Set Of Constraints Research Articles

Homogeneous search for helium in the atmosphere of 11 gas giant exoplanets with SPIRou

The metastable helium triplet in the near-infrared (10 833 Å) is among the most important probes of exoplanet atmospheres. It can trace their extended outer layers and constrain mass loss. We used the near-infrared high-resolution spectropolarimeter SPIRou on the CFHT to search for the spectrally resolved helium triplet in the atmospheres of eleven exoplanets, ranging from warm mini-Neptunes to hot Jupiters and orbiting G, K, and M dwarfs. Observations were obtained as part of the SPIRou Legacy Survey and complementary open-time programs. We applied a homogeneous data reduction to all datasets and set constraints on the presence of metastable helium, despite the presence of systematics in the data. We confirm published detections for HAT-P-11 b, HD 189733 b, and WASP-69 b and set upper limits for the other planets. We applied the p–winds open source code to set upper limits on the mass-loss rate for the nondetections and to constrain the thermosphere temperature, mass-loss rate, line-of-sight velocity, and the altitude of the thermosphere for the detections. We confirm that the presence of metastable helium correlates with the stellar mass and the extreme-ultraviolet flux received by the planets. We investigated the correlation between the mass-loss rate and the presence of metastable helium, but it remains difficult to draw definitive conclusions. Finally, some of our results are in contradiction with previous results in the literature, and therefore we stress the importance of repeatable, homogeneous, and larger-scale analyses of the helium triplet to obtain robust statistics, study temporal variability, and better understand how the helium triplet can be used to explore the evolution of exoplanets.

Artificial neural networks for the prediction of mechanical properties of CGNP/PLGA nanocomposites

The mechanical properties of composite-based components have an impact on both their quality and efficacy. As a result, a scientific framework is needed to discover the practical set of progression constraints that will result in exceptional mechanical qualities. The possibilities of an artificial neural network (ANN) for predicting mechanical properties—specifically, the density and hardness of a carboxyl decorated graphene Nanocomposites (CGNP)/Poly Lactic-co-Glycolic Acid (PLGA) nanocomposite created under various spark plasma sintering (SPS) operating conditions—are explored in this work. A back-propagation with a 2–10-2 structural design and Levenberg-Marquardt algorithm was advanced to anticipate the automated performance of CGNP/PLGA nanocomposites in expressions of density and hardness parameters. The hypothesized outcomes of the models were compared to the actual experimental value. Density (0.9842) and hardness (0.9737) were found to be near to one; the archetypal accomplished well, with a good correlation coefficient (R) for both outputs and a low root-mean-squared error. The anticipated data findings were found to be quite compatible with the values acquired from the investigational assessment results. Our findings demonstrate the utility of a well-trained ANN method in assessing the density and hardness properties of SPSed CGNP/PLGA nanocomposites. As a result, the ANN system provides an unswerving tool for making decisions that can reduce the high costs involved with the experimental characterization of newly manufactured polymer composites.

Sequential Load Restoration With Soft Open Points and Time-Dependent Cold Load Pickup for Resilient Distribution Systems

Global changes in climate conditions and increasing number of weather-related disasters in the world have elevated the quest for better load restorations in power distribution systems. This paper proposes a distribution system restoration (DSR) model where non-exponential time-dependent cold load pickup (CLPU) characteristics and soft open point operations are simultaneously addressed. We develop a time-dependent CLPU model for attaining a more realistic and secure DSR in which we demonstrate that the traditional delayed exponential model (DEM) for a long outage duration is a special case of the adopted CLPU model. The current soft open point (SOP) models would typically relax the operation constraints without devising any tightening procedures, which might lead to inaccurate or even infeasible SOP states. In this paper, we analyze the reasons traditional SOP constraints would lead to unsecure operations and present a set of novel SOP operation constraints for recovering bindings. Also, we adopt the sequential second order cone programming (SSOCP) algorithm to establish the sequential framework of DSR and solve the proposed model effectively. Our presented case studies discuss the rationality of the proposed DSR model and algorithm.

Safe and Stable Secondary Voltage Control of Microgrids Based on Explicit Neural Networks

This paper proposes a novel safety-critical secondary voltage control method based on explicit neural networks (NNs) for islanded microgrids (MGs) that can guarantee any state inside the desired safety bound even during the transient. Firstly, an integrator is introduced in the feedback loop to fully eliminate the steady-state error caused by primary control. Then, considering the impact of secondary control on the stability of the whole system, a set of transient stability and safety constraints is developed. In order to achieve online implementation that requires fast computation, an explicit NN-based secondary voltage controller is designed to cast the time-consuming constrained optimization in the offline NN training phase, by leveraging the local Lipschitzness of activation functions. Specially, instead of using the NN as a black box, the explicit representation of NN is substituted into the closed-loop MG for transferring the stability and safety constraints. Finally, the NN is trained by safe imitation learning, where an optimization problem is formulated by maximizing the imitation accuracy and volume of the stable region while satisfying the stability and safety constraints. Thus, the safe and stable region is approximated that any trajectory initiates within will converge to the equilibrium while bounded by safety conditions. The effectiveness of the proposed method is verified on a prototype MG with detailed dynamics.

Efficacious Models of Discrete Control Devices for Convex Optimization in Distribution Systems

Distribution system optimizations often lead to NP-hard programming problems. Although convex relaxations have proved effective for treating most of these problems, solution time increases drastically when discrete control devices (DCDs), e.g., tap-changers, and capacitor banks, are involved. The models of DCDs are categorized in two categories. The models of the first category rely on auxiliary binary variables to achieve a set of convex constraints. The first proposed model in this paper expedites the models of the first category by upgrading the coding of variables. The second category includes the models that elude the computational burden of handling the auxiliary binary variables by relaxing the equations of DCDs and restricting the relaxation region through a sequential bound tightening. Though this approach expedites solving large problems, the risk of encountering a suboptimal/infeasible solution is considerable. The second propounded model resolves such issues while being kept computationally competitive, by designing a parameter-free stopping criterion, a systematic mechanism to reduce the step-size in bound tightening, and an error-cognizant convex model selection function. A wide range of studies demonstrates that the proposed models resolve the issues of those available. The paper concludes with broad recommendations on the suitability of these models.

Condition-based Design of Variable Impedance Controllers from User Demonstrations

This paper presents an approach to ensure conditions on Variable Impedance Controllers through the off-line tuning of the parameters involved in its description. In particular, we prove its application to term modulations defined by a Learning from Demonstration technique. This is performed through the assessment of conditions regarding safety and performance, which encompass heuristics and constraints in the form of Linear Matrix Inequalities. Latter ones allow to define a convex optimisation problem to analyse their fulfilment, and require a polytopic description of the VIC, in this case, obtained from its formulation as a discrete-time Linear Parameter Varying system. With respect to the current state-of-art, this approach only limits the term definition obtained by the Learning from Demonstration technique to be continuous and function of exogenous signals, i.e. external variables to the robot. Therefore, using a solution-search method, the most suitable set of parameters according to assessment criteria can be obtained. Using a 7-DoF KinovaGen3 manipulator, validation and comparison against solutions with relaxed conditions are performed. The method is applied to generate Variable Impedance Controllers for a pulley belt looping task, inspired by the Assembly Challenge for Industrial Robotics in World Robot Summit 2018, to reduce the exerted force with respect to a standard (constant) Impedance Controller. These controllers fulfil a set of safety constraints, namely stability, bounds on task variables and maximum response overshooting; and their performance is determined by the User Preference heuristic, which allows to intuitively define the desired compliant behaviour along the task. In the context of the task, this is used to generate new controllers for one-off modifications of the nominal belt looping task setup without new demonstrations.

Data-Driven Tabulation for Chemistry Integration Using Recurrent Neural Networks.

Due to the wide range of time scales involved in the ordinary differential equations (ODEs) describing chemical reaction kinetics, multidimensional numerical simulation of chemical reactive flows using detailed combustion mechanisms is computationally expensive. To confront this issue, this article presents an economic data-driven tabulation algorithm for fast combustion chemistry integration. It uses the recurrent neural networks (RNNs) to construct the tabulation from a series of current and past states to the next state, which takes full advantage of RNN in handling long-term dependencies of time series data. The training data are first generated from direct numerical integrations to form an initial state space, which is divided into several subregions by the K-means algorithm. The centroid of each cluster is also determined at the same time. Next, an Elman RNN is constructed in each of these subregions to approximate the expensive direct integration, in which the integration routine obtained from the centroid is regarded as the basis for a storing and retrieving solution to ODEs. Finally, the alpha-shape metrics with principal component analysis (PCA) are used to generate a set of reduced-order geometric constraints that characterize the applicable range of these RNN approximations. For online implementation, geometric constraints are frequently verified to determine which RNN network to be used to approximate the integration routine. The advantage of the proposed algorithm is to use a set of RNNs to replace the expensive direct integration, which allows to reduce both the memory consumption and computational cost. Numerical simulations of a H₂/CO-air combustion process are performed to demonstrate the effectiveness of the proposed algorithm compared to the existing ODE solver.

Event-Triggered Distributed Stochastic Mirror Descent for Convex Optimization.

This article is concerned with the distributed convex constrained optimization over a time-varying multiagent network in the non-Euclidean sense, where the bandwidth limitation of the network is considered. To save the network resources so as to reduce the communication costs, we apply an event-triggered strategy (ETS) in the information interaction of all the agents over the network. Then, an event-triggered distributed stochastic mirror descent (ET-DSMD) algorithm, which utilizes the Bregman divergence as the distance-measuring function, is presented to investigate the multiagent optimization problem subject to a convex constraint set. Moreover, we also analyze the convergence of the developed ET-DSMD algorithm. An upper bound for the convergence result of each agent is established, which is dependent on the trigger threshold. It shows that a sublinear upper bound can be guaranteed if the trigger threshold converges to zero as time goes to infinity. Finally, a distributed logistic regression example is provided to prove the feasibility of the developed ET-DSMD algorithm.

Modified Gradient Projection Neural Network for Multiset Constrained Optimization

To solve nonlinear optimization problems under multiple set constraints, a modified gradient projection neural network (MGPNN) is proposed and investigated. Different from existing approaches specialized for linear constrained optimizations, such as the gradient-based recurrent neural network or dynamic-parameter zeroing neural network, the MGPNN is intrinsically designed from the perspective of the multiple set constrained optimization (MSCO), which is a more generalized form for the linear constrained optimization. The MGPNN is able to efficiently and conveniently provide a feasible solution to the MSCO problem. Ultimately, compared with existing solution methods, numerical simulations and applications to the control of an underactuated portal crane system are provided for verifications of the robust stability and preponderance of the proposed MGPNN model.

Current Fault Tolerance Control Strategy for 3-Phase Switched Reluctance Motor Combined With Position Signal Reconstruction

In this paper, a fault-tolerant control method for switched reluctance motors is proposed for the faults of power converters that occur during the operation of switched reluctance motors. The method combines position reconstruction strategies based on fault tolerant control. The fault detection scheme can accurately determine the faults component and faults type, and then and then maintain the motor drive through the current reconstruction. The position reconstruction scheme improves the traditional inductance detection method, which can improve the situation that the position signal fault leads to the current reconstruction fault. In addition, this paper sets constraints on the reconstruction of position signals to enhance the stability and robustness of position signals. Finally, the stability of the proposed method is verified by experiments.

Variations in the intermediate wind region of the blue supergiant 55 Cygni

Context. The quantitative near-infrared (NIR) spectroscopic synthesis is an important technique for determining wind properties of massive stars. The Brα line is an excellent mass-loss tracer and provides valuable information on the physical conditions of intermediate-wind regions. The knowledge of the wind properties gained by studying the NIR lines could provide extra ingredients to the theory of line-driven winds, mainly because the standard theory does not predict observed properties of blue supergiants, such as high values for the β parameter (β> 2), low terminal velocities, and mass-loss variability. Aims. We seek to enhance our understanding of the wind properties of B supergiants. To this end, we propose analysing their NIR spectra over different epochs to study wind variability and its connection with phenomena arising from regions close to the photosphere. Methods. We present the first sets of multi-epoch high-resolution K- and L-band spectra of 55 Cyg acquired with the Gemini Near-InfraRed Spectrograph (GNIRS). We measured line equivalent widths and modelled the Brα line to derive (unclumped) mass-loss rates. Synthetic line profiles were computed for a homogeneous spherical wind by solving the radiative transfer equations in the co-moving frame for a multi-level atom in non-local thermodynamic equilibrium (NLTE). Results. We observe variations in the spectral lines originating in the upper photosphere and the wind. The perturbations, on average, have periods of ~13 and ~23 days; the latter is similar to that found previously from optical data (22.5 days). The NIR lines observed in 2013 are described with the same wind structure used to model a quasi-simultaneous observation in Hα. By contrast, from observations taken in 2015, we derived a higher mean mass-loss rate. Variations in the mass-loss rate are also detected within a few weeks. Interestingly, we find that the profile shape of the Hu14 line sets constraints on the mass loss. Moreover, we find the Mg II doublet in emission, which suggests a tenuous circumstellar gas ring or shell. Conclusions. The variability detected in the NIR H emission lines of 55 Cyg is related to changes in the mass-loss rate, which doubled its value between 2013 and 2015. Furthermore, the short-term variability (within three weeks) in the spectral lines and mass loss supports the hypothesis of strange-mode oscillations. This pilot project demonstrates the importance of comprehensive monitoring of blue supergiants’ variability to deeply understand the physical properties of their stellar winds and the role of pulsations in recurrently enhancing mass loss.

Bi-level programming and multi-objective optimization for the distribution of resources in hierarchical organizations

The decision regarding the hierarchical distribution of resources is crucial for many organizations that want to allocate such resources efficiently. At the upper level of the hierarchy, a decision-maker may have an objective and a set of feasible solutions partially determined by the lower level. Nevertheless, the upper level can influence but not control the decision-maker at the lower level. Moreover, the objectives of the upper and lower levels are conflicting. Hence, the hierarchical distribution of resources can be formulated as a bi-level programming model. This paper reformulates the hierarchical allocation of resources. The Hooke-Jeeves (HJ) algorithm and the hybrid scatter search–Nelder-Mead (SSNM) method are proposed to solve the newly formulated problem. The problem is also analyzed from a multi-objective perspective to equally prioritize the upper and lower levels while subjecting each to an interdependent set of constraints; a multi-objective scatter search algorithm (MSSA) was developed for that purpose. Tailor-made instances were also designed for the implementation of the developed algorithms. The findings demonstrated that the HJ algorithm provides the best solution according to the upper-level objective function. However, it does not guarantee the best result for the lower level. In comparison, MSSA has a set of best possible solutions for both objectives. The hybrid SSNM method could not match the results provided by the former methods.

A fair consensus adjustment mechanism for large-scale group decision making in term of Gini coefficient

With social media and e-democracy development, decision-making environments and problems have become increasingly complex. Traditional group decision-making, including a small number of decision makers, cannot deal with many practical decision-making problems. Large-scale group decision-making has received extensive attention as an efficient measure to address this issue. This paper proposes a two-stage optimization consensus mechanism for large-scale group decision-making by the Gini coefficient, which considers both the consensus among decision makers’ opinions and the fairness of consensus adjustment. First, we introduce an improved density peak clustering method to cluster decision makers into subgroups that can avoid different clustering centers being too close. Then, we define the subgroups' weights by the silhouette coefficient, the subgroup judgment distance, and their sizes. Respecting the importance of minority opinions, we design a new method to identify and adjust the weights of the minority opinions, which considers more aspects than previous ones. Further, we use the Gini coefficient to measure the equity of consensus adjustment of the group and subgroup. Moreover, we present a two-stage Gini coefficient-based consensus mechanism to obtain the adjusted consensus individual decision matrices, which can ensure the minimization and fairness of consensus adjustment and allocation under the set constraints. Based on these results, we give a new large-scale group decision-making method. Notably, it is the first method that fully considers the opinions of minor subgroups and the fairness of allocation results. These characteristics are essential in decision-making problems such as resource allocation and urban public construction. Finally, we show the utility and validity of the new method through a case study and make the comparative analysis from numerical and principle aspects.

Soft Semi-Supervised Deep Learning-Based Clustering

Semi-supervised clustering typically relies on both labeled and unlabeled data to guide the learning process towards the optimal data partition and to prevent falling into local minima. However, researchers’ efforts made to improve existing semi-supervised clustering approaches are relatively scarce compared to the contributions made to enhance the state-of-the-art fully unsupervised clustering approaches. In this paper, we propose a novel semi-supervised deep clustering approach, named Soft Constrained Deep Clustering (SC-DEC), that aims to address the limitations exhibited by existing semi-supervised clustering approaches. Specifically, the proposed approach leverages a deep neural network architecture and generates fuzzy membership degrees that better reflect the true partition of the data. In particular, the proposed approach uses side-information and formulates it as a set of soft pairwise constraints to supervise the machine learning process. This supervision information is expressed using rather relaxed constraints named “should-link” constraints. Such constraints determine whether the pairs of data instances should be assigned to the same or different cluster(s). In fact, the clustering task was formulated as an optimization problem via the minimization of a novel objective function. Moreover, the proposed approach’s performance was assessed via extensive experiments using benchmark datasets. Furthermore, the proposed approach was compared to relevant state-of-the-art clustering algorithms, and the obtained results demonstrate the impact of using minimal previous knowledge about the data in improving the overall clustering performance.

Generation of Achievable Three-Dimensional Trajectories for Autonomous Wheeled Vehicles via Tracking Differentiators

Planning an achievable trajectory for a mobile robot usually consists of two steps: (i) finding a path in the form of a sequence of discrete waypoints and (ii) transforming this sequence into a continuous and smooth curve. To solve the second problem, this paper proposes algorithms for automatic dynamic smoothing of the primary path using a tracking differentiator with sigmoid corrective actions. Algorithms for setting the gains of the differentiator are developed, considering a set of design constraints on velocity, acceleration, and jerk for various mobile robots. When tracking a non-smooth primary path, the output variables of the differentiator generate smooth trajectories implemented by a mechanical plant. It is shown that the tracking differentiator with a different number of blocks also generates derivatives of the smoothed trajectory of any required order, taking into account the given constraints. Unlike standard analytical methods of polynomial smoothing, the proposed algorithm has a low computational load. It is easily implemented in real time on the on-board computer. In addition, simple methods for modeling a safety corridor are proposed, taking into account the dimensions of the vehicle when planning a polygon with stationary obstacles. Confirming results of numerical simulation of the developed algorithms are presented.

Constrained Density Peak Clustering

Clustering is a commonly used tool for discovering knowledge in data mining. Density peak clustering (DPC) has recently gained attention for its ability to detect clusters with various shapes and noise, using just one parameter. DPC has shown advantages over other methods, such as DBSCAN and K-means, but it struggles with datasets that have both high and low-density clusters. To overcome this limitation, the paper introduces a new semi-supervised DPC method that improves clustering results with a small set of constraints expressed as must-link and cannot-link. The proposed method combines constraints and a k-nearest neighbor graph to filter out peaks and find the center for each cluster. Constraints are also used to support label assignment during the clustering procedure. The efficacy of this method is demonstrated through experiments on well-known data sets from UCI and benchmarked against contemporary semi-supervised clustering techniques.

Sensitivity analysis of point neuron model simulations implemented on neuromorphic hardware.

With the ongoing growth in the field of neuro-inspired computing, newly arriving computational architectures demand extensive validation and testing against existing benchmarks to establish their competence and value. In our work, we break down the validation step into two parts-(1) establishing a methodological and numerical groundwork to establish a comparison between neuromorphic and conventional platforms and, (2) performing a sensitivity analysis on the obtained model regime to assess its robustness. We study the neuronal dynamics based on the Leaky Integrate and Fire (LIF) model, which is built upon data from the mouse visual cortex spanning a set of anatomical and physiological constraints. Intel Corp.'s first neuromorphic chip "Loihi" serves as our neuromorphic platform and results on it are validated against the classical simulations. After setting up a model that allows a seamless mapping between the Loihi and the classical simulations, we find that Loihi replicates classical simulations very efficiently with high precision. This model is then subjected to the second phase of validation, through sensitivity analysis, by assessing the impact on the cost function as values of the significant model parameters are varied. The work is done in two steps-(1) assessing the impact while changing one parameter at a time, (2) assessing the impact while changing two parameters at a time. We observe that the model is quite robust for majority of the parameters with slight change in the cost function. We also identify a subset of the model parameters changes which make the model more sensitive and thus, need to be defined more precisely.

A black box approach to fitting smooth models of mortality

Actuaries have long been interested in the forecasting of mortality for the purpose of the pricing and reserving of pensions and annuities. Most models of mortality in age and year of death, and often year of birth, are not identifiable so actuaries worried about what constraints should be used to give sensible estimates of the age and year of death parameters, and, if required, the year of birth parameters. These parameters were then forecast with an ARIMA model to give the required forecasts of mortality. A recent article showed that, while the fitted parameters were not identifiable, both the fitted and forecast mortalities were. This result holds if the age term is smoothed with P-splines. The present article deals with generalized linear models with a rank deficient regression matrix. We have two aims. First, we investigate the effect that different constraints have on the estimated regression coefficients. We show that it is possible to fit the model under different constraints in R without imposing any explicit constraints. R does all the necessary booking-keeping ‘under the bonnet’. The estimated regression coefficients under a particular set of constraints can then be recovered from the invariant fitted values. We have a black box approach to fitting the model subject to any set of constraints.

A Literature Review on the Optimal Placement of Static Synchronous Compensator (STATCOM) in Distribution Networks

The existing distribution networks were designed at a time when there was virtually no embedded generation. The design methods ensured the voltage at various parts of the network remained within the limits required by standards, and for the most part, this was very successfully achieved. As Distributed Energy Resources (DERs) started to grow, the rise in voltage due to injected currents and the local impedances started to push network voltages toward, and even above, the desired upper limits. Voltage limits are based on typical appliance requirements, and long-term over-voltages will ultimately result in unacceptably short appliance life spans. Distribution Static Compensators (dSTATCOMs) are shunt-connected devices that can improve low-voltage networks’ performance by injecting currents that do not transfer real power. The currents can be reactive, negative or zero sequence, or harmonic. System performance can be improved by reducing conduction loss, improving voltage profile and voltage balance, or reducing Total Harmonic Distortion (THD). To obtain these benefits, optimal sizes of dSTATCOMs need to be placed at optimal locations within the distribution network. This paper has considered seventy research articles published over the past years related to the optimal placement and sizing of dSTATCOMs. In this study, minimization of power losses, voltage profile improvement, loadablity factor, voltage sag mitigation, and reduction in annual operating costs are considered fitness functions that are subjected to multiple constraint sets. The optimization algorithms found in the literature are categorized into six methods: analytical methods, artificial neural network-based methods, sensitivity approaches, metaheuristic methods, a combination of metaheuristic and sensitivity analysis, and miscellaneous. This study also presents a comparison among distribution network types, load flow methods optimization tools, etc. Therefore, a comprehensive review of optimal allocation and sizing of dSTATCOMs in distribution networks is presented in this paper, and guidance for future research is also provided.

Exergy-Optimum coupling of radiant panels with heat pumps for minimum CO2 emission responsibility

This paper introduces an exergy-based optimization model for radiant heating or cooling panels coupled with heat pumps for minimum carbon dioxide emissions responsibility. The general idea that there exists an optimum tube spacing in radiant panels for minimum carbon dioxide emission responsibility in terms of simple life-cycle analysis and the first law of thermodynamics is challenged. Using the Rational Exergy Management Model, it is shown that a real optimum is unavailable according to the second law with or without using heat pumps or boilers for temperature peaking or chillers for cooling. It is hypothesized that the use of temperature-adjusting heat pumps is not an exergy-optimum choice. Through twelve sets of optimization constraints, identified in this paper, it is shown that potential optimality conditions concerning the increase of emission embodiments of the radiant panel material and labor with respect to narrover tube spacing versus an increase in the coefficient of performance of the heat pump, thus a decrease in operational emissions, lies outside the feasible design range. This paper also reminds that heat pumps destroy part of the high unit exergy of the electrical power demand by generating only low-exergy thermal energy. It is further shown that delegating solar photo-voltaic panels for thermal and electrical power supply as well as temperature peaking provides optimal solutions both for radiant panel heating and cooling, enabling wide use of low-enthalpy energy sources. Three case studies are presented. Case 1: Hydronic floor heating with temperature-peaking heat pump. This case has the highest emissions responsibility among other cases. It is + 0.095 kg CO2/kW-hexergy of heat delivered per unit panel area. Case 2: Oversized radiant floor heating with closer tube spacing without a heat pump. This case is responsible for + 0.035 kg CO2/kW-hexergy of heat delivered per unit panel area. 3-a: Heat-piped solar photo-voltaic-thermal panels with floor heating with heat pipes. This case has a negative emissions responsibility of −0.010 kg CO2/kW-hexergy of heat delivered. 3-b: This case is the radiant ceiling cooling option of Case 3-a. It is responsible for + 0.0010 kg CO2/kW-hexergy of sensible cooling. Results show that negative-carbon solar heating with radiant panels may be achieved with real roots available for optimum tube spacing. Optimal tube spacings for ceiling cooling panels also exist.