Abstract
Due to the increased cost of energy sources and related environmental problems, systems with higher efficiency such as combined heat and power (CHP) units are getting more popular. Renewable energy sources can be another alternative solution for the above mentioned problems. Scheduling of renewable-based systems are getting more complicated due to the intermittent behavior of these sources. In this chapter, a stochastic programming framework is utilized to model uncertainties in dynamic economic dispatch (DED) problem of CHP based systems integrating wind energy. Forecast errors of electrical load and wind power are assumed as the two sources of uncertainty. A heuristic method called particle swarm optimization (PSO) is used to attain optimal solution of the problem due to non-linearity, non-convexity, and complexity of the problem. The stochastic programming provides more comprehensive and realistic viewpoint about dispatch problem by considering a variety of most probable scenarios compared to a single scenario.
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Abbreviations
- \(P_{d,t,s}\), \(P_{w,t,s}\):
-
Electrical load demand and output power of wth wind unit at time t in scenario s, respectively
- \(P_{d,t}^{forecasted}\), \(P_{w,t}^{forecasted}\):
-
Forecasted values for electrical load demand and output wind units at time t, respectively
- \(\Delta P_{d,t,s}\), \(\Delta P_{w,t,s}\):
-
Forecast errors related to electrical load demand and output power of wind unit w at time t in scenario s, respectively
- \(N_{s}\) :
-
Total number of scenarios
- \(B_{{({\text{interval}},t,s)}}^{L}\), \(B_{{({\text{interval}},t,s)}}^{W}\):
-
Binary parameters of intervals at time t in scenario s for electrical load demand and wind power, respectively
- \(\pi_{s}\) :
-
Probability of scenario s
- \(\alpha_{i,t}\), \(\beta_{j,t}\):
-
Probabilities of electrical load demand and wind power for intervals i and j at time t, respectively
- \(C_{i} (P_{i}^{p} )\) :
-
Operation cost of ith power-only unit for producing \(P_{i}^{p}\) MW
- \(C_{j} (P_{j}^{c} ,H_{j}^{c} )\) :
-
Operation cost for jth co-generation unit for producing \(P_{j}^{c}\) MW electricity power and \(H_{j}^{c}\) MWth heat power
- \(C_{k} (P_{k}^{h} )\) :
-
Operation cost of heat-only unit while producing \(H_{k}^{h}\) MWth heat power
- \(N_{p}\), \(N_{c}\), \(N_{h}\):
-
Total number of power-only, CHP and heat-only units, respectively
- \(i\), \(j\), \(k\):
-
Indices for power-only, CHP and heat-only units, respectively
- \(\alpha_{i}\), \(\beta_{i}\), \(\gamma_{i}\), \(\lambda_{i}\), \(\rho_{i}\):
-
Constant cost coefficients for ith power-only unit
- \(a_{j}\), \(b_{j}\), \(c_{j}\), \(d_{j}\), \(e_{j}\), \(f_{j}\):
-
Coefficients of cost function related to jth CHP unit
- \(a_{k}\), \(b_{k}\), \(c_{k}\):
-
Coefficients for calculating the operation cost of heat-only units
- \(P_{d,t,s}\) :
-
Electrical power demand at time t in scenario s
- \(H_{d,t}\) :
-
Heat power demand at time t
- \(P_{i}^{p,{\min} }\), \(P_{i}^{p,{\max} }\):
-
Lower and upper generation limits for power-only units, respectively
- \(P_{j}^{c,{\min} }\),\(H_{j}^{c,{\min} }\), \(P_{j}^{c,{\max} }\), \(H_{j}^{c,{\max} }\):
-
Minimum and maximum electric and heat powers outputs for CHP units, respectively
- \(H_{k}^{h,{\min} }\), \(H_{k}^{h,{\max} }\):
-
Limits for heat-only units
- \(V^{CO}\), \(V^{CI}\), \(V^{R}\):
-
Cut-off, cut-in and rated speed of wind turbine
- \(P^{{\max} }\) :
-
Rated power of the wind turbine
- \(V_{t}\) :
-
Forecasted wind speed at time t
- N :
-
Total number of decision variables in the problem
- \(\omega\) :
-
The inertia weight for PSO
- \(r_{1}^{n}\), \(r_{2}^{n}\):
-
Random numbers in the interval [0, 1]
- \(p_{{best_{i,n} }}^{iter - 1}\), \(g_{best,n}^{iter - 1}\):
-
Best position of ith particle in previous iteration and best position of entire swarm
- \(C_{1}\), \(C_{2}\):
-
Learning factors of PSO
- \(x_{n}^{{\max} }\), \(x_{n}^{{\min} }\):
-
Maximum and minimum limits of variables
- \(r\) :
-
Parameter to control the amount of change in velocity in PSO
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MATLAB Codes
MATLAB Codes
The main m-file of the program is as follows:
The function of PSO algorithm that is used for optimization procedure is provided in the following. The number of particles and total number of iterations are inputs of this function, while the optimal cost, best solutions and best cost of each iteration are outputs of this function:
Fitness function for calculating the objective function that receives particles as input and returns total cost as output is as follows:
The MATLAB code for scenario generation process is as follows:
The function that is used for producing binary scenarios is provided in the following:
The function for converting binary scenarios to scenarios with real values is as follows:
The fast forward scenario reduction MATLAB code is as follows:
The function for creating electrical and heat demands as forecasted values is coded as:
The function to model the wind turbine and forecasted wind speed can be coded as:
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Pourghasem, P., Sohrabi, F., Jabari, F., Mohammadi-Ivatloo, B., Asadi, S. (2020). Combined Heat and Power Stochastic Dynamic Economic Dispatch Using Particle Swarm Optimization Considering Load and Wind Power Uncertainties. In: Pesaran Hajiabbas, M., Mohammadi-Ivatloo, B. (eds) Optimization of Power System Problems . Studies in Systems, Decision and Control, vol 262. Springer, Cham. https://doi.org/10.1007/978-3-030-34050-6_7
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