A 3D finite element vibration model of water turbine generator set is constructed considering the coupling with hydropower house foundation. The method of determining guide bearing dynamic characteristic coefficients ...A 3D finite element vibration model of water turbine generator set is constructed considering the coupling with hydropower house foundation. The method of determining guide bearing dynamic characteristic coefficients according to the swing of the shaft is proposed, which can be used for studying the self-vibration characteristic and stability of the water turbine generator set. The method fully considers the complex supporting boundary and loading conditions; especially the nonlinear variation of guide bearing dynamic characteristic coefficients and the coupling effect of the whole power-house foundation. The swing and critical rotating speed of an actual generator set shaft system are calculated. The simulated results of the generator set indicate that the coupling vibration model and calculation method presented in this paper are suitable for stability analysis of the water turbine generator set.展开更多
A novel cooling system with cooling channels is proposed for the stator of 3MW wind-driven generator.An experimental platform is built to investigate the performance of the cooling system with different loads.At30%,50...A novel cooling system with cooling channels is proposed for the stator of 3MW wind-driven generator.An experimental platform is built to investigate the performance of the cooling system with different loads.At30%,50% or 80% generator loads,the temperatures meet the design requirement.However,it is a little over the requirement at 100%load,duo to experimental errors and some unknown thermal resistances.In the test at 100%load,the developing trends of the parameters of these two generators are similar and only minor differences occurs when they reach steady state our work can be benefit for the design and improvement of MW wind-driven generator cooling solutions.展开更多
Lagrange-Maxwell's equation is extended firstly. With the theory of electromechanical analytical dynamics, the magnetic complement energy in air gap of generator is acquired. The torsional vibration differential equa...Lagrange-Maxwell's equation is extended firstly. With the theory of electromechanical analytical dynamics, the magnetic complement energy in air gap of generator is acquired. The torsional vibration differential equations with periodic coefficients of rotor shafting of generator which is in the state of.magnetic saturation are established. It is shown that the magnetic saturation may cause double frequency electromagnetic moment. By means of the averaging method, the first approximate solution and corresponding solution of the primary parametric resonance is obtained. The characteristics and laws of the primary parametric resonance excited by the electromagnetism are analyzed and some of new phenomena are revealed.展开更多
As an energy generating equipment, the engine-generator set supplies power to the electric transmission. Therefore, its control is one of the key technologies of electric vehicles. Based on the discussion about the de...As an energy generating equipment, the engine-generator set supplies power to the electric transmission. Therefore, its control is one of the key technologies of electric vehicles. Based on the discussion about the demands to the engine-generator set in tracked vehicles, the detailed function of engine-generator and the control strategy are determined. The hardware and software of the control system are also developed and tested in a prototype vehicle. The experiment results show that the control system has good reliability and can satisfy the power requirements of vehicles under all operating conditions.展开更多
A type of remote monitoring and diagnosis system is brought forward which based on Matlab Web Server.Firstly,wavelet packet decomposition is introduced to acquire energy features of which reflect hydrogenerator sets p...A type of remote monitoring and diagnosis system is brought forward which based on Matlab Web Server.Firstly,wavelet packet decomposition is introduced to acquire energy features of which reflect hydrogenerator sets performance to be Feature Parameter.Then these Feature Parameters can be adopted as BP Neural Network input variable to realize fault diagnosis.Most of all,it is the first time to adopt Matlab Web Server to hydro-generator sets faults diagnosis field to implement distributed remote monitoring and diagnosis system.Therefore,remote diagnosis application is independent from the OS used on server side.There is no need for software maintenance by clients.And clients can finish remote diagnosis by Web Browser and without installation of Matlab-software.Client users can monitor and diagnose hydro-generator sets by Browser.Finally,further research work is pointed out such as hydro-generator sets fault modeling,accelerating BP Neural Network learning speed and convergence property,improving data transfer speed of Matlab Web Server to meet the needs of real-time diagnosis for hydropower generator sets.展开更多
This study analyzes the structural characteristics of wind-driven generator,concludes its comment malfunctions and proposes effective methods by general fault analysis methods,so as to design online detection and faul...This study analyzes the structural characteristics of wind-driven generator,concludes its comment malfunctions and proposes effective methods by general fault analysis methods,so as to design online detection and fault diagnosis system of wind-driven generator in virtual instrument.This work will realize real-time detection,help engineers to proceed remote fault diagnosis,reduce maintenance time and increase production efficiency.This study is meaningful and practical to develop a fault diagnosis system for wind-driven generators,which shows professionalization of fault diagnosis system.展开更多
对2007年版《IEEE Guide for AC Generator Protection》中失磁保护定值整定的几个问题提出不同意见。明确指出:发电机失磁故障必有滑差(s),其平均异步阻抗Xd(js)Xd(s=0),失磁保护中的阻抗元件下端定值为-Xd,已有足够余度,不应该是-(1.1...对2007年版《IEEE Guide for AC Generator Protection》中失磁保护定值整定的几个问题提出不同意见。明确指出:发电机失磁故障必有滑差(s),其平均异步阻抗Xd(js)Xd(s=0),失磁保护中的阻抗元件下端定值为-Xd,已有足够余度,不应该是-(1.1~1.25)Xd;在P-Q坐标与R-X坐标的转换公式中,U值应由"失磁保护动作滞后于低励限制线(UEL)"原则来确定,不应主观选择0.95UN。展开更多
To improve the performance of multiple classifier system, a knowledge discovery based dynamic weighted voting (KD-DWV) is proposed based on knowledge discovery. In the method, all base classifiers may be allowed to ...To improve the performance of multiple classifier system, a knowledge discovery based dynamic weighted voting (KD-DWV) is proposed based on knowledge discovery. In the method, all base classifiers may be allowed to operate in different measurement/feature spaces to make the most of diverse classification information. The weights assigned to each output of a base classifier are estimated by the separability of training sample sets in relevant feature space. For this purpose, some decision tables (DTs) are established in terms of the diverse feature sets. And then the uncertainty measures of the separability are induced, in the form of mass functions in Dempster-Shafer theory (DST), from each DTs based on generalized rough set model. From the mass functions, all the weights are calculated by a modified heuristic fusion function and assigned dynamically to each classifier varying with its output. The comparison experiment is performed on the hyperspectral remote sensing images. And the experimental results show that the performance of the classification can be improved by using the proposed method compared with the plurality voting (PV).展开更多
The shafting vibration is closely related to the rotational angular speed.The angular speed of hydro turbine generating sets(HTGS)is rapidly change in fault transient,it maybe reduce the shafting damage.By means of en...The shafting vibration is closely related to the rotational angular speed.The angular speed of hydro turbine generating sets(HTGS)is rapidly change in fault transient,it maybe reduce the shafting damage.By means of energy analysis,the differential equation of shafting vibration for the HTGS is derived,in which include the equation of generator rotor and hydro turbine runner,it can be applied to transient analysis.Shafting model is transformed into first order differential equation groups,and is combined with the motion equation of HTGS to build integrated model.Various additional forces of shafting are taken as input inspire in proposed model,the generality of model is good.At last,the shafting vibration in emergency stop transient is simulated.展开更多
We constructed a class of generalized statistically self-similar set.S and give the necessary and sufficent conditions to ensure a random recursive set being a generalized statistically self-similar set. The statist...We constructed a class of generalized statistically self-similar set.S and give the necessary and sufficent conditions to ensure a random recursive set being a generalized statistically self-similar set. The statistically self-similar sets defined by Hutchinson,Falconer,Graf are the special cases of ours.展开更多
A generalized Lyapunov function was employed to investigate the ultimate bound and positively invariant set of a generalized Lorenz system.We derived an ellipsoidal estimate of the ultimate bound and positively invari...A generalized Lyapunov function was employed to investigate the ultimate bound and positively invariant set of a generalized Lorenz system.We derived an ellipsoidal estimate of the ultimate bound and positively invariant set for the generalized Lorenz system,for all the positive values of system parameters a,b,and c.Our results extend the related result of Li,et al.[Li DM,Lu JA,Wu XQ,et al.,Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system,Journal of Mathematical Analysis and Application,2006,323(2):844-653].展开更多
a-Input resolution and a-unit resolution for generalized Horn clause set are discussed in linguistic truth-valued lattice-valued first-order logic ( Lv( n × 2) F(X) ), which can represent and handle uncerta...a-Input resolution and a-unit resolution for generalized Horn clause set are discussed in linguistic truth-valued lattice-valued first-order logic ( Lv( n × 2) F(X) ), which can represent and handle uncertain linguistic values-based information. Firstly the concepts of a-input resolution and a.unit resolution are presented, and the equivalence of them is shown. Then α-input (a-unit) resolution is equivalently transformed from Lv( n × 2) F(X) into that of LnP(X), and their soundness and completeness are also established. Finally an algorithm for a-unit resolution is contrived in LnP( X).展开更多
The model of grey multi-attribute group decision-making (MAGDM) is studied, in which the attribute values are grey numbers. Based on the generalized dominance-based rough set approach (G-DR- SA), a synthetic secur...The model of grey multi-attribute group decision-making (MAGDM) is studied, in which the attribute values are grey numbers. Based on the generalized dominance-based rough set approach (G-DR- SA), a synthetic security evaluation method is presented. With-the grey MAGDM security evaluation model as its foundation, the extension of technique for order performance by similarity to ideal solution (TOPSIS) integrates the evaluation of each decision-maker (DM) into a group's consensus and obtains the expected evaluation results of information system. Via the quality of sorting (QoS) of G-DRSA, the inherent information hidden in data is uncovered, and the security attribute weight and DMs' weight are rationally obtained. Taking the computer networks in a certain university as objects, the example illustrates that this method can effectively remove the bottleneck of the grey MAGDM model and has practical significance in the synthetic security evaluation.展开更多
Difficulties encountered in studying generators of semigroup of binary relations defined by a complete X -semilattice of unions D arise because of the fact that they are not regular as a rule, which makes their invest...Difficulties encountered in studying generators of semigroup of binary relations defined by a complete X -semilattice of unions D arise because of the fact that they are not regular as a rule, which makes their investigation problematic. In this work, for special D, it has been seen that the semigroup , which are defined by semilattice D, can be generated by the set .展开更多
In this paper, category GIFS of generalized intuitionistic fuzzy sets(GIF) is built up. Topoi properties of category GIFS are studied. Firstly, it is proved that the category GIFS has all topoi properties except that ...In this paper, category GIFS of generalized intuitionistic fuzzy sets(GIF) is built up. Topoi properties of category GIFS are studied. Firstly, it is proved that the category GIFS has all topoi properties except that it has no subobject classifiers, Secondly, it is proved that the category GIFS has middle object and consequently GIFS is a weak topos. Thirdly, by the use of theory of weak topos GIFS, the power object of an object in GIFS is studied.展开更多
Several equivalent statements of generalized subconvexlike set-valued map are established in ordered linear spaces. Using vector closure, we introduce Benson proper efficient solution of vector optimization problem. U...Several equivalent statements of generalized subconvexlike set-valued map are established in ordered linear spaces. Using vector closure, we introduce Benson proper efficient solution of vector optimization problem. Under the assumption of generalized subconvexlikeness, scalarization, multiplier and saddle point theorems are obtained in the sense of Benson proper efficiency.展开更多
Let W^-(t)(t∈R+^N) be the d-dimensional N-parameter generalized Brownian sheet. We study the polar sets for W^-(t). It is proved that for any α∈ R^d, P{W^-(t) = α, for some t∈ R〉^N} = {1, if βd 〈 2N ...Let W^-(t)(t∈R+^N) be the d-dimensional N-parameter generalized Brownian sheet. We study the polar sets for W^-(t). It is proved that for any α∈ R^d, P{W^-(t) = α, for some t∈ R〉^N} = {1, if βd 〈 2N ,0 if αd〉 2N and the probability that W^-(t) has k-multiple points is 1 or 0 according as whether 2kN〉d(k-1)β or 2kN 〈 d(k - 1)α. These results contain and extend the results of the Brownian sheet, where R〉^N = (0,+∞)U,R+^N = [0,+∞)^N,0〈 α ≤1and β〉1.展开更多
The current researches on the tooth surface mathematical equations and the theory of gearing mainly pay attention to the ordinary type worm gear set(e.g., ZN, ZA, or ZK). The research of forming mechanism and three-...The current researches on the tooth surface mathematical equations and the theory of gearing mainly pay attention to the ordinary type worm gear set(e.g., ZN, ZA, or ZK). The research of forming mechanism and three-dimensional modeling method for the double pitch worm gear set is not enough. So there are some difficulties in mathematical model deducing and geometry modeling of double pitch ZN-type worm gear set based on generation mechanism. In order to establish the mathematical model and the precise geometric model of double pitch ZN-type worm gear set, the structural characteristics and generation mechanism of the double pitch ZN-type worm gear set are investigated. Mathematical model of the ZN-type worm gear set is derived based on its generation mechanism and the theory of gearing. According to the mathematical model of the worm gear set which has been developed, a geometry modeling method of the double pitch ZN-type worm and worm gear is presented. Furthermore, a geometrical precision calculate method is proposed to evaluate the geometrical quality of the double pitch worm gear set. As a result, the maximum error is less than 6′10–4 mm in magnitude, thus the model of the double pitch ZN-type worm gear set is available to meet the requirements of finite element analysis and engineering application. The derived mathematical model and the proposed geometrical modeling method are helpful to guiding the design, manufacture and contact analysis of the worm gear set.展开更多
In this paper, two kinds of parametric generalized vector quasi-equilibrium problems are introduced and the relations between them are studied. The upper and lower semicontinuity of their solution sets to parameters a...In this paper, two kinds of parametric generalized vector quasi-equilibrium problems are introduced and the relations between them are studied. The upper and lower semicontinuity of their solution sets to parameters are investigated.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 50679009)
文摘A 3D finite element vibration model of water turbine generator set is constructed considering the coupling with hydropower house foundation. The method of determining guide bearing dynamic characteristic coefficients according to the swing of the shaft is proposed, which can be used for studying the self-vibration characteristic and stability of the water turbine generator set. The method fully considers the complex supporting boundary and loading conditions; especially the nonlinear variation of guide bearing dynamic characteristic coefficients and the coupling effect of the whole power-house foundation. The swing and critical rotating speed of an actual generator set shaft system are calculated. The simulated results of the generator set indicate that the coupling vibration model and calculation method presented in this paper are suitable for stability analysis of the water turbine generator set.
文摘A novel cooling system with cooling channels is proposed for the stator of 3MW wind-driven generator.An experimental platform is built to investigate the performance of the cooling system with different loads.At30%,50% or 80% generator loads,the temperatures meet the design requirement.However,it is a little over the requirement at 100%load,duo to experimental errors and some unknown thermal resistances.In the test at 100%load,the developing trends of the parameters of these two generators are similar and only minor differences occurs when they reach steady state our work can be benefit for the design and improvement of MW wind-driven generator cooling solutions.
文摘Lagrange-Maxwell's equation is extended firstly. With the theory of electromechanical analytical dynamics, the magnetic complement energy in air gap of generator is acquired. The torsional vibration differential equations with periodic coefficients of rotor shafting of generator which is in the state of.magnetic saturation are established. It is shown that the magnetic saturation may cause double frequency electromagnetic moment. By means of the averaging method, the first approximate solution and corresponding solution of the primary parametric resonance is obtained. The characteristics and laws of the primary parametric resonance excited by the electromagnetism are analyzed and some of new phenomena are revealed.
文摘As an energy generating equipment, the engine-generator set supplies power to the electric transmission. Therefore, its control is one of the key technologies of electric vehicles. Based on the discussion about the demands to the engine-generator set in tracked vehicles, the detailed function of engine-generator and the control strategy are determined. The hardware and software of the control system are also developed and tested in a prototype vehicle. The experiment results show that the control system has good reliability and can satisfy the power requirements of vehicles under all operating conditions.
基金Sponsored by the National Pandeng Project(Grant No.PD9521907)
文摘A type of remote monitoring and diagnosis system is brought forward which based on Matlab Web Server.Firstly,wavelet packet decomposition is introduced to acquire energy features of which reflect hydrogenerator sets performance to be Feature Parameter.Then these Feature Parameters can be adopted as BP Neural Network input variable to realize fault diagnosis.Most of all,it is the first time to adopt Matlab Web Server to hydro-generator sets faults diagnosis field to implement distributed remote monitoring and diagnosis system.Therefore,remote diagnosis application is independent from the OS used on server side.There is no need for software maintenance by clients.And clients can finish remote diagnosis by Web Browser and without installation of Matlab-software.Client users can monitor and diagnose hydro-generator sets by Browser.Finally,further research work is pointed out such as hydro-generator sets fault modeling,accelerating BP Neural Network learning speed and convergence property,improving data transfer speed of Matlab Web Server to meet the needs of real-time diagnosis for hydropower generator sets.
基金supported by the following funding projects:Scientific Research Project of Jieyang Polytechnic(Project No.2019JYPCQB02)Science and Technology Project of Jieyang(Project No.sdzx002)。
文摘This study analyzes the structural characteristics of wind-driven generator,concludes its comment malfunctions and proposes effective methods by general fault analysis methods,so as to design online detection and fault diagnosis system of wind-driven generator in virtual instrument.This work will realize real-time detection,help engineers to proceed remote fault diagnosis,reduce maintenance time and increase production efficiency.This study is meaningful and practical to develop a fault diagnosis system for wind-driven generators,which shows professionalization of fault diagnosis system.
文摘对2007年版《IEEE Guide for AC Generator Protection》中失磁保护定值整定的几个问题提出不同意见。明确指出:发电机失磁故障必有滑差(s),其平均异步阻抗Xd(js)Xd(s=0),失磁保护中的阻抗元件下端定值为-Xd,已有足够余度,不应该是-(1.1~1.25)Xd;在P-Q坐标与R-X坐标的转换公式中,U值应由"失磁保护动作滞后于低励限制线(UEL)"原则来确定,不应主观选择0.95UN。
基金This project was supported by the National Basic Research Programof China (2001CB309403)
文摘To improve the performance of multiple classifier system, a knowledge discovery based dynamic weighted voting (KD-DWV) is proposed based on knowledge discovery. In the method, all base classifiers may be allowed to operate in different measurement/feature spaces to make the most of diverse classification information. The weights assigned to each output of a base classifier are estimated by the separability of training sample sets in relevant feature space. For this purpose, some decision tables (DTs) are established in terms of the diverse feature sets. And then the uncertainty measures of the separability are induced, in the form of mass functions in Dempster-Shafer theory (DST), from each DTs based on generalized rough set model. From the mass functions, all the weights are calculated by a modified heuristic fusion function and assigned dynamically to each classifier varying with its output. The comparison experiment is performed on the hyperspectral remote sensing images. And the experimental results show that the performance of the classification can be improved by using the proposed method compared with the plurality voting (PV).
基金financially supported by the National Natural Science Foundation of China under Grant No.51179079
文摘The shafting vibration is closely related to the rotational angular speed.The angular speed of hydro turbine generating sets(HTGS)is rapidly change in fault transient,it maybe reduce the shafting damage.By means of energy analysis,the differential equation of shafting vibration for the HTGS is derived,in which include the equation of generator rotor and hydro turbine runner,it can be applied to transient analysis.Shafting model is transformed into first order differential equation groups,and is combined with the motion equation of HTGS to build integrated model.Various additional forces of shafting are taken as input inspire in proposed model,the generality of model is good.At last,the shafting vibration in emergency stop transient is simulated.
基金the National Natural Science Foundation of China
文摘We constructed a class of generalized statistically self-similar set.S and give the necessary and sufficent conditions to ensure a random recursive set being a generalized statistically self-similar set. The statistically self-similar sets defined by Hutchinson,Falconer,Graf are the special cases of ours.
文摘A generalized Lyapunov function was employed to investigate the ultimate bound and positively invariant set of a generalized Lorenz system.We derived an ellipsoidal estimate of the ultimate bound and positively invariant set for the generalized Lorenz system,for all the positive values of system parameters a,b,and c.Our results extend the related result of Li,et al.[Li DM,Lu JA,Wu XQ,et al.,Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system,Journal of Mathematical Analysis and Application,2006,323(2):844-653].
基金National Natural Science Foundations of China (No. 60875034,No. 61175055)
文摘a-Input resolution and a-unit resolution for generalized Horn clause set are discussed in linguistic truth-valued lattice-valued first-order logic ( Lv( n × 2) F(X) ), which can represent and handle uncertain linguistic values-based information. Firstly the concepts of a-input resolution and a.unit resolution are presented, and the equivalence of them is shown. Then α-input (a-unit) resolution is equivalently transformed from Lv( n × 2) F(X) into that of LnP(X), and their soundness and completeness are also established. Finally an algorithm for a-unit resolution is contrived in LnP( X).
文摘The model of grey multi-attribute group decision-making (MAGDM) is studied, in which the attribute values are grey numbers. Based on the generalized dominance-based rough set approach (G-DR- SA), a synthetic security evaluation method is presented. With-the grey MAGDM security evaluation model as its foundation, the extension of technique for order performance by similarity to ideal solution (TOPSIS) integrates the evaluation of each decision-maker (DM) into a group's consensus and obtains the expected evaluation results of information system. Via the quality of sorting (QoS) of G-DRSA, the inherent information hidden in data is uncovered, and the security attribute weight and DMs' weight are rationally obtained. Taking the computer networks in a certain university as objects, the example illustrates that this method can effectively remove the bottleneck of the grey MAGDM model and has practical significance in the synthetic security evaluation.
文摘Difficulties encountered in studying generators of semigroup of binary relations defined by a complete X -semilattice of unions D arise because of the fact that they are not regular as a rule, which makes their investigation problematic. In this work, for special D, it has been seen that the semigroup , which are defined by semilattice D, can be generated by the set .
文摘In this paper, category GIFS of generalized intuitionistic fuzzy sets(GIF) is built up. Topoi properties of category GIFS are studied. Firstly, it is proved that the category GIFS has all topoi properties except that it has no subobject classifiers, Secondly, it is proved that the category GIFS has middle object and consequently GIFS is a weak topos. Thirdly, by the use of theory of weak topos GIFS, the power object of an object in GIFS is studied.
文摘Several equivalent statements of generalized subconvexlike set-valued map are established in ordered linear spaces. Using vector closure, we introduce Benson proper efficient solution of vector optimization problem. Under the assumption of generalized subconvexlikeness, scalarization, multiplier and saddle point theorems are obtained in the sense of Benson proper efficiency.
基金the National Natural Science Foundation of China (10471148)the Natural Science Foundation of Shaanxi Province (2005A08, 2006A14)
文摘Let W^-(t)(t∈R+^N) be the d-dimensional N-parameter generalized Brownian sheet. We study the polar sets for W^-(t). It is proved that for any α∈ R^d, P{W^-(t) = α, for some t∈ R〉^N} = {1, if βd 〈 2N ,0 if αd〉 2N and the probability that W^-(t) has k-multiple points is 1 or 0 according as whether 2kN〉d(k-1)β or 2kN 〈 d(k - 1)α. These results contain and extend the results of the Brownian sheet, where R〉^N = (0,+∞)U,R+^N = [0,+∞)^N,0〈 α ≤1and β〉1.
基金Supported by Major National Basic Research Program of China(973Program,Grant No.2011CB013400-05)PhD Programs Foundation of Ministry of Education of China(Grant No.20110191110005)
文摘The current researches on the tooth surface mathematical equations and the theory of gearing mainly pay attention to the ordinary type worm gear set(e.g., ZN, ZA, or ZK). The research of forming mechanism and three-dimensional modeling method for the double pitch worm gear set is not enough. So there are some difficulties in mathematical model deducing and geometry modeling of double pitch ZN-type worm gear set based on generation mechanism. In order to establish the mathematical model and the precise geometric model of double pitch ZN-type worm gear set, the structural characteristics and generation mechanism of the double pitch ZN-type worm gear set are investigated. Mathematical model of the ZN-type worm gear set is derived based on its generation mechanism and the theory of gearing. According to the mathematical model of the worm gear set which has been developed, a geometry modeling method of the double pitch ZN-type worm and worm gear is presented. Furthermore, a geometrical precision calculate method is proposed to evaluate the geometrical quality of the double pitch worm gear set. As a result, the maximum error is less than 6′10–4 mm in magnitude, thus the model of the double pitch ZN-type worm gear set is available to meet the requirements of finite element analysis and engineering application. The derived mathematical model and the proposed geometrical modeling method are helpful to guiding the design, manufacture and contact analysis of the worm gear set.
基金The NSF(10871226) of Chinathe NSF(ZR2009AL006) of Shandong Province
文摘In this paper, two kinds of parametric generalized vector quasi-equilibrium problems are introduced and the relations between them are studied. The upper and lower semicontinuity of their solution sets to parameters are investigated.