Optimal operation and planning of hybrid AC/DC power systems using multi-objective grasshopper optimization algorithm

Optimal power flow (OPF) in a hybrid alternating current and multi-terminal high-voltage direct current (AC-MTHVDC) grid is currently one of the most popular optimization problems in modern power systems. The critical necessity of addressing global warming and reducing generation costs is encouraging the integration of eco-friendly renewable energy sources (RESs) into the OPF problem. In this direction, the present research has centred on the formulation and solution of the multi-objective (MO) AC-MTHVDC-OPF problem incorporating RESs such as wind, solar, small-hydro, and tidal power. The available power of RESs is calculated by means of the Weibull, lognormal, and Gumbel probability density functions. The proposed MO-OPF optimizes the double and triple configurations of various objective functions, including total cost, the total cost with the valve-point effect, the total cost with emission and carbon tax, voltage deviation, and power loss. Multi-objective grasshopper optimization algorithm (MOGOA) is applied to find non-dominated Pareto-optimal solutions of the non-convex, nonlinear and high-dimensional MO/AC-MTHVDC-OPF problem. The obtained results are compared with the results of MSSA, MODA, MOALO, and MO_Ring_PSO_SCD algorithms. The comparison of results gives that MOGOA outperforms competitive optimizers with respect to the quality of Pareto-optimal solutions and their distribution.

OPF:

Optimal power flow

MTHVDC:

Multi-terminal high-voltage direct current

AC:

Alternating current

MO:

Multi-objective

RESs:

Renewable energy sources

MOGOA:

Multi-objective grasshopper optimization algorithm

MSSA:

Multi-objective salp swarm algorithm

MODA:

Multi-objective dragonfly algorithm

MOALO:

Multi-objective ant lion optimizer

MO_Ring_PSO_SCD:

Multi-objective particle swarm optimization using ring topology and special crowding distance

IEEE:

The institute of electrical and electronics engineers

VSC:

Voltage source converter

HVDC:

High-voltage direct current

PDFs:

Probability density functions

MO/AC-MTHVDC-OPF:

Multi-objective alternating current multi-terminal high voltage direct current optimal power flow

\({S}_{ab}\) :

MVA injected into the VSCs of the AC grid

\({R}_{\mathrm{dc}}\) :

Resistance of DC line

\({V}_{{c}_{b}}\) :

Magnitude of the controlled voltage source that is linked to AC buses

\({V}_{{s}_{a}}\) :

Voltage magnitude of AC bus

\({I}_{ab}\) :

Injected current into the VSCs of the AC grid

\({N}_{\mathrm{AC}-b}\) :

Number of AC buses that is linked to VSC

\({N}_{\mathrm{VSC}}\) :

Number of VSC

\({Pc}_{b}+{jQc}_{b}\) :

Active and reactive powers at each VSC

\({Ps}_{a}+{jQs}_{a}\) :

Active and reactive powers at AC side

\({\delta }_{ab}\) :

Phase angle difference between VSC and the attached AC side

\({P}_{{\text{TG}}_{1}}\) :

Active power output of slack generator

\({V}_{\text{L}}\) :

Voltage magnitude of load bus

\({Q}_{\mathrm{TG}}\) :

Reactive power of thermal generator

\({Q}_{\mathrm{WS}}\) :

Reactive power of wind generator

\({Q}_{\mathrm{PVSH}}\) :

Reactive power of combined solar PV/small-hydro system

\({Q}_{\mathrm{WS}+\mathrm{TDL}}\) :

Reactive power of combined wind-tidal energy system

\({S}_{L}\) :

Transmission line loading

\(\mathrm{NTG}\) :

Number of thermal generators

\(\mathrm{NPQ}\) :

Number of load buses

\(\mathrm{NWS}\) :

Number of wind power units

\(\mathrm{NPVSH}\) :

Number of combined solar PV/small-hydro power units

\(\mathrm{NWSTDL}\) :

Number of combined wind-tidal power units

\(\mathrm{NTL}\) :

Number of AC transmission lines

\({P}_{\mathrm{TG}}\) :

Power output from thermal generator

\({P}_{\mathrm{WS}}\) :

Power output from wind generator

\({P}_{\mathrm{PVSH}}\) :

Power output from combined solar PV/small-hydro energy unit

\({P}_{\mathrm{WS}+\mathrm{TDL}}\) :

Power output from combined wind-tidal energy unit

\({V}_{\text{G}}\) :

Voltage magnitude of generator buses

\(T\) :

Transformer tap setting ratio

\(\mathrm{NG}\) :

Number of generator buses

\(\mathrm{NT}\) :

Number of transformers

\({V}_{{\mathrm{dc}}_{i}}\) :

Voltage magnitude of i-th bus on DC grid

\({V}_{{c}_{i}}\) :

Voltage magnitude at the AC terminal of i-th VSC

\({P}_{{s}_{i}}\) :

Active power output from i-th connected converter to the AC side

\({Q}_{{s}_{i}}\) :

Reactive power output from i-th connected converter to the AC side

\({CF}_{0}\) :

Generation cost (in $/h) of thermal generators

\({m}_{i},{l}_{i}, {k}_{i}\) :

Cost coefficients of i-th thermal generator

\(CF\) :

Generation cost (in $/h) of thermal generator with valve-point effect

\({n}_{i}, {r}_{i}\) :

Valve-point effect coefficients for i-th thermal generator

\({{P}_{TGi}}^{\mathrm{min}}\) :

Minimum active power output from i-th thermal generator

\({E}_{\mathrm{total}}\) :

Total emission value

\({C}_{\mathrm{tax}}\) :

Carbon tax value

\({C}_{\mathrm{E}}\) :

Emission cost

\({\mu }_{i}\), \({\varphi }_{i}\), \({\acute{\alpha}}_{i}\), \({\gamma }_{i}\), \({\grave{\varepsilon}}_{i}\) :

Emission coefficients for i-th thermal generator

\({DC}_{\mathrm{W}}\) :

Direct cost for wind power

\(\mathrm{wp}\) :

Direct cost coefficient of the wind power

\({DC}_{\mathrm{PVSH}}\) :

Direct cost for combined solar PV/small-hydro power

\({P}_{\mathrm{PVSH},s}\) :

Scheduled power of solar PV unit

\({P}_{\mathrm{PVSH},sh}\) :

Scheduled power of small-hydro unit

\(\mathrm{pv}\) :

Cost coefficient of the solar PV power

\(\mathrm{sh}\) :

Cost coefficient of small-hydro power

\({DC}_{\mathrm{WSTDL}}\) :

Direct cost for combined wind-tidal power

\({P}_{\mathrm{TDLS}}\) :

Scheduled power of tidal energy

\(tdl\) :

Cost coefficient for tidal power

\({OC}_{\mathrm{W}}\) :

Over-estimation cost for wind power

\({UC}_{\mathrm{W}}\) :

Under-estimation cost for wind power

\({C}_{\mathrm{Ow}}\) :

Over-estimation cost coefficient for wind power

\({C}_{\mathrm{Uw}}\) :

Under-estimation cost coefficient for wind power

\({P}_{\mathrm{wav}}\) :

Available power from wind power plant

\({P}_{\mathrm{wr}}\) :

Rated power of wind power plant

\({OC}_{\mathrm{PVSH}}\) :

Over-estimation cost for combined solar PV/small-hydro power

\({UC}_{\mathrm{PVSH}}\) :

Under-estimation cost for combined solar PV/small-hydro power

\({C}_{\mathrm{Opvsh}}\) :

Over-estimation cost coefficient for combined solar PV/small-hydro power

\({C}_{\mathrm{Upvsh}}\) :

Under-estimation cost coefficient for combined solar PV/small-hydro power

\({P}_{\mathrm{PVSHav}}\) :

Available power from combined solar PV/small-hydro power plant

\({OC}_{\mathrm{TDL}}\) :

Over-estimation cost for tidal power

\({UC}_{\mathrm{TDL}}\) :

Under-estimation cost for tidal power

\({C}_{\mathrm{Otdl}}\) :

Over-estimation cost coefficient for tidal power

\({C}_{\mathrm{Utdl}}\) :

Under-estimation cost coefficient for tidal power

\({P}_{\mathrm{TDLav}}\) :

Available power from tidal power plant

\({F}_{\mathrm{obj}}\) :

Objective function

\({N}_{\mathrm{DC}}\) :

Number of buses on DC side

\({P}_{\mathrm{loss}\_\mathrm{AC}}\) :

AC grid active power loss

\({P}_{\mathrm{loss}\_\mathrm{DC}}\) :

DC grid active power loss

\({P}_{\mathrm{loss}\_\mathrm{VSC}}\) :

Active power loss at VSC

\({\psi }_{1}\), \({\psi }_{2}\), \({\psi }_{3}\) :

Power loss coefficients related to each VSC

\({I}_{c,i}\) :

The current of i-th VSC

\({N}_{\mathrm{AC}}\) :

Number of AC bus

\({P}_{Gi}\) :

Active power at generation bus i

\({P}_{Di}\) :

Active power demand at generation bus i

\({Q}_{Gi}\) :

Reactive power of the i-th generator

\({Q}_{Di}\) :

Reactive power of the i-th load bus

\({Q}_{ci}\) :

Reactive power compensation at bus i

\({G}_{ij}\) :

Conductance between bus i and bus j

\({B}_{ij}\) :

Susceptance between bus i and bus j

\({\delta }_{i}\), \({\delta }_{j}\) :

Voltage angle of the i-th and j-th AC bus

\({P}_{dc,i}\) :

Active power flow through a DC line (leaving DC bus i)

\({G}_{dc,ij}\) :

Conductance of the DC line between bus i and bus j

\({S}_{\mathrm{DC}}\) :

Transmission line loading of DC line

\({d}_{i}^{\mathrm{min}}, {d}_{i}^{\mathrm{max}}\) :

Minimum and maximum limits of the circle diameter related to the i-th VSC P-Q capability

\({P}_{0}, {Q}_{0}\) :

Circle center related to the i-th VSC P-Q capability

\({N}_{f}\) :

Number of DC transmission line

\({v}_{w}\) :

Wind speed (m/s)

\(\acute{\omega}\) , \(\lambda\) :

Weibull PDF scale and shape factors

\({f}_{v}\left({v}_{w}\right)\) :

Probability of wind speed

\({v}_{\mathrm{in}}\), \({v}_{{\rm r}},\) \({v}_{\mathrm{out}}\) :

Cut-in, rated and cut-out wind speeds

\({p}_{w}\) :

Output power from a wind turbine

\({p}_{\mathrm{wr}}\) :

Rated output power of wind turbine

\({f}_{w}\left({p}_{w}\right)\) :

Wind power probability

\(\xi , { \vartheta }\) :

Lognormal PDF mean and standard deviation parameters

\({G}_{\text{pv}}\) :

Solar irradiance

\({f}_{{G}_{\mathrm{pv}}}\left({G}_{\mathrm{pv}}\right)\) :

Probability of solar irradiance

\({P}_{\mathrm{pv}}\) :

Power output from solar PV unit

\({R}_{\mathrm{C}}\) :

Certain irradiance

\({Q}_{\mathrm{wsh}}\) :

River flow rate

\({f}_{Q}\left({Q}_{\mathrm{wsh}}\right)\) :

Probability of river flow rate

\(\ddot{\Upsilon}\), ψ:

Gumbel PDF scale and location parameters

\({Q}_{\mathrm{TDL}}\) :

Discharge rate

\({f}_{\mathrm{QTDL}}\left({Q}_{\mathrm{TDL}}\right)\) :

Probability of discharge rate

\(\zeta\), :

Gumbel PDF parameters

\(A\) :

Archive vector

\(P\) :

Population

\({O}_{\mathrm{A}}\) :

Objective function value of archive members

\({O}_{\mathrm{P}}\) :

Objective function value of solution candidates in population

\(\mathrm{PS}\) :

Pareto set

\(\mathrm{PF}\) :

Pareto front

MVAr:

Megavolt-ampere reactive

MW:

Megawatt

\({P}_{\mathrm{RES}}^{\mathrm{max}}\) :

Maximum active power output from renewable energy source

maxFEs:

Maximum number of fitness function evaluations

\({n}_{\mathrm{obj}}\) :

Number of objective functions

p.u:

Per unit

$/h:

Dollar per hour

Authors and Affiliations

1. Department of Electronics and Automation, Vocational School, Dogus University, 34775, Istanbul, Turkey

   Huseyin Bakir

2. Department of Electrical and Electronics Engineering, Engineering Faculty, Duzce University, 81620, Duzce, Turkey

   Ugur Guvenc

3. Department of Software Engineering, Of Technology Faculty, Karadeniz Technical University, 61080, Trabzon, Turkey

   Hamdi Tolga Kahraman

Corresponding authors

Correspondence to Huseyin Bakir or Ugur Guvenc.

Conflict of interest

The authors declare that they have no conflict of interest.

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