensional thermodynamic knowledge of gas turbines with three-dimensional fluid dynamics simulation knowledge. Secondly, an optimal data-driven model is proposed through conducting a systematic comparative study on four popular machine learning models with two types of input variables feature extraction. Then, a novel hybrid model is proposed based on the improved physical-based model, optimal data-driven model, and the principle of minimizing hybrid model errors. The average relative prediction error of the hybrid model over the complete operation range is 0.68%. Compared with the physical-based model and the data-driven model, the proposed hybrid model can improve the prediction accuracy by 63% and 29%, respectively. As the high-calorific fuel fired gas turbine system is relatively simple and has a small range of fuel fluctuations, the method established in this paper can be extended to high-calorific fuel fired gas turbines, which is of great significance for promoting the dynamic balance of loads in a new type of electric power system based on new energy.

Susceptibility of a Fe-Mn-Al-Ni-Cr shape memory alloy to environment-assisted corrosion cracking

In recent years, iron-based shape memory alloys (SMAs) have become increasingly interesting for application in civil engineering structures. Promising candidates in the field are iron-based Fe-Mn-Al-Ni-X (X = Ti, Cr) SMAs. These alloys have demonstrated the potential to be used in various types of applications. While the fully recoverable martensitic phase transformation can be used for structures where high damping capacities are required, i.e. for high rise buildings in earthquake prone areas, it has been successfully demonstrated that the material can also be used as prestressing element in concrete structures. However, for the application in civil engineering structures the problem of stress corrosion cracking and environment-assisted cracking has to be considered. The simultaneous occurrence of tensile stresses and unfavorable environmental conditions, such as carbonation of concrete or the penetration of salt, can lead to spontaneous failure of the building structure. In the present study, the susceptibility of Fe-Mn-Al-Ni-Cr to stress corrosion cracking was studied in neutral chloride containing solutions. For that purpose, tensile tests under constant load control at different stress levels have been conducted. The pitting corrosion that occurred during immersion in the electrolytic solution has a decisive influence on the martensitic phase transformation and the crack evolution. The high density of dislocations at the martensite-to-austenite interfaces promotes the formation of transgranular cracks as well as stress-induced corrosion cracks originating from corrosion pits. Assessment of the fracture surfaces revealed dominating intergranular crack growth, most likely induced by hydrogen embrittlement. The detrimental mechanisms lead to failure of the specimens well before the targeted test time of 1000 h.

Biomechanical analysis of the tandem spinal external fixation in a multiple-level noncontiguous lumbar fractures model: a finite element analysis.

This study aimed to investigate the biomechanical characteristics of the tandem spinal external fixation (TSEF) for treating multilevel noncontiguous spinal fracture (MNSF) using finite element analysis and provide a theoretical basis for clinical application. We constructed two models of L2 and L4 vertebral fractures that were fixed with the TSEF and the long-segment spinal inner fixation (LSIF). The range of motion (ROM), maximum stresses at L2 and L4 vertebrae, the screws and rods, and the intervertebral discs of the two models were recorded under load control. Subsequently, the required torque, the maximum stress at L2 and L4 vertebrae, the screws and rods, and the intervertebral discs were analyzed under displacement control. Under load control, the TSEF model reserved more ROM than the LSIF model. The maximum stresses of screws in the TSEF model were increased, while the maximum stresses of rods were reduced compared to the LSIF model. Moreover, the maximum stresses of L2 and L4 vertebrae and discs in the TSEF model were increased compared to the LSIF model. Under displacement control, the TSEF model required fewer moments (N·mm) than the LSIF model. Compared to the LSIF model, the maximum stresses of screws and rods in the TSEF model have decreased; the maximum stresses at L2 and L4 in the TSEF model were increased. In the flexion condition, the maximum stresses of discs in the TSEF model were less than the LSIF model, while the maximum stresses of discs in the TSEF model were higher in the extension condition. Compared to LSIF, the TSEF has a better stress distribution with higher overall mobility. Theoretically, it reduces the stress concentration of the connecting rods and the stress shielding of the fractured vertebral bodies.

Urban Stormwater Phosphorus Export Control: Comparing Traditional and Low-impact Development Best Management Practices.

Data from the International Stormwater Best Management Practices (BMP) Database were used to compare the phosphorus (P) control performance of six categories of stormwater BMPs representing traditional systems (stormwater pond, wetland basin, and detention basin) and low-impact development (LID) systems (bioretention cell, grass swale, and grass strip). Machine learning (ML) models were trained to predict the reduction or enrichment factors of surface runoff concentrations and loadings of total P (TP) and soluble reactive P (SRP) for the different categories of BMP systems. Relative to traditional BMPs, LIDs generally enriched TP and SRP concentrations in stormwater surface outflow and yielded poorer P runoff load control. The SRP concentration reduction and enrichment factors of LIDs also tended to be more sensitive to variations in climate and watershed characteristics. That is, LIDs were more likely to enrich surface runoff SRP concentrations in drier climates, when inflow SRP concentrations were low, and for watersheds exhibiting high impervious land cover. Overall, our results imply that stormwater BMPs do not universally attenuate urban P export and that preferentially implementing LIDs over traditional BMPs may increase TP and SRP export to receiving freshwater bodies, hence magnifying eutrophication risks.

Role of Al2O3 reinforcements in polymer-based nanocomposites for enhanced nanomechanical properties: Time-dependent modeling of creep and stress relaxation

Alumina (Al2O3) based nanocomposites mark a paradigm shift in materials science, offering unprecedented opportunities for multifunctional applications. This study meticulously examines the mechanical properties of polymer nanocomposites (PNCs), with a specific focus on the impact of varying sizes and concentrations of Al2O3 nanoparticles. By investigating these parameters, the research enables the tailoring of mechanical properties to meet specific requirements. The study reveals remarkable enhancements in the reduced modulus (Er), reaching up to 350%, 175%, and 100% under load and displacement control modes for 25 nm, 80 nm, and 200 nm average particle sizes, respectively. Beyond mechanical strength, the research delves into the nanocomposites resistance to long-term deformation (creep) and their ability to maintain consistent performance under load (stress relaxation). Additionally, the tribological performance lays the groundwork for the development of high-performance materials. The insights gained from this study represent more than an incremental improvement; they signify a transformative leap forward in addressing the multifaceted challenges of advanced material technology, with the potential to revolutionize various industries.

A lightweight method of integrated local load forecasting and control of edge computing in active distribution networks

The strong resource constraints of edge-computing devices and the dynamic evolution of load characteristics put forward higher requirements for forecasting methods of active distribution networks. This paper proposes a lightweight adaptive ensemble learning method for local load forecasting and predictive control of active distribution networks based on edge computing in resource constrained scenarios. First, the adaptive sparse integration method is proposed to reduce the model scale. Then, the auto-encoder is introduced to downscale the model variables to further reduce computation time and storage overhead. An adaptive correction method is proposed to maintain the adaptability. Finally, a multi-timescale predictive control method for the edge side is established, which realizes the collaboration of local load forecasting and control. All cases can be deployed on an actual edge-computing device. Compared to other benchmark methods and the existing researches, the proposed method can minimize the model complexity without reducing the forecasting accuracy.

Development of an integrated energy management system for off-grid solar applications with advanced solar forecasting, time-of-use tariffs, and direct load control

Effectively managing and maximizing the integration of renewable energy sources is essential for a sustainable power grid due to the stochastic and intermittent nature of renewable energy generation. This study develops a comprehensive Integrated Energy Management System incorporating supply-demand side management in the form of time-of-use credit, direct load control, and generator control to enhance photovoltaic utilization in off-grid applications. A novel three-step solar energy forecasting approach is proposed in this paper, utilizing low-level data fusion and regression models to predict next-day photovoltaic generation with improved accuracy, and a rule-based decision algorithm is developed to correct forecast errors and manage loads dynamically. A techno-economic analysis covering a 20-year duration is carried out for scenarios with and without the integrated energy management system; three configurations are investigated for supplying an off-grid residential home, including diesel generator, diesel generator/photovoltaic system, and diesel generator/photovoltaic system/integrated energy management system. Results reveal that the hybrid configuration with integrated energy management system achieved 44 % and 46 % reductions in costs and carbon dioxide emissions compared to the diesel generator alone, and 8 % and 9 % compared to the diesel generator/photovoltaic setup respectively. The Integrated Energy Management System further enhanced photovoltaic utilisation rate by over 113 % when compared to the diesel generator/photovoltaic system. Further evaluations include customer behaviour impacts, demonstrating that a fully automated system with 100 % compliance significantly outperforms systems with manual customer control, highlighting the detrimental effect of overrides on the efficiency of direct load control. The flexibility of the Integrated Energy Management System framework allows potential adaptation for on-grid applications, showcasing its utility in diverse operational contexts.

P300 Modulates Endothelial Mechanotransduction of Fluid Shear Stress.

P300 is a lysine acetyltransferase that plays a significant role in regulating transcription and the nuclear acetylome. While P300 has been shown to be required for the transcription of certain early flow responsive genes, relatively little is known about its role in the endothelial response to hemodynamic fluid stress. Here we sought to define the role of P300 in mechanotransduction of fluid shear stress in the vascular endothelium. To characterize cellular mechanotransduction and physical properties after perturbation of P300, we performed bulk RNA sequencing, confocal and Brillouin microscopy, and functional assays on HUVEC. Inhibition of P300 in HUVEC triggers a hyper-alignment phenotype, with cells aligning to flow sooner and more uniformly in the presence of the P300 inhibitor A-485 compared to load controls. Bulk transcriptomics revealed differential expression of genes related to the actin cytoskeleton and migration in cells exposed to A-485. Scratch wound and bead sprouting assays demonstrated that treatment with A-485 increased 2D and 3D migration of HUVEC. Closer examination of filamentous actin revealed the presence of a perinuclear actin cap in both P300 knockdown HUVEC and HUVEC treated with A-485. Interrogation of cell mechanical properties via Brillouin microscopy demonstrated that HUVEC treated with A-485 had lower Brillouin shifts in both the cell body and the nucleus, suggesting that P300 inhibition triggers an increase in cellular and nuclear compliance. Together, these results point to a novel role of P300 in modulating endothelial cell mechanics and mechanotransduction of hemodynamic shear stress. The online version contains supplementary material available at 10.1007/s12195-024-00805-2.

Efficient energy management of domestic loads with electric vehicles by optimal scheduling of solar-powered battery energy storage system

The increasing adoption of electric vehicles (EVs) and variable energy usage patterns substantially strain the electrical grid; indeed, optimal energy management, monitoring, and utilization are required for the reliable operation of the grid. This paper introduces a novel model design of a solar-powered battery energy storage system (SPBESS) as a viable substitute for conventional demand-side management (DSM) and time of use (ToU) pricing schemes, intending to optimize energy management and utilization with IoT monitoring. In addition, the IoT-based prototype has been developed to monitor and control the proposed system. To validate the proposed SPBESS model, the study examines two distinct cases: one encompassing domestic loads and the other integrating EV loads alongside household demand. Using ToU pricing, it determines the optimal charging and discharging strategies for the SPBESS, evaluates their implications for the system configuration and grid ToU pricing, and quantifies the annual reductions in power purchases, electricity bills, energy costs, and carbon emissions. Moreover, the study comprehensively analyzes the optimal power exchanges between the photovoltaic system, battery energy storage system, and the grid, precisely considering the specific load requirements and grid ToU pricing. The conducted research findings manifest considerable decreases in energy costs, with the domestic load condition witnessing a reduction from $0.312 per kilowatt-hour to $0.245 per kilowatt-hour and the scenario involving EV and residential loads experiencing a decline from $0.27 per kilowatt-hour to $0.197 per kilowatt-hour. Furthermore, the annualized cost savings for cases 1 and 2 amount to $1,336 and $4,544, respectively, yielding substantial reductions in polluting gas emissions. Besides, the IoT-based constructed prototype model design for real-time power parameters monitoring and control shows the importance of IoT-based monitoring for remote load control, allowing users to maximize energy efficiency, resource utilization, and system visualization.

Malicious mode attack on electric vehicle coordinated charging and its defense strategy

Power systems exploit Internet of Things (IoT) to coordinate the huge charging demands of numerous electric vehicles (EVs) in intelligent transportation systems (ITSs). However, since connected with public networks, the coordinated charging control system is in a low-level cyber security and greatly vulnerable to malicious attacks. The malicious mode attack (MMA), a new cyber-attack pattern, generates high amplitude forced oscillations, which seriously threats the stability of power systems and the power supply of charging stations. However, the MMA model and attack process is not clear. Moreover, it is difficult to actively eliminate the MMA risk through the existing cyber defense technologies. Therefore, this paper analyzes the characteristics of MMA and provides a multi-information active distribution rejection control (MIADRC) method for the physical smart grid to defense MMA. First, the potential threat of MMA is clarified by investigating the vulnerability of the IoT-based coordinated charging load control system. And an MMA process like Mirai is given as an example, which affects the electricity charging system by penetration, infection and attack. Then, an MMA model is established for impact analysis, where expressions of the mean response (i.e. expected value) and stochastic response (i.e. standard deviation) of the MMA forced oscillation are derived to discover main impact factors. Further, to mitigate the impact of MMA, a defense strategy based on multi-index information active disturbance rejection control is proposed to improve the stability and anti-disturbance ability. Finally, simulations are conducted to verify the existence and characteristics of MMA threats. And it is verified that the proposed MIADRC can suppress 91.8 % of the MMA oscillation amplitude. As far as the authors know, it is the first time that the characteristics of MMA are investigated and multi-information-based defensive strategy for MMA is proposed, which enhances system damping through compensating for the attack disturbance.

Real-time outage management in active distribution networks using reinforcement learning over graphs

Self-healing smart grids are characterized by fast-acting, intelligent control mechanisms that minimize power disruptions during outages. The corrective actions adopted during outages in power distribution networks include reconfiguration through switching control and emergency load shedding. The conventional decision-making models for outage mitigation are, however, not suitable for smart grids due to their slow response and computational inefficiency. Here, we present a graph reinforcement learning model for outage management in the distribution network to enhance its resilience. The distinctive characteristic of our approach is that it explicitly accounts for the underlying network topology and its variations with switching control, while also capturing the complex interdependencies between state variables (along nodes and edges) by modeling the task as a graph learning problem. Our model learns the optimal control policy for power restoration using a Capsule-based graph neural network. We validate our model on three test networks, namely the 13, 34, and 123-bus modified IEEE networks where it is shown to achieve near-optimal, real-time performance. The resilience improvement of our model in terms of loss of energy is 607.45 kWs and 596.52 kWs for 13 and 34 buses, respectively. Our model also demonstrates generalizability across a broad range of outage scenarios.

Idealitas Weight and Balance pada Pesawat Tipe Comac ARJ21-700 Maskapai TransNusa di Bandar Udara Internasional Soekarno-Hatta

The calculation of weight and balance requires high accuracy so that the flight can be carried out safely to the destination airport. A number of obstacles that load control officers experience can affect the ideality of weight and balance and must be properly calculated. The purpose of this research is to determine the ideal weight and balance of Comac ARJ21-700 aircraft of TransNusa airline at Soekarno-Hatta International Airport. This research uses qualitative and quantitative methods using primary data obtained through observation and interviews, secondary data using documentation and literature studies. Data collection techniques used by observation, interviews and documentation. Interviews were conducted with supervisors and senior load control narrow body and junior load control. Data analysis techniques by means of data collection, data reduction, data presentation, and conclusion drawing. The results of this study are the obstacles of load control officers in calculating the weight and balance of Comac ARJ21-700 aircraft of TransNusa airline, namely the delay of check-in officers in inputting data on the number of passengers and luggage. Determination of the location of passengers and luggage is carried out by check-in officers, if there is a mismatch, the load control officer makes a request to the check-in officer to maximize the front compartment. The average center of gravity of the Comac ARJ21-700 shows at -7.29UP which states that the center of gravity of the Comac ARJ21-700 is -7.29UP.

Neural network predictive controller based on the improved TPA-LSTM model for ultra-supercritical units

To mitigate the impact of large-scale renewable energy power on the national grid in China, it is imperative to enhance the flexible peaking capability of coal-fired thermal power units. The coordinated control system, central to the load control of coal-fired units, faces challenges such as multivariable coupling, sluggish response, and uncertain coal quality parameters. This paper introduces a neural network predictive controller based on the improved TPA-LSTM model, aimed at addressing these issues. Initially, a data-driven control model is established to break through the limitations of traditional linear predictive control and effectively handle disturbance uncertainties. Then, a multivariable coordinated control strategy based on the neural network controller is designed, achieving effective decoupling of multiple parameters and ensuring high adaptability across all load conditions. Additionally, by integrating an automatic model updating mechanism, the system can recalibrate in real-time when model mismatches occur due to equipment aging, maintenance, or changes in coal quality, thereby enhancing overall control performance. Simulation results demonstrate that this strategy has excellent control effectiveness, meeting the flexible peaking demands of 1000 MW ultra-supercritical units. The calibration feature of the data-driven model significantly improves control performance following model mismatches.

Design method and flow analysis of a mixed flow pump based on blade load control

Abstract Mixed flow pumps are commonly used due to their wide operating range. Load distribution is the key parameter of the inverse design of mixed flow pumps. In this study, an inverse design method using a quadratic function to control the blade load distribution is introduced to design a mixed flow pump. The hydraulic performance of the pump is calculated using numerical simulation, and then the internal flow is analyzed. The results show that the pump designed by this method has a high efficiency at the designed point and a wide operating range. The flow field is symmetrical along the circumference and varies uniformly along the impeller inlet to the impeller outlet. No flow separation and no unfavourable low-pressure area appear.

The effect of rail crown film hole injection angle and blowing ratio on the flow and cooling performance of the squealer tip in a turbine blade

Squealer tip is considered as an efficient and reliable tip geometry design in leakage loss control and thermal load reduction in heavy-duty gas turbine blade. The cooling and aerodynamic characteristics of a squealer tip with different film hole parameters are investigated after validating the numerical approach. Firstly, superior aerodynamic and cooling performance of the rail crown film hole (RCFH) scheme is proved by comparing three squealer tip structures. Subsequently, two kinds of injection angles (streamwise injection α and normal injection β) and three angle values (35°, 90°, and 145°) are studied in RCFH schemes. Finally, a compound angle scheme consisting of optimal α and β is investigated at blowing ratios (M) varying from 0.5 to 2.0. In varied α hole schemes, the velocity of the coolant can be divided into the radial and streamwise components, the effects of which on leakage control are complementary, leading to minimal sensitivity of total leakage flow rate (LFR) to α. Increasing and decreasing α can both enhance the coolant attachment on the rail crown surface compared with the radial injected scheme. In varied β hole schemes, more coolant momentum is used to resist the leakage flow as β increases, which results in remarkable LFR reduction. Moreover, the back-press effect of leakage flow on the coolant is markedly enhanced, improving the utilization of coolant. In compound angle scheme, increasing M leads to enhanced impingement on clearance flow, and the maximum values of average film cooling efficiency (η) on the rail crown and cavity floor appear at M = 1.0 and M = 1.5, respectively.