Static optimal load flow of combined heat and power system with valve point effect and prohibited operating zones using Krill Herd algorithm

Abstract

Cogeneration of heat as well as electrical energy is an efficient and matured method which consumes less primary energy and reduces emission level significantly. In this process, heat production is reliant on power production and is restricted by the feasible operation region (FOR). This paper considered prohibited operating zones and valve point effect in its maiden formulation of static load flow problem of power system involving cogeneration units for maintaining voltage level of buses within specified limits while satisfying other linear and non-linear constraints in order to make the production cost effective. Cost of production and transmission loss is optimized using Krill Herd algorithm. This algorithm is based on the foraging activities of individual Krill. The distance of Krill individual from food and the denser swarm constitutes the objective function. The efficacy of the algorithm is showed through results of two test systems. Both test systems have been adapted from IEEE 30 bus 6 generators standard test system.

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Abbreviations

CHP:

Combined heat and power (Power-MW; Heat-MWTh)

TCT:

Tap changing transformer

VPE:

Valve point effect

POZ:

Prohibited operating zone

KH:

Krill Herd

POU:

Power only unit

FOR:

Feasible operation region

P UB :

Upper bound of poz (MW)

P LB :

Lower bound of poz (MW)

P Gi, P Gi max, P Gi min :

Generation of i-th generator, its limits (MW)

NG :

No of POUs

Nq :

No of CHPs

Q G :

Reactive power produced by generators (MVAR)

PV :

No of PV buses

TL :

No of transmission lines

NT :

No of TCTs

NC :

No of shunt VAR compensators

T :

Tap setting of TCT

Q C :

Output of shunt VAR compensators (MVAR)

P Di :

Injected active power at bus ‘i’ (MW)

Q Di :

Injected reactive power at bus ‘i’ (MVAR)

θ ik :

Phase angle between buses i and k (rad)

G ik :

Transfer conductance between i and k buses (mho)

B ik :

Susceptance between i and k buses (Siemens)

NB :

Total no of buses

V Li , V Li max , V Li min :

Voltage of each PQ bus, its maximum and minimum limits [pu (Volt)]

L li , L li max :

Line limit (MVA)

S m :

Motion induced by surrounding Krills (m/s)

A m :

Foraging Motion of m-th Krill (m/s)

P m :

Diffusion Motion of m-th Krill (m/s)

λ :

Direction of motion

F, F m , F worst , F best :

Fitness of m-th Krill, its worst and best value (m)

Y, Y m , Y j :

Position of a Krill (m)

ε :

A very small predetermined number

d s,m :

Sensing distance of a Krill (m)

N :

Number of neighbours

I, Imax :

Iteration number, maximum iteration no

Vf :

Foraging speed (m/s)

A m old :

Last foraging motion (m/s)

P, P max :

Diffusion speed, its maximum value (m/s)

:

Random direction of motion

ω n :

Inertia weight factor of induced motion

ω f :

Inertia weight factor of foraging motion

ϒ m food , ϒ m best :

Food attractive, effect of best fitness

C t :

Constant (value between 0 and 2)

U B , L B :

Maximum and minimum limit of search space (m)

X food :

Virtual centre of food location (m)

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Appendix

Appendix

Test System II: cost function of CHP units:

$$ Cost_{CHP5} = 2650 + 34.5 \, \mathop P\nolimits_{5} + \, 0.1035 \, \mathop P\nolimits_{5}^{2} + \, 2.203h_{5} + \, 0.025 \, \mathop h\nolimits_{5}^{2} + \, 0.051h_{5}P_{5} $$
$$ Cost_{CHP8} = \, 2650 \, + \, 34.5 \, \mathop P\nolimits_{8} + \, 0.1035 \, \mathop P\nolimits_{8}^{2} + \, 2.203 \, \mathop h\nolimits_{8} + \, 0.025 \, \mathop h\nolimits_{8}^{2} + \, 0.051 \, \mathop P\nolimits_{8} \mathop h\nolimits_{8} $$
$$ Cost_{CHP11} = \, 2650 \, + \, 34.5 \, \mathop P\nolimits_{11} + \, 0.1035 \, \mathop P\nolimits_{11}^{2} + \, 2.203 \, \mathop h\nolimits_{11} + \, 0.025 \, \mathop h\nolimits_{11}^{2} + \, 0.051 \, \mathop P\nolimits_{11} \mathop h\nolimits_{11} $$
$$ Cost_{CHP13} = \, 2650 \, + \, 34.5 \, \mathop P\nolimits_{13} + \, 0.1035 \, \mathop P\nolimits_{13}^{2} + \, 2.203 \, \mathop h\nolimits_{13} + \, 0.025 \, \mathop h\nolimits_{13}^{2} + \, 0.051 \, \mathop P\nolimits_{13} \mathop h\nolimits_{13} $$

Subject to fulfillment of the following constraints:

$$ \mathop P\nolimits_{5} + \, 0.2727272727 \, \mathop h\nolimits_{5} {-} \, 60 \le 0;\begin{array}{*{20}c} {} \\ \end{array} 20 \, {-} \, \mathop P\nolimits_{5} \, {-} \, 0.25 \, \mathop h\nolimits_{5} \le 0;\begin{array}{*{20}c} {} \\ \end{array} 2.333333333 \, \mathop h\nolimits_{5} {-} \, \mathop P\nolimits_{5} {-} \, 83.3333333332 \le 0 $$
$$ \mathop P\nolimits_{8} + \, 0.2727272727 \, \mathop h\nolimits_{8} {-} \, 60 \le 0;\begin{array}{*{20}c} {} \\ \end{array} 20 \, {-} \, \mathop P\nolimits_{8} \, {-} \, 0.25 \, \mathop h\nolimits_{8} \le 0;\begin{array}{*{20}c} {} \\ \end{array} 2.333333333 \, \mathop h\nolimits_{8} {-} \, \mathop P\nolimits_{8} {-} \, 83.3333333332 \le 0 $$
$$ \mathop P\nolimits_{11} + \, 0.2727272727 \, \mathop h\nolimits_{11} {-} \, 60 \le 0;\begin{array}{*{20}c} {} \\ \end{array} 20 \, {-} \, \mathop P\nolimits_{11} \, {-} \, 0.25 \, \mathop h\nolimits_{11} \le 0;\begin{array}{*{20}c} {} \\ \end{array} 2.333333333 \, \mathop h\nolimits_{11} {-} \, \mathop P\nolimits_{11} {-} \, 83.3333333332 \le 0 $$
$$ \mathop P\nolimits_{13} + \, 0.2727272727 \, \mathop h\nolimits_{13} {-} \, 60 \le 0;\begin{array}{*{20}c} {} \\ \end{array} 20 \, {-} \, \mathop P\nolimits_{13} \, {-} \, 0.25 \, \mathop h\nolimits_{13} \le 0;\begin{array}{*{20}c} {} \\ \end{array} 2.333333333 \, \mathop h\nolimits_{13} {-} \, \mathop P\nolimits_{13} {-} \, 83.3333333332 \le 0 $$
$$ 0 \, \le {\text{ h}}_{31} \le \, 2695.2 MWTh $$

Cost function of power only generators with VPE (coefficient from Table 2):

$$ C_{G} = a + b \times P + c \times P^{2} + \left| {d \times \sin ((P_{\min } - P) \times e)} \right| $$

where CG are the cost of generators, P are the power production of generators and a, b, c, d, e are cost coefficients of respective generator as given in Table 2.

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Adhvaryyu, P.K., Adhvaryyu, S. Static optimal load flow of combined heat and power system with valve point effect and prohibited operating zones using Krill Herd algorithm. Energy Syst 12, 133–156 (2021). https://doi.org/10.1007/s12667-020-00378-9

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Keywords

  • Optimal load flow
  • Valve point effect (VPE)
  • Prohibited operating zone (POZ)
  • Krill
  • Krill Herd algorithm (KHA)
  • Feasible operation region (FOR)