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A Two-Stage Approach for Combined Heat and Power Economic Emission Dispatch: Combining Multi-Objective Optimization with Integrated Decision Making
Yang Li, Jinlong Wang, Dongbo Zhao, Guoqing Li, Chen Chen
TL;DR
CHPEED requires identifying best compromise solutions that reflect decision makers’ different, potentially conflicting preferences. The paper proposes a two-stage approach combining θ-DEA with FCM-GRP, and reports effectiveness and superiority on three test cases from simple to complex.
Problem
The paper addresses how to identify best compromise solutions representing decision makers’ different, even conflicting, preferences.
Method
A two-stage methodology uses θ-DEA to obtain Pareto-optimal solutions, then applies fuzzy c-means clustering and grey relation projection to identify compromise solutions.
Results
Simulation results on three test cases from simple to complex demonstrate the approach’s effectiveness and superiority over other state-of-the-art methods.
Takeaways & Limitations
The approach incorporates integrated decision making into CHPEED to determine compromise solutions for different decision-maker preferences.
Takeaways & Limitations
The grey relation projection analysis sets the weights of the two objectives equal, while dynamic dispatch imposes ramp-rate limits on each unit.
Abstract
from arXiv · showhide
To address the problem of combined heat and power economic emission dispatch (CHPEED), a two-stage approach is proposed by combining multi-objective optimization (MOO) with integrated decision making (IDM). First, a practical CHPEED model is built by taking into account power transmission losses and the valve-point loading effects. To solve this model, a two-stage methodology is thereafter proposed. The first stage of this approach relies on the use of a powerful multi-objective evolutionary algorithm, called θ-dominance based evolutionary algorithm (θ-DEA), to find multiple Pareto-optimal solutions of the model. Through fuzzy c-means (FCM) clustering, the second stage separates the obtained Pareto-optimal solutions into different clusters and thereupon identifies the best compromise solutions (BCSs) by assessing the relative projections of the solutions belonging to the same cluster using grey relation projection (GRP). The novelty of this work is in the incorporation of an IDM technique FCM-GRP into CHPEED to automatically determine the BCSs that represent decision makers' different, even conflicting, preferences. The simulation results on three test cases with varied complexity levels verify the effectiveness and superiority of the proposed approach.
Acronyms
The paper uses abbreviations for cogeneration dispatch, optimization methods, decision analysis, feasible operation, and valve-point effects.
- CHP, CHPED, and CHPEED denote combined heat and power, CHP economic dispatch, and CHP economic emission dispatch, respectively.
- MOO, MOEAs, MOPSO, NSGA-II, MLCA, SPEA 2, RCGA, PSO, EP, AIS, DE, BCO, and EFA name optimization approaches.
- TVA-PSO, NLP, and NBI denote time-varying acceleration particle swarm optimization, non-linear programming, and normal boundary intersection.
- VPLE and FOR denote valve-point loading effects and the feasible operation region.
- IDM, FCM, GRP, and BCSs refer to integrated decision making, fuzzy c-means, grey relation projection, and best compromise solutions.
Symbols
The symbols define unit cost and emission quantities, system demands and losses, dispatch timing, optimization geometry, clustering, and grey-relation assessment.
- Total C and C_i^p, C_j^c, and C_k^h denote total fuel cost and the costs of power-only, CHP, and heat-only units.
- a_i, b_i, d_i, φ_k, η_k, λ_k, α_j, β_j, γ_j, δ_j, ε_j, and ξ_j are unit cost coefficients.
- P_i, O_j, H_j, and T_k describe generating capacities or calorific values for power-only, CHP, and heat-only units.
- N_p, N_h, N_c, E_Total, E_S, E_C, and the listed μ, κ, π, σ, υ, τ, ρ, ϑ, ψ, and ϖ terms describe unit counts, emissions, and emission coefficients.
- P_D, P_L, B_ij, B_0i, B_00, and H_D represent power demand, transmission loss, loss coefficients, and total heat demand.
- a_t, N_T, UR_i, and DR_i specify the time interval, number of intervals, and ramp-up and ramp-down limits.
- F(x), Λ, λ_j, L, u, and the distance terms define normalized objectives, reference-point geometry, projections, and distances.
1. Introduction
The introduction motivates CHPEED as a constrained multi-objective dispatch problem and presents a two-stage approach for generating and selecting compromise solutions.
- Motivation: CHP improves energy utilization by recovering waste heat, with reported efficiency of 90% and above.
- Motivation: CHP generation is reported to save 10%~40% of generation cost and reduce greenhouse-gas emissions by nearly 13%~18%.
- Problem: CHPEED seeks heat-power operating points with reasonable fuel costs and emissions while satisfying heat, electricity, equality, and inequality constraints.
- Problem: The problem is computationally difficult because it is nonlinear, non-convex, and non-smooth, and is hard to solve directly.
- Literature gap: Existing methods obtain Pareto-optimal solutions, but decision makers face difficulty identifying BCSs under different or changing preferences.
- Literature gap: The paper frames identifying BCSs representing different, even conflicting, preferences as a pressing CHPEED task.
- Contributions: The proposed model includes valve-point loading effects and power transmission losses to coordinate economic and environmental objectives.
- Contributions: A two-stage method uses θ-DEA for multiple Pareto-optimal solutions and FCM-GRP to automatically identify best compromise solutions.
2. Problem formulation
The formulation models fuel cost and gas emissions for power-only, CHP, and heat-only units, while enforcing demand and transmission-loss constraints.
- 2.1.1 Fuel costs: Valve-point loading effects model increased consumption when a valve suddenly opens, producing pulsating effects in the unit consumption curve.
- 2.1.1 Fuel costs: The VPLE change function uses each power-only unit’s lower output limit and cost-change coefficients.
- 2.1.1 Fuel costs: Total fuel cost combines the costs of power-only, CHP, and heat-only units, with separate coefficients for each unit type.
- 2.1.1 Fuel costs: The cost formulation uses generating capacities, calorific values, and the numbers of power-only, heat-only, and CHP units.
- 2.1.2 Gas emission: Gas emissions include SO2, NOx, and CO2 because these pollutants are particularly harmful to environments.
- 2.1.2 Gas emission: Total emissions combine SO2 and NOx emissions with CO2 emissions using coefficients for power-only, CHP, and heat-only units.
- 2.2.1. Power demand constraint: The power-demand constraint requires generation to meet demand plus power transmission loss.
- 2.2.1. Power demand constraint: Transmission loss is calculated from unit productions using pairwise, unit-specific, and constant loss coefficients.
The amount of heat required for the system is shown in Eq. (7):
Equation (7) expresses the system’s total heat requirement, denoted by D H.
- D H denotes the total system heat demand in Eq. (7).
The constraints of each unit are as follows [8, 13]:
The model constrains power-only, CHP, and heat-only units, while CHP heat and power outputs are interrelated within a feasible operating region.
- Power-only unit outputs are bounded between their minimum and maximum limits.
- CHP unit power and heat outputs are constrained by coupled feasible-region relationships.CHP heat and power generations limit each other within the heat-power feasible operation region.
- Heat-only unit outputs are bounded by minimum and maximum heat-generation limits.
- The CHP feasible operating region is represented by the closed area surrounded by curve ABCDEF.
- Dynamic dispatch additionally imposes ramp-up and ramp-down rate limits over time intervals.URi and DRi denote the ramp-up and ramp-down rate limits of unit i.
3. Proposed approach
The proposed approach first obtains Pareto-optimal CHPEED solutions with θ-DEA, then clusters them with FCM and selects best compromise solutions using GRP.
- MOO generates multiple solutions to accommodate differently weighted or conflicting objectives and decision-maker preferences.
- Stage 1 uses θ-DEA to seek Pareto-optimal solutions, while Stage 2 uses FCM-GRP to determine BCSs.
- θ-DEA: θ-dominance balances convergence and diversity through a fitness-evaluation scheme.The objective vector is associated with reference points and distances in objective space.
- θ-DEA: The θ-DEA workflow generates reference points and populations, iterates through offspring and clustering steps, and outputs final nondominated solutions.
- FCM-GRP: FCM separates Pareto-optimal solutions into two clusters and provides cluster centroids for the CHPEED problem.The clustering number cl N is set to 2 to reflect preferences over economy and environmental protection.
- FCM-GRP: GRP ranks schemes by relative projection, with higher values indicating schemes closer to the positive ideal and farther from the negative ideal; highest values define BCSs.The two objective weights are set equally for analysis.
- Performance evaluation: IGD and Spread evaluate convergence and distribution, with smaller IGD indicating better convergence and diversity and higher Spread indicating worse distribution.
- Solving process: The solving process applies θ-DEA, evaluates IGD and Spread against MOPSO and other alternatives, then performs FCM-based decision making.
4. Case Studies
Three test cases evaluate the approach from simple to complex settings, progressively incorporating valve-point effects, transmission losses, and dynamic dispatch.
- Three test cases are designed with varied complexity, from simple to complex.
- Case 1 considers only valve-point loading effects.
- Cases 2 and 3 include valve-point loading effects and power transmission losses, with Case 3 formulated as dynamic dispatch.
- The proposed approach is compared extensively with other advanced algorithms using equal maximum iteration numbers and population sizes of 100.
- The test case contains one power-only unit, three CHP units, and one heat-only unit.
Appendix A.1.
The appendix evaluates θ-DEA and the FCM-GRP decision process across CHPEED cases, showing strong Pareto-front quality and multiple preference-sensitive compromise solutions. Results also indicate advantages over comparison algorithms in convergence, distribution, extreme solutions, and dispatch practicality.
- Case 1: θ-DEA generates nearly complete, uniformly distributed Pareto-optimal solutions and generally outperforms NSGA-II and MOPSO.Its Pareto front is reported as well distributed, and it dominates comparison fronts in most cases.
- Case 1: Fuel cost decreases as gas emission increases, demonstrating a trade-off between the two conflicting objectives.This conflict complicates selecting compromise solutions that reflect different decision-maker preferences.
- Case 1: FCM-GRP clusters Pareto-optimal solutions and extracts multiple BCSs representing different decision-maker preferences.In Case 1, FCM clustering separates the solutions into two groups before grey relation projection identifies two BCSs.
- Case 1: Compared with NSGA-II and SPEA 2, BCS 1 reduces cost by $ 504.5 and $ 460.1 while increasing emission by 1.4 kg and 1.1 kg, respectively.For BCS 2, cost increases by $ 128.6 and $ 173.0 while emission decreases by 1.0 kg and 1.3 kg, respectively.
- Case 2: In Case 2, θ-DEA achieves better IGD and Spread metrics, with well-distributed and well-separated solutions that generally dominate other algorithms.The reported results indicate stronger convergence and distribution performance than MOPSO and NSGAII.
- Case 2: For Case 2 extreme solutions, θ-DEA outperforms other algorithms in cost and transmission loss, while its emission extreme solution is superior without transmission-loss degradation.The reported comparisons include RCGA, PSO, EP, AIS, DE, BCO, and MOPSO.
- Case 2: Case 2 cost dispatch places power-only units near maximum output while CHP units provide support, reflecting their lower operating costs.The stated explanation attributes this pattern especially to the lower costs of power-only unit 4.
- Case 2: In Case 2, BCS 1 reduces cost by $ 1236.9, $ 255.1, and $ 227.8 versus NSGA-II, MLCA, and NBI, while BCS 2 reduces emissions by 16.1 kg, 1.4 kg, and 1.2 kg.The two BCSs also have better power transmission losses than the other methods.
5. Conclusion
The proposed two-stage approach combines a practical CHPEED model with θ-DEA and FCM-GRP to generate Pareto-optimal solutions and identify best compromise solutions. Results across test cases indicate that it coordinates economic and environmental objectives while representing diverse decision-maker preferences.
- The practical CHPEED model accounts for valve-point loading effects and power transmission losses.
- The two-stage methodology uses θ-DEA to obtain multiple Pareto-optimal solutions and FCM-GRP for integrated decision making.
- Across three test cases, the approach yields multiple complete and well-distributed Pareto-optimal solutions and automatically identifies BCSs representing different preferences.
- In Case 1, BCS 1 has cost $14504.2 and emission 7.5 kg, whereas BCS 2 has cost $15137.3 and emission 5.1 kg.
- Compared with NSGA-II and SPEA2, BCS 1 reduces cost by $504.5 and $460.1, while BCS 2 reduces emissions by 1.0 kg and 1.3 kg, respectively.
- Future work will extend the study toward smart integrated energy systems and incorporate load, renewable-generation, and energy-storage considerations.