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This paper presents a real options approach to handling uncertainties associated with the long-term planning of water distribution system development. Furthermore, carbon emissions associated with the installation and operation of water distribution networks are considered. These emissions are computed by taking an embodied energy approach to the different materials used in water networks. A simulated annealing heuristic is used to optimise a flexible eco-friendly design of water distribution systems for an extended life horizon. This time horizon is subdivided into different time intervals in which different possible decision paths can be followed. The proposed approach is applied to a case study and the results are presented according to a decision tree. Lastly, some comparisons and results are used to demonstrate the quality of the results of this approach.

Water supply and distribution systems represent a major investment for a society, whether it is in the construction of new systems or the maintenance and rehabilitation of ageing infrastructure. For example, the cost of replacing ageing water infrastructure in the USA could reach more than $1 trillion over the next few decades (AWWA 2012). These systems also have to cope with future uncertainties, including growing populations, shifting consumption patterns and climate change. Therefore, constructing and maintaining water infrastructure with the aim of improving reliability and reducing costs is a difficult task and this is compounded by a number of associated environmental issues that should be addressed. Concern about global warming is increasing. Nations will need to act to dramatically reduce greenhouse gas emissions (GHG), specifically those countries that have signed and ratified the Kyoto Protocol of 2009. One hundred and ninety-two countries follow this protocol and have to limit and reduce carbon emissions over the coming decades. In Portugal, the most polluting industry is the electricity generation sector, based on ERSE (2012). Between 2005 and 2010, this sector was responsible for 55% of total carbon emissions. In this paper, we propose an approach that both handles environmental impacts, and tries to find appropriate flexible solutions for the design and operation of water distribution systems. McConnell (2007) defined system flexibility as ‘the ability for a system to actively transform, or facilitate a future transformation, to better anticipate or respond to changing internal or external conditions’. These problems are challenging and very difficult to solve. The real options (ROs) approach could be very useful in this field. Black & Scholes (1973) and Merton (1973) are the works that define and solve the financial option valuing problem. Inspired by them, Myers (1977) introduced ROs. This approach permits flexible planning, thus allowing decision makers to adjust investment according to new future information. ROs have already been utilised for: designing maritime security systems (Buurman et al. 2009); finding the optimal capacity for hydropower projects (Bockman et al. 2008); dam project investments (Michailidis & Mattas 2007); constructing a parking garage (De Neufville et al. 2006); and designing satellite fleets (Hassan et al. 2005). However, there are very few papers where ROs concepts are applied to water infrastructure: Woodward et al. (2011) used ROs for flood risk management and Zhang & Babovic (2012) used ROs for decision support in the design and management of a flexible water resources framework through innovative technologies. We propose a ROs approach to define the design of water distribution networks under different possible future conditions and taking carbon emissions into account. Several definitions are being used for direct and indirect carbon emissions. Alker et al. (2005) make the distinction between direct emissions, i.e., those from sources that are owned or controlled by water companies, and indirect emissions, which are a consequence of the activities of the water company but that occur at sources owned or controlled by another company and generated away from the water infrastructure site. In water supply systems, the source of a direct emission would be the excavation works for traditional pipe installation, because this process is under the water company's direct control. An indirect emission source would be the pipe manufacturing process, because this is controlled by another company. In the last decade, objectives focused on environmental issues have started to feature in water distribution networks optimisation works. The key work by Filion et al. (2004) has been followed by a vast body of literature. Some works analysed and compared the carbon emissions with different pipe material installation (e.g. Dandy et al. (2006) and Shilana (2011)) in a single objective framework. Wu et al. (2008) were the first work to introduce the goal of minimising GHG into the multiobjective optimal design of water networks. The works of Wu et al. (2010, 2011, 2013) report some developments and comparisons based on the multiobjective approach. Herstein et al. (2009) take the idea of concentrating different environmental impacts in a single measure and present an index-based method to evaluate the environmental impacts of water distribution systems. This environmental index aims to aggregate multiple environmental measures calculated by an economic input–output life-cycle assessment model. However, some criticism of this methodology has emerged (Herstein & Filion 2011). Herstein et al. (2010) and Herstein et al. (2011) include different optimisation models to minimise this index. Water distribution networks are usually planned and constructed to be operated over a long planning horizon and so annual operating costs should be discounted. MacLeod & Filion (2011) and Roshani et al. (2012) study the effect of reducing carbon emission pricing and discount rates on the design and operation of water distribution networks. Finally, Oldford & Filion (2013) have reviewed the policy and research initiatives that have been used to incorporate environmental impacts in the design and optimisation of water distribution systems. The aim is to develop a regulatory framework to limit these impacts during the design and operation of a water distribution system. Our approach calculates carbon emissions using a different procedure. In the literature, carbon emissions associated with pipe installation only include those related to pipe manufacturing. In our work, emissions are calculated by considering the manufacturing of pipes and by computing the emissions of other materials required for pipe installation. The emissions from tank construction are also computed and carbon emissions from energy consumption are calculated for the whole of the planning horizon. A simulated annealing heuristic has been used to solve the optimisation model. The problem addressed in our work is large, non-linear and complex and involves discrete decision variables; therefore modern heuristics (simulated annealing, genetic algorithms, particle swarm optimisation and so on) are suitable for its resolution. A literature review shows that simulated annealing has been used in various fields and good performances were observed. It has been successfully implemented in areas such as aquifer management (Cunha 1999), water treatment plants (Afonso & Cunha 2007), wastewater systems (Zeferino et al. 2012), rail network planning (Costa et al. 2013) and water distribution design (Cunha & Sousa 2001; Reca et al. 2007, 2008). Simulated annealing is an iterative process based on the Monte Carlo method and inspired by an analogy made between the annealing process as a metal cools into a minimum energy crystalline structure and a search for a global minimum solution in an optimisation problem. The simulated annealing approach used is based on Cunha & Sousa (1999, 2001), where a more detailed analysis of the parameterisation of this method and its application to the optimisation of water distribution networks can be found. In brief, the basic idea of simulated annealing rests on the analogy made between the temperature reduction of physical systems and the minimisation problem. The simulated annealing temperature is used in the Metropolis criterion (Metropolis et al. 1953) to accept uphill moves in terms of cost. The temperature starts at a high value so that a high proportion of attempted changes can be accepted. As the iterative process progresses, the temperature is reduced according to an annealing schedule, defined in our work by a geometric progression with a cooling factor of 0.90. A minimum number of iterations are required to reduce the temperature. In each temperature reduction, the proportion of accepted moves goes down until, finally, no uphill moves (in cost) are accepted. If the simulated annealing has been performed slowly enough the final solution could be the global minimum. The remainder of this paper is organised as follows: the following section sets out a methodology to compute the carbon emissions of a water network; next, the decision model is built, and then a case study is presented to examine the application of the methodology and to show some results. Finally, some comparisons are made and conclusions drawn. CARBON EMISSIONS OF WATER DISTRIBUTION SYSTEMS To incorporate carbon emission costs in the design and operation of the water networks, it is necessary to quantify emissions from the very beginning of the extraction of the materials that are used until their final disposal. Water distribution infrastructure is built from and maintained with a range of materials. The most common are the steel and cast iron used in pipes, accessories and pumps; reinforced concrete in civil construction works like tanks, manholes and anchorages; plastic in pipes and accessories; aggregates in pipeline backfill and asphalt for repaving. The carbon emissions of these materials can only be evaluated if the whole life cycle is involved, which includes the extraction of the raw material, transport, manufacturing, assembling, installation, dismantling, demolition and/or decomposition. The embodied energy is determined by the sum of the energy sources (fuels, materials, human resources and so on) that are used for product manufacturing and its use. The embodied energy tries to compute the sum of the total energy expended during all the life cycle of the product. Hammond & Jones (2008) present the embodied energy for the life cycle of some materials. Table 1 shows the embodied energy of the most common materials used in water distribution infrastructure. Table 1 Embodied energy of some materials used in water infrastructure Embodied energy MaterialMJ/kgkWh/kg Ductile iron for pipes 34.40 9.56 Aggregates 0.11 0.03 Asphalt 6.63 1.84 Concrete 2.91 0.81 Structural steel 28.67 7.96 Embodied energy MaterialMJ/kgkWh/kg Ductile iron for pipes 34.40 9.56 Aggregates 0.11 0.03 Asphalt 6.63 1.84 Concrete 2.91 0.81 Structural steel 28.67 7.96 From the data collected by Hammond & Jones (2008) and presented in Table 1, it is possible to compute the total amount of embodied energy needed to build new pipes and reservoirs. The quantities of materials needed for pipeline installation are computed based on the scheme in Figure 1. Some simplifications are assumed. The embodied energy to build the water network is determined from five materials: pipe material; aggregates to backfill pipes; asphalt for repaving, concrete and structural steel to build tanks. The units are expressed in KWh of energy per kg of material used. Figure 1 Scheme to compute quantities of materials (dimensions in metres). Figure 1 Scheme to compute quantities of materials (dimensions in metres). To determine the embodied energy of pipe construction in the traditional way, the quantity of energy per metre of pipe is considered. The weight of the materials used to settle 1 m of pipe must, therefore be determined. Given the scheme in Figure 1, we can calculate the volume of aggregates and asphalt needed for the settlement of each metre of pipe. The quantity of materials per metre is a function of the pipe's external diameter (ED), since the excavation and repaving volumes increase the higher the pipe diameter ED. We assume ductile iron pipes and Equation (1) is used to compute the embodied energy of the material: 1 where is the embodied energy of the pipe with commercial diameter Dc (kWh/m); is the weight of the commercial diameter Dc (kg/m); and is the embodied energy of the ductile iron for pipes (kWh/kg). The quantities of aggregate are a function of the commercial diameter that is to be used. The width of the trench is the same as the external diameter of the pipes plus 0.5 m. The walls of the trench are assumed to be vertical and the entire trench is filled with aggregates. Based on this, the quantity of embodied energy of aggregates is computed by Equation (2): 2 where is the embodied energy of aggregates to backfill a pipe with diameter Dc (kWh/m); is the external diameter of the pipe with diameter Dc (m); is the weight of aggregates, equal to 2,240 kg/m3; and is the embodied energy of the material (kWh/kg). Finally, the last material is asphalt; 0.2 m is assumed for the extra paving of each side of the trench. The embodied energy is computed by Equation (3): 3 where is the embodied energy of asphalt (kWh/m); is the weight of the asphalt, equal to 2,300 kg/m3; and is the embodied energy of asphalt (kWh/kg). To determine the total embodied energy (Equation (4)) per metre of installed pipe, Equations (1)–(3) are added together: 4 where is the total embodied energy of pipe installation (kWh/m). Now the embodied energy can be computed for the different commercial diameters, considering the contribution of the ductile iron pipes, aggregates to backfill the pipe and asphalt for repaving. The carbon emissions related to the total embodied energy can be computed through Equation (5): 5 where is the carbon emissions of installing pipes with commercial diameter Dc (tonCO2/m); and CET is the total carbon emissions from energy generation (tonCO2/kWh). Carbon emissions are computed assuming a value of CET = 0.637 × 10–3 tonCO2/kWh of energy produced by non-renewable resources and obtained by a fuel mix of 58% coal, 20% natural gas, 13% oil, 5% diesel and 4% of other resources. This is a mean value of the carbon emissions of electricity generation sector by non-renewable resources between 2005 and 2010 in Portugal (ERSE 2012). This work also considered the carbon emissions related to the installation of new tanks in the network. New tanks are assumed to be cylindrical and have the same transversal area of 500 m2. For simplification, the walls and the slabs of the tanks are assumed to have the same thickness (see Figure 2). Figure 2 Scheme for computing the concrete used in tank construction. Figure 2 Scheme for computing the concrete used in tank construction. The amount of concrete is a function of the volume of the tank. The thickness of the slabs and the walls is taken to be and the inner radius of the tank is . Based on these conditions, the quantity of embodied energy of concrete is computed by Equation (6): 6 where is the embodied energy of concrete of the tank t (kWh); is the radius of the slab of the tank, 12.62 m; is the thickness of the walls of the tank, 0.35 m; is the thickness of the slabs of the tank, 0.35 m; is the height of the tank (m); is the weight of concrete, 2,500 kg/m3; and is the embodied energy of concrete (kWh/kg). The embodied energy of reinforcing steel bars for the concrete of the tanks is also considered. For this study, the quantity of steel is taken to be a percentage of the cubic metres of concrete used in civil construction works, so the embodied energy of this material is given by Equation (7): 7 where is the embodied energy of steel bars to build the tank t (kWh); is the quantity of steel per cubic metre of concrete, 100 kg/m3; and is the embodied energy of steel bars (kWh/kg). Summing the values given by Equations (6) and (7), the carbon emissions derived from constructing the tanks are determined through Equation (8): 8 where CETKt is the carbon emissions of the tank t (tonCO2). In addition to the above, significant carbon emissions arise from generating the electric energy consumed during the water infrastructure operation. Large amounts of energy are consumed resulting in important carbon emissions that should be measured by Equation (9): 9 where CEop is the carbon emissions from energy used in the operation of the network (tonCO2); and is the energy consumption of the network during the operation (kWh). Equation (9) computes carbon emissions generated by network operation. This work does not take into account carbon emissions related to other network elements that are negligible when compared with pipe and tank construction. By adding together the individual contributions of pipes, tanks and energy consumption we can determine the cost in terms of total carbon emissions of the water network life cycle. This cost is included in the optimisation model presented in the next section. OPTIMISATION MODEL Many scenarios are possible over the life cycle of a water distribution infrastructure. The future operating conditions of the water networks are uncertain. However, decisions have to be made and there are some constraints that further increase the complexity of the problem. The optimisation of a water distribution network is very complex because the objective is to find a good solution within an enormous solution space. Furthermore, the decision variables are normally discrete, which makes it even harder to find optimum solutions. The approach we describe uses ROs to handle different possible scenarios that can occur during the life cycle of the infrastructure. According to Wang et al. (2004), the ROs approach has two stages: option identification and option analysis. Option identification consists of trying to find all possible scenarios for the lifetime horizon. The option analysis stage can use an optimisation model to find possible solutions. This formulation enables decision makers to include additional possible situations simultaneously and to develop different decision plans throughout the life cycle. The objective function, OF, includes the minimisation of the costs and carbon emissions resulting from implementing and operating the network. The objective function is presented in Equation (10): 10 where Cinitial is the cost of the initial solution to be implemented in year zero; NS is the number of scenarios; NTI is the number of time intervals into which the life cycle is subdivided; is the future design costs for time t in scenario s; Probnt,s is the probability of future design in time nt in scenario s; CEinitial is the carbon emissions of the initial solution to be applied in year zero; is the carbon emissions for time t in scenario s; and CEC is the carbon emissions cost. The objective function given by Equation (10) has to find the first stage solution, T = 1, and future decisions to be implement. The objective function is given by the sum of different terms. The initial solution cost is given by Equation (11): 11 where NPI is the number of pipes in the network; is the unit cost of pipe i as function of the diameter Di,1 adopted; Di,1 is the diameter of pipe i installed in time interval T = 1; Li is the length of pipe i; NT is the number of new tanks in the network; CTt is the cost of tank t; is the unit cost to rehabilitate existing pipe i as a function of diameter Di,1; NPU is the number of pumps in the network; is the equipment cost of pump j for time interval T = 1; NDC is the number of demand conditions considered for the design; Ced is the cost of energy for demand condition d; γ is the specific weight of water; QPj,d,1 is the discharge of pump j for demand condition d; and time interval T = 1; HPj,d,1 is the head of pump j for demand condition d and time interval T = 1; ηj is the efficiency of pump j; Δtd is the time in hours for demand condition d; IR is the annual interest rate for updating the costs; and NYt is the number of years under the same conditions considered for time interval T = 1. The term Cinitial (Equation (11)) computes the network cost for the first stage. This term is given by the sum of the cost of pipes, the cost of tanks, the rehabilitation cost of the existing pipes, the cost of new pumps and the present value energy cost. The pump cost is given by Equation (12): 12 where is the cost of the pump; Q is the flow of pump (m3/s); and is the head of pump (m). The other term of the objective function is given by the weighted sum of the future costs. The future cost is computed by Equation (13): 13 The future cost is computed for all time intervals beginning at T = 2 (the cost is already computed for the first time interval) and is given as the sum of three terms. The first term computes the present value cost of the pipes to be laid in the different time intervals and scenarios, the second term computes the present value equipment cost of the pumps for the different time intervals and for the different scenarios, and finally, the third term computes the present value of energy cost for each scenario. The sum of the initial and the future costs give the network cost for the entire time horizon, considering future uncertainty. Looking at events on statistically independent decision nodes, the probabilities for the different scenarios can be computed by the product of the probabilities of the decision nodes in each path for all the time periods. Finally, a term to compute the environmental impacts of the water supply system is also added. This term is computed as the sum of two terms multiplied by the carbon emission cost, CEC. These terms are introduced in Equations (14) and (15). 14 15 Equation (14) computes the total carbon emissions for the first operation period and Equation (15) computes the carbon emissions for the different future scenarios weighted by their probability of occurrence. The initial carbon emissions are calculated by adding together the carbon emissions related to the pipe installation, tank construction and energy consumption. The carbon emissions in the future scenarios are computed using a similar procedure. These emissions are multiplied by the carbon emission cost for each tonCO2 (CEC). It should be noted that the carbon emissions costs are not updated. A zero discount rate should be used for carbon emissions (Wu et al. 2010). This complies with the recommendation of the Intergovernmental Panel on Climate Change (IPCC). High carbon emissions degrade air quality and thus it seems prudent and ethical to think about future generations and assign the same importance (or value) to the carbon emissions of today as well as those in future. A zero discount rate implies the same weight for current and future costs. The objective function proposed in (Equation (10)) aims at minimising the initial and the future costs (Equations (11) and (13)) and initial and future carbon emissions (Equations (14) and (15)). The constraints of the model are those commonly used in the optimal design of water distribution networks and can be consulted in the work of Cunha & Sousa (2001). Some decisions have to be taken now, but others can be delayed until new information is available. The ROs framework enables water infrastructure to be designed with some decisions postponed to a future date. CASE STUDY A well-known water network was used to demonstrate the application of the ROs approach. The case study was based on a hypothetical network inspired by Walski et al. (1987). The network aims to represent an old town, small in size (see Figure 3). Figure 3 Scheme of the network (inspired from Walski et al. 1987). Figure 3 Scheme of the network (inspired from Walski et al. 1987). Figure 3 shows a water distribution network planned for the next 60 years. However, this planning horizon is subdivided into three time intervals of 20 years. In the first 20 years of operation, some decisions have to be made. The water company is held to need to improve the network capacity to satisfy future demand during the first 20-years time interval. However, eight different possible future scenarios could be considered, as shown in Figure 4. Figure 4 Decision tree and probabilities of occurrence for the life cycle. Figure 4 Decision tree and probabilities of occurrence for the life cycle. This work considers a number of expansion areas. For T = 2, the authorities are planning to build a new industrial area (NIA) and a new public services area (NPA) with some facilities near the river, so in this time interval the network may be extended to those two areas. For T = 3, it is predicted that a new residential area (NRA) may be developed close to the industries and public services, because of the labour required by the new industries and the public services facilities. However, if these areas are not built the area near the river may see a decline in population and the water consumption could fall to 75%. The areas in question are shown in Figure 3. Finally, the probabilities for each path of the different scenarios should be indicated. The probabilities for the different paths of the systems for the case study are shown in Figure 4. The probabilities of the scenarios are computed by the product for all the time periods of the decision node probabilities in each path. The network has two tanks operating with water levels between the elevations of 65.53 and 77.22 m and each with a capacity of 1,136 m3, but according to the original case study the company wants to operate the tanks between 68.58 and 76.20 m. The volume between 65.53 and 68.58 m is used for emergency needs and amounts to a volume of 284 m3 in each tank. A minimum pressure of 28.14 m is required at all nodes for average daily flow conditions, and the instantaneous peak flow is given as the average nodal demand multiplied by 1.8. The system is also subject to three different firefighting conditions, each lasting 2 h. The minimum nodal pressures under firefighting conditions are 14.07 m. The firefighting conditions are: 157.73 L/s at node 9; 94.64 L/s at nodes 18, 20, 21; and 63.09 L/s at nodes 12 and 16. These fire flows should be met simultaneously with a daily peak flow 1.3 times the average flow. All the pressure requirements should be assured when one pump is out of service and the tanks are at the minimum levels after a normal operating day. This problem is solved by considering the design and operation of the network simultaneously. The city has grown up around an old centre located to the southeast of link 14. Excavations in this area cost more than in other areas. There is an adjacent residential area with some industries near node 16. The reinforcement possibilities are to duplicate existing pipes, clean and line existing pipes, install new pumps and build new tanks. The city is supplied from a water treatment plant and three identical pumps connected in parallel. Pumps have to be replaced every 20 years, but according to the original case study, there are already pumps in the first time interval and there is no cost associated with installation. The possibility of installing two additional pumps in parallel is considered if additional capacity is required. The water treatment plant is maintained at a fixed level of 3.048 m. The characteristics of the links are given in Table 2. Table 2 Characteristics of the pipes PipeInitial nodeFinal nodeLength (m)Existing diameterArea 3,657.60 406.4 Urban 3,657.60 304.8 Residential 11 3,657.60 304.8 Urban 2,743.20 304.8 Residential 10 1,828.80 304.8 Urban 1,828.80 254.0 Urban 1,828.80 304.8 Urban 1,828.80 254.0 Urban 1,828.80 304.8 Urban 10 1,828.80 254.0 Urban 11 15 1,828.80 254.0 Urban 12 10 1,828.80 254.0 Urban 13 10 15 1,828.80 304.8 Urban 14 15 1,828.80 254.0 Urban 15 1,828.80 254.0 Residential 16 1,828.80 254.0 Residential 17 2,743.20 254.0 Residential 18 1,828.80 254.0 Residential 19 1,828.80 254.0 Residential 20 14 1,828.80 254.0 Residential 21 14 15 1,828.80 203.2 Residential 22 15 16 1,828.80 203.2 Residential 23 10 16 1,828.80 203.2 Residential 24 10 11 1,828.80 203.2 Urban 25 11 16 1,828.80 254.0 Residential 26 11 12 1,828.80 203.2 Residential 27 12 16 2,743.20 New 28 12 13 1,828.80 203.2 Residential 29 13 16 1,828.80 254.0 Residential 30 13 17 1,828.80 203.2 Residential 31 14 16 1,828.80 203.2 Residential 32 14 17 3,657.60 203.2 Residential 33 14 3,657.60 203.2 Residential 34 23 30.48 762.0 Urban 35 19 30.48 304.8 Urban 36 16 22 30.48 304.8 Residential 37 23 Pump 38 23 Pump 39 23 Pump 40 14 21 1,828.80 New 41 14 20 1,828.80 New 42 20 21 1,828.80 New 43 18 1,828.80 New 44 18 20 1,828.80 New 45 24 1,828.80 New 46 24 25 1,828.80 New 47 25 1,828.80 New 48 25 26 1,828.80 New 49 26 1,828.80 New 50 26 27 1,828.80 New 51 27 18 1,828.80 New PipeInitial nodeFinal nodeLength (m)Existing diameterArea 3,657.60 406.4 Urban 3,657.60 304.8 Residential 11 3,657.60 304.8 Urban 2,743.20 304.8 Residential 10 1,828.80 304.8 Urban 1,828.80 254.0 Urban 1,828.80 304.8 Urban 1,828.80 254.0 Urban 1,828.80 304.8 Urban 10 1,828.80 254.0 Urban 11 15 1,828.80 254.0 Urban 12 10 1,828.80 254.0 Urban 13 10 15 1,828.80 304.8 Urban 14 15 1,828.80 254.0 Urban 15 1,828.80 254.0 Residential 16 1,828.80 254.0 Residential 17 2,743.20 254.0 Residential 18 1,828.80 254.0 Residential 19 1,828.80 254.0 Residential 20 14 1,828.80 254.0 Residential 21 14 15 1,828.80 203.2 Residential 22 15 16 1,828.80 203.2 Residential 23 10 16 1,828.80 203.2 Residential 24 10 11 1,828.80 203.2 Urban 25 11 16 1,828.80 254.0 Residential 26 11 12 1,828.80 203.2 Residential 27 12 16 2,743.20 New 28 12 13 1,828.80 203.2 Residential 29 13 16 1,828.80 254.0 Residential 30 13 17 1,828.80 203.2 Residential 31 14 16 1,828.80 203.2 Residential 32 14 17 3,657.60 203.2 Residential 33 14 3,657.60 203.2 Residential 34 23 30.48 762.0 Urban 35 19 30.48 304.8 Urban 36 16 22 30.48 304.8 Residential 37 23 Pump 38 23 Pump 39 23 Pump 40 14 21 1,828.80 New 41 14 20 1,828.80 New 42 20 21 1,828.80 New 43 18 1,828.80 New 44 18 20 1,828.80 New 45 24 1,828.80 New 46 24 25 1,828.80 New 47 25 1,828.80 New 48 25 26 1,828.80 New 49 26 1,828.80 New 50 26 27 1,828.80 New 51 27 18 1,828.80 New The average daily water demand for nodes is presented in Table 3 along with the elevation of the nodes and tanks. Table 3 Characteristics of the nodes NodeElevation (m)Average day demand (L/s)NodeElevation (m)Average day demand (L/s) 3.05 WTP 15 36.58 24.236 6.10 31.545 16 36.58 63.090 15.24 12.618 17 36.58 25.236 15.24 12.618 18 24.38 37.854 15.24 37.854 19 65.53 Tank 15.24 31.545 20 24.38 37.854 15.24 31.545 21 24.38 37.854 15.24 31.545 22 65.53 Tank 15.24 63.090 23 3.05 0.000 10 15.24 31.545 24 15.24 37.854 11 15.24 31.545 25 15.24 37.854 12 36.58 24.236 26 15.24 12.618 13 36.58 24.236 27 15.24 12.618 14 24.38 24.236 NodeElevation (m)Average day demand (L/s)NodeElevation (m)Average day demand (L/s) 3.05 WTP 15 36.58 24.236 6.10 31.545 16 36.58 63.090 15.24 12.618 17 36.58 25.236 15.24 12.618 18 24.38 37.854 15.24 37.854 19 65.53 Tank 15.24 31.545 20 24.38 37.854 15.24 31.545 21 24.38 37.854 15.24 31.545 22 65.53 Tank 15.24 63.090 23 3.05 0.000 10 15.24 31.545 24 15.24 37.854 11 15.24 31.545 25 15.24 37.854 12 36.58 24.236 26 15.24 12.618 13 36.58 24.236 27 15.24 12.618 14 24.38 24.236 Demand varies during an operating day. Table 4 shows the demand variation in 24 h. For example, between 0 and 3 h the demand is 70% of the average daily demand. Table 4 Variation of demand during 24 h operation Daily period (h)Demand 0–3 0.7 3–6 0.6 6–9 1.2 9–12 1.3 12–15 1.2 15–18 1.1 18–21 1.0 21–24 0.9 Daily period (h)Demand 0–3 0.7 3–6 0.6 6–9 1.2 9–12 1.3 12–15 1.2 15–18 1.1 18–21 1.0 21–24 0.9 It is possible to duplicate or clean and line 35 pipes. There are also 13 new links in the expansion areas. The commercial diameters and the unit cost of new pipes, cleaning and lining, as function of the network area, are given in Table 5. Table 5 Diameters and unit cost Unit cost Installation of pipesCleaning and lining existing pipes Pipe diameter (mm)Urban ($/m)Residential ($/m)New ($/m)Urban ($/m)Residential ($/m)
152.4 85.958 46.588 41.995 55.774 39.370
203.2 91.207 64.961 58.399 55.774 39.370
254.0 111.877 82.349 73.819 55.774 39.370
304.8 135.827 106.299 95.801 55.774 42.651
355.6 164.698 131.890 118.766 59.711 46.588
406.4 191.929 159.121 143.045 64.961 50.853
457.2 217.192 187.664 168.963 70.866 56.102
508.0 251.969 219.160 197.178 77.100 66.273
609.6 358.268 280.512 252.625 98.753
762.0 467.520 380.906 346.129 135.499
Unit cost
Installation of pipesCleaning and lining existing pipes
Pipe diameter (mm)Urban ($/m)Residential ($/m)New ($/m)Urban ($/m)Residential ($/m) 152.4 85.958 46.588 41.995 55.774 39.370 203.2 91.207 64.961 58.399 55.774 39.370 254.0 111.877 82.349 73.819 55.774 39.370 304.8 135.827 106.299 95.801 55.774 42.651 355.6 164.698 131.890 118.766 59.711 46.588 406.4 191.929 159.121 143.045 64.961 50.853 457.2 217.192 187.664 168.963 70.866 56.102 508.0 251.969 219.160 197.178 77.100 66.273 609.6 358.268 280.512 252.625 98.753 762.0 467.520 380.906 346.129 135.499 If a pipe has been cleaned and lined, the Hazen–Williams coefficient is then C = 125, and if there is a new pipe, it is C = 130. Over the life cycle, pipes age and wall roughness increases. Based on the DWSD (2004) report, the Hazen–Williams coefficients of ductile iron pipes decrease at a fixed rate of 2.5 per decade. Obviously this rate depends on all kinds of different conditions and is also time dependent. However, to simplify the problem we have assumed a fixed rate for the life cycle. The 24 h operation of the network is subdivided into 1 h time steps. Three pumps have to supply the daily needs. This work considers the possibility of installing two extra parallel pumps because of planned building of new areas. The number of the pumps used in the 24 h results in additional variables to solve in the optimisation problem, in each time interval and for each scenario. Table 6 gives five points of the characteristic curves for each pump. These curves are the same as in the original case study. Table 6 Function points of each pump Flow (L/s)Pump head (m)Efficiency (%) 91.5 126.2 89.1 50 252.4 82.4 65 378.5 70.2 55 504.7 55.2 40 Flow (L/s)Pump head (m)Efficiency (%) 91.5 126.2 89.1 50 252.4 82.4 65 378.5 70.2 55 504.7 55.2 40 The energy costs are$0.12 per kWh. The present value costs are computed using a discount rate of 4% over the life cycle. According to Wu et al. (2010), defining discount rates is a very complex issue and they normally vary from 2 to 10%. This work takes a 4% rate to emphasise the importance of the future costs in the decision-making process. There is also the possibility of installing new tanks at the nodes in the network. Tanks are connected to nodes by a short pipe 30.48 m long whose pipe diameters varies. Tank cost is a function of the volume and is given in Table 7. These data are the same as in the original case study.

Table 7

Tank cost

Volume (m3)Cost × 103 ($) 227.3 115 454.6 145 1,136.5 325 2,273.0 425 4,546.0 600 Volume (m3)Cost × 103 ($)
227.3 115
454.6 145
1,136.5 325
2,273.0 425
4,546.0 600

Finally, it is held that the tank installation and rehabilitation of the existing pipes can only occur in the first time interval and has to perform well relative to all the possible future conditions given in Figure 5. Based on Equation (4), the embodied energy is calculated for different commercial diameters used in this work and is shown in Table 8.

Table 8

Embodied energy and carbon emissions arising from installing commercial diameters

Diameters (mm)Ductile iron pipes (kWh/m)Aggregates (kWh/m)Asphalt (kWh/m)Embodied energy (kWh/m)Total emissions (tonCO2/m)
152.4 269.88 44.91 445.38 760.17 0.48
203.2 406.20 49.95 466.87 923.03 0.59
254.0 575.89 55.07 488.37 1,119.33 0.71
304.8 705.15 60.26 509.87 1,275.27 0.81
355.6 776.37 65.52 531.37 1,373.26 0.87
406.4 890.32 70.86 552.87 1,514.05 0.96
457.2 1,004.37 76.27 574.37 1,655.01 1.05
508.0 1,118.33 81.75 595.87 1,795.95 1.14
609.6 1,346.24 92.95 638.86 2,078.05 1.32
762.0 1,688.10 110.30 703.36 2,501.77 1.59
Diameters (mm)Ductile iron pipes (kWh/m)Aggregates (kWh/m)Asphalt (kWh/m)Embodied energy (kWh/m)Total emissions (tonCO2/m)
152.4 269.88 44.91 445.38 760.17 0.48
203.2 406.20 49.95 466.87 923.03 0.59
254.0 575.89 55.07 488.37 1,119.33 0.71
304.8 705.15 60.26 509.87 1,275.27 0.81
355.6 776.37 65.52 531.37 1,373.26 0.87
406.4 890.32 70.86 552.87 1,514.05 0.96
457.2 1,004.37 76.27 574.37 1,655.01 1.05
508.0 1,118.33 81.75 595.87 1,795.95 1.14
609.6 1,346.24 92.95 638.86 2,078.05 1.32
762.0 1,688.10 110.30 703.36 2,501.77 1.59
Figure 5

Decision tree design of Anytown network.

Figure 5

Decision tree design of Anytown network.

Table 8 shows the embodied energy computed for the different commercial diameters, considering the contribution of the ductile iron pipes, aggregates for pipe bedding and asphalt for repaving works. The last column (right) of the table shows the carbon emissions of the total embodied energy. The optimisation model described here is intended to minimise the installation cost of pipes, pumps and tanks, the energy cost and the carbon cost. The carbon emission costs are calculated assuming a carbon tax given by a value associated with each carbon tonne emitted. This study takes $5 as reference value and is defined according to European Union allowances market, but different values can be easily accommodated by the model. RESULTS The approach described here uses ROs to minimise the life cycle costs of water distribution systems, taking uncertainty into consideration. When a long time horizon is considered, the future is unknown. The water demand will certainly vary considerably. New urban developments can be built and others can become depopulated. The ROs approach can handle these uncertainties and give decision makers good design solutions for flexible water networks. This work uses a decision tree with eight possible different scenarios that may occur over the 60-years life cycle. However, it is only necessary to decide the configuration of the network for the first time period of 20 years. The solution of this period should not only work well in the first stage, but also take into account future (uncertain) needs. This is a robust solution that will be adapted in the subsequent time intervals as circumstances evolve. The hydraulic simulator EPANET (Rossman2000) has been used in the optimisation process whenever hydraulic constraints have to be verified. Figure 5 gives the solution achieved by the approach described. The results are represented in a life cycle tree that has the same shape as the decision-making alternatives exposed in Figure 4. Figure 5 summarises the design achieved for the case study. A table is presented for each node with the results of the design, starting by showing the pipe rehabilitation decisions, the new parallel pipes and the tank locations and capacities. The present value costs are subdivided into the cost of the pipes, tanks, pumps, energy, carbon emissions and total costs. The last branches of the decision tree represent the total life cycle cost for each of the scenarios. It can be concluded from the results that the life cycle cost depends on the decisions that are taken in the time intervals. However, the first time interval of 0–20 years accounts for most of the investment costs. In this time interval, the network will be reinforced with some new parallel pipes, new tanks and the cleaning and lining of existing pipes. The total cost takes the carbon emissions arising from the installation of pipes and tanks and from energy consumption into account. The solution for scenario 1 is schematised in Figure 6. Figure 6 Scheme of the network for the last time interval of scenario 1. Figure 6 Scheme of the network for the last time interval of scenario 1. For scenario 1, the water distribution network will be expanded in the second time interval to cope with the NIA and the NPA. Furthermore, the network will be expanded for the NRA in the last time interval. Figure 6 shows the pipes that will be cleaned, the diameters of the new parallel pipes and the diameters of the pipes installed in the new areas. The location of the new tanks and the inclusion of two additional parallel pumps are also shown. These interventions will result in a total life cycle cost of$46,975,016, including the carbon emissions cost of the construction and operation of the water distribution network. This is the most expensive solution. However, if the life cycle does not follow the decision path of scenario 1 then other interventions will occur. In the case of scenario 8, the network does not need to expand to new areas, so the life cycle cost is approximately 10% lower than for scenario 1. The ROs solution can handle uncertainties according to the life tree and adapt the solution to new requirements.

The ROs solution for the first time interval has to be implemented at year zero. To show that considering carbon emissions in the optimisation model has an impact on the final solution, a comparison is made of the first time interval solution with and without carbon emissions costs. If the carbon emission costs are taken as zero, different results are obtained. Table 9 shows some comparisons regarding costs.

Table 9

Comparison of solutions with and without carbon emission costs

CostsWith CO2 costsWithout CO2 costs
Pipes 8,931,410 8,010,350
Tanks 1,650,783 1,324,100
Pumps 3,118,800 3,118,800
Energy 12,125,541 13,393,570
CO2 1,073,035
Total 26,899,569 25,846,820
CostsWith CO2 costsWithout CO2 costs
Pipes 8,931,410 8,010,350
Tanks 1,650,783 1,324,100
Pumps 3,118,800 3,118,800
Energy 12,125,541 13,393,570
CO2 1,073,035
Total 26,899,569 25,846,820

If carbon emission costs are taken into account, the total cost is high, but it can be seen that the difference is practically accounted for by the carbon emission costs. However, other conclusions can also be drawn. Most of the carbon emissions are derived from the energy consumed by the pumps. If carbon costs are not included, the optimisation model will find solutions that have high energy costs with some reduction in pipe and tank costs. Table 9 shows that if the total cost of the pipes, tanks, pumps and energy are kept practically the same, the consideration of carbon emissions implies allocating the costs in a different way, i.e., by increasing the cost of the pipes and tank and decreasing the energy cost. Larger diameter pipes allow the energy expenditure to be cut, with a consequent reduction in the total carbon emissions.

CONCLUSIONS

The scientific community has made efforts in recent years to find tools to optimise water network design and operation. Water distribution infrastructure has a high cost and is essential to people's well-being. This work has tried to find good solutions for water distribution networks that may operate under uncertain future scenarios, and considering the carbon emission costs generated by installation and operation works.

The application of the ROs approach has been examined in the search for a flexible, robust solution to a water distribution network design and operation problem that includes the carbon emission costs. The problem consisted of finding the minimum cost solution for a design whose variables included additional new pipes, cleaning and lining existing pipes, replacement of existing pipes, siting and sizing of new tanks and installing and operating pumps. The optimisation algorithm was based on simulated annealing, a method that can be successfully applied to solve such problems.

The results indicate that the ROs approach is able to identify good solutions for flexible networks. The simultaneous optimisation of the network and carbon emission costs achieves solutions that take into account the environmental impacts of the networks. The solution presented provides flexibility to the network and automatically minimises the carbon emissions. The solution was obtained using the life cycle decision tree. It can also be concluded that if carbon emission costs are considered it is possible to find solutions with practically the same investment costs but with lower carbon emissions. This is achieved by higher investment cost and lower spending on energy. Further improvements can still be achieved by considering better carbon emission estimations and comparing the results for real networks.

ACKNOWLEDGEMENTS

This work was financed by FEDER funds through the Programa Operacional Factores de Competitividade – COMPETE, and by national funds from FCT – Fundação para a Ciência e Tecnologia under grant PTDC/ECM/64821/2006. The participation of the first author in the study is supported by FCT – Fundação para a Ciência e Tecnologia through Grant SFRH/BD/47602/2008.

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text:
The one-bit-matching conjecture for independent component analysis (ICA) has been widely believed in the ICA community. Theoretically, it has been proved that under the assumption of zero skewness for the model probability density functions, the global maximum of a cost function derived from the typical objective function on the ICA problem with the one-bit-matching condition corresponds to a feasible solution of the ICA problem. In this note, we further prove that all the local maximums of the cost function correspond to the feasible solutions of the ICA problem in the two-source case under the same assumption. That is, as long as the one-bit-matching condition is satisfied, the two-source ICA problem can be successfully solved using any local descent algorithm of the typical objective function with the assumption of zero skewness for all the model probability density functions.

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text:
# I give up, I am embracing pie charts

November 27, 2012
By

(This article was first published on Simply Statistics » R, and kindly contributed to R-bloggers)

Most statisticians know that pie charts are a terrible way to plot percentages. You can find explanations here, here, and here as well as the R help file for the pie function which states:

Pie charts are a very bad way of displaying information. The eye is good at judging linear measures and bad at judging relative areas. A bar chart or dot chart is a preferable way of displaying this type of data.

I have only used the pie R function once and it was to make this plot (R code below):

So why are they ubiquitous? The best explanation I've heard is that they are easy to make in Microsoft Excel. Regardless, after years of training, lay people are probably better at interpreting pie charts than any other graph. So I'm surrendering and embracing the pie chart. Jeff's recent post shows we have bigger fish to fry.

 N=10 for(i in 0:(N-1)){ x=cos(2*pi/N*i);y=x+1;z=(y-2)*22.5 bitmap(sprintf("Rplot%02d.png",i)) pie(c(y,8-y),col=c("white","yellow"),init.angle=135-z,border=FALSE,labels=NA) dev.off() } system("convert Rplot*.png pacman.gif") ##system("rm *.png") edited to save caffo's pngs (see comments) system("rm Rplot*.png")


text:
## Thursday, October 1, 2009

### Latex Tip: Author Affiliations

I've been writing in LaTeX a lot recently, and found a nice little package to use when composing a document with authors with different affiliations.  Often, author order is affiliation independent, so the authblk package makes all this possible.  Here's the sample code:

\documentclass[10pt]{article}

\usepackage{authblk}
\renewcommand\Affilfont{\itshape\small}

\begin{document}

\title{Author Affiliations in \LaTeX}
\author[,1]{Arthur A. Filliation\footnote{aaf@mail.com}}
\author[,2]{Arthur A. Filliation Jr.\footnote{aaf2@mail.com}}
\author[1]{Con T. Ributor}
\affil[1]{Department of Latex Studies.}
\affil[2]{Typesetters of America}
\date{\today}
\maketitle

\end{document}

A few notes.  First, renewing the command \Affilfont allows you to change the look of the affiliation list.   However, the small italic font seems pretty standard, so this is what I've always used.  Second, the option following the \author command adds a superscript corresponding to one of the affiliations stated by the \affil command. When using this command while also adding a footnote for contact information, the two superscripts will need to be separated by a comma.  You'll have to manually put this comma in after the \author command.  When all is said and done, you should get this:

1. Thanks for the tip

2. The footnotes don't show up for me. Any ideas? Otherwise cool.

3. DJ P: The only thing I could think of is that you have something suppressing all footnotes... does the regular \footnote{} command work for you? Perhaps you are using \pagestyle{empty} in your preamble?

4. 一個人就像一個分數，他的實際才能是分子，他對自己的評價是分母。分母越大，則分數的價值越小。..................................................

5. 「不可能」這個字詞，在聰明人的字典中是找不到的。......................................................

6. 人生最重要的一件事，就是從生活中認識你自己。 ..................................................

7. 向著星球長驅直進的人，反比踟躕在峽路上的人，更容易達到目的。............................................................

8. 快樂與滿足的秘訣，就在全心全意投注於現在的每一分，每一秒上..................................................

9. Riches serve a wise man but command a fool.............................................................

10. 一個人的價值，應該看他貢獻了什麼，而不是他取得了什麼....................................................

11. 愛情是一種發明，需要不斷改良。只是，這種發明和其他發明不一樣，它沒有專利權，隨時會被人搶走。.................................................................


text:
# Problem 19. Prove that there exists a Jordan measurable set contained in IR which is not...

###### Question:

Problem 19. Prove that there exists a Jordan measurable set contained in IR which is not a Borel set.

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What is the implicit derivative of 1=ytanx-y^2?...
##### Price discrimination O a. increases total profit for the firm. O b. increases consumer surplus. O...
Price discrimination O a. increases total profit for the firm. O b. increases consumer surplus. O c. Both a) and b) Od. None of the above....


text:
# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
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Landau’s theorem for biharmonic mappings. (English) Zbl 1156.31001
The article ‘Landau’s theorem for biharmonic mappings’, having as authors Z. Abdulhadi and Y. Abu Muhanna, contains: - Introduction - Schwarz lemma - Theorem 1 and Theorem 2 of the authors. The Introduction states the necessary and sufficient condition that the continuously differentiable complex-valued function must be biharmonic in a simply connected domain $D$, using an article of S. Khuri and the authors [J. Inequal. Appl. 2005, No. 5, 469–478 (2005; Zbl 1100.30006)]. This theorem reduces the discussion of the ‘biharmonic mapping’ condition to harmonic functions (or to a certain linear combination of harmonic functions). The Schwarz lemma determines the boundary of function $f$ a harmonic mapping of the unit disk. Theorem 1 and Theorem 2 of the present article’s authors show that Landau’s theorem extends to the bounded biharmonic mapping of the unit disk. Thus, Theorem 1 shows that if is a biharmonic mapping in (unit disk), boundary conditions take place for the module of harmonic functions and , where the function is supposed to be univalent in a disk with ray . The size of ray depends on the boundary of the module of functions. Theorem 2 only considers function having the module bounded by and univalent in the disk with ray. The equation which determines the size of ray is cubic and it depends on boundary. The demonstrations for Theorem 1 and Theorem 2 consider functions and to be analytical and use their developing into power series in the interior of a disk completely included into the unit disk. The variation of function in both theorems is calculated using, the line-segment in the disk. The authors state at every given theorem that: ‘this result is not sharp’.

##### MSC:
 31A05 Harmonic, subharmonic, superharmonic functions (two-dimensional)
##### Keywords:
biharmonic; harmonic; univalent


text:
#### From Quantity to Quality: Massive Molecular Dynamics Simulation of Nanostructures under Plastic Deformation in Desktop and Service Grid Distributed Computing Infrastructure

##### Olexander Gatsenko, Lev Bekenev, Evgen Pavlov, Yuri G. Gordienko

The distributed computing infrastructure (DCI) on the basis of BOINC and EDGeS-bridge technologies for high-performance distributed computing is used for porting the sequential molecular dynamics (MD) application to its parallel version for DCI with Desktop Grids (DGs) and Service Grids (SGs). The actual metrics of the working DG-SG DCI were measured, and the normal distribution of host performances, and signs of log-normal distributions of other characteristics (CPUs, RAM, and HDD per host) were found. The practical feasibility and high efficiency of the MD simulations on the basis of DG-SG DCI were demonstrated during the experiment with the massive MD simulations for the large quantity of aluminum nanocrystals ($\sim10^2$-$10^3$). Statistical analysis (Kolmogorov-Smirnov test, moment analysis, and bootstrapping analysis) of the defect density distribution over the ensemble of nanocrystals had shown that change of plastic deformation mode is followed by the qualitative change of defect density distribution type over ensemble of nanocrystals. Some limitations (fluctuating performance, unpredictable availability of resources, etc.) of the typical DG-SG DCI were outlined, and some advantages (high efficiency, high speedup, and low cost) were demonstrated. Deploying on DG DCI allows to get new scientific $\it{quality}$ from the simulated $\it{quantity}$ of numerous configurations by harnessing sufficient computational power to undertake MD simulations in a wider range of physical parameters (configurations) in a much shorter timeframe.

arrow_drop_up


text:
# Singularity problem at Inverse kinematic solver

I am struggling with this problem for days. Really hope that someone could give me a hint what the problem is.

The robot consists of 5 axes. The first axis rotates around the z-axis and other 4 axes rotate around the y-axis. And the solver basically works, as you can see below:

The problem occurs when the goal position contains y-movement also as below:

Here is what I have done so far:

1. I calculate the manipulability factor with my jacobian matrix (only translational part, since only the position is tracked here. Actually, I also tried with a combined jacobian matrix, so not only the translational part but also the rotational part. But jerky motion was there anyway):

2. Then the damping factor is :

3. The damping factor is then integrated into the pseudo inverse calculation :

As you can see, this is just a classical pseudo inverse kinematic solver with damped least square method. The manipulability factor according the second (problem) movement is : The manipulability drops in the beginning of the video. But why? As far as I know, this manipulability factor indicates the linear dependance of the axes. To me the axes dont seem to be linear dependant in the beginning part.

This jerky motion drives me crazy. As you can see in the first animation, the solver seems to work properly. What am I missing here?

UPDATE -----------------------------

Two more animations, the first one is with very high alpha in the damping factor calculation below and the second is one is with alpha =0, which means without any damping around singularity area.

As you can see above, with very high alpha the movement stops around the singularity area (the manipulability value is in that position 0.12 ! At the home configuration the manipulability value is around 15.)

Below you can see how the robot moves, if the damping effec is equal to zero (alpha =0).

UPDATE 2 -----------------------------

I might find the error, but not sure because this leads to another error... So I changed the unit_vectors of my 2-5 axes so that they rotate around x-axis, instead of y-axis:

the problematic point can be reached now :

But the point that could be reached before is not reachable now....

To me it doesnt make sense that the unit_vectors should be around x-axis. What am I missing here?

• Welcome to Robotics, Joe! This is a fantastic question, but unfortunately I don't think there's quite enough detail to answer the question. When a system goes to a singularity, I would expect some axis of motion to make a wild swing, like one axis swinging 180 degrees because another one is trying to go 89 to 91 degrees (if 90 degrees were the singularity). I'm not really seeing that behavior here. I'm wondering if this is more of an implementation issue, such as using something like atan instead of atan2, etc. Could you please edit your question to post the code you're using? – Chuck Feb 1 at 13:59
• @ Chuck, I edited the question so that the "wild swing" can be seen. (look at the 2. updated animation ) – Joe Feb 1 at 15:51

As others have already pointed out, there must be an issue with your implementation of the IK algorithm since there shall not be any singular behavior in the descriptions you gave.

Now you have two alternatives: either you start off debugging the code or you might want to exploit the fact that the problem can be easily broken down in two subproblems for which you can readily employ the most part of the code written so far.

Given the desired 3D target $$\left( x_d,y_d,z_d \right)$$, it's straightforward to observe that the desired value of the first joint is: $$\theta_{1d}=\arctan\left( \frac{y_d}{x_d}\right)$$.

The control law to drive the first joint of the manipulator to $$\theta_{1d}$$ can be as simple as:

$$\dot{\theta_1}=K_1 \cdot \left( \theta_{1d}-\theta_{1} \right).$$

Then, let $$R \in SO(3)$$ be the matrix that accounts for the rotation of $$\theta_{1d}$$ around the $$z$$ axis:

$$R = \left( \begin{array}{cccc} \cos{\theta_{1d}} & -\sin{\theta_{1d}} & 0 \\ \sin{\theta_{1d}} & \cos{\theta_{1d}} & 0 \\ 0 & 0 & 1 \\ \end{array} \right).$$

Through $$R$$, you'll get the new target $$\left( x_d,0,z_d \right)_1=R^T \cdot \left( x_d,y_d,z_d \right)^T$$ that will establish a new IK 2D planar problem in the $$xz$$ plane.

At this point, you can solve for $$\left( x_d,z_d \right)_1$$ by using the Jacobian of the remaining 4-DOF manipulator.

• I think there is no need to break down the problem in subproblems. Since the declaration of the unit vector already indicates what you have just described. – Joe Feb 15 at 20:22
• The break down is a mere suggestion to let you reuse the 4 DOF Jacobian, which is the part of your code working smoothly, as you reported at the beginning of your post. – Ugo Feb 15 at 20:30

I think you have introduced an algorithmic singularity at the first wrist axis. It appears to me that, when it reaches 90 degrees “down,” that instead of going to 91 it tries to flip back through zero to -269 degrees.

Of course this is speculative without seeing the code.


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The virtue of high variance approaches

Suppose you’re trying to do something. You can adopt a variety of strategies – some of them are likely to produce better results on average, some of them are more variable and could either produce great or terrible results, etc.

Lets adopt an overly simplistic mathematical model and say that how “good” the result of the strategy is is determined by a normal distribution $$N(\mu, \sigma^2)$$.

Lets further simplify and say that all we’re actually interested in to determine whether a strategy is good or not is the expected value of this goodness.

So how should we choose the best strategy?

Well we pick the one with the highest value of $$\mu$$. The $$\mu$$ parameter is exactly the expected value.

Now, suppose there are multiple people all working independently at the same task. What strategy should they adopt? Should they just all adopt the individually best strategy?

Well… it depends.

Specifically it depends on how these values combine.

If they just add together (e.g. say you’re doing something whose output is measured in quality adjusted life years and there are more than enough people you should be helping that you’re not meaningfully going to start interfering with eachother) then yes, you should adopt the same strategy. The sum of independent normal random variables is another normal random variable with expectation equal to the sum of the expectations of its components.

But not everything is additive. For example, suppose you’re all trying to come up with a new type of invention, or to write the best essay, or anything where you can simply take the best result and replicate it as many times as you want.

What strategy should you adopt then?

Suppose that each of $$N$$ people adopt a normally distributed strategy as above. Say the results of person $$n$$ are $$X_n$$. Then the overall result is $$R = \mathrm{max} X_n$$.

By scaling, the expected value of this is $$\mathbb{E}(R) = \mu + \sigma \mathbb{E}(\mathrm{max} Z_n)$$ where the $$Z_n$$ are standard normal random variables.

And it turns out that for $$N \geq 4$$, $$\mathbb{E}(\mathrm{max} Z_n) > 1$$. So for groups of at least 4 people, increasing the standard deviation actually improves the best result more than increasing the mean by the same amount does. This shouldn’t be terribly surprising – at 4 people there’s just over a 50% chance that someone will get at least one standard deviation above the mean.

So for groups of more than a couple people who are working independently and whose results can simply be replicated, the optimal strategy is not in fact to optimise your local strategy but to pursue higher variance ones. You should still try to produce good strategies of course, but by pursuing higher variance at some cost of optimality you may significantly improve the results.

Obviously this model is overly simplistic – quality isn’t scalar, it’s probably not normally distributed, and independence can be hard to achieve in practice, but I hope it is at least suggestive.

This entry was posted in Uncategorized on by .


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# What is the difference between an Authorization ID and a Authentication ID?¶

Authentication is the act of proving who you are. “Hello, I’m Dave. To prove it, here’s my password: Foo.”

Authorization is the act of deciding whether to grant access to resources.

"I'd like to read Kellie's mail for her."

In the example, I’m trying to read my wife’s mail. I supply my own username as the “authentication identifier”, my own password (Or biometric scan, or whatever else is required to prove I’m really me, with whichever mechanism is in use), and my wife’s username as the “authorization identifier”.

At no point need I know my wife’s password - instead, either Kellie or an administrator needs to explicitly state that I am allowed in “as Kellie”. Once I’ve logged in, all the access checks are done against Kellie, not against Dave, because I’m acting for her. To all intents and purposes, after the authentication exchange itself, the server can simply forget about who authenticated - it’s not important any more - and concentrate on who needs to be authorized.

Another, more common example of the use of differing authentication identifiers and authorization identifiers is in the design of many proxy systems. You authenticate perfectly normally to the proxy, authorizing as yourself. The proxy then authenticates to the master as itself, but supplies you as the authorization identifier, thus getting all the right access checks done at source, but not having to have access to your authentication credentials. Finally, some mechanisms don’t support passing a distinct authorization identifier, and for most its optional, and defaults to the case that most people are familiar with, where authorization and authentication identifiers are the same.


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# How do you simplify  x^3 + x^4 ?

Jun 15, 2018

See a solution process below:

#### Explanation:

First, rewrite each term as:

$\left(\textcolor{red}{{x}^{3}} \times 1\right) + \left(\textcolor{red}{{x}^{3}} \times x\right)$

Now, factor out the common term:

$\textcolor{red}{{x}^{3}} \left(1 + x\right)$

Or

$\textcolor{red}{{x}^{3}} \left(x + 1\right)$

Jun 15, 2018

Hmm...

#### Explanation:

We can't really simplify this equation anymore, as adding it just gives out the same result: ${x}^{3} + {x}^{4}$.

However, we can factor this expression, and the highest common factor is ${x}^{3}$, and so we get:

$= {x}^{3} \left(x + 1\right)$

or

$= {x}^{3} \left(1 + x\right)$


text:
# ICHEP 2020

28 July 2020 to 6 August 2020
virtual conference
Europe/Prague timezone

## The CMS Tracker Upgrade for the High Luminosity LHC

28 Jul 2020, 15:30
15m
virtual conference

#### virtual conference

Talk 13. Detectors for Future Facilities (incl. HL-LHC), R&D, Novel Techniques

### Speaker

Katja Klein (RWTH, I. Physik. Inst.)

### Description

The LHC machine is planning an upgrade program which will smoothly bring the luminosity to about $5-7.5\times10^{34}$cm$^{-2}$s$^{-1}$, to possibly reach an integrated luminosity of $3000-4000\;$fb$^{-1}$ over about a decade. This High Luminosity LHC scenario, HL-LHC, starting in 2027, will require an upgrade program of the LHC detectors known as Phase-2 upgrade. The current CMS Outer Tracker, already running beyond design specifications, and CMS Phase-1 Pixel Detector will not be able to survive HL-LHC radiation conditions and CMS will need completely new devices, in order to fully exploit the highly demanding conditions and the delivered luminosity. The new Outer Tracker should have also trigger capabilities. To achieve such goals, R\&D activities have explored options for both the Outer Tracker and for the Inner Tracker. The solutions developed will allow to include tracking information in the first level trigger stage. The design choices for the Tracker upgrades are discussed along with some highlights on technological approaches and R\&D activities.

 Secondary track (number) 12

### Primary author

Katja Klein (RWTH, I. Physik. Inst.)


text:
Theorem (The Radon-Riesz Theorem): Let $H$ be a Hilbert space. Then a sequence $(h_n)$ in $H$ norm converges to $h \in H$ if and only if $(h_n)$ weakly converges to $h$ and $\displaystyle{\lim_{n \to \infty} \| h_n \| = \| h \|}$.
• Proof: $\Rightarrow$ Suppose that $(h_n)$ norm converges to $h$. Since the norm convergence implies weak convergence we have that $(h_n)$ weakly converges to $h$. Furthermore, since $(h_n)$ norm converges to $h$ we have that $\displaystyle{\lim_{n \to \infty} \| h_n - h \| = 0}$. Since $|\| h_n \| - \| h \|| \leq \| h_n - h \|$ we conclude that $\displaystyle{\lim_{n \to \infty} \| h_n \| = \| h \|}$ as well.
• $\Leftarrow$ Suppose that $(h_n)$ weakly converges to $h$ and $\displaystyle{\lim_{n \to \infty} \| h_n \| = \| h \|}$. Observe that:
• Since $(h_n)$ weakly converges to $h$, for all $f \in H^*$ we have that $\displaystyle{\lim_{n \to \infty} f(h_n) = f(h)}$. But since $H$ is a Hilbert space, by The Riesz Representation for Hilbert Spaces, we have that $\displaystyle{\lim_{n \to \infty} \langle h_n, h \rangle = \langle h, h \rangle = \| h \|^2}$. Hence:
• Therefore $(h_n)$ norm converges to $h$. $\blacksquare$


text:
## Elementary Technical Mathematics

Published by Brooks Cole

# Chapter 10 - Section 10.3 - Finding Binomial Factors - Exercise - Page 350: 78

#### Answer

$a(x-4)(x-11)$

#### Work Step by Step

$ax^2-15ax+44a\longrightarrow$ a is a common factor. $a(x^2-15x+44)$ =$a(\ \ \ \ \ \ \ \ \ )(\ \ \ \ \ \ \ \ \ )$ =$a(x\ \ \ \ \ \ \ )(x\ \ \ \ \ \ \ )$ =$a(x-\ \ )(x-\ \ )$ Since the 2nd term is negative and the 3rd is positive, both factors or 44 must be negative =$a(x-4)(x-11)$ -4 and -11 have a sum of -15 and a product of 44.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# Proving trigonometric identities Calculator

## Get detailed solutions to your math problems with our Proving trigonometric identities step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here!

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$\cot\left(x\right)\cdot\sec\left(x\right)=\csc\left(x\right)$
2

Applying the secant identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$

$\cot\left(x\right)\frac{1}{\cos\left(x\right)}=\csc\left(x\right)$
3

Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$

$\cot\left(x\right)\frac{1}{\cos\left(x\right)}=\frac{1}{\sin\left(x\right)}$
4

Apply the formula: $\cot\left(x\right)$$=\frac{\cos\left(x\right)}{\sin\left(x\right)}$

$\frac{\cos\left(x\right)}{\sin\left(x\right)}\cdot\frac{1}{\cos\left(x\right)}=\frac{1}{\sin\left(x\right)}$
5

Multiplying fractions

$\frac{\cos\left(x\right)}{\sin\left(x\right)\cos\left(x\right)}=\frac{1}{\sin\left(x\right)}$
6

Simplify the fraction by $\cos\left(x\right)$

$\frac{1}{\sin\left(x\right)}=\frac{1}{\sin\left(x\right)}$
7

Both expressions are equal

true

### Struggling with math?

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text:
# Repossession 1 answer below »

Newton Co. had installment sales of $1,000,000 and cost of installment sales of$650,000 in 2012. A 2012 sale resulted in a default in 2014, at which time the balance of the installment receivable was $40,000. The repossessed merchandise had a fair value of$21,000.
Instructions
(a) Calculate the rate of gross profit on 2012 installment sales.
(b) Make the entry to record the repossession

shubham m

I assumed here you need gross profit ,
The gross profit = Sales -...

## Plagiarism Checker

Submit your documents and get free Plagiarism report

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text:
### On Secure Two-party Integer Division

Morten Dahl, Chao Ning, and Tomas Toft

##### Abstract

We consider the problem of {\it secure integer division}: given two Paillier encryptions of $\ell$-bit values $n$ and $d$, determine an encryption of \intdiv{n}{d} without leaking any information about $n$ or $d$. We propose two new protocols solving this problem. The first requires $\Oh(\ell)$ arithmetic operation on encrypted values (secure addition and multiplication) in $\Oh(1)$ rounds. This is the most efficient constant-rounds solution to date. The second protocol requires only $\Oh \left( (\log^2 \ell)(\kappa + \loglog \ell) \right)$ arithmetic operations in $\Oh(\log^2 \ell)$ rounds, where $\kappa$ is a correctness parameter. Theoretically, this is the most efficient solution to date as all previous solutions have required $\Omega(\ell)$ operations. Indeed, the fact that an $o(\ell)$ solution is possible at all is highly surprising.

Note: Extending the bit-length protocol to base-m and hybrid-base digit-length protocol.

Available format(s)
Category
Cryptographic protocols
Publication info
Published elsewhere. A shorten version can be seen in Proc. FC' 2012
Keywords
Secure two-party computationSecure integer divisionConstant-roundsBit-Length
Contact author(s)
ncnfl @ 163 com
History
2015-10-16: last of 3 revisions
See all versions
Short URL
https://ia.cr/2012/164

CC BY

BibTeX

@misc{cryptoeprint:2012/164,
author = {Morten Dahl and Chao Ning and Tomas Toft},
title = {On Secure Two-party Integer Division},
howpublished = {Cryptology ePrint Archive, Paper 2012/164},
year = {2012},
note = {\url{https://eprint.iacr.org/2012/164}},
url = {https://eprint.iacr.org/2012/164}
}

Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.


text:
markovRing -- ring of joint probability distributions on several discrete random variables

Synopsis

• Usage:
markovRing(d)
• Inputs:
• d, , with positive integer entries $(d_1,\dots ,d_r)$
• Optional inputs:
• Coefficients => ..., default value QQ, optional input to choose the base field in markovRing or gaussianRing
• VariableName => ..., default value p, symbol used for indeterminates in a ring of discrete joint probability distributions
• Outputs:
• a ring, a polynomial ring with $d_1*d_2*\dots *d_r$ variables $p_{i_1,\dots ,i_r}$, with each $i_j$ satisfying $1\leq i_j \leq d_j$.
• Consequences:
• Information about this sequence of integers is placed into the ring, and is used by other functions in this package. Also, at most one ring for each such sequence is created since the ring is cached.

Description

The sequence $d$ represents the number of states of each discrete random variable. This example creates a ring of joint probability distributions on 4 random variables with 2, 3, 4, and 5 states. This ring has a total of 120 indeterminates.

 i1 : d=(2,3,4,5); i2 : R = markovRing d; i3 : numgens R o3 = 120 i4 : R_0, R_1, R_119 --here are some of the variables in the ring o4 = (p , p , p ) 1,1,1,1 1,1,1,2 2,3,4,5 o4 : Sequence

If no coefficient choice is specified, the polynomial ring is created over the rationals.

 i5 : coefficientRing R o5 = QQ o5 : Ring

The optional argument Coefficients allows to change the base field.

 i6 : R2 = markovRing (d,Coefficients=>CC); i7 : coefficientRing R2 o7 = CC 53 o7 : ComplexField

The indeterminates are labeled with the letter ''p'' suggesting probability distributions. However, it is useful to be able to create a new ring where the indeterminates are labeled different (for example, they may represent marginal probabilities). This can be accomplished with the VariableName option.

 i8 : d=(1,2); i9 : markovRing (d,VariableName => q); i10 : gens oo o10 = {q , q } 1,1 1,2 o10 : List

The routines hiddenMap, inverseMarginMap, marginMap, markovMatrices require the ring to be created by this function. The routines conditionalIndependenceIdeal, discreteVanishingIdeal require the ring to be created by this function or the method gaussianRing.

Caveat

As opposed to gaussianRing, this method does not store information about a graph or the names of the random variables. In case these random variables are not numbered $1, 2, \dots, n$, then the methods conditionalIndependenceIdeal and markovMatrices require an additional input in the form of a list of the random variable names. This list must be in the same order as the implicit order used in the sequence $d$. The user is encouraged to read the caveat on the method conditionalIndependenceIdeal regarding probability distributions on discrete random variables that have been labeled arbitrarily.

• conditionalIndependenceIdeal -- ideal of a list of conditional independent statements
• discreteVanishingIdeal -- vanishing ideal of a discrete graphical model
• gaussianRing -- ring of Gaussian correlations on n random variables or a graphical model
• hiddenMap -- linear map between the ring of a model with one hidden variable and the ring of the corresponding fully observed model
• inverseMarginMap -- inverse of the marginMap
• marginMap -- linear map on joint distributions for discrete random variables replacing marginals for indeterminates
• markovMatrices -- matrices whose minors form the ideal of a list of independence statements

Ways to use markovRing :

• "markovRing(Sequence)"

For the programmer

The object markovRing is .


text:
## anonymous 5 years ago what is the sum of all even numbers between 1 and 1000

1. Owlfred

Hoot! You just asked your first question! Hang tight while I find people to answer it for you. You can thank people who give you good answers by clicking the 'Good Answer' button on the right!

2. anonymous

sum of all even number between 1 and 1000 is 2+4+6.......................................998 a=2 d=2 an = a+ (n-1)d 998= 2+n-1)2 998-2=(n-1)2 996=(n-1)2 488=n-1 n=489 s489= 489/2[2+489] =489/2[491]=120049

3. anonymous

That's the sum of all terms of the arithmetic progression, with a=2 and d=2. last term is 998 a=2 d=2 nth term = 998 998=a+(n-1)d 998=2+(n-1)2 998=2n n=499 sum=n/2(2a+(n-1)d) therefore sum of 499 terms = 249500

4. anonymous

@shrishti, I think you made a slight error while calculating 996/2, which should equal 498, instead you have 488.

5. anonymous

There are 500 even numbers first is 2 last is 1000 second is 4 last but one is 998 You will see (just like Gauss did ) that they both sum to 1002 So as there are 250 of these 'sums' ... 250 x 1002 = 250 500 This is a correct answer

6. anonymous

Another way of seeing this is sum from k =1 to k = 500 (2k) sorry about the lack of mathematics text.

7. anonymous

$\sum_{1}^{500}$ 2k hmmm still not good is it?

8. anonymous

1002 * 250 = 250,500

9. anonymous