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Solving the combined heat and power economic dispatch problem in multi-zone systems by applying the imperialist competitive Harris hawks optimization

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Abstract

Combined heat and power economic dispatch (CHPED) problem is one of the challenging issues in power system control, and operation, that searches for the optimal values of power and heat generation of different units to minimize the total operation cost. The interdependency of heat and power outputs of CHP units and valve point loading effects of power-only units make this problem very nonlinear and non-convex. So far, a wide variety of optimization algorithms based on simulation of various natural phenomena have been proposed and applied to solve this complex problem, claiming that the proposed algorithm is better than the similar optimization algorithms in terms of accuracy and run-time. In this paper, the CHPED problem is investigated in different case studies by using the imperialist competitive Harris hawks optimization (ICHHO) algorithm, as a combination of the imperialist competitive algorithm and Harris hawks optimization. The effectiveness of the proposed algorithm is first evaluated by testing it on the standard cases of 5, and 7 units as small-scale systems and 48-unit as the large-scale test system, considering the electrical losses. Also for the first time, the multi-zone CHPED (MZCHPED) problem is solved by a metaheuristic algorithm. The obtained results show that ICHHO algorithm reduces the operating costs in the range of 0.02–1.1%, compared to the best-reported results, in different single-zone CHPED (SZCHPED) problems. Also, the MZCHPED results, prove the effective performance of the algorithm compared to the similar SZCHPED problem.

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Data availability

Enquiries about data availability should be directed to the authors.Enquiries about data availability should be directed to the authors.

Abbreviations

ACS-DEM:

Adaptive cuckoo search with differential evolution mutation

B:

Matrix of Kron’s loss formula

BCO:

Bee colony optimization

CHP:

Combined heat and power

CHPED:

Combined heat and power economic dispatch

DE:

Differential evolution

EMA:

Exchange market algorithm

EP:

Evolutionary programming

FOR:

Feasible operating region

FRCSA-ABC:

Fuzzy adaptive ranking- based crow search algorithm with modified artificial bee colony

GA:

Genetic algorithm

GSA:

Gravitational search algorithm

HBOA:

Heap-based optimizer algorithm

HHO:

Harris hawks optimization

HOUs:

Heat-only units

HS:

Harmony search

ICA:

Imperialist competitive algorithm

ICHHO:

Imperialist competitive Harris hawks optimization algorithm

IMPOA:

Improved marine predators’ optimization algorithm

LF:

Levy flight

LR:

Lagrangian relaxation

MAs:

Meta-heuristic algorithms

MGOA-IHHO:

Modified grasshopper optimization algorithm and the improved Harris hawks optimizer

MGSO:

Modified group search optimizer

MICA:

Modified ICA

MIQP:

Mixed integer quadratic programming

MZCHPED:

Multi-zone CHPED

OF:

Objective function

OPF:

Optimal power flow

POUs:

Power-only units

PRD:

Progressive rapid dives

PSO:

Particle swarm optimization

RCGA:

Real-coded GA

SGWO:

Society-based grey Wolf optimizer

SZCHPED:

Single-zone CHPED

TVAC-PSO-GSA:

Time-varying acceleration coefficients PSO algorithm combined with GSA

VPLE:

Valve point loading effect

\(i,j,k,l\) :

Indices defining the different unit numbers

\(N_{{\text{p}}}\) :

Number of POUs

\(N_{{\text{c}}}\) :

Number of CHP units

\(N_{{\text{h}}}\) :

Number of HOUs

\(N_{{{\text{p}}.z}}\) :

Number of thermal units, located in the zth zone

\(N_{{{\text{c}}.z}}\) :

Number of CHP units located in the zth zone

\(N_{{{\text{h}}.z}}\) :

Number of HOUs in the zth zone

N imp :

Number of the strongest countries

N country.:

Number of countries

NHawks :

Number of hawks

NGroups :

Number of groups

\(d\) :

Number of variables

\(C_{{{\text{p}}i}} \left( {P_{i} } \right)\) :

Cost function of POU, ($/h)

\(C_{{{\text{c}}i}} \left( {O_{i} \cdot H_{i} } \right)\) :

Cost function of CHP unit, ($/h)

\(C_{{{\text{h}}i}} \left( {T_{i} } \right)\) :

Cost function of HOU, ($/h)

\(P_{{{\text{loss}}}}\) :

Electrical transmission loss, (MW)

\(E\) :

Escaping energy of the rabbit

\(f_{i}^{Hawks}\) :

Power of ith Hawks

\(f_{i}^{rabbit}\) :

Power of ith rabbit

\({\text{TC}}_{i}\) :

Total power, (or cost) of ith group

\(P_{i}\) :

Electrical power generated by the ith POU, (MW)

\(O_{i}\) :

Electrical power generated by the ith CHP unit, (MW)

\(H_{i}\) :

Generated heat by the ith CHP unit, (MWth)

\(T_{i} \) :

Generated heat by the ith HOU, (MWth)

\(P_{i.z}\) :

Generated power by the ith POU in the zth zone, (MW)

\(O_{i.z}\) :

Generated power by the ith CHP unit in the zth zone, (MW)

\( P\left( {z_{t} .z_{u} } \right)\) :

Exchange power between zone \({t}\), and zone \({u}\), (MW)

\(H_{i.z}\) :

Heat generated by the ith unit of CHP in the zth zone, (MWth)

\(T_{i.z}\) :

Heat generated by the ith unit of HOU in the zth zone, (MWth)

\(X\left( t \right)\) :

Position vector of search agents (i.e., hawks) at iteration t

\(X\left( {t + 1} \right)\) :

Position vector of search agents (i.e., hawks) at iteration t + 1

\(X_{{{\text{rabbit}}}} \left( t \right)\) :

Position of the rabbit, at iteration t

\(X_{m} \left( t \right)\) :

Mean state of the population of hawks, at iteration t

\(X_{i} \left( t \right)\) :

Position of the ith hawks in tth iteration

\(P_{{{\text{error}}}}\) :

Difference between demand power and total generated powers

\(H_{{{\text{error}}}}\) :

Difference between demand heat and total generated heat

\(\Delta X\left( t \right)\) :

Difference vector between the vectors of rabbit position and hawks position in the iteration of t

\(Z,Y\) :

Updated positions of hawks based on applying the LF

\(H_{{{\text{demand}}.z}}\) :

Heat demand of the zth zone, (MWth)

\(P_{i}^{\min }\) :

Minimum limit of generated power for ith POU, (MW)

\(P_{i}^{\max }\) :

Maximum limit of generated power for ith POU, (MW)

\(O_{i}^{\min } \left( {H_{i} } \right) \) :

Minimum limit of the generated power by ith CHP unit, (MW)

\(O_{i}^{\max } \left( {H_{i} } \right)\) :

Maximum limit of the generated power by ith CHP unit, (MW)

\(H_{i}^{\min } \left( {O_{i} } \right) \) :

Minimum limit of the produced heat by ith CHP unit, (MWth)

\(H_{i}^{\max } \left( {O_{i} } \right)\) :

Maximum limit of the produced heat by ith CHP unit, (MWth)

\(T_{i}^{\min }\) :

Minimum limit for produced heat by ith HOU, (MWth)

\(T_{i}^{\max }\) :

Maximum limit for produced heat by ith HOU, (MWth)

\({\text{LB}}\) :

Lower bound limit of the problem decision variables

\({\text{UB}}\) :

Upper bound limits of the problem decision variables

\(x_{i}^{\max }\) and \(x_{i}^{\min } \) :

Upper and lower limits of the ith variable

\(a_{i}\)($/MW2 h), \(b_{i}\)($/MW h), and \(c_{i}\)($/h):

Cost coefficients of the ith POU

\(d_{i}\)($/h) and \(e_{i}\)(rad/MW):

VPLE coefficients of the thermal unit i

\(\alpha_{i}\) ($/MW2 h), \(\beta_{i}\)($/MW h), \(\gamma_{i}\)($/h), \(\delta_{i}\)($/MWth2 h), \(\varepsilon_{i}\)($/MWth h) and \(\xi_{i}\)($/MW MWth h):

Cost coefficients of ith CHP unit

\(\eta_{i}\)($/MWth2 h), \(\theta_{i}\)($/MWth h) and \(\lambda_{i}\)($/h):

Cost coefficients of the ith HOU

\(B_{ij}\) :

Loss coefficient regarding the generating units i and j (1/MW)

\(P_{{\text{d}}}\) :

Electrical demand, (MW)

\(P_{{{\text{demand}}.z}}\) :

Power demand of the zth zone, (MW)

\(H_{{\text{d}}}\) :

System heat demand, (MWth)

\(E_{0}\) :

Initial energy state of rabbit strength

ε :

Accuracy parameter, equal to 0.05

\(\sigma ,u,v,\beta\) :

LF coefficients

\(q\) :

Random number in the range of (0,1), the chance for escaping the prey (rabbit)

\(u,v,r_{1} ,r_{2} ,r_{3} ,r_{4} ,{\text{rand}}\left( {} \right)\) :

Random numbers in the range of (0,1)

\(S\) :

Random vector with an equal dimension of size 1 × \(d\)

\(\Gamma\) :

Gamma function

\(\beta\) :

LF coefficient, and is equal to 1.5

\(x_{i.j}^{\left( 0 \right)}\) :

Initial value of the ith variable in jth country

\(\xi\) :

Positive number less than one.

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Funding

This study was not funded by any organization, agency or grant project.

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Authors and Affiliations

  1. Electrical Engineering Department, Engineering Faculty, Razi University, Kermanshah, Iran

    Amir Nazari & Hamdi Abdi

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  1. Amir Nazari
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  2. Hamdi Abdi
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Contributions

AN: Conceptualization, Software simulation, Validation, Formal analysis, Investigation, Writing-Original Draft, Resources. HA: Conceptualization, Methodology, Validation, Checking, Editing, Writing-Review & Editing, Supervision, Final Preparing.

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Correspondence to Hamdi Abdi.

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Nazari, A., Abdi, H. Solving the combined heat and power economic dispatch problem in multi-zone systems by applying the imperialist competitive Harris hawks optimization. Soft Comput 26, 12461–12479 (2022). https://doi.org/10.1007/s00500-022-07159-9

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