Pump Turbine Research Articles

Numerical investigation on the transient gas–liquid flow in the rapid switching process of pump turbine

AbstractThe frequent switching of pump mode and turbine mode of the pump turbine leads to frequent transient phenomena. To ensure the safe and stable operation of the unit, a detailed study on the exhaust and pressurization process when the unit switches from turbine mode to the pump mode has been carried out. Based on the Shear Stress Transfer model (SST) k–ω turbulent model, the numerical simulations are processed both in a steady and unsteady state. The visualization results of gas–liquid two phases distribution in the dynamic process of exhaust and pressurization are given, and the characteristic references of each stage are also carried out. The transient characteristics of the torque, axial, and radial force of the runner and guide vane are analyzed by combining the short‐time Fourier transform. The results show that the main frequencies in this transient process are the blade passing frequency and its harmonic frequency. This process also presents a high amplitude band at all frequency values, which may be caused by the entrainment and centrifugal action of the runner on the free liquid surface.

Exploring the Mechanism of Strong-Pressure Fluctuation under Partial Load in the Turbine Mode of Pump Turbines for Hydro and Marine Power Storage

As a core component of pumped storage power plants for hydro and marine power storage, this paper investigates the mechanism of pressure pulsation fluctuations under different load conditions to improve the efficiency and operational stability of the storage units. The results of a combination of experiments and numerical simulations showed that the pressure pulsation fluctuations in the pump turbine under strong and weak loads were significantly different at different monitoring points. The three-dimensional flow lines diagram of the pump turbine unit from the CFD numerical simulation showed that the flow line of the pump turbine was relatively chaotic and the vortex existed under weak load conditions. Pressure clouds and flow lines in the cross-section and longitudinal section of the pump turbine are shown. Both showed high-pressure values and a chaotic flow line with a vortex under weak load conditions. To a certain extent, it revealed the pressure pulsation fluctuation mechanism of the pump turbine and provides some guidance for solving practical problems in engineering.

Transient simulation of reversible pump turbine during pump mode's starting up

A pumped storage power station is an electric power system that utilizes low-peak electricity to pump water into a high-altitude reservoir and releases water to generate electricity during high-peak electricity demand. The pump mode's starting up impacts the operational stability and lifespan of the reversible pump turbine (RPT). Especially, the pressure pulsation signal during the pump mode's starting up is a typical non-stationary signal in RPT, which is an important basis for judging the operation status of the unit. In this study, the time-varying boundary conditions are used to calculate the pressure fluctuation and flow pattern evolution for a prototype RPT. This paper analyzed and compared the time-frequency characteristics of pressure fluctuations by continuous wavelet transform (CWT) and by variational mode decomposition (VMD). And the internal flow of RPT was also visualized using entropy generation theory. In this study, the effect of the penalty coefficient on the VMD algorithm is investigated in the form of a signal-noise ratio (SNR). With the increase of the penalty coefficient, the SNR of the signal gradually decreases and tends to be stable. Several modal components with strong physical significance and center frequency separation are obtained. Influenced by the boundary conditions that change with time, the pressure fluctuation presents a five-stage trend of “descending - ascending - descending - ascending - descending”. There are three main types of high-flow energy dissipation (FED) areas during this process of the RPT. The research and analysis in this paper will provide a good reference for the stable and safe operation of RPT and pumped storage power stations.

Numerical Study of the Internal Fluid Dynamics of Draft Tube in Seawater Pumped Storage Hydropower Plant

Pumped storage hydropower plants are renewable energy systems that are effective in saving energy and solving electricity peak-on shortage. Seawater pumped storage hydropower plants are a novel type of pumped storage hydropower plant specifically supplying electric power for ocean islands with the support of solar energy and wind energy. Compared with traditional pumped storage hydropower plants that are constructed on the mainland, seawater pumped storage hydropower plants should take the influence of the complex marine environment, such as extreme waves and winds, into consideration. Taking the characteristics of waves near islands in the East China Sea as an example, we explored the transient hydraulic characteristics in the draft tube of a pump turbine under wave disturbance using a sliding grid interface and the detached eddy simulation (DES) turbulence model. By analyzing the characteristics of unsteady flow in the draft tube, the vortex characteristics under the Q criterion, the frequency characteristics of the pressure pulsation, the evolution law of the internal fluid, and the propagation law of the pressure pulsation were explored. For the situation without wave disturbance, an obvious eccentric vortex in the straight cone section of the draft tube was observed in the case where the opening of the guide vane was small. With the increase in the opening of the guide vane, the eccentric vortex gradually dissipated. For the situation with wave disturbance, the main frequency of the draft tube equaled the frequency of the wave disturbance, the maximum pressure pulsation at the selected monitoring points increased 5 to 15 times, and the superposition of the wave pressure pulsation signals and the draft tube pressure pulsation produced more low-frequency, high-amplitude pressure pulsation signals. Even though the pressure pulsation frequency spectrum varied a lot, the frequency domain of the pressure pulsation without wave disturbance still existed. In addition, the wave disturbance merely varied with the pressure of the draft tube. The influence of wave disturbance on the pressure distribution in the draft tube was relatively small. The results can provide a reference for the operation of seawater pumped storage hydropower plants.

Research on Two-Way Contra-Rotating Axial-Flow Pump–Turbine with Various Blade Angles in Pump Mode

In tidal two-way contra-rotating units, significant differences in performance often occur when arranging the front and rear impellers, which requires an optimized design of the impellers. To solve this problem, by reducing the blade inlet and outlet angles, the impact of different blade angles on the performance of two-way pump–turbines and the internal flow was explored, and the effects of the blade inlet angle of the impellers on the performance of the counter-rotating pump were obtained. Afterward, the streamline and vorticity of the two-stage impeller at different angles were analyzed. The results show that different blade angles will have a certain impact on the internal flow of the two-way pump–turbine. Different blade outlet angles have a significant impact. The variation in different inlet blade angles is not significant for the vorticity changes in the front impeller and rear impeller. In addition, changes in the outlet blade angle will have an impact on the location of LE impact water of the rear impeller, which in turn affects the contours of vorticity of the rear impeller near LE, which also means that the vorticity in this area is mainly dominated by the vortex stretching term.

Discussion on Fatigue Life Analysis Methods for Connecting Bolts of the Top Cover of Water Pump Turbines

Under the joint action of dynamic and static loads, the connecting bolts of the top cover of the pump turbine are prone to cumulative damage in local areas such as threads, resulting in cracks or development, and fatigue fracture may occur in severe cases, which will cause serious safety accidents. Therefore, it is necessary to carry out accurate fatigue life estimation analysis for connecting bolts, which has always been a hot spot and difficulty in the industry. This paper introduces several commonly used fatigue life analysis methods of parts, combined with the force characteristics and structural characteristics of the connecting bolts of the top cover, and analyzes and discusses the fatigue life analysis methods suitable for the top cover bolts. Finally, it is believed that the application of stress-strain field strength method can better describe the actual state of the bolt danger area, and the current damage state has an influence on the amount of damage generated by further loading, and the fatigue life analysis of connecting bolts can be combined with the stress-strain field strength method and variable damage linear cumulative damage theory, and the fatigue life and residual fatigue life can be estimated in the design stage or after the phased service of the bolt.

Analysis on the Mechanism of Rotating Stall Inner a Pump Turbine in Pump Mode Based on the Proper Orthogonal Decomposition

Abstract An in-depth understanding of stall behavior is essential to improve the safe and stable operation of the pump turbine. In this research, the detached eddy simulation (DES) is used to investigate the various characteristics and propagation mechanism of the rotating stall of the hump instability region. The frequency characteristics and spatial intensity distribution of the stall cells are studied by adopting the proper orthogonal decomposition (POD) method. The results indicate that the energy loss at 0.80QDES and 0.75QDES are induced by the rotating stall with three cells that occur in the diffuser flow channel, and the corresponding high amplitude frequencies are St0.0074 and St0.0111, respectively. The effect of the stall cells is particularly severe near the hub. When the flow rate is further reduced to 0.68QDES, the three rotating stall cells develop into two symmetrically distributed stable stall cells with no definite characteristic frequencies, and the blocking phenomenon is concentrated near the shroud. With the stall cells losing their rotational characteristics, they have less impact on the flow field and contribute less energy. This paper also reveals that the energy loss in the diffuser contributes significantly to the hump formation on the performance curve.

Floating Photovoltaic Systems Coupled with Pumped Hydroplants under Day-Ahead Electricity Market Conditions: Parametric Analysis

The intermittent nature of the solar resource together with the fluctuating energy demand of the day-ahead electricity market requires the use of efficient long-term energy storage systems. The pumped hydroelectric storage (PHS) power plant has demonstrated its technical and commercial viability as a large-scale energy storage technology. The objective of this paper is to analyse the parameters that influence the mode of operation in conjunction with a floating photovoltaic (FPV) power plant under day-ahead electricity market conditions. This work proposes the analysis of two parameters: the size of the FPV power plant and the total process efficiency of the PHS power plant. Five FPV plant sizes are analysed: 50% (S1), 100% (S2), 150% (S3), 350% (S4) and 450% (S5) of the PHS plant. The values of the total process efficiency parameter analysed are as follows: 0.77 for old PHS plants, and 0.85 for more modern plants. The number of daily operating hours of the PHS plant is 4 h. These 4 h of operation correspond to the highest prices on the electricity market. The framework of the study is the Iberian electricity market and the Alto Rabagão dam (Portugal). Different operating scenarios are considered to identify the optimal size of the FPV power plant. Based on the measured data on climatic conditions, an algorithm is designed to estimate the energy production for different sizes of FPV plants. If the total process efficiency is 0.85, the joint operation of both plants with FPV plant sizes S2 and S3 yields a slightly higher economic benefit than the independent mode of operation. If the total process efficiency is 0.77, there is always a higher economic benefit in the independent operation mode, irrespective of the size of the FPV plant. However, the uncertainty of the solar resource estimation can lead to a higher economic benefit in the joint operation mode. Increasing the number of operating hours of the PHS plant above 4 h per day decreases the economic benefit of the joint operation mode, regardless of the total process efficiency parameter and the size of the FPV plant. As the number of operating hours increases, the economic benefit decreases. The results obtained reveal that the coupling of floating photovoltaic systems with pumped hydroelectric storage power plants is a cost-effective and reliable alternative to provide sustainable energy supply security under electricity market conditions. In summary, the purpose of this work is to facilitate decision making on the mode of operation of both power plants under electricity market conditions. The case studies allow to find the optimal answer to the following practical questions: What size does the FPV power plant have to be in order for both plants to be better adapted to the electricity market? What is the appropriate mode of operation of both plants? What is the economic benefit of changing the turbine pump of the PHS power plant? Finally, how does the installation of the FPV power plant affect the water volume of the upper reservoir of the PHS plant? Knowledge of these questions will facilitate the design of FPV power plants and the joint operation of both plants.

Novel Pumping System based on Micro-Hydro Turbine and Centrifugal Pump coupled using EVT Induction Machine. Application in Rural Area Irrigation

In this article, a topology for energy conversion oriented to water pumping for irrigation in remote areas is proposed. A micro-hydraulic turbine working at fixed speed, in a "run-off-river" configuration, drives a centrifugal pump through a variable electric transmission (EVT). An EVT based on induction machines was selected, with the stator directly connected to a weak AC grid but the internal rotor fed by a variable frequency drive, allowing an adjustable pump speed. The grid supplies a base power for residential consumption, while the hydro turbine increases the available power for irrigation and thus augmenting productivity of the land uphill and beyond the vicinity of the watercourse. The grid also provides an assistance function, absorbing/delivering a reduced amount of power in case the hydraulic resource is excessive/insufficient for the requested water pumping. As the turbine delivery is constant, the water pumping requirement establishes the power exchanged between turbine, AC grid and pump. For this reason, an analysis to quantify the power flow in the system components for different pump speeds has been carried out. The system modeling is described and computer simulations, using a specific pump speed profile, are presented to validate the performed theoretical analysis.

Comparative studies on the propagation of rotating stall in a liquefied natural gas cryogenic submerged pump-turbine in both pump and turbine mode

With the tightening of liquefied natural gas (LNG) supply, unsteady stall flow under partial flow rate will likely cause significant difficulties with respect to the application of a cryogenic submerged pump-turbine (PT) at LNG-receiving terminals. In this study, the unsteady propagation characteristics of stall cells in pump mode (PM) and turbine mode (TM) of the PT are investigated numerically using the timescale-based hybrid turbulence model. The predicted performance curves show good consistency with on-site experimental data. As the rotating speed and fluid undercooling increase, the triggering of stall flow tends to a lower flow rate in PM, but this is reversed in TM. Under the superimposed action of separation and backflow vortex, stall flow in the impeller suffers from dynamic emergence to dissolution in PM, but quasi-static stall vortex is dominant in TM, benefiting from the rectification of the guide vane. By comparison, running in PM is prone to severe deep stall, and the stall cells have a higher propagation frequency and strength, which also induces greater local energy loss than in TM.

Numerical Investigation of Flow and Structural Characteristics of a Large High-Head Prototype Pump–Turbine during Turbine Start-Up

Transient processes that occur in pumped storage power plants can cause high-pressure conditions, which in turn can result in vibrations in the pump–turbine structure and even damage to structural components. It is therefore crucial to research the transient process of the large pump–turbine units and the flow-induced vibrations of the structural components. The three-dimensional flow field and structural field models of a high-head prototype pump–turbine were constructed to study its flow characteristics and structural characteristics under the turbine start-up. Calculations and analyses were performed on the pressure variation and the flow-induced stress concentrations of the pump–turbine during start-up in turbine mode. The simulated pressure distributions during the turbine start-up were mapped onto the finite element calculation model of the structures of the pump–turbine to calculate the flow-induced stress concentrations. This study provides a reference to improve the design and operation of high-head prototype pump–turbines based on the findings of the flow and structural characteristics.

Cavitation mechanism and effect on pump power-trip transient process of a pumped-storage unit

Backflow vortexes (BFV) and cavitation are the main sources of pressure fluctuations (PF) in pump-turbine (PT) transitions. However, their interaction mechanism and effect on the transitions of pumped-storage power systems remain unclear. In the present work, the guide vane closing process (GVCP) after the pump power-trip (PPT) of a pumped-storage power system was investigated adopting the one- and three-dimensional coupled two-phase cavitation flow simulation approaches and time-frequency combined signal processing approach. Based on the study findings, large-scale vapor cavities in the draft tube and flow-separated vortexes in the guide/stay vane blocked the PT flow passage. Further, their dynamic evolution induced instantaneous flow rate pulses and further caused pressure pulses, which are equivalent to the water hammer effects. Large amounts of BFV and vapor cavities were found on the impeller high-pressure (HP) side. The unsteady evolution of BFV and vapor cavities and their interactions induced complex-frequency PF near the maximum reverse discharge (Qmax_r) condition in the pump brake (PB) mode. Herein, the flow-separated vortexes were related to the low matching degree between the guide vane (GV) and stay vane. Overall, our findings can provide an important theoretical basis for avoiding the vibration of pumped-storage power systems.

Research on energy‐saving technology of loader hydraulic torque converter based on MATLAB dynamic characteristic simulation

AbstractAt present, hydraulic transmission loaders have high automatic adaptability and convenient operation, but their transmission efficiency is low, and the intelligent degree of energy consumption regulation is limited. Strengthening the research of locking device in torque converter is an important measure to effectively reduce energy consumption. Therefore, the research uses MATLAB software and torque converter parameters to build the external characteristic curves of the engine and torque converter, and builds the dynamic model based on the working principle and locking process of the lockup clutch. The research and experimental analysis are carried out from three aspects of the lockup dynamic process of the torque converter, the lock‐up and unlocking conditions, and the change of the lockup oil pressure of the clutch. The results show that the speed difference range between pump wheel and turbine (233.08–365.44 rpm) is much lower than that of other cases (409.68–616.23 rpm) when the engine and torque converter are matched. And the fuel rate is the highest when the throttle opening of the loader is 70%, about 38%. The above results show that the locking structure of the hydraulic torque converter of the loader can effectively reduce the working vibration and improve the service life and fuel rate. And its energy‐saving application effect has an important enlightening effect on the performance optimization and resource utilization of construction machinery, and provides new ideas for improvement.

Investigation on thermo-economic performance of shipboard waste heat recovery system integrated with cascade latent thermal energy storage

Waste heat recovery (WHR) from engine exhaust gas is a viable solution to improve energy utilization and reduce greenhouse gas emissions for marine power plants. However, the WHR system will operate inefficiently and even be shut down due to the variations of waste heat availability. In this study, a CO2-based WHR system integrated with cascade latent thermal energy storage (CLTES) is presented to ensure the efficient and continuous operation throughout the ship's voyage. The objective of the current work is to assess the thermo-economic performance of the proposed WHR system in a case cruise through theoretical analysis. The thermodynamic models under both design and off-design conditions are developed, and the strategy of exhaust gas assignment for WHR system is identified. Also the effects of main design parameters on the system performance are discussed. In addition, the performance comparison between the proposed WHR system and the conventional one is carried out. The results show that there is a trade-off between pump efficiency and turbine efficiency for the desired power capacity, and turbine efficiency has greater impact on the thermo-economic performance enhancement of the proposed system than pump efficiency. The total net power output and heating capacity from the proposed system during a daily voyage are respectively 5356 kWh and 6958 kWh, which are 19.1 % and 7.9 % higher than those of the traditional WHR system. And the levelized exergy cost of the proposed system decreases by 17.9 % compared with the reference cycle.

Research on configuration and performance of water ramjet pressurized water intake system

Water ramjet is the most ideal power propulsion device for high-speed underwater weapons. However, failure to start it at low-speed makes it hard for the engine to be widely used. In order to solve the problem of starting water ramjet at low-speed, a configuration scheme of water ramjet pressurized water intake system of the micro-turbine driving micro-mixed-flow pump is proposed. The design method of micro-turbine and micro-mixed-flow pump is perfected. And the simulation model of turbine and mixed-flow pump is established. The rationality of the design results was verified by numerical simulation. The matching design of the turbine and the mixed-flow pump is completed. And the working performance of the pressurized water inlet system is analyzed. The results show that the maximum relative deviation between the design results of the turbine and the pump and the target value is less than 3.2%, and the booster system can increase the fluid pressure by 2.0 MPa, reducing the starting speed of the ramjet from 90 to 63 m/s. The research results of this paper have reference value for the engineering application of water ramjet technology.

Pemakaian Metode ASME PTC 4.1 Pada Evaluasi Performa (Efisiensi) Boiler Pulverized Firing di Pembangkit Listrik Tenaga Uap (PLTU)

The rankine cycle is widely used for thermal generators that use steam as a turbine driver in power generation plant. There are 4 main equipments in power plant with rankine cycle system that is: Boiler, Turbine, Condenser and Feed pump. Boiler is a tool to heat water up to steam with high temperature and pressure. Then steam goes into the turbine to drive the generator. The steam coming out of the turbine is cooled inside the condenser to be changed into liquid again. Condenser coolers are widely used in steam power plants is sea water, but some are using river water. Water becomes a working fluid in the rankine cycle and undergoes closed cycle (close loop cycle) meaning continuously water at the end of the cycle process re-entry to the early process cycle. There are several methods to calculate boiler efficiency by direct method (input method) and heat loss method (energy balanced method). The calculation by the direct method by dividing the resulting steam energy divided by the fuel energy coming into the boiler. While the calculation of boiler efficiency with heat loss method is the fuel energy that enter the boiler reduced by losses that occur in the boiler. ASME PTC 4.1 method can be used to calculate boiler efficiency at large boiler capacity, so PTC 4.1 method is often used to evaluate boiler in power plant. The ASME PTC 4.1 method is recommended for the calculation of heat-loss methods in the boiler and is often performed to review the efficiency according to the design as a benchmark.

Effect of Unbalanced Magnetic Pull of Generator Rotor on the Dynamic Characteristics of a Pump—Turbine Rotor System

In pumped storage units, the rotor-bearing electromagnetic system is under the joint influence of hydraulics, mechanics, and electromagnetics, and the mechanism of unit vibration problems is very complex to investigate. ANSYS software is used to establish a three-dimensional model of a pumped storage power plant’s rotor-bearing electromagnetic system, and the stiffness coefficient of the unbalanced magnetic traction forces is calculated using the Fourier series of the magnetic conductivity of the air gap. This shows that the nonequilibrium magnetic attraction increases non-linearly with increasing excitation current and eccentricity of the rotor. At each order, the critical velocity of the rotor system increases as the stiffness factor of the bearing increases, with the greatest increase in critical velocity at the third and fourth orders. In the first-order mode-oscillation pattern, the unbalanced magnetic attraction has an effect on the intrinsic frequency of the transverse oscillation, with a reduction in the amplitude of the intrinsic frequency by 34.65%. Axial and transverse modal vibrations manifest themselves as upward and downward motions and transverse oscillations in different portions of the rotor system, respectively, whereas torsional modal vibrations manifest as a radial broadening or reduction in the generator rotor, runner, and coupling portions of the rotor system. The results of the study provide a theoretical foundation and a computational method for the dynamic analysis and design of the rotor system of pumped storage power stations.

Dynamic characteristics of a running away pump-turbine with large head variation: 1D + 3D coupled simulation

The dynamic characteristics of running away pump turbines (PTs) with a large head variable amplitude have not been understood thus far, primarily because of two difficulties in simulation and analysis. The first is how to provide accurate time-varying boundary conditions for transient simulation of the turbine runaway process (TRP). The other is how to determine the specific appearance time of each frequency component of the complex pressure fluctuations. This study presented a one- and three-dimensional (1D-3D) coupled approach considering waterway dynamics to provide accurate unsteady boundary conditions for the transient flow simulation of a PT with a large head variable amplitude. The short-time Fourier transformation (STFT) approach was adopted to analyse the time-frequency characteristics of the transient pressures and impeller forces. The study found that the fluctuations of pressures and impeller forces during the TRP of the PT with a large head variable amplitude contained two exclusive fluctuation frequency components. The former was approximately three times the rated rotational frequency of the impeller. The later was a series of integer fold transient rotational frequencies of the impeller, which was irrelevant to the rotor-stator interactions. The findings have important value for controlling the pressure fluctuations during the TRP of PTs.

Study on the Service Performance of a Two-Stage Floating-Ring Isolation Seal for a High-Speed Turbopump with a Cryogenic Medium

The reliability and stability of the seal at the fuel-supply end of a rocket-engine turbopump are important factors in determining safety. Conventional single-stage floating rings used for the isolation-sealing of cryogenic media are highly susceptible to operational instability during startup and shutdown, which places demands on the seals’ structure, size, and material properties. In this study, a two-stage floating-ring isolation seal with a non-slotted main sealing surface was designed using a tangential air-intake mode. Based on the full-size three-dimensional finite-volume model, the leakage characteristics of the floating ring during operation were calculated, taking into account the effect of the “inlet effect” on the seal’s performance. Combining the temperature and pressure distributions of the sealing system under cryogenic operating conditions, calculated using a numerical simulation, a reliability analysis of various inlet directions and two kinds of floating-ring schemes was carried out on a self-constructed service-performance test bench. The results indicate that the main wear location of the non-slotted floating ring occurs on the auxiliary sealing surface, with stable working performance. When the inlet direction and spindle-rotation direction are the same, this is more conducive to ensuring the stability of seal performance in practical applications. The results of the current research are instructive for designing floating-ring isolation seals for turbine pumps.

Optimization of Setting Angle Distribution to Suppress Hump Characteristic in Pump Turbine

Pump turbines play a quite important role of peak-valley shifting in the grid, and the hump margin is a critical criterion related to the safety and stability of operation in pump mode. Aiming at investigating the influence of runner outlet setting angle distribution on hump performance of a pump turbine, three runners with different linear distributions of setting angle at outlet were proposed, and the corresponding hump performance comparison was analyzed numerically through the SST k-ω turbulent model. The numerical result shows that, compared to the experiment, the relative errors of all simulated performances (energy characteristic, torque characteristic, and efficiency) were within 3%. Moreover, it was found that setting angle distribution modes could lead to a remarkably different performance in the hump region and, for the runner whose setting angle at shroud was 10° larger than that at hub, the hump safety margin could be increased from 4% to 4.5%. Thereafter, the corresponding mechanisms including energy input and hydraulic loss were investigated through the Euler head theory and the entropy method, respectively. It was found that hydraulic loss distribution played a more important role than the input energy on controlling hump performance. Moreover, for the runner with the largest hump margin, the hydraulic loss was distributed more evenly in the decreasing discharge direction, contributing to the elimination of hump performance. In addition, hydraulic loss distribution was calculated through local entropy production rate (LEPR) method. For all proposed runners, when the pump turbine entered the hump region from a normal operation point, the hydraulic loss was mainly concentrated in vaneless areas and guide/stay vane channels, while the runner with a large setting angle at shroud could better control the hydraulic loss distribution in both the spatial location and the discharge varying direction, increasing the hump margin. The design method presented in our paper is more likely to be applied in engineering applications.