Two-stage stochastic programming of steam power system based on Markov chain

doi:10.11949/0438-1157.20221430

Abstract

Abstract:

The existing optimization methods to solve the uncertainty of steam demand in steam power system include stochastic programming and robust optimization. However, these methods can not trade off the stability and economy at the same time. This paper proposes a two-stage stochastic programming based on Markov chain to solve this problem. In the first stage, uncertain variables are divided based on spatial distance expression and divided into different working conditions by clustering algorithm. In the second stage, the Markov chain is constructed based on the state transition probability, and the demand value of steam is predicted by the method of scenario generation and reduction. The steam power system of a coal-to-gas enterprise is taken as an example to establish the corresponding optimization model, and the predicted steam value is brought into the optimization model to solve. The optimal operation scheme obtained is compared and analyzed with stochastic programming and robust optimization. The results show that the proposed optimization method combines the advantages of high economy of stochastic programming and high stability of robust optimization, both stability and economy are intermediate between stochastic programming and robust optimization, and provides a new idea for solving uncertain optimization problems of steam power system.

Key words: steam power system, steam demand prediction, Markov chain, two-stage stochastic programming, scenario generation and reduction

摘要：

现有的解决蒸汽动力系统蒸汽需求不确定性的优化方法有随机规划和鲁棒优化，但二者不能同时兼顾稳定性和经济性。本文提出一种基于马尔可夫链的两阶段随机规划去解决这个问题。第一阶段基于空间距离表达划分不确定变量，通过聚类算法划分成不同工况。第二阶段基于状态切换概率构建马尔可夫链，通过场景生成和削减的方法预测蒸汽的需求值。以某煤制气企业蒸汽动力系统为实例建立相应的优化模型，将预测的蒸汽值带入优化模型求解，得出的最优操作方案与随机规划和鲁棒优化法进行对比和分析。结果表明，本优化方法综合了随机规划经济性高和鲁棒优化稳定性高的优点，稳定性和经济性都介于随机规划和鲁棒优化的中间，为解决蒸汽动力系统的不确定优化问题提供了新思路。

关键词: 蒸汽动力系统, 蒸汽需求预测, 马尔可夫链, 两阶段随机规划, 场景生成和削减

CLC Number:

TQ 062.2

Kenian SHI, Jingyuan ZHENG, Yu QIAN, Siyu YANG. Two-stage stochastic programming of steam power system based on Markov chain[J]. CIESC Journal, 2023, 74(2): 807-817.

史克年, 郑景元, 钱宇, 杨思宇. 基于马尔可夫链的蒸汽动力系统两阶段随机规划[J]. 化工学报, 2023, 74(2): 807-817.

Figures/Tables 22

Fig.1 Annual steam demand fluctuation curve

Fig.2 Results of working condition division of three levels of uncertain parameters

Fig.3 Results of operating condition division of annual steam demand

Table 1 Distribution of each working condition

工况	时长/h	概率
合计	8784	1.000
0	483	0.055
1	535	0.060
2	2153	0.240
3	1743	0.200
4	3302	0.380
5	568	0.065

Fig.4 State transition probability matrix

Fig.5 Scenario reduction results

Fig.6 Prediction steam flow of 8.8, 5.5, 2.0 MPa

Fig.7 Comparison between the number of different generated scenarios and Sp=5000

Fig.8 Mean absolute error between the number of different generated scenarios and Sp=5000

Fig.9 8.8 MPa prediction steam flow results

Fig.10 5.5 MPa prediction steam flow results

Fig.11 2.0 MPa prediction steam flow results

Table 2 Each steam evaluation index

蒸汽等级	稳定性/%	MAPE/%	RMSE
8.8 MPa	87.1	1.62	7.25
5.5 MPa	79.4	3.79	15.93
2.0 MPa	76.8	6.83	16.23

Fig.12 Schematic diagram of multistage steam turbine

Fig.13 Steam power system model

Table 3 Parameters of boilers

Boilers	Maximum load / (t/h)	Minimum load / (t/h)	Start-stop cost/CNY	Depreciation cost/(CNY/h)
B1—B4	470	330	105	150

Table 4 Parameters of turbines

Turbines

Rated power/MW

Maximum admission

flow/(t/h)

Maximum extraction flow/(t/h)

Table 6 Resource consumption of each optimization method before and after optimization

Method	Coal consumption/(t/h)	Water consumption/(t/h)
before optimization	270.39	1094.84
this paper optimization	263.49	1053.03
robust optimization	268.42	1084.60
stochastic programming	262.25	1044.19

Table 7 Cost of each optimization method before and after optimization

Item	Cost/(10⁴ CNY)
Item	Before optimization	This paper optimization	Robust optimization	Stochastic programming
coal	5871	5721	5828	5694
water	867	834	859	827
purchased electricity	0	0	0	0
purchased steam	0	0	0	0
the total cost	6738	6555	6687	6521
cost saving	—	183	51	217

Fig.14 Stability comparison of different optimization methods

Fig.15 Economic comparison of different optimization methods

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