The need for circular textiles has led to an interest in the production of biologically derived materials, generating new research into the bioproduction of textiles through design and interdisciplinary approaches. Bacterial cellulose has been produced directly from fermentation into sheets but not yet investigated in terms of producing filaments directly from fermentation. This leaves a wealth of material qualities unexplored. Further, by growing the material directly into filaments, production such as wet spinning are made redundant, thus reducing textile manufacturing steps. The aim of this study was to grow the bio-material, namely bacterial cellulose directly into a filament. This was achieved using a method of co-designing with the characteristics of biological materials. The method combines approaches of material-driven textile design and human-centred co-design to investigate co-designing with the characteristics of living materials for biological material production. The project is part of a wider exploration of bio-manufacturing textiles from waste. The practice-based approach brought together biological sciences and material design through a series of iterative experiments. This, in turn, resulted in designing with the inherent characteristics of bacterial cellulose, and by doing so filaments were designed to be fabricated directly from fermentation. In this investigation, creative exploration was encouraged within a biological laboratory space, showing how interdisciplinary collaboration can offer innovative alternative bioproduction routes for textile filament production.  The textile industry is one of the main carbon polluters and generators of waste. In order to reduce carbon emissions and align with the 2050 UN/EU goals, the textile industry must change from linear to circular models . For thiThe European Union has stated that waste should be regarded as a resource to transition material consumption from linear to circular modals; this is specifically with a view to reducing the reliance on non-renewables as raw materials ,3. In reBacterial cellulose is one of the main materials being investigated within the biomaterial field due to its unique characteristics such as its ability to self-assemble, its high mechanical strength and how it can be produced from accessible food sources or waste . Most inOur background review provides context in four parts: the relation between biomaterials and waste, how bioproduction innovations investigate co-design when translating biomaterials into textiles, a theoretical investigation of bacterial cellulose and how bacterial cellulose is explored in textile design. In order to reduce the reliance on non-renewable resources, waste has been highlighted as an important circular feedstock for textiles and apparel ,10,11. RSeveral themes of bioproduction for textiles have been explored through the interdisciplinary interaction of design, science and textile manufacturing knowledge. Examples of innovative bio-manufacturing and design will be discussed, in particular the impact of co-creating with biomaterial characteristics.Camere; Karana (2018), discuss theories of designing with materials and how the practices of working with materials can implement cleaner production systems . AlthougCollet’s ‘bio lace’ (2012) is an example of leading with a conceptualisation of a design future, with synthetic biology emerging as a form of bio-production . SituateThe trajectory of designing with living materials has expanded across multiple fields of interest. Alima, Snookand and McCormack (2022) work primarily with mycelium, furthering the concepts of co-creations and merging the boundaries of technology and biological material . Their wA hybrid process is also clear in Modern Synthesis’s interaction of biological matter, textile manufacturing and design knowledge . The reiThe commonality across these innovations is twofold. Firstly, the success of interdisciplinary collaborations and knowledge shows the necessity for designers and scientists to directly interact. Proving this theory, Ribul et al. 2021) discuss the material-driven textile design  was the primary material under investigation in this study. The following section will facilitate an understanding of BC as a material.Bacterial cellulose was first discovered by Brown in (1886), and by the 1990s, 100–150 tons were being produced per year ,23. SincBacterial cellulose is the purest form of cellulose found within nature and is produced during fermentation by specific bacteria ,27. DuriThere are different fermentation methods for growing bacterial cellulose such as static and shaken fermentations. For static fermentation, the liquid media is incubated in shallow trays, and the cellulose grows as an intertwined mat of cellulose microfibres at the surface . Shaken Chemically, bacterial cellulose is the same as plant-based cellulose; however, the crystallinity of the material is higher . BacteriBy understanding the biological characteristics of this growth, bio-production opportunities can be explored. If the growth of the material is a core factor in the production, such as growing the material directly to product, manufacturing steps could be reduced. The more steps in the manufacturing that are removed, the higher the sustainable prospects.Agricultural waste is a source of interest for alternative material production and has been reviewed as a viable feedstock for bacterial cellulose ,14,31. BThis section discusses how bacterial cellulose is being investigated as a textile. Bacterial cellulose was suggested as a new material textile for the apparel industry by Suzanne Lee ,35. The There is evidence that bacterial cellulose can be extruded into filaments for yarns through wet spinning ,38. By aThis collaborative research formed part of the BBSRC Bio-Manufacturing Textiles from Waste project. The interdisciplinary design and biological sciences aspects of the research project are described in this methodology, which informed the investigation for co-designing with the inherent characteristics of bacterial cellulose. We first describe the method of co-designing with biology, informed by both the material-driven textile design (MDTD) methodology and human-centred co-design. We then describe the materials and methods for the production of the bacterial cellulose filaments.In order to inform the interdisciplinary investigation, the research involved a design placement in the laboratory at the University of the York over six months. This laboratory was set up solely for bacterial cellulose production. The first stage of the research consisted of the designer and lead author of this article to observe the static bacterial cellulose production methods and to learn about standard processes within a laboratory setting. As the experiments were developed iteratively, a broader understanding of the material’s characteristics was formed. From this, it emerged that in designing the production of filaments, the material’s characteristics are a key factor.This paper combines approaches of two design methods to encourage designers to interact with the characteristics of biological materials. These methods have been selected as part of this research as they both offer approaches for working in interdisciplinary design–science collaboration and could be applied to biomaterial development. The first one is the material-driven textile design (MDTD) methodology . The MDTBacterial cellulose was first considered from a theoretical perspective that employed a literature review in order to gain understanding of its inherent characteristics and current production methods, which were then explored in the laboratory. By working directly with the material within the lab, we explored factors such as the fermentation process, the different requirements and conditions for growth, as well as opportunities for design intervention to edit the process. The interdisciplinary approach allowed for dynamic discussion around questions throughout the experimentation, as well as during planning and the review of results. Co-designing with the material highlighted three physical designable characteristics: first, the interaction of oxygen and nutrients allowing for self-assembly; second, the physical parameters of growth and vessel; and third, the ability to reduce in weight and size when dried. These themes are discussed in more detail in Fermentation of bacterial cellulose was prepared at the Centre for Novel Agricultural Products (CNAP) at the University of York. Bacterial cellulose is formed by bacteria secreting ribbons of cellulose. These ribbons form together at the surface of a nutrient-filled liquid and oxygen. Because of this, bacterial cellulose grows to the shape of the vessel it is contained in while maintaining its nonwoven-like structure. This method of forming BC is known as static fermentation.2HPO4 and 1.5 g/L citric acid with the later addition of 20 g/L glucose. The glucose solution was sterile-filtered and added to the HS medium after it had been autoclaved to prevent the Maillard reaction, which occurs when autoclaving sugars and amino acids together. Cultures were statically incubated at 30 °C. Stocks of ATCC 53524 were prepared as previously described and stored at −70 °C with 25% glycerol [Within this project, bacterial cellulose was produced from the strain Komagataeibacter xylinus ATCC 53524, obtained from the American Type Culture Collection (ATCC). The bacteria were grown as advised by ATCC. In brief, the medium used was the Hestrin-Schramm (HS) medium that contains 5 g/L bacterial peptone, 5 g/L yeast extract, 2.7 g/L Naglycerol . This meThe following sections describe the experiments and co-design method building on approaches of MDTD and co-design that led to the bioproduction of bacterial cellulose filaments.The initial conceptualisation of transforming bacterial cellulose growth into a filament began with research into established textile filament and yarn manufacturing processes. The textile manufacturing process of vortex spinning takes staple fibres and spins them into a yarn through a spiralled air flow in a vortex motion. First experiments explored how the micro fibril and ribbon growth of bacterial cellulose could be spun into filament emulating a similar vortex process. However, using a spinning method to form the bacterial medium in a liquid vortex lacked the control required for bacterial growth into a continuous filament shape, and the outcomes were irregular spherical shapes as the cellulose formed around itself, as it occurs in the shaken fermentation method of bacterial cellulose production. The physical characteristics and growth of bacterial cellulose had to be evaluated and understood further for design purposes. The second stage of experiments investigated approaches for co-designing with the bacterial cellulose characteristics. The material characteristics informed the next experiments.The second stage of experiments involved co-design with the material characteristics of bacterial cellulose. The initial experiments discussed above highlighted three physical designable characteristics: Due to the bacterial cellulose forming in a randomised web of interlocking cellulose ribbons, this set of experiments guided the growth into a filament shape by applying bacteria, separate from the nutrients to a specific pattern. First, the nutrient medium was solidified into a gel-like state by adding agar. Then, a filament shape was engraved into the agar, and the bacteria was added into the engraved shape. The growth across the agar medium was uneven, and often spores would bleed out of the contained engraving. The bacterial cellulose that formed adhered to the agar and thus could not be removed. This did, however, show that the bacteria would grow within a defined and small space with access to nutrients and oxygen.By interacting with the material within a laboratory, the material characteristics were integrated in the co-design process. The bacterial cellulose requires a surface to form into a fibrous structure through access to nutrients and oxygen. However, the growth formation could be encouraged into a designed shape to change the physical parameters of growth. The following sections discuss the successful experiment in which the design was informed by the characteristics of the biological material.v/v) and then transferred to the spiral using a serological pipette. We prepared the inoculum as described by Florea et al., 2016, and measured the OD600 of the inoculum. We tested a range of loading and obtained optimised results using 3% (v/v) loading [A reusable 3D-printed fermentation spiral was designed by Roberta Morrow and made by Mohamid H Hassen at the 3D printing and bioengineering Lab at the University of Manchester. The spiral was made from polylactic acid (PLA) with the intention of reuse in different experiments. With a 2 mm growth gap, 20 cm in width and 2 cm in height, this was designed to allow for shrinkage. Within the drying process, bacterial cellulose shrinks up to 80% in weight and size once the fluid has been removed. The aim was to initially create a yarn-like diameter with a length of 7 m . The ves loading .Static fermentation was chosen to produce continuous filament growth to not disturb the entanglement of the microfilaments. A Petri dish lid was placed on top of the spiral to maintain sterile conditions before being transferred to a 30 °C room to incubate statically for 48 h. During the fermentation process, the bacterial cellulose formed at the air–liquid interface of the media, producing a cellulose pellicle which is filament-like. When wet, the filament has a diameter of 2 mm, and when dried this is reduced to 0.5 mm.The pellicle was lifted from the spiral using fine tweezers and carefully wrapped around a glass cylinder to prevent the cellulose string from intertwining. The glass cylinder was submerged in a glass beaker containing 0.1 M NaOH and heated at 80 °C for 5 h. This removed the bacteria from the pellicle. The cellulose was thoroughly washed in deionised water before air drying at room temperature. The spiral was thoroughly washed between each experiment. The filaments were lightly twisted to mimic a two-ply twisted yarn. This was then hand knitted into a small swatch using 2 mm knitting needles to create a small 3 cm by 5 cm swatch shown in the next section. The small, knitted textile test swatch demonstrated how the filament can be introduced into textile form.The aim of this project was to investigate whether filaments could be produced directly from the fermentation of bacterial cellulose using approaches of co-design with the material characteristics. The successful results of the filament prototype show that bacterial cellulose can directly produce filaments. By allowing the inherent characteristics of the material to have an influence on the design of the production process, an efficient production of filaments is demonstrated.As a result of co-designing with the material, three designable characteristics were identified: firstly, the ability to self-assemble at the interface of oxygen and the nutrient medium; secondly, the link of physical growth and structural characteristics; and thirdly, the ability to reduce in weight and size depending on liquid content. It was found that these three functions point to the designability of the shape towards the bioproduction of filaments directly from fermentation.The process of bacterial cellulose production was kept the same as pellicle growth to understand how the material interacts with physical changes. The filament production was successful in producing an even amount of cellulosic filament that could be replicated through the repeated use of the spiral. The filament can be seen dried in Bacterial cellulose is commonly produced in sheet form from static fermentation, which results in a biological nonwoven-like material . DesigneA method of co-designing with material characteristics was applied to the production of bacterial cellulose filaments directly from fermentation. The outcomes of co-designing with bacterial cellulose are threefold: firstly, the research highlights three physical characteristics of bacterial cellulose that are inherently designable; secondly, this research shows that by co-designing with a biomaterial, efficient new production can be designed; thirdly, this paper has shown there is further space to investigate using co-design as a method for biomaterial production, bypassing conventional textile production steps and potentially shortening value chains.Although bacterial cellulose has a multitude of characteristics , this diThree physical elements that can be co-designed were stated above. The first characteristic was bacterial cellulose’s ability to grow within a confined environment but requiring access to nutrients and oxygen. The spiral restricts the formation of bacterial cellulose from a large surface area to a smaller surface area to change its physical form. This was achieved using a maze-like spiral structure. Secondly, it was highlighted that the process of the growth and the mechanical material characteristics such as strength cannot be separated. Thus, the ribbons of bacterial cellulose growth must be allowed to form in order to maintain the inherent qualities. The thickness of the growth determines the width of the filament; therefore, sufficient space is needed for the bacterial cellulose ribbons to entangle while controlling the thickness of the resulting filament. This leads to the third characteristic: the material changes in weight, texture and form when in a wet state, and this affects a reduction in size. This worked well combined with the above characteristic of requiring sufficient space to grow as it allowed the material to be produced with a larger diameter of 2 mm, which then reduced to 0.5 mm in diameter after drying. The inherent mechanical strength of bacterial cellulose was therefore maintained. These characteristics point to implementing a physical design of a spiral to navigate the production. This also highlights the relationship between the growth of the material and its characteristics. The outcome shows how co-designing with the material characteristics forms an understanding of the physical conditions that can be designed.Further research could investigate the tensile strength of the filaments and characterisation towards their application in textile contexts. The filaments produced in this research were not fine enough to be tested on the filament tensile tester. This points to possibilities for further investigation of finer filaments in the bioproduction process. Finer filaments were not produced within this study as the paper focuses on demonstrating the production process and method of co-design. This can be expanded in future research. In addition to this, the filaments do not have the twist of a yarn to produce comparative results. Future research could produce filaments for evaluation against filaments and yarns for textile forms in a further investigation of the application of the filaments through testing.This research investigated how co-designing with the characteristics of bacterial cellulose can inform new bioproduction processes and resulted in growing filaments directly from fermentation. The background section of this paper shows how innovative bioproduction sits at the intersection of biological sciences and material design. In order to utilise the biomaterial to the best of its ability, the material in question, namely bacterial cellulose, must be an integral part of the production decisions. This highlights how the material becomes an active part of the production method. A method for co-designing with the material’s characteristics was explored to form bacterial cellulose filaments directly from growth. This method was inspired by successful bioproduction outcomes found within the background review and built on material-driven textile design and human-centred co-design approaches.Bacterial cellulose has successfully been translated into nonwoven materials by designing novel bio-manufacturing systems combining biology, design and textile manufacturing. These investigations have been designed around the fact that the bacterial cellulose grows like a biological nonwoven, consisting of micro filaments that entangle. This, however, leaves a gap to investigate, namely, introducing bacterial cellulose into a wealth of alternative textile elements and production processes such as textile filaments and yarns.The characteristics of bacterial cellulose were explored through interdisciplinary collaboration, a literature review and physical experiments, which then directly informed a co-design method of filament production. The resulting filament prototype that was produced shows that the characteristics of bacterial cellulose can inform the bioproduction of textiles. This paper’s results highlight three designable characteristics: the ability to self-assemble, the link of growth with structural characteristics and the ability to reduce in weight and size after drying. By co-designing with these characteristics, filaments were produced directly from fermentation, making the need for further wet spinning processing for fibres and filament manufacturing steps redundant. The designable characteristics of bacterial cellulose discussed within this paper can be tuned further and developed to investigate the capabilities of the filaments. For example, textile testing to understand future applications, texturising of the filaments, finer filament growth and scaling of production are all future areas for investigation.


The performance requirements for power devices in airborne equipment are increasingly demanding, while environmental and working stresses are becoming more diverse. The degradation mechanisms of devices subjected to multiple stresses become more complex. Most proposed degradation mechanisms and models in current research only consider a single stress, making it difficult to describe the correlation between multiple stresses and the correlation of failures. Then, a multi-physical field coupling model based on COMSOL is proposed. The influence relationship between temperature, moisture, electrical load, and vibration during device operation is considered, and a three-dimensional finite element model is built to investigate the multi-stress degradation mechanism under multi-physical field coupling. The simulation results show that, compared with single-stress models, the proposed multi-stress coupled model can more accurately simulate the degradation process of SiC MOSFET. This provides references for improving the reliability design of power device packaging. The electric aircraft power system comprises many power converters, and as a vital component, the reliability of power devices directly determines the safety of the power system. Power devices operate in a highly complex environment that involves temperature, moisture, vibration, etc. Under the combined effects of these multiple stresses, the degradation mechanism of power devices is no longer single.Numerous scholars have studied the degradation mechanism of electronic components and obtained valuable results. Haojie Wang et al. studied the effect of random vibration on component structure and suggested that increasing substrate thickness can extend device lifespan . MayssamAirborne power devices are often characterized by rapid temperature changes and high stress. Their working is affected by both aspects of the natural environment, such as temperature and moisture, as well as the operating environment, such as temperature, electrical load, and vibration. Most degradation mechanisms and models proposed in current research only apply to single stress, which fails to describe the correlation between multiple stresses and failures under multiple stresses. Based on the above, this paper establishes a theoretical and finite element analysis model of power devices under multiple stresses, considering the coupling relationship between temperature, moisture, random vibration, and electrical load. The proposed model enables us to comprehensively analyze the influence of various factors on power device degradation and, more accurately, to describe the degradation process. It can provide references for the packaging design of the device.The device is uniformly and securely soldered at each package layer, remaining flat and free from warping in its initial state.Under the influence of temperature and moisture, the device expands uniformly in an isotropic manner.Some material properties of each package layer are invariable constants.This research focuses on a SiC MOSFET transistor and is based on several assumptions, including:Using the principles of energy conservation, current conservation, generalized Hooke’s law, and Fick’s mass transfer law, a mathematical model is established to describe the coupling between different physical fields within the power.SiC MOSFET continuously turns on and off during the conversion of electrical energy. The heat is generated due to switching loss and conducted through the physical layers of the device from the chip to the copper substrate, resulting in temperature changes in each layer. The temperature change will cause changes in the conductivity of the chip and bonding line, which will affect the heat generation of the device. Thus, relationships between electrical and thermal properties can be expressed as follows:Some packaging materials used in SiC MOSFET are hygroscopic. Changes in either the internal or external temperature of the device will affect the moisture distribution within it. The temperature dependence of this moisture transfer process is described by vapor saturation pressure The change in material properties caused by the diffusion of water vapor is represented by the change in effective thermal conductivity as well as the heat source:The existence of a temperature gradient inside the device causes thermal stress and thermal expansion, both of which will affect the strain situation inside the device, and the thermoelasticity theory can be described as:As moisture expansion has a similar effect on material as thermal expansion, the relationship between hygroscopic stress is established by applying the temperature effect theory to the material. Assuming no stresses or strains are generated during the expansion process, the strain induced by the moisture change can be expressed as:The stress and strain of the device under the influence of temperature and humidity result from changes in temperature and humidity within the material. Thermoelastic mechanics principles are employed to calculate the stress and strain of the device during exposure to temperature and humidity. The thermo-hydroelastic equation can be expressed as follows:During an aircraft’s flight, the power device can be affected by mechanical vibration due to changes in acceleration and deceleration. Its motion equation is:For a damped system, the eigenfrequency is Under the influence of temperature and moisture, the device experiences stress and strain not only from mechanical vibrations but also due to variations in temperature and humidity within the material. The stress and strain caused by mechanical vibrations are determined using principles of elastic mechanics, while the stress and strain resulting from changes in temperature and humidity are calculated using thermoelastic theory. The mathematical model for thermal–moisture stress and vibration is further analyzed using Equations (11) and (12).According to the mathematical model presented in Equations (1)–(13), we further obtain the multi-stress field coupling mathematical model of SiC MOSFET.The influence relationship between the fields in the mathematical modeling process is shown in Under the action of electrical signals, heat is generated due to loss. Changes in junction temperature will, in turn, affect the chip conductivity. At the same time, changes in the internal temperature of the module will affect the water vapor saturation pressure, which will lead to changes in the saturated moisture content of the encapsulated material, affecting the humidity distribution within the module. As the moist air invades the device, the mode of heat conduction in the material changes from a single solid phase to a coexistence of solid, liquid, and gas. With the accumulation of time, the water content inside the module continues to increase, resulting in a certain change in the effective thermal conductivity and heat. The uneven distribution of temperature and humidity in various encapsulation materials, as well as the mismatched thermal and humidity expansion coefficients, can cause module deformation due to expansion. This deformation subsequently impacts the heat conduction and humidity transfer processes within the module. Temperature and humidity can impact the Young’s modulus of the material, consequently affecting the vibration mode and frequency of the devices, leading to variations in device stress.In Referring to the electrical parameters in the C3M0016120K datasheet, the gate voltage value is set to 15 V, the drain current amplitude is 115 A, and the source is grounded in the electric field simulation. The electrical conduction of bonding wires is ignored, so the upper surface of the chips is grounded.The heat generated internally by the device due to electricity is transferred through heat conduction from the chip, passing through various physical layers and reaching the copper substrate. The copper substrate then transfers heat to the radiator. Forced heat dissipation is typically used in the device. To enhance simulation convergence and the heat dissipation effect of the equivalent radiator, the convective heat transfer coefficient on the lower surface of the copper substrate is set to 5000 To simulate water vapor diffusion behavior, one surface of the air domain is set as an inflow interface with a flow rate of 0.2 m/s, while the other surfaces are open boundaries to simulate the flow of wet air during operation.−18 s and a duty cycle of 50%.To realistically simulate the switching process of SiC MOSFET, the drain current is set as a rectangular waveform with a switching time of 1 × 10The thermal stress in the solder layer follows a similar pattern as that of temperature changes. As shown in The thermal effect affects not only the device packaging but also the electrical parameters of the chip. The definition of the switching time in Wet air invades the device from outside to inside. Due to the poor hygroscopicity of the copper substrate, wet air is primarily concentrated within the silicone encapsulation layer. Since the moisture absorption rate of the encapsulation material is slower than that of heat conduction, only qualitative simulations are performed.The temperature and moisture of the external environment are atmospheric data from a particular location. −4 MPa, and that of the solder layer is 2.21 × 10−3 MPa. The diffusion behavior of wet air in two packaging layers is similar. However, the stress is different due to the coefficient of moisture expansion (CME) between the two packaging layers. Furthermore, significant warpage occurs at the corners of the passivation layer and chip solder layer under the effect of moisture stress, which may be related to the diffusion behavior of humid air from outside to inside. Similar to thermal stress, this phenomenon may cause the solder layer to peel off.Just like thermal stress, moisture stress is also generated within hygroscopic material after environmental moisture air invades the device. To realistically simulate the process of moisture stress generation, the electric–thermal–moisture coupling model data are used to realize the electric–thermal–moisture–stress coupling. When wet air invades the device from outside to inside, the silicone layer is the first to be affected. Apart from thermal and moisture stresses, mechanical vibrations can also influence the device’s reliability. During testing, the acceleration power spectral density is provided with reference to the random vibration requirement specified in RTCA/DO-160G, as shown in The stability of the structure is closely related to the previous order modes. The natural frequency and the first three order modes of the device under the combined influence of thermal–moisture stress and vibration are shown in Thermal stress is caused by uneven temperature distribution inside the device and mismatched CTEs of each package layer. Temperature changes will affect the parasitic capacitance, resulting in a shorter turn-on time but a longer turn-off time.Moisture diffusion slightly decreases internal temperature, but it generates fluctuation peaks during temperature changes, and at the same time, the wet fluctuations inside the device will be suppressed when the device works.Moisture stress is caused by uneven moisture distribution inside the device and mismatched CMEs of each package layer. In the thermal–moisture coupling field, moisture changes the thermal time constant. More stress is generated by temperature change-related fluctuation spikes. Additionally, the moisture expansion of moisture-absorbing materials can aggravate stress generation and then accelerate device failure.The mechanism of thermal and moisture effects on the vibration characteristics of the device is consistent, which is all due to the change in material performance leading to the change in the material stiffness matrix. However, the influence of temperature on the vibration characteristics of the device is greater than that of humidity. Under the combined action of thermal–moisture stress and vibration, the device is more susceptible to failure.The finite element method was used to simulate the SiC MOSFET’s operational environment. A multi-physical field coupling model was established, and the following conclusions were drawn on the multi-physical field coupling degradation mechanism:In summary, the analysis results demonstrate that the proposed multi-physics coupling modeling method using COMSOL is reasonable and effective. The multi-stress coupling model can simulate the degradation process of SiC MOSFET more comprehensively and accurately, providing references for improving the reliability design of power device packaging.


The coronavirus infection COVID-19 killed millions of people around the world in 2019-2022. Hospitals were in the forefront in the battle against the pandemic. This paper proposes a novel approach to assess the effectiveness of hospitals in saving lives. We empirically estimate the production function of COVID-19 deaths among hospital inpatients, applying Heckman’s two-stage approach to correct for the bias caused by a large number of zero-valued observations. We subsequently assess performance of hospitals based on regression residuals, incorporating contextual variables to convex quantile regression. Data of 187 hospitals in England over a 35-week period from April to December 2020 is divided in two sub-periods to compare the structural differences between the first and second waves of the pandemic. The results indicate significant performance improvement during the first wave, however, learning by doing was offset by the new mutated virus straits during the second wave. While the elderly patients were at significantly higher risk during the first wave, their expected mortality rate did not significantly differ from that of the general population during the second wave. Our most important empirical finding concerns large and systematic performance differences between individual hospitals: larger units proved more effective in saving lives, and hospitals in London had a lower mortality rate than the national average. The coronavirus infection (henceforth COVID-19) started to spread from Wuhan, China, at the end of 2019 , thus faeffectiveness of outcomes. Instead of trying to model the production function of hospitals, which is a highly complex task with multiple parallel operations that interact creating spillovers [A large stream of operational research and management science literature considers efficiency, productivity and the returns to scale and scope in hospital operations –9). Whilillovers , we consEmpirical implementation of the proposed conceptual idea involves at least two methodological challenges. The first challenge concerns the appropriate specification of the functional form of the production function and the modelling of contextual variables to capture the observed heterogeneity of hospitals and their operating environments. Building on the insights from the nonparametric frontier estimation literature , 13, theanalysis  that itementclasspt{minimaanalysis  to allowc to all observations  c., 21 to cWere there systematic performance differences in the COVID-19 care at the regional or hospital levels?Were there increasing or decreasing returns to scale in the COVID-19 care?Did performance of COVID-19 care improve over time due to learning by doing?Were there significant structural differences between the first and the second waves of the COVID-19 pandemic?From the management point of view, the methodological and empirical results of this study could help the NHS and other national health service providers to develop systematic monitoring and benchmarking routines to identify and disseminate best practices more efficiently. In the event of possible future pandemics, more efficient identification and dissemination of best practices has potential to save thousands of lives.With more than 127 thousand confirmed COVID-19 deaths, more than 1,800 deaths per million inhabitants, the United Kingdom (UK) ranks among the most hard-hit nations in per capita terms. The empirical contribution of this study is to study the performance of English hospitals during the first and second waves of the COVID-19 pandemic using the unique data of the National Health Service (NHS) of England. More specifically, we focus on a 35-week period from April to December 2020 to investigate the performance of 187 hospitals in dealing with this virus, to shed light on the following empirical questions: The general approach developed in this paper could be readily applied in other countries and jurisdictions for which similar data are available at the hospital or regional level, not only in the context of the COVID-19 pandemic, but also for other hospital and health care operations such as emergency care or ambulance services. Indeed, the novel approach to performance analysis focusing on effectiveness of outcomes rather than resource efficiency forms the main contribution of this paper.https://github.com/ds2010/Covid-19).The structure of the paper will be organized as follows. Section th January 2020, when two Chinese nationals fell ill in York. The earliest known person to contract COVID-19 within the UK is believed to be a 75-year-old woman from Nottinghamshire, who tested positive on the 21st February 2020 [The first confirmed cases of COVID-19 in the UK were on the 29ary 2020 . She is rd March 2020. As a result, the epidemic started to slow down, and by the end of May there was a notable decline in the daily new positive cases and the death toll. The easing of the lockdown began on the 23rd June. From the beginning of September, however, there was again a substantial increase in the number of daily positive cases. Fortunately, the mortality rate has been lower than the one of the first wave of the epidemic in the Spring 2020. During the second wave, the most affected areas include such cities as Liverpool, Manchester, Sheffield and Leeds in the Northern part of England. Since the number of COVID-19 cases was rising alarmingly, the government decided to issue another national lockdown on the 5th November 2020. The NHS started its vaccination programme in December 2020, and more than 34 million had received the first dose by the end of April 2021. The first steps in easing the UK’s lockdown begun in March by allowing children to return to school. In England, all restrictions were lifted on the 19th July 2021. However, travel restrictions and the mandatory use of face masks on public transport and at shops were reintroduced in November 2021 – January 2022 due to the wildly contagious Omicron variant.After some initial hesitation, the UK government decided to issue a national lockdown, starting on the 23The key objective of the lockdown and other restrictions introduced during the pandemic has been to slow down the spread of the COVID-19 virus in order to spare the hospital capacity to provide sufficient care to the COVID-19 infected patients and hence reduce the mortality of COVID-19 patients. Of course, the restrictions come at an enormous social cost, affecting not only the economic growth, but also human wellbeing in general. How effectively the hospitals can facilitate the recovery of the patients has a decisive impact on how harsh measures are required to slow down the spread of the Covid-19 virus.Established in 1948, the NHS is a governmental health care service that provides care to all the UK citizens: health care is tax financed and free at the point of use. The funding for health services in England comes from the department for Health and Social Care’s budget. The planned spending for 2021/22 was £190.3 billion, this included 33.8 billion extra funding in response to the COVID-19 pandemic.The NHS consists of a number of organizations that work both at the national and local levels. The Secretary of State for Health and Social Care is responsible for setting relevant policies for the NHS, including the waiting times, funding and staffing targets. Almost two thirds of the total NHS budget is controlled by the commissioning groups, which are run by general practitioners, nurses, and consultants. The commissioning groups have the responsibility of commissioning health care services for their local areas based on the assessed needs of the people, including primary care services, mental health, ambulance, social care, and hospital services. The budget of the commissioning groups is overseen by the NHS commission board, which has a number of regional offices around England. To promote competition, in 2006 the NHS mandated that all patients requiring treatment can choose between five different hospitals, and adopted a payment system in which hospitals are paid fixed, regulated prices for treating patients . In 2012The care commissioned by the commissioning groups is provided by the NHS foundation trusts. As of April 2020, there are totally 219 foundation trusts. Each foundation trust is further divided into acute trust, mental health trust and ambulance trust. The responsibility of the acute trust is to make sure that they provide high-quality healthcare and resources are allocated in an efficient way. In comparison, the mental health trust is responsible for providing health and social care services to people with mental health problems. Finally, the ambulance trusts are mainly responsible for providing emergency access to healthcare.Since the early stages of the COVID-19 pandemic, there has been a lot of research on the mortality rate of this new virus  , 25. It To model the dynamics of the COVID-19 mortality among hospital inpatients, we adapt the notion of production function from economics as follows. Firstly, we consider the stock of inpatients as an input factor, analogous to the capital stock. We measure this input by the average number of beds occupied by the confirmed COVID-19 patients during a week, drawing a distinction between the regular hospital beds and the MV beds. The outflow of patients from the hospital can occur only through their discharge or death. Therefore, we consider the following two flow variables: i) the number of COVID-19 infected patients discharged from the hospital (weekly sum), and ii) the number of COVID-19 infected patients who died (weekly sum). The former can be regarded as a desirable output and the latter as an undesirable output. Note that in contrast to the input variables that measure the stock of inpatients during a given week, the outputs are flow variables that capture the outflow of inpatients from the stock. We therefore quantify the inputs by the weekly average, whereas the weekly sum is used as the flow variables.The, NHS data also allows us to control for some contextual variables that represent the observed heterogeneity of hospitals and their operating environments. Most importantly, it is important to control for the share of senior patients in the stock of inpatients because the senior patients are known to be associated with a higher mortality rate when infected by COVID-19.Finally, we also want to control for the staff absence in the hospitals, which was a serious concern during the peak of the first wave of the pandemic .Step 1: Define the binary variable s ratios  \documenD, predict the number of deaths in hospital i in week t by i in week t can be measured in multiplicative form as Step 2: Include i in week t by using i in week t by O the multiplicative performance measure Step 3: For the subsample More specifically, the stepwise estimation procedure can be stated as follows:MPerf is equal to zero by construction, and hence, we propose to use the additive performance measure APerf for the zero-valued observations.This procedure allows one to discard the problematic zero-valued observations in the estimation of production function without adding an arbitrary constant term. We correct for the resulting truncation bias by using the Heckman two-step procedure. Following Heckman, we use the probit regression in step 1, but one could alternatively apply the logistic regression (logit), or the panel data variants of probit or logit. Step 2 can be modified for estimating the convex regression, expectiles, or quantiles. One can also impose further properties such as constant returns to scale, or relax assumptions such as monotonicity or concavity . Finallynd of April 2020 to 2nd December 2020 was obtained from the NHS website,The data of the English hospitals over the period from the 2Table nd April are unavailable.To illustrate the development of the pandemic during the study period, Fig. Figure th July- 5th August), and refer to weeks 1-18 as the first wave, and weeks 18-35 as the second wave, respectively. This division is justified by two observations. First, the average MV bed occupancy reached its minimum level during the week 18, which is included in both sub-periods in our empirical analysis. Second, by comparing the two subperiods of equal length, we effectively avoid the possibility that differences in the sample sizes of the subperiods would add bias to our estimates.To assess the possible structural changes between the first and the second waves of the pandemic, we partition the sample period in half at week 18 .Having excluded the zero-valued observations and included the inverse Mills ratio as a contextual variable, we next apply the CNLS estimator, which is the special case of the CER formulation (2) obtained by setting f. The bottom part reports the regression coefficients of the contextual variables Table Recall that coefficients The estimated intercepts ficiency . This woThe estimated coefficients of the contextual variables also reveal interesting patterns. During the first wave, the elderly inpatients of age 65–84 years or more than 85 years were at significantly higher risk of dying than the younger patients. During the second wave, the shares of elderly inpatients are no longer significant predictors of death. Possible explanations include the improved hospital practices, but also the mutated straits of COVID-19 that are considered to be more contagious among younger people.We find a significant negative time trend during the first wave. The absolute value of the time trend coefficient is relatively large, which points to rapid learning by doing during this period  = 1.62), we find that during the first wave in the North East and North West, the expected mortality rate was more than 60% higher than in London, keeping all the other factors constant. The comparison of the two subsamples representing the first and the second waves indicates a positive result that the performance gaps to London decreased over time, but the North East and North West had still the highest expected mortality rates.The best performance of the London hospitals is not surprising as such, given the large number of hospitals in close proximity to one another, and the high concentration of top hospitals and physicians, cf., . HoweverTo gain deeper understanding of the possible structural differences between the best performing hospitals and the weakest performers, we next apply the convex expectile regression formulation  with sixTable by Efron . For exaConsider first the shares of the elderly inpatients, the coefficients of which are reported on the first two rows of Table Staff absence has a positive effect on expected mortality, but its impact is considerably lower among the best performing observations than in the worst ones. The weekly time trend has a relatively stable negative effect across all expectiles, which suggests relatively homogenous rate of learning by doing. However, the worst performers have a slightly smaller coefficient, which may be associated with catching up the best performing units.Considering the regional performance differences, Table When the first cohorts of COVID-19 infected patients were admitted to hospitals, all medical teams were inexperienced in the care of this unprecedented disease. During the first wave of the pandemic, our results indicate a sharp decrease in the expected mortality of COVID-19 inpatients: mortality decreased 8% per week on average based on the estimated time trend. This impressive performance improvement can be attributed to learning by doing. Unfortunately, in the absence of systematic performance management and benchmarking, there was a large gap between the best performing and the weakest performing hospitals. Presence of such large performance gaps, both at the regional and hospital levels, would suggest that inefficient practices and slow diffusion of information caused loss of life that could had been avoidable by better management. The empirical results of our study have three key managerial implications with a view towards more effective management of possible future pandemics, but also to better management of other hospital operations such as emergency care or ambulance services.First, the performance assessment approach proposed in this study would enable NHS and similar health care organizations around the world to monitor hospital performance virtually in real time, utilizing the data that is already collected and published by NHS. Identifying the best performing hospitals forms the first step of best practice benchmarking, which could help medical teams to identify specific practices that help to save lives of COVID-19 inpatients, and disseminate information about these practices across all hospitals. During the first weeks of the COVID-19 pandemic, medical teams gained valuable experience on how to best utilize the MV bed capacity and in which position to lay the COVID-19 inpatients in hospital beds.Second, our results show that larger COVID-19 care units were more effective in saving lives of COVID-19 patients than smaller units. While the volume-outcome effect on mortality is well-established in the literature , to our Third, we find large systematic regional differences in hospital performance, both during the first and the second waves of the pandemic. In particular, expected mortality of COVID-19 patients was significantly lower in London hospitals than in all other regions. Even after controlling for the size of the COVID-19 unit, the share of senior patients, and staff absence, hospitals in the North East and North West had 60% higher expected mortality rate than London hospitals during the first wave of the pandemic, decreasing to 40% during the second wave. Such large regional performance differences indicate significant inequalities between the patients hospitalized in different NHS regions. These results also suggest that more efficient identification and dissemination of information on best practices from the best performing London hospitals to the other NHS regions might had saved thousands of lives. While the significant performance differences during the first wave of the pandemic were to some extent unavoidable, there was sufficient time to identify best practices and disseminate information prior to the second wave of the pandemic. While the reforms of NHS over the past decade have given greater freedom and independence for the local authorities, the policy reform may have reduced coordination and oversight at the national level.In conclusion, based on our empirical results we would encourage NHS and other national health service providers to 1) establish systematic performance monitoring and benchmarking procedures to identify and disseminate best practices; 2) utilize economies of scale by allocating patients to larger units when feasible; and 3) ensure sufficient coordination between the regional care providers to facilitate more efficient dissemination of best practices not only locally but nationwide.Hospitals around the world were in the forefront in the battle against the COVID-19 pandemic, however, thus far there has been little attention on the performance of hospitals in saving lives. This study proposed to assess performance from a novel perspective, introducing the conceptual notion of the production function of death that approaches hospital performance from the perspective of the effectiveness of outcomes instead of the conventional notions of efficiency and productivity.To apply the proposed approach to the empirical data, this study addressed two methodological challenges. First, we incorporated contextual variables to the convex quantile regression to gain further insights on the impacts of the contextual variables at different levels of performance. Second, we developed a theoretically new approach to model the zero-valued observations, making use of Heckman’s two-step approach to correct for the sample selection bias. While the zero-valued outputs present a major challenge in this specific application, we would argue that the zero-valued outputs are rather common in the empirical studies and that inappropriate modelling of zero outputs can cause serious bias in the empirical results. The two-stage bias correction proposed in this study could be readily applied in other production studies to alleviate such bias.Our empirical findings reveal a significant negative trend in the production function of death during the first wave of the pandemic. We also find significant decreasing returns to scale, which implies that the larger COVID-19 units are more effective in saving lives. The mortality rate of inpatients is also significantly and positively associated with the share of senior patients aged 65 and above. Comparing the hospital performance among different areas of England, we find that the hospitals in London had lower mortality than the national average, while the ones in the North East and North West showed weakest performance. Finally, there are large and systematic performance differences between individual hospitals, which would warrant further investigation. We found the quantile approach a useful complementary tool to gain deeper insight on the structural differences between the best performing and the worst performing hospitals. In particular, the best performing hospitals excelled in saving lives of elderly patients, but also managed to cope with staff absence better than the worst performing hospitals. Interestingly, the regional performance differences were also more pronounced for the worst performing hospitals. This seems to suggest that the performance differences may actually occur at the hospital level: there are not necessarily any inherent regional differences, however, the best performing hospitals are located in London. Our empirical analysis revealed several structural differences between the first and the second waves of the pandemic in England. It is fortunate that the expected mortality rates have decreased notably over time, especially for the elderly inpatients and those with the most severe symptoms. The doctors, nurses and other hospital staff have demonstrated impressive ability to improve performance through learning by doing. Unfortunately, there are significant performance differences both across the NHS regions and individual hospitals, which cannot be explained by the staff absence or the share of senior patients.th January 2021. As a result, the UK issued another national lockdown from the 6th January 2021 to March 2021. The NHS started its COVID-19 vaccination programme on the 8th December 2020. The approved vaccines are expected to be effective against the UK strain of COVID-19, but while writing this, it is too early to declare the battle against COVID-19 to be over at the global level.We hope that the findings of this line of research could help the NHS to identify and disseminate more broadly the best hospital practices in saving lives. After the study period considered in this paper, the daily number of the positive COVID-19 cases kept increasing, peaking at more than 68 thousand on the 8Considering possible future pandemics, it would be critically important to gain better understanding of the factors influencing hospital performance during the COVID-19 pandemic to be better prepared when another unprecedented virus appears. While our empirical analysis focused on the NHS hospitals in England, the general approach developed in this paper could be readily applied to other countries and jurisdictions for which similar data are available at the hospital or regional level. While the approach has been developed in the context of the COVID-19 pandemic, it is more broadly applicable to analysing performance of hospitals or other healthcare providers in saving lives, for example, in emergency care or ambulance services.


Federated learning is an effective means to combine model information from different clients to achieve joint optimization when the model of a single client is insufficient. In the case when there is an inter-client data imbalance, it is significant to design an imbalanced federation aggregation strategy to aggregate model information so that each client can benefit from the federation global model. However, the existing method has failed to achieve an efficient federation strategy in the case when there is an imbalance mode mismatch between clients. This paper aims to design a federated learning method guided by intra-client imbalance degree to ensure that each client can receive the maximum benefit from the federation model. The degree of intra-client imbalance, measured by gain of a class-by-class model update on the federation model based on a small balanced dataset, is used to guide the designing of federation strategy. An experimental validation for the benchmark dataset of rolling bearing shows that a 23.33% improvement of fault diagnosis accuracy can be achieved in the case when the degree of imbalance mode mismatch between clients is prominent. As a key component of a motor, fault diagnosis research on rolling bearings plays an important role in ensuring the safe and stable operation of the motor ,2,3. As Deep learning is a powerful feature extraction tool since it can approximate complex functions by using layer-by-layer feature extraction . AccordiThe effectiveness of deep learning fault diagnosis relies on massive balanced data. However, rolling bearings usually operate under normal conditions and fault samples are difficult to obtain, so the data collected are often imbalanced. The data imbalance problem reduces the effectiveness of deep learning fault diagnosis ,13. To rFederated learning is a distributed machine learning algorithm that can realize the joint optimization of individual clients to obtain a robust global fault diagnosis model under the premise of ensuring data privacy ,20,21. TDealing with the data imbalance problem in federated learning can improve the fault diagnosis accuracy. Zhao et al.  pre-traiA federated learning framework driven by intra-client imbalance degree was designed in this paper to establish a federation strategy for the case where there is an imbalance mode mismatch between clients.The degree of intra-client imbalance was used to guide the design of the federation strategy related to cost-sensitivity. Using the imbalance data of the local client, an inter-class imbalance degree for each client was computed by the gain of a class-by-class model update on the federation model of the small balanced dataset.In the case where there is a significant mismatch of imbalance mode between clients, the federated learning-based fault diagnosis proposed in this paper can well overcome the problem arisen by both intra-client imbalance and inter-client imbalance to ensure the accuracy of fault diagnosis for each client.In this work, we designed a mechanism to accurately measure the degree of imbalance for each client, using the model gain when updating the balanced small sample model with the imbalance dataset as the unique imbalance degree for each client, and developed a federation strategy between clients through cost-sensitive learning. Using the proposed method, we can better solve the problem of identifying the optimal federation strategy construction when there is an imbalance mode mismatch between clients. The contribution of this paper is as follows: An autoencoder (AE) is an unsupervised neural network model that can learn the hidden features of the input data by encoding and reconstructing the original input data by decoding . The feaFederated learning is a distributed machine learning algorithm that enables the joint training of multiple clients without sharing data directly. The schematic of federated learning is shown in Designing a federation strategy that considers the degree of intra-client imbalance is particularly important for improving the performance of federated learning. This is because the degree of intra-client imbalance can measure the contribution of each class sample to the federation model, and this can be used to guide the design of federation strategy related to cost-sensitivity. Therefore, this section outlines a federated learning method driven by intra-client imbalance degree (Fed_ICID) to measure the unique inter-class imbalance degree of each client as the contribution to the federation model. Additionally, the federation strategy between clients related to cost-sensitivity is established by using the degree of intra-client imbalance to ensure that the local client can benefit from the federation model to the maximum extent and achieve fault diagnosis accuracy. The specific steps are as follows.When there is an imbalance mode mismatch between clients, the degree of intra-client imbalance can be used to drive the design of local model training and the federation strategy related to cost-sensitivity so that each client can receive the maximum benefit from the federation model and improve the fault diagnosis accuracy. Therefore, we designed a federated learning framework driven by the degree of intra-client imbalance to weaken the negative effects of imbalanced data on federated learning, as shown in Since the data on each client are imbalanced, the local model training of each client directly on the global model will affect the federated learning. Therefore, the balanced data with a small sample size for each client are constructed by means of a sample balancing. The imbalanced data of the local client are used to compute the inter-class imbalance degree by the gain of a class-by-class model update on the federation aggregation model using balanced data. The inter-class imbalance degree of each client can be used for local cost-sensitive learning. By aggregating the cost-sensitive loss value of each client, the federated center loss function is constructed to drive the joint learning of local model parameters and aggregation weights for each client, thereby ensuring that each client can achieve the maximum benefit from the federation global model.Step 1: Define the training dataset for each clientThe degree of intra-client imbalance aims to measure the contribution degree of each client to the federation model, which can be used to guide the design of the federation strategy related to cost-sensitivity. This is because the degree of the intra-client imbalance can accurately measure the contribution of each class sample to the federation model, which is important for improving the effectiveness of federated learning. The detailed steps of the proposed federated learning fault diagnosis method driven by intra-client imbalance degree are shown below. The block diagram of the proposed Fed_ICID method is shown in Step 2: Establish the federation model based on the small balanced datasetStep 3: Measure the intra-client imbalance degree by gain of the class-by-class model update of the federation modelThe federation model established by the small balanced dataset of each client can be formulated as: Step 4: Federation aggregation strategy driven by intra-client imbalance degreeUsing the imbalanced data of the local client, the inter-class imbalance degree for each client can be computed by the gain of the class-by-class model update of the federation model The intra-client imbalance degree will inevitably affect the efficiency of the federation. A federation aggregation strategy related to cost-sensitivity driven by intra-client imbalance degree is proposed in this part, as shown in Equation (7).The detailed update process of joint optimization for local model parameters and aggregation weights can be computed in Equations (10) and (11), respectively.Once the federation model and the local model are both well trained by joint optimization, the online fault diagnosis can be conducted. The flowchart of the federated learning-based fault diagnosis method guided by intra-client imbalance degree is shown in Rolling bearing is important component in the motor driven system . In thisThe time domain vibration data used in this paper is collected by the drive-side acceleration sensor with sampling frequency 12 kHz and motor load 1 HP with 1772 rpm speed. There are four types of faults in total: normal, inner race, outer race, and ball. Motor bearings were implanted with faults through electrical discharge machining techniques with a fault diameter of 0.021 inches. Outer race faults were located at 6 o’clock. The collected vibration signals were used to construct the dataset with a sliding window of 400 and step size of 30, and the fault labels are assigned as shown in To verify the effectiveness of the proposed method, the fault diagnosis experiments shown in The test set of each client is a balanced dataset containing 1000 samples. Experiments 1–4 are designed to verify the effectiveness of the proposed method when the degree of imbalance mode mismatch between clients changes.DNN is chosen as the network model for each client, and the number of neurons in each layer is 400/600/300/100/4, respectively. The hidden layer uses the relu activation function. The output layer uses the softmax activation function. The number of federations is 100. The exponential decay learning rate is used in this paper; the initial learning rate is 0.05, the decay steps are 50 rounds, and the decay rate is 0.98. The proposed method is compared with existing methods. The model and explanation used for the comparison are shown in It is difficult to obtain an effective fault diagnosis model when the sample size of a single client is limited and there is imbalanced data. Federated learning can achieve a joint update of multiple clients without sharing data directly. However, when there is a data imbalance intra-client, each class sample has a unique contribution degree to the federation model. Existing methods ignore the contribution of the intra-client imbalance to the federation model, affecting the effectiveness of federated learning. Therefore, this paper outlines a federated learning method that can measure the imbalance degree intra-client and guide the design of federation aggregation strategy by cost-sensitive learning. Experiments 1–4 were designed based on the mismatch degree of the imbalance mode between clients, and the experimental results are listed in Comparing columns 2–4 in row 2 of Comparing Comparing Comparing To further present the diagnosis effect of the proposed method, the confusion matrix of each model’s fault diagnosis result when the intra-client imbalance degree is 23:1 is shown in Each subfigure in Federated learning can realize the joint optimization of different client models by the federation aggregation strategy without sharing data directly. The existing federation aggregation strategy tends to reflect the imbalance degree of inter-client based on the sample size or the model performance metrics of each client to obtain federation aggregation weights. However, it only considers the whole model performance of each client rather than how the inter-class imbalance of the intra-client will affect the effectiveness of the federation aggregation model for the case when there is imbalance mode mismatch between clients. Therefore, it is significant to design a federated learning framework driven by intra-client imbalance degree to guide the construction of federation aggregation strategy for the case when there is an imbalance mode mismatch between clients.The degree of inter-class imbalance for each client can be determined by the gain of each class sample of imbalanced data when updated separately on the federation model of balanced samples. By aggregating the cost-sensitive loss function computed by inter-class imbalance degree to obtain the federated center loss function, which is used to drive the joint updating of local model parameters and federation model parameters so that each client can benefit maximally from the federation aggregation model. Experimental results show that the average fault diagnosis accuracy of the proposed Fed_ICID method is 23.33% higher than existing method when there is significant mismatch of imbalance mode between clients.This study only focuses on solving the problem of labeled samples with imbalance without considering the massive amounts of unlabeled data. So, the question of how to design a semi-supervised federated learning method using a combination of labeled and unlabeled data to solve the problem of sample imbalance is a promising research direction.