Sequential DOE via dynamic programming

Development of a framework for sequential Bayesian design of experiments: Application to a pilot-scale solvent-based CO2 capture process

In this paper, a methodology is developed for sequential design of experiments (SDoE) for process systems and applied to a solvent-based CO2 capture system. In this approach, the prior knowledge of the system is used to prioritize process data collection at specific operating conditions. These data are then incorporated into a Bayesian inference methodology for updating a stochastic model by refining estimations of its underlying parameters, and the updated model is then used to generate the next set of test runs. Thus, the new knowledge obtained from the data is used to guide subsequent iterations of the experimental runs, ensuring that the overall data collection is maximally informative given that most experimental campaigns, especially at pilot or higher-scale plants, are costly, time-consuming, and resource-limited. The test run objective for this work was to minimize the maximum model prediction uncertainty for key output variables, but the methodology is generic and can be readily applied to other test run objectives. This methodology is applied to an aqueous monoethanolamine (MEA) pilot plant campaign at the National Carbon Capture Center (NCCC) in Wilsonville, Alabama, USA. The SDoE framework was utilized for two iterations, while collecting 18 sets of data representing different process conditions, and this resulted in an overall average reduction in uncertainty of approximately 50% in the prediction of CO2 capture percentage. Moreover, 11 additional data sets were obtained with variation of absorber packing height for further model validation. This work shows the capability of the SDoE framework to maximize learning given limited resources, allowing for the reduction of model uncertainty, which is of great importance for many applications including reduction of technical risk associated with scale-up and economic analysis.

Compared with the conventional energy power generation investments, the uncertainty of wind power investment will bring greater risk. For wind power investors, wind power investment strategy-makers, even for wind power equipment manufacturers, all urgently need decision-support tools to minimize the risk and obtain optimal investment opportunities. In this paper, firstly we analyze the current problems and risks of wind power industry in China. Secondly, based on stochastic theory and dynamic programming methods, the paper establishes model about wind power investment decision-making. It considers the wind power project investment uncertainties risk factors. Example analysis shows that the model can be used for wind power developers to choose the right investment opportunities and to provide decision-making reference.

In this study, a dual interval robust stochastic dynamic programming (DIRSDP) method is developed for planning water resources management systems under uncertainty. As an extension of the existing interval stochastic dynamic programming (ISDP) method, DIRSDP can deal with two-stage stochastic programming (TSP)-based planning problems associated with dynamic features, input uncertainties, and multistage concerns. Compared with other optimization methods dealing with uncertainties, the developed DIRSDP method has advantages in addressing uncertainties with complex presentations and reflecting decision makers’ risk-aversion attitudes within its optimization process. Parameters in the DIRSDP model can be represented as probability distributions as well as single and/or dual intervals. Decision makers’ risk-aversion attitudes can be reflected through restricting the deviation of the recourse costs to a tolerance level. Water-allocation plans can then be developed based on the analysis of tradeoffs between the system benefit and solution robustness. The developed method is applied to a case of water resources management planning. The solutions are reasonable, indicating applicability of the developed methodology.

This paper studies a class of renewable resource allocation problems for the processing of dynamically arriving tasks with deterministic deadlines. This class of problems has many applications, however, it conforms to neither the standard resource allocation model nor the standard optimal control model. A new problem formulation has to be developed and analyzed to provide a satisfactory solution. In this paper, a new formulation is presented. By augmenting the state variabls, the problem is converted into a Markovian decision problem. It can then be treated, at least in principle, by using the stochastic dynamic programming (SDP) method. However, since the system dynamics involves the evolution of sets (of tasks and resources), the implementation of the dynamic programming equation is by no means straightforward. For a problem with infinite planning horizon, the optimal strategy is shown to be stationary under mild conditions. An SDP algorithm based on a successive approximation technique is developed to obtain the optimal stationary strategy. The implementation of the algorithm employs a special coding scheme to handle set variables, and utilizes a dominance property for computational efficiency. Effects of key system parameters on optimal decisions are investigated and analyzed through numerical examples. As the computational complexity of the algorithm is of exponential increase, practical applications of the algorithm is limited to problems of moderate size. Two heuristic rules are therefore investigated and compared to the optimal policy.

Risk neutral and risk averse Stochastic Dual Dynamic Programming method

Sequential Design of Experiments to Maximize Learning from Carbon Capture Pilot Plant Testing

This study describes the use of an analytical model, constructed using sequential design of experiments (DOEs), to optimize and quantify the uncertainty of a Diesel engine operating point. A genetic algorithm (GA) was also used to optimize the design. Three engine parameters were varied around a baseline design to minimize indicated specific fuel consumption (ISFC) without exceeding emissions (NOx and soot) or peak cylinder pressure constraints. An objective merit function was constructed to quantify the strength of designs. The engine parameters were start of injection (SOI), injection duration, and injector included angle. The engine simulation was completed with a sector mesh in the commercial computational fluid dynamics (CFD) software CONVERGE, which predicted the combustion and emissions using a detailed chemistry solver with a reduced mechanism for n-heptane. The analytical model was constructed using the SmartUQ software using DOE responses to construct kernel emulators of the system. Each emulator was used to direct the placement of the next set of DOE points such that they improve the accuracy of the subsequently generated emulator. This refinement was either across the entire design space or a reduced design space that was likely to contain the optimal design point. After sufficient emulator accuracy was achieved, the optimal design point was predicted. A total of 5 sequential DOEs were completed, for a total of 232 simulations. A reduced design region was predicted after the second DOE that reduced the volume of the design space by 96.8%. The final predicted optimum was found to exist in this reduced design region. The sequential DOE optimization was compared to an optimization performed using a GA. The GA was completed using a population of 9 and was run for 71 generations. This study highlighted the strengths of both methods for optimization. The GA (known to be an efficient and effective method) found a better optimum, while the DOE method found a good optimum with fewer total simulations. The DOE method also ran more simulations concurrently, which is an advantage when sufficient computing resources are available. In the second part of the study, the analytical model developed in the first part was used to assess the sensitivity and robustness of the design. A sensitivity analysis of the design space around the predicted optimum showed that injection duration had the strongest effect on predicted results, while the included angle had the weakest. The uncertainty propagation was studied over the reduced design region found with the sequential DoE in the first part. The uncertainty propagation results demonstrated that for the relatively large variations in the input parameters, the expected variation in the ISFC and NOx results were significant. Finally, the predictions from the analytical model were validated against CFD results for sweeps of the input parameters. The predictions of the analytical model were found to agree well with the results from the CFD simulation.

A proposed stochastic dynamic programming (SDP) method with uncertainties in stream flow and water demand is developed to calculate optimal reservoir operation rules. The SDP method extends the classic SDP method that considers only one uncertainty in its operation rules. Time series of reservoir inflow and agricultural water demand for the Aydoghmoush Reservoir in eastern Azarbayejan, Iran, were obtained from meteorological data, available climate parameters, hydrologic data, and crop water demand. The application of the developed SDP with two uncertainties was evaluated with operation rules corresponding to four different scenarios, and optimal reservoir releases were determined for a drought year. Reservoir operation results were evaluated with different performance indices. This study's results demonstrate the advantage of considering uncertainties in reservoir inflow and water demand with the SDP method. The developed SDP method has general applicability under a range of climatic conditions, and the calculated operation rules cover the expected ranges of streamflow and water demand during the operating years.

Stochastic dual dynamic programming is a cutting plane type algorithm for multi-stage stochastic optimization originated about 30 years ago. In spite of its popularity in practice, there does not exist any analysis on the convergence rates of this method. In this paper, we first establish the number of iterations, i.e., iteration complexity, required by a basic dynamic cutting plane method for solving relatively simple multi-stage optimization problems, by introducing novel mathematical tools including the saturation of search points. We then refine these basic tools and establish the iteration complexity for both deterministic and stochastic dual dynamic programming methods for solving more general multi-stage stochastic optimization problems under the standard stage-wise independence assumption. Our results indicate that the complexity of some deterministic variants of these methods mildly increases with the number of stages $T$, in fact linearly dependent on $T$ for discounted problems. Therefore, they are efficient for strategic decision making which involves a large number of stages, but with a relatively small number of decision variables in each stage. Without explicitly discretizing the state and action spaces, these methods might also be pertinent to the related reinforcement learning and stochastic control areas.

Reservoir management is a highly complex problem, which includes the uncertainty in the inflows as well as in the objectives. Stochastic dynamic programming (SDP) has been widely used to develop the optimal operating policies of multipurpose reservoirs. In the recent years, fuzzy optimization has been used to incorporate the uncertainty caused due to the imprecise nature of objectives. In the present study, a comparison is made between a conventional SDP model and fuzzy dynamic programming models for optimal operation of a multipurpose reservoir. The two models are developed for Hirakud Reservoir on River Mahanadi in the State of Orissa in India. The performance of the reservoir is evaluated under the operating policies derived from these models through simulation. The results indicate that the fuzzy dynamic programming model performs better in achieving the flood control objective and thus in the overall performan1ce of reservoir.

Multi-stage planning of clean resources and energy storage assets with hybrid uncertainty modeling for low-carbon resilient distribution systems

The increased water demand and the increased capital construction costs of water resource development projects force engineers and water planners to utilize efficiently the existing and proposed water resources systems. Hence, it is considered necessary to find the optimum real time operating rules for systems of multiple reservoirs.Stochastic dynamic programming (SDP) has been used as the tool for deriving the real time operating rules for a system of multiple reservoirs, in preference to simulation analysis and linear programming. This is because the operation of the storages is considered as a sequential decision problem and dynamic programming is well suited to handle sequential, multi-period type decision problems. Further, the stochastic nature of streamflow is considered explicitly through a multivariate distribution for streamflow. The method theoretically produces the global optimum solution.The computational difficulties in terms of computer time and memory space have prohibited the realistic application of this method to systems of multiple reservoirs in the past. In this study, two methods have been introduced to reduce the computational difficulties of the method, thus making SDP more attractive to use in the derivation of the real time operating rules for a system of multiple reservoirs.The first method, the assumption of cross correlation of streamflow, reduces the number of states in the SDP method which reduces computer time and memory space. Further, it reduces the number of conditional probability estimates of streamflow which are used in recursive relation of the SDP algorithm. The second method, the “corridor” approach, reduces computer time involved in the SDP method by eliminating the computation of infeasible and/or inferior (suboptimal) solutions. These two methods have resulted in significant savings in the computer effort of the SDP method.The conditional probabilities of streamflow are computed by considering the hypervolume under the conditional probability density function between the integration limits using the computationally efficient Gaussian Legendre quadrature method. The conditional probability density function is derived from the historical streamflow data.Real time operating rules are expressed as a function of the storage volume(s) at the beginning of the current time interval and the forecast inflow into the storage(s) during the current time interval.The Melbourne water supply system is used as the example. The derived operating rules have been tested by operating a simulation model and the conclusion is made that the SDP method is a satisfactory technique for objectively deriving real time operating rules for a system of multiple reservoirs.Finally, a combined stochastic dynamic programming – statistical disaggregation approach is introduced to determine the real time operating rules for a system of multiple reservoirs where operational data are available. Releases from individual storages are a function of the total system storage volume and inflow. The method uses SDP to determine the release rules from the equivalent single reservoir of the multiple reservoir system; these releases are then disaggregated to individual releases using the statistical release disaggregation model whose parameters are derived from the historical operational data. The approach was used to derive real time operating rules for the Melbourne water supply system and the operating rules were validated using a simulation model of the system.

Optimal High-Frequency Trading in a Pro-Rata Microstructure with Predictive Information

마이크로 유전알고리즘은 적은 수의 개체 사용 및 무작위 개체 구성을 통한 돌연변이 기능 대체의 특징을 갖는 진화연산을 수행하여 일반적인 유전알고리즘이 갖는 각 세대당 많은 계산 량이 요구되는 단점을 극복하고자 하였다. 이러한 마이크로 알고리즘은 특히 설계변수가 3~5 개를 갖는 문제에 효율적이라는 것이 많은 연구자들에 의하여 알려졌다. 따라서 본 연구의 목적은 순차적 실험계획법과 마이크로 유전알고리즘을 이용한 최적화 알고리즘을 개발하는 것이며, 이를 수학예제와 구조물 문제에 적용하여 실용성을 확인하고자 한다. 순차적 실험계획법은 저자들의 선행연구에서 제안되었으며, 실험계획법과 반응표면법을 이용하는 근사최적화 기법에 의한 시행착오적인 반복과정을 최소화하고자 하는 방법으로써, 행렬실험과 평균분석을 반복 적용하는 개념이다.