Detailed Non-Linear Constrained Multi-Objective Optimal Operation of Power Systems Including Renewable Energy Sources

Abstract

A modified cuckoo search (MCS) algorithm is proposed in this paper for an optimal operation problem of power systems with renewable energy sources including solar and wind energy sources. A non-linear constrained multi-objective optimal operation problem is formulated and detailed for an integrated power system in order to make it more realistic. Furthermore, the cuckoo search (CS) algorithm is modified to increase the convergence rate that is called the MCS algorithm. This variant mentions the step size of the Lévy flight. The modified IEEE 10-generator power system with integrated solar and wind power sources is considered in this paper. The numerical results on the above power system confirm the effectiveness of the proposal for the optimal operation of the integrated power system. A comparison with the CS algorithm and variants of the particle swarm optimization (PSO) algorithm indicates the superiority of the MCS algorithm for resolving complicated optimal operation problems of integrated power systems.

1 Introduction

Normally, the main purpose of resolving the optimal operation problem is to find out a fuel cost minimized for the power system [1,2,3,4]. Due to emissions produced by thermal power plants, the classical optimal operation problem can no longer be only considered to minimize the fuel cost recently. In addition, the emission optimization problem is proposed to minimize the emissions of SO_x and NO_x caused by thermal power plants [5,6,7, 11].

Additionally, the demand of electricity utilization has significantly increased in many countries, especially in Industry 4.0 which led to an energy crisis. In order to resolve this issue, there are a lot of research moving towards renewable energy sources. Compared to fossil-fuel energy sources, solar and wind energy sources are more sustainable, never going to run out as well as less polluted. These two power sources are considered to integrate with the existing power system to resolve the energy crisis and environmental pollution [1, 3, 4, 8, 9]. The integrated power system obviously depends on the solar irradiation, temperature and wind speed creating challenges in the optimal operation problem. In order to resolve this problem, there are many approaches mentioned such as a lagrangian relaxation with incremental proximal approach [1]; a granular computing approach [2]; a PSO algorithm [3, 7, 11]; a CS algorithm [4]; a bat algorithm with weighted sum [5]; a modified differential evolutionary with sine function [7], an ant colony optimization algorithm [9] and et cetera. It can be realized that there are recently more and more modern meta-heuristic algorithms proposed for resolving the optimal operation problem. These algorithms are inspired by nature and used popularly; however, they also have few drawbacks such as a premature convergence phenomenon, a speed-efficiency trade-off, selection of algorithm parameters and implementation in real-time which make them inflexible and complicated.

In this paper, a detailed non-linear constrained multi-objective optimal operation problem is presented in order to make the operation problem more realistic. In addition, the proposed MCS algorithm is to improve the searching ability of the CS algorithm as the solutions get closer to the optimal result. The proposed algorithm is apply to optimize the fuel cost and emission of the modified IEEE 10-generator power system with integrated solar and wind power sources.

2 Renewable Power Sources

The solar and wind power sources are considered in this paper where solar power is produced by either solar PV panels, solar thermal plants, or both; and wind power is produced by wind turbines. The solar power is presented in Sect. 2.1 followed by a description of the wind power.

2.1 Solar Power

The obtained power of solar PV panels at a maximum power point (MPP) is [8]:

$$P_{PV} = \left[ {P_{PV,STC} \times \frac{{G_{T} }}{1000} \times \left[ {1 - \gamma \times \left( {T - 25} \right)} \right]} \right] \times N_{PVs} \times N_{PVp}$$

(1)

where

P_PV:

The obtained PV power at the MPP (W);

P_{PV, STC}:

The rated PV power at the MPP considered in the standard testing condition (STC) (W);

G_T:

The irradiation level (W/m²);

γ:

The power temperature coefficient at the MPP (%/ °C);

T:

The cell temperature (°C);

N_PVs and N_PVp:

The number of PV modules in series and parallel composing the PV generator, respectively.

The obtained power of a solar thermal plant is [8]:

$$P_{Ther} = \eta \times A_{c} \times G_{T}$$

(2)

where

P_Ther:

The obtained power of a solar thermal plant (W);

η:

The collector efficiency;

A_c:

The collector area (m²).

2.2 Wind Power

The obtained power of a wind turbine is [8]:

$$P_{wind} = \frac{1}{2} \times C_{e} \times \rho \times A_{s} \times V_{wind}^{3}$$

(3)

where

P_wind:

The obtained power of a wind turbine (W);

C_e:

The efficiency factor;

ρ:

The air density, ρ = 1.225 kg/m³;

A_s:

The surface area traversed by the wind (m²);

V_wind:

The wind speed (m/s).

3 Optimal Operation of Integrated Power Systems

The optimal operation problem is to minimize the objective functions of the fuel cost and emission with the equality and inequality constraints established as follows.

3.1 Objective Function

* Fuel cost

The objective function of the fuel cost, C(P_G) is the quadratic function based on the fuel cost curves of thermal generators. The valve loading effects is considered through a sine component. Then, the objective function of the fuel cost is [9]:

$$f_{1} = C\left( {P_{G} } \right) = \sum\limits_{i = 1}^{{N_{g} }} {a_{i} + b_{i} P_{Gi} + c_{i} P_{Gi}^{2} + \left| {d_{i} \sin \left[ {e_{i} \left( {P_{Gi} - P_{Gi}^{\hbox{min} } } \right)} \right]} \right|}$$

(4)

where

C(P_G):

The fuel cost ($/h);

a_i, b_i and c_i:

The cost coefficients of the ith generator;

d_i and e_i:

The cost coefficients of the ith generator reflecting valve-point effects;

P_Gi:

The output active power of the ith generator, i = 1, …, N_g;

P_Gj:

The output active power of the jth generator, j ≠ i and j = 1, …, N_g;

P ^min_Gi :

The lower limit of the output active power of the ith generator;

P ^max_Gi :

The upper limit of the output active power of the ith generator;

P_G:

The vector of the output active powers of N_g generators defined as follow:

P_G:

$\left[ {P_{G1} ,P_{G2} , \ldots ,P_{GNg} } \right]^{T} ;$

N_g:

The total number of thermal generators in a power system;

T:

The time of one day.

* Emission

The objective function of the emission, E(P_G) is [9]:

$$f_{2} = E\left( {P_{G} } \right) = \sum\limits_{i = 1}^{{N_{g} }} {\alpha_{i} + \beta_{i} P_{Gi} + \gamma_{i} P_{Gi}^{2} + \xi_{i} \exp \left( {\omega_{i} P_{Gi} } \right)}$$

(5)

where

E(P_G):

The emission (ton/h);

α_i, β_i, γ_i, ξ_i and ω_i:

The emission coefficients of the ith generator.

* Objective function of fuel cost and emission

It is obvious that the optimal operation problem of the fuel cost and emission is to minimize f₁ = C(P_G) and f₂ = E(P_G) where the objective functions of the fuel cost and emission can be weighted according to their relative importance described as follows:

$$f = \left\{ {\begin{array}{*{20}l} {wC\left( {P_{G} } \right)} \\ {\left( {1 - w} \right)E\left( {P_{G} } \right)} \\ \end{array} } \right.$$

(6)

where

w:

The weighting coefficient, w∈ [0, 1];

If w = 0 then f is the objective function of the emission and if w = 1 then f is the objective function of the fuel cost.

A trade-off can be obtained as the weighting coefficient, w is varied from zero to one in order to ensure the optimization in operating.

Additionally, the uncertainty of solar and wind powers makes the optimal operation problem of the integrated power system more complicated. Therefore, these powers are treated as negative loads in order to make it more simple.

Then, the actual load power is:

$$P_{Actual\_load} = P_{Total\_load} - \left( {P_{solar} + P_{wind} } \right)$$

(7)

where

P_{Actual_load} and P_{Total_load}:

The actual and total load powers, respectively (W);

P_solar and P_wind:

The solar and wind powers, respectively (W).

3.2 Constraint Condition

The active power balance, power generation limits and ramp rate limits are the constraints considered in this paper.

* Active power balance

The total generation power must cover the actual load power, P_{Actual_load} and the active power loss in transmission lines, P_loss. Then, the constraint of the active power balance is:

$$\sum\limits_{i = 1}^{{N_{g} }} {P_{Gi} } = P_{Actual\_load} + P_{loss}$$

(8)

where

P _loss :

The active power loss in transmission lines (W).

$$P_{loss} = \sum\limits_{i = 1}^{{N_{g} }} {\sum\limits_{j = 1}^{{N_{g} }} {P_{Gi} B_{ij} P_{Gj} } + \sum\limits_{i = 1}^{{N_{g} }} {B_{i0} P_{Gi} } } + B_{00}$$

(9)

where

B_ij, B_i0 and B₀₀:

The B-coefficients of the power loss of the power system depending on the impedance parameters of transmission lines.

* Power generation limits

The generation powers should be within the limits as follows:

$$P_{Gi}^{\hbox{min} } \le P_{Gi} \le P_{Gi}^{\hbox{max} } ,\quad i = 1, \ldots ,N_{g}$$

(10)

$$0 \le P_{solar} \le P_{solar}^{\hbox{max} }$$

(11)

$$0 \le P_{wind} \le P_{wind}^{\hbox{max} }$$

(12)

where

P ^max_solar :

The upper limit of the solar power (W);

P ^max_wind :

The upper limit of the wind power (W).

*Ramp rate limits

The ramp rate limits for thermal power plants are described as follows:

$$P_{{G_{i} (t)}} - P_{{G_{i} \left( {t - 1} \right)}} \le P_{{G_{i} up}}$$

(13)

$$P_{{G_{i} \left( {t - 1} \right)}} - P_{{G_{i} (t)}} \le P_{{G_{i} down}}$$

(14)

where

P_Giup and P_Gidown:

The up and down ramp rate of the power of the ith generator, respectively (W).

4 Optimal Operation of an Integrated Power System Using a MCS Algorithm

The CS algorithm is reviewed in Sect. 4.1 followed by a description of the MCS algorithm. The MCS algorithm is proposed to define the optimal operation solutions for an integrated power system.

4.1 CS Algorithm

The CS algorithm is a stochastic global search algorithm based on the interesting breeding behaviors of cuckoos through the following principles [10].

Each cuckoo only lays one egg at a time and dumps its egg in the randomly chosen nest, each egg is a solution. The best nests with high quality of eggs will carry over to the next generation. The availability of host nests is fixed and a probability, p_a∈ [0, 1] represents the possibility of an alien egg to be discovered by host bird.

In this application, the cuckoo selects randomly the nest position to lay an egg which is a new generation power, P ^iter+1_Gi,k through the Lévy flight behavior.

$$P_{Gi,k}^{iter + 1} = P_{Gi,k}^{iter} + \varepsilon \oplus Levy(\sigma )$$

(15)

where

ε > 0:

The step size;

k :

The kth host nest, k = [1, …, m];

l :

The lth random nest, l = [1, …, m];

m :

The number of the host nest;

iter :

The iterth iteration;

Iter ^max :

The number of maximum iteration;

⊕:

The entry wise multiplications.

The Lévy flight behavior (15) is essentially the stochastic description for a random walk which is a Markov chain. Its next location depends on the current location, $P_{Gi,k}^{iter}$ and the transition probability, $\varepsilon \oplus Levy(\sigma )$.

The Lévy distribution is [10]:

$${\text{L}}{\acute{\text{e}}}{\text{vy}}(\upsigma) = t^{ - \sigma } ,\sigma \in \left( {1,3} \right].$$

(16)

4.2 MCS Algorithm

With the aim of improving the searching ability as the solutions get closer to the optimal result, the MCS algorithm is proposed and applied for the optimal operation problem of the integrated power system.

It is realized that one of the CS algorithm parameters, affecting the searching ability of the CS algorithm, is the Lévy flight step size, ε. Normally, this step size is assumed constant in this CS algorithm whereas it is proposed to decrease as the number of generations increases for improving the searching efficiency in the MCS algorithm.

Therefore, the Lévy flight step size at each generation is:

$$\varepsilon_{i} = \frac{{\varepsilon_{0} }}{{\sqrt {iter} }}$$

(17)

where

ε₀:

The initial value of the Lévy flight step size.

The flowchart of the MCS algorithm, applied for the optimal operation problem is shown as in Fig. 1.

Fig. 1

Flowchart of the MCS algorithm applied for the optimal operation problem

5 Numerical Results

The numerical results of the non-linear constrained multi-objective optimal operation problem are achieved by the MCS algorithm on the modified IEEE 10-generator power system with the fuel cost coefficients and active generation limits; the emission coefficients; and the values of the B-coefficients matrix [11].

Table 1 is the total load demand; the solar power, P_solar including the PVs’ power, P_pv and the power of solar thermal plants, P_Ther; as well as the wind power, P_wind in 24 h.

Table 1 Total and actual load demands in day and night of 24 h

Figure 1 obviously shows that the total power of traditional thermal generators has been cut down by the solar and wind powers. The average reduction is 22.64%. In the period of 8–12 h, the total solar and wind powers generated is highest, 170 MW including the solar power, P_solar = 90 MW and the wind power, P_wind = 80 MW. The percentage of the total power of traditional thermal generators cut down is largest in the period of 5–8 h, 26.79%. These positively impact on the factors concerning on the fuel cost and emission minimization of the power system. This also confirms a tendency towards power systems with renewable energy sources in the future.

Table 2 is the parameters of the CS and MCS algorithms. The difference between these two algorithms is the chosen value of the Lévy flight step size, ε. It is constant, ε = 0.5 in the CS algorithm and is a generation-varying variable in the MCS algorithm. This is to increase the convergence ability including the convergence speed and value of the CS algorithm.

Table 2 Parameters of the CS and MCS algorithms

Tables 3 and 4 are the best solution of the fuel cost and emission of the IEEE 10-generator power system with the objective function of the fuel cost, w = 1 and the objective function of the emission, w = 0, respectively. The optimal operation problem is considered with and without the valve point effect using the time varying acceleration based PSO (PSO-TVAC), Chaos PSO, CS and MCS algorithms. The fuel cost and emission are always improved using the MCS algorithm compared with other algorithms in Tables 3 and 4. Without considering the valve point effect, the fuel cost is 1.0803 × 10⁵  $/h and the emission is 4533.6 ton/h by using the MCS algorithm which are less than 1.1105 × 10⁵ $/h and 4541.2 ton/h by using the PSO-TVAC algorithm; 1.1103 × 10⁵ $/h and 4540.8 ton/h by using the Chaos PSO algorithm; and 1.1002 × 10⁵ $/h and 4539.1 ton/h by using the CS algorithm. Similarly, with considering the valve point effect, the fuel cost and emission by using the MCS algorithm are less than those by using the PSO-TVAC, Chaos PSO and CS algorithm.

Table 3 Comparison of the best solution for the fuel cost minimization with and without the valve point effect

Table 4 Comparison of the best solution for the emission minimization with and without the valve point effect

Furthermore, Tables 5 and 6 are the best solution of the fuel cost and emission of the IEEE 10-generator power system with the objective function of the fuel cost, w = 1 and the objective function of the emission, w = 0, respectively. In Table 5, the MCS algorithm is proposed to solve the optimal operation problem of the power system with and without the solar and wind power sources based on the fuel cost minimization. The obtained results by using the MCS algorithm are compared with those by using the PSO-TVAC, Chaos PSO and CS algorithms. Obviously, the fuel cost and emission are also improved using the MCS algorithm compared with other algorithms in Tables 5 and 6. Especially, the improvement percentages of the fuel cost and emission, 21.02 and 22.15% by using the MCS algorithm are always higher than 19.82 and 20.98% by using the PSO-TVAC algorithm; 19.87 and 21.01% by using the Chaos PSO algorithm; and 20.04 and 21.06% by using the CS algorithm. Obviously, the reduction percentages are always higher than 19% by integrating the power system with the solar and wind power sources. Additionally, the modifications of the CS algorithm have improved more the reduction. Then, the reduction percentages are highest by using the MCS algorithm. Similarly, in Table 6, the optimal operation problem is based on the emission minimization. The PSO-TVAC, Chaos PSO, CS and MCS algorithms are alternately applied for this problem. The improvement percentages of the fuel cost and emission, 23.11 and 21.34% by using the MCS algorithm are always higher than 21.03 and 18.12% by using the PSO-TVAC algorithm; 19.17 and 20.43% by using the Chaos PSO algorithm; and 22.36 and 21.27% by using the CS algorithm. Obviously, the reduction percentages are always higher than 18% by integrating the power system with the solar and wind power sources and the reduction percentages are highest by using the MCS algorithm.

Table 5 Comparison of the best solution for the fuel cost minimization with and without the solar and wind powers considering the valve point effect

Table 6 Comparison of the best solution for the emission minimization with and without the solar and wind powers considering the valve point effect

Figures 2, 3, 4 and 5 are the convergence characteristics of the fuel cost and emission for the fuel cost and emission minimization without and with the solar and wind power sources considering the valve point effect using the MCS algorithm. It is obvious that the convergence speed and value of the MCS algorithm are good. The MCS algorithm always converges at the iteration steps which are less than 20. Furthermore, the convergence value of the MCS algorithm is also better than this of the PSO-TVAC, Chaos PSO and CS algorithms (Fig. 6).

Fig. 2

Total and actual load powers; and the solar and wind powers obtained in day and night

Fig. 3

Convergence characteristic of the fuel cost for the fuel cost minimization without the solar and wind powers considering the valve point effect using the MCS algorithm

Fig. 4

Convergence characteristic of the emission for the fuel cost minimization without the solar and wind powers considering the valve point effect using the MCS algorithm

Fig. 5

Convergence characteristic of the fuel cost for the fuel cost minimization with the solar and wind powers considering the valve point effect using the MCS algorithm

Fig. 6

Convergence characteristic of the emission for the fuel cost minimization with the solar and wind powers considering the valve point effect using the MCS algorithm

6 Conclusion

The solar and wind powers have been integrated more popular into the existing power system for resolving the issues of the energy crisis and environmental pollution. This made the optimal operation problem more complicated. The MCS algorithm has been proposed to find out the optimal operation strategy based on the minimization of the fuel cost and emission for the integrated power system. The MCS algorithm is the variant of the CS algorithm with the generation varying Lévy flight step size proposed to improve the ability of searching as the solutions get closer to the optimal result. The achieved results by using the MCS algorithm are always better than those by using the PSO-TVAC, Chaos PSO and CS algorithms.