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Det går inte att visa den länkade bilden. Filen kan ha flyttats, fått ett nytt namn eller tagits bort. Kontrollera att länken pekar på rätt fil och plats. PROCESSTYRNING 1262 Beslutsstöd för produktionsplanering av fjärrvärme med hjälp av olinjär programmering Per-Ola Larsson, Stéphane Velut, Håkan Runvik, Sara Modarres Razavi, Andreas Nilsson, Markus Bohlin, Jonas Funkquist Beslutsstöd för produktionsplanering av fjärrvärme med hjälp av olinjär programmering Decision Support for Short-Term Production Planning of District Heating using Non-linear Programming Per-Ola Larsson, Modelon AB Håkan Runvik, Modelon AB Stéphane Velut, Modelon AB Sara Modarres Razavi, SICS Andreas Nilsson, SICS Markus Bohlin, SICS Jonas Funkquist, Vattenfall AB. P Värmeforsk Serviceaktiebolag STOCKHOLM Tel December 2014 ISSN Abstract The short term production planning optimization problem for a district heating system is solved in two steps by integrating physics-based models into the standard approach. The first optimization step solves for the discrete variables of the unit commitment problem (UCP) using mixed integer linear models and standard mixed-integer solvers. The second step, the economic dispatch problem (EDP), considers dynamic optimization using physics-based non-linear models that utilize the unit statuses from the first step. All optimizations aim at maximizing production profit using fuel, electricity and heat prices as well as maintenance and start-up/stop costs. Through the physics-based models in the EDP, it is feasible to optimize over and consider constraints on power flows as well as important physical variables such as supply temperature, supply flow rate, pump speeds and condenser pressures which is not available in today s standard methods. The modeling has focused on distributed consumption and production. The goal has been to represent the most important units and network distribution of the Uppsala district heating network. The distribution yields that the total heat demand is distributed and the delay times from production to customers are customer individual. The district heating net has been modelled using physics-based pipes, including mass flow dependent delays and temperature dependent (water and outdoor temperature) heat losses. Comparisons between optimizations with and without distribution net models have been performed, showing that careful modeling of the net impacts the production planning in form of reduction of costly production peaks and delay of costly unit startups, production compensation for heat losses and time delays as well as usage of the net for heat storage (accumulation). The optimizations also results in production plans where supply temperature and flow is minimized and maximized, respectively, and there is a balance between heat production and heat consumption. The physics-based modeling and dynamic optimization techniques provides a flexible way to formulate the optimization problem and include constraints of physically important variables such as supply temperature and flow, pressures, heat loads, and start-up/stop trajectories for production units. Two examples of stochastic optimization have shown that taking probabilities of heat demand predictions into account, the expected profit may be increased. i ii Sammanfattning Optimeringsproblemet för den kortsiktiga produktionsplaneringen i ett fjärrvärmenät löses i två steg genom integrering av fysikaliska modeller och standardmetoden. I det första optimeringssteget behandlas start/stop problemet (SSP), där de diskreta variablerna bestäms med hjälp av diskrettidsmodeller och standardlösare för linjärprogrammering. I det andra steget, det optimala lastfördelningsproblemet (LFP), används dynamisk optimering med fysikaliska olinjära modeller där statussignalerna från steg ett ingår. Vid alla optimeringar är målet att maximera den ekonomiska vinsten, genom att el-, värme- och bränslepriser tas i beaktande, liksom kostnader för underhåll, uppstart och nedstängning. Med hjälp av den fysikaliska modelleringen i LFP är det möjligt att optimera effektflöden och inkludera bivillkor för dessa, liksom för andra viktiga variabler såsom framledningstemperatur, massflöden, pumphastigheter och kondensortryck. Detta är inte möjligt med dagens standardmetoder. Modelleringen har fokuserat på distribuerad konsumtion och produktion. Målet har varit att representera de viktigaste enheterna, samt distributionsnätet i Uppsala. Distributionen leder till att det totala värmebehovet är fördelat och fördröjningstiden från produktion till kunder är kundspecifik. Fjärrvärmenätet har modellerats med hjälp av fysikaliska rör, med massflödesberoende tidsfördröjning och temperaturberoende (vatten- och utomhustemperatur) värmeförluster. Jämförelser mellan optimeringar med och utan den distribuerade nätmodellen har genomförts och resultaten visar att noggrann modellering av nätet påverkar produktionsplaneringen genom att dyra produktionstoppar kan undvikas och dyra uppstarter av produktionsenheter kan fördröjas, produktionen kompenseras för värmeförluster och fördröjningstider och nätet kan användas för värmelagring (ackumulation). Optimeringarna resulterar också i att framledningstemperaturen och massflödet minimeras respektive maximeras och att det är balans mellan värmeproduktion och värmekonsumtion. Den fysikaliska modelleringen och dynamiska optimeringstekniken ger en flexibel metod för att formulera optimeringsproblemet och inkludera bivillkor på fysikaliskt viktiga variabler såsom framledningstemperatur och -massflöde, tryck, värmelaster och uppstarts-/nedstängningsprofiler för produktionsenheter. Två exempel med stokastisk programmering har visat att man kan uppnå en större förväntad vinst genom att beakta sannolikheten för olika värmebehovsprognoser. iii Executive Summary The standard approach for production planning of a district heating network today typically involves highly simplified models where most of the physical descriptions are removed. This makes it possible to solve the resulting optimization problem using simple linear optimization methods, but significant physical constraints and degrees of freedom are then excluded from the model. In this study the production planning problem is divided into two separate optimization problems; the Unit Commitment Problem (UCP) and the Economic Dispatch Problem (EDP). In the UCP the modeling of the system is simplified to only include piecewise linear dependencies between the variables. The system is furthermore discretized in time, creating a Mixed Integer Linear Programming (MILP) problem. This kind of optimization problem can be solved using a numerical solver. The EDP model includes much more detailed representations of the producers, customers and distribution net in the district heating network, compared to the UCP model. The modeling is physical, which has benefits in terms of accuracy, interpretation, and possibility to impose constraints corresponding to the limitations of the physical system. In order to use the EDP model for optimization purposes it is necessary to remove the integer variables from the formulation. The status signals from the UCP optimization results are therefore used as input signals in this optimization. When the EDP optimization formulation is discretized a Non-Linear programming (NLP) problem is created. The district heating network of Uppsala is modeled in this study. The most important production unit in this network is the cogeneration plant denominated KVV (Kraft- och värmeverk), situated at the production site Boländerna. This plant is represented in detail in the EDP, where a large part of the vapor cycle is modeled. The main components in this model are high and low pressure turbine stages, condensers, and a reheater. For the UCP a polytope in the space of electricity, heat and return temperature is used to describe the KVV. Other production units are not modeled physically in either the UCP or the EDP, but are instead simply adding heat to the district heating water proportionally to their load. An important unit in the Uppsala network is the accumulator. This is modeled using a finite volume approximation in the EDP, while it works as an integrator for the stored energy in the UCP. The physical distribution of the customers and producers in the Uppsala district heating network is represented using a one dimensional approach based on the delay time for the customers relative to Boländerna. In order to represent the delay time a simple delay is used in the UCP, while a more complex pipe model is implemented in the EDP. This model consists of a fixed delay combined with a finite volume pipe implementation. In iv both the UCP and the EDP a heat loss model based on the outdoor temperature is included in the pipe modeling. The Pyomo modeling language is used for creating the UCP optimization formulation. Two different solvers are used to solve the problem; Gurobi and GLPK. The Gurobi Optimizer is a commercial solver which can handle several different types of optimization problems and modeling languages. GLPK is an open source software package. The EDP optimization problem is formulated using the Optimica language. The open source platform JModelica.org is used to solve the problem. This tool translates the problem into an NLP which is solved by the Interior Point Optimizer (IPOPT). For both the EDP and the UCP the optimization problem is formulated so that the economic profit is maximized. The revenue comes from selling heat and electricity, while fuel and maintenance costs are the main expenses. Only constant prices are considered. Thanks to the physical modeling in the EDP, the optimization formulation in this case includes constraints on critical variables such as mass flows and customer inlet temperatures. Five optimization cases have been created in order to study the implemented method for solving the production planning problem. The first three consider only deterministic data and are of increasing complexity. In the last two uncertain signals with discrete probability distributions are included in the formulation. The base optimization scenario is a 24 hour period with a demand profile roughly corresponding to the heat demand expected from a residential area on a week day. In the first optimization case only one producer, the KVV, and one customer is present. A comparison between having no pipes in the model and having supply and return pipes between the production unit and the customer model is conducted. The results show that the optimal solution in both cases involves maximizing the electricity production. In the EDP this is achieved by maximizing the mass flow through the KVV. Constraints on the pump speed, condenser pressure and customer inlet temperature are limiting. When there are pipes in the model the EDP solution involves using the heat stored in the pipes to satisfy a part of the heat demand. This lowers the customer supply temperature, which again maximizes the electricity production. In case II there are three customers connected in parallel. The KVV is the production unit in the system and there are pipes between the KVV and the customers, and also between the customers. The effect of adding dissipation of heat in the pipe models is investigated. The main feature of the results is the lowering of the main production peak due to the distributed customer network. The delay times between the customers distributes the demand peak in time for the producer. Another observation that can be made for this case is that the delay time does not have any influence when heat load changes are handled by only changing the mass flow. The incompressibility of the district heating v water makes the heat flow changes instantaneous in this case. The introduction of heat losses has a surprisingly large influence on the optimal solution. The reason for this is that the condenser steam pressure is sensitive to changes in return temperature and mass flow. With heat losses the steam pressure constraint gets active for lower mass flows, resulting in a different production profile. Case III is a more realistic case with three production units and an accumulator in the system. The optimization interval is four days, making the integration between the UCP and EDP important. A comparison between a distributed and point-wise network is conducted. The UCP results indicate a clear advantage of considering the distribution of the network in production planning. By doing this it is possible to delay the start-up of additional units, due to the reduced production peaks caused by the distribution. It is also notable that the accumulator is used to handle all load variations and that the production units therefore can be run with constant loads. The EDP results are very similar to the UCP results without the distributed network. The differences are greater when pipes and delays are added to the system; the reason for this is that is the more detailed pipe modeling in the EDP, which introduces additional possibilities such as using the network as an accumulator. In case IV a comparison between stochastic programming and robust optimization is conducted. The setup includes three different production units: the KVV, an oil boiler and a solid fuel boiler. The heat demand profile is uncertain for the second half of the optimization time. A higher expected profit can be achieved with stochastic programming compared to the robust formulation. The reason for this is that the stochastic programming scheme avoids starting additional units unnecessarily. This is highly beneficial when the heat demand turns out to be low. For higher heat demands it becomes necessary to use the more expensive production unit, but due to the probability distribution between the cases the extra cost this introduces is smaller than the gain at lower demands. In case V the results from stochastic programming are again compared with a robust optimization results. In this model there are two production units and both the heat demand and the electricity price are uncertain. The results show that a higher expected profit can be achieved with stochastic programming. A scaling test indicates that the commercial solver is clearly superior when optimization problems with many possible scenarios are considered. For less complex problems, such as the UCPs in the deterministic optimization cases, the open-source solver seems sufficient in terms of convergence speed. vi Contents 1 INTRODUCTION BACKGROUND PROJECT GOAL DISTRIBUTION OF WORK OVERVIEW OF UPPSALA DISTRICT HEATING NETWORK SYSTEM MODELING AND OPTIMIZATION OVERVIEW UNIT AND NET MODELS MODELING FOR THE DISCRETE OPTIMIZATION MODELING FOR THE CONTINUOUS OPTIMIZATION CUSTOMER DISTRICT HEATING NETWORK MODEL OPTIMIZATION TOOLS DISCRETE OPTIMIZATION CONTINUOUS OPTIMIZATION OPTIMIZATION FORMULATIONS PLANT ECONOMICS DISCRETE OPTIMIZATION CONTINUOUS OPTIMIZATION INTEGRATION OF DISCRETE AND CONTINUOUS OPTIMIZATION OPTIMIZATION EXAMPLES OVERVIEW OF CASES CASE I: POINT-WISE NETWORK CASE II: DISTRIBUTED NETWORK CASE III: OPTIMIZATION OVER SEVERAL DAYS CASE IV: STOCHASTIC PROGRAMMING CASE V: STOCHASTIC PROGRAMMING WITH UNCERTAIN ELECTRICITY PRICE SCALING TEST SUMMARY GENERAL MODELING OPTIMIZATION FUTURE WORK MODELING OPTIMIZATION BIBLIOGRAPHY vii 1 Introduction 1.1 Background Short-Term Production Planning Running the production of heat and power in a cost efficient manner is desirable both from a producer and consumer perspective. Production planning does not only consider the cost when optimizing unit schedules, but also network heat load demand and operational constraints. The scheduling includes the status (on/off, discrete variable) of each production unit as well as each unit s produced electric power and heat (continuous variables). To the authors best knowledge, there exists no robust algorithm for solving the resulting optimization problem that involves both the discrete and continuous variables, known as a mixed-integer non-linear programming (MINLP) problem. For tractability reasons, it is therefore necessary to divide the optimization formulation into two separate optimization problems: 1. The Unit Commitment Problem (UCP), where the statuses of the units are optimized and the main difficulty lies in the combinatorial nature of the problem. 2. The Economic Dispatch Problem (EDP), which considers the result from the UCP and optimizes the load level for each active unit. The main difficulty of this optimization problem are the non-linearity of the units and the non-convexity of the optimization problem (local minima may be present). The major differences between the two optimization problems are the model complexities and resulting optimization problem types. The UCP contains only linear or piecewise linear models and the optimization problem is a mixed integer linear problem (MILP) that can be used to solve both UCP and EDP. The EDP contains non-linear models with higher level of detail than the UCP models and the resulting optimization problem does not contain any discrete variables and is casted as a non-linear program (NLP) to be solved Common Approaches Approaches do not involve solving both the discrete and continuous variables simultaneously due to the difficulty of the MINLP. Instead, a related MILP problem (e.g., Outer Approximation or General Bender Decomposition) or a related NLP problem (eg., Branch and Bound, see [1]), is iterated over. A few approaches solve both UCP and EDP and most of them are based on Lagrangian relaxation (LR) or on mixed integer linear programs (MILP). The LR approach can handle some non-linearities by using relaxation, but network topology is not considered. The most appealing feature of the LR approach is the decomposition of the global optimization problem into a global master-problem and a small unit-specific problem, which may be beneficial for large networks (over e.g., 100 units) The MILP formulation 1 results in large-scale integer optimization problems, but due to the progress of efficient solvers for such problem, this strategy has increased in popularity. Typical for today s scheduling of units is that the unit models are highly simplified such that all physical descriptions are almost removed and the resulting optimization problems can be solved by using simple linear optimization methods. In [2], it is mentioned that the following commercial software uses this approach: - Planner [3] - Energy Optima 2000 [4] - OPTIMAX PowerFit [5] A survey of available approaches for short-term production planning can be found in [6] and [7]. The Värmeforsk-reports [2] and [8] focuses on the effect of an uncertain load prediction on the optimization results and the effect of integrating a model of the distribution network in the MILP formulation, respectively. In [9], the effects of a detailed physics-based model of the EDP are seen together with an integration of UCP and EDP Limitations Current scheduling algorithms performed using MILP formulation of the UCP/EDP problems contain heavily simplified models, often only modelled as algebraic equations with exceptions for storing dynamics (heat and fuel) and delays in the distribution net. The continuous optimization variables are typically heat flows and the effects of the supply temperature and flow as well as return temperature are not directly considered. This is limiting as these temperatures and flows affect many critical parameters such as amount of energy that can be stored in the net, heat losses in the net and efficiencies for electricity production in steam turbines. The simplified modeling approach can be expanded to include supply temperature, see [10], but this strategy introduces several difficulties in the formulation. The distribution of heat is in many cases point wise

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