August 15, 2022

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Greenhouse gases emission reduction for electric power

The growing demand of electricity and power generation from fuel contribute significantly to greenhouse gases emissions and global climate change1,2. This detrimental role is becoming more pronounced as the economic and industrial advancements are accelerating throughout the world. To this end, there has been a tendency for emissions reduction in the energy industry via the integration of renewable energy sources (RES) such as wind and solar alongside the traditional thermal plants. Energy emission load dispatch (EELD) is a critical problem for optimizing the power systems energy along with climate advantages under various equality and inequality constraints3,4. Economic load dispatch (ELD) is currently classified into static ELD and dynamic energy emission dispatch (DEED); the later one takes both cost and emission objectives into account. Additionally, there is another high-performance composite energy emission dispatch (CPEED) problem, which integrates the renewable energy production into DEED. CPEED considers both renewable sources of energy (such as wind or solar) and the multi-objective economic emission function of the conventional thermal power plants. Further, complex constraints associated with the CPEED optimization problem are more difficult to handle due to their complex and multi-dimensional nature. When addressing the CPEED issue, it is vital to consider the limits imposed by thermal power plants such as network load demand constraint, generator operating capacity limits, ramp-rate limit of generating units, and forbidden operating regions (FOR) in addition to those imposed by renewable energy sources. As a result, CPEED is a highly difficult optimization problem with multiple objectives. Many experts are currently engaged in exploring the CPEED problem exclusively through the lens of wind and photo voltaic generation5,6.

To investigate the effect of renewable energy production on power systems, the research in7 considers both wind and photo-voltaic renewable energy production in the CPEED objective function. Further, implementation of a novel solution, based on state of the art ABWO computing approach, has been investigated for an optimum solution of fuel cost and air emissions by considering RES. The numerical method and artificial intelligence algorithm are the two types of techniques that have been developed so far to solve EELD and DEED nonlinear objective functions, respectively. Numerical methods are generally iterative techniques, such as gradient iterative optimization algorithm8, lambda incremental cost iteration approach9, linear programming10, quadratic programming11 and Newton-Raphson schema12 to solve convex energy load dispatch (CELD) problems. Various artificial intelligence and evolutionary programming techniques such as particle swarm optimization13, enhanced sailfish approach (ESFA)14, genetic algorithm15,16, real-coded chemical reaction optimization17, bio-geography-based optimization18 and hybrid stochastic optimization (HSO)19,20 have been developed to resolve complex-constrained multi-objective real-world engineering problems.

The EELD issue has been extensively researched throughout the world, and numerous methods for investigating EELD problem have been proposed in the recent literature. Qiao et al.21 proposed a novel integrated system for electric car and wind power that combines charging and discharging of electric cars in order to mitigate the renewable power penalty costs involved and to deal with underestimated wind energy accessibility. To demonstrate the implemented model’s feasibility and efficiency, the authors proposed a bi-objective differential evolution technique that uses non-dominant sorting. However, the study has some deficiencies because the optimization strategy’s generalisation and convergence stability are needed to be improved. Moreover, CPEED problem ignores the effects of wind power variability on energy costs. The authors in22 suggested exponential moving average (EMA) scheme to resolve the wind energy emissions dispatch problem. The EMA algorithm works well for nonlinear problems with many variables. The study ignores systems with high wind turbine penetration, assuming idealistic wind power curves. Chen et al.23 proposed a conditional value credibility (CVaC) model for hedging multi-period energy emissions dispatch, integrated with random fuzzy wind power.

The largest portion of the world’s energy needs is met by carbon and hydrogen-based fuels, which are in limited supply24. These pollution sources cause global warming, extreme weather, and the ozone hole breach25. In contrast, renewable systems with abundant availability and zero carbon footprint are ideal for the aforementioned issues. However, they have their problems, including low energy density, instability, and unpredictability26. In order to develop a robust, sustainable, and cost-effective energy system, the integration of diverse energy sources into an electric power grid has shown to have a contribution to address fossil fuel and RES-related concerns27. Renewable energy is unpredictable and often unavailable. Wind and solar energies are affected by changing weather conditions. RES sources are also impacted by the climate change, and the modification of climatic influences can have a variety of consequences on different types of RES. For example, there is a chance that biomass crop yields can fluctuate in water streams, which will ultimately have severe consequences for hydroelectric plants. The increase in temperature reduces the photovoltaic energy system’s output. In addition, a spike in temperature can alter wind speed, currents, and patterns, ultimately affecting the production of wind power plants28. The bi-objective cross-entropy approach was developed in29 to tackle the uncertainty about intermittent power sources and practical constraints for various renewable energy systems, such as wind, hydro, thermal, and photo-voltaic systems. The same model with the intermittent nature of wind energy is solved by a chaotic sine-cosine algorithm in30. Due to the increasing penetration of renewable energy, a modified multi-objective cross entropy algorithm is proposed in31 to solve the EELD problem with uncertain RES. On the basis of the DEED problem, the CPEED problem considers the impact of renewable energy production on fuel costs and greenhouse gases emission. However, the associated convex and non-convex constraints of the CPEED problem are much more complicated than those of the static ELD and EELD problems, necessitating the use of algorithms with significantly greater solving potentials. Wu et al. in32 proposed a grid-connected integrated energy system for the synergistic interactions between electric powers, temperature, and gas energy flows, in which the flange between bio-fuels, solar, and wind energy is properly explored. Further, anaerobic digestion heating is used to maintain a proper heat for anaerobic digestion bio-gas. A new bi-objective virus colony optimizer (VCO) method was proposed, based on anti-filtering theory and fuzzy strategic planning, by Zou et al.33 for finding the best possible solution between the Pareto sets. At various points throughout the working day, wind power generators and plug-in electric mobility (PEM) are being integrated into the bi-objective DEED problem, which seeks to minimize the cost of wind-thermal electrical energy while also minimizing the emissions produced by fossil-fueled power plants.

There has been a considerable increase in the green energy integration with the conventional energy systems around the world particularly in western countries. This has brought new operational issues such uncertain behavior of climate conditions. A lot of theoretical and empirical investigations have been conducted on the uncertainty modelling for RES. Monte Carlo Simulation (MCS) is widely-applied as a stochastic method that offers high precision. Researchers in34 proposed a modified Metropolis-coupled Markov chain MCS scheme to anticipate the stochastic behaviour of numerous unpredictable elements in the planning of RES integrated energy hubs. Solar irradiance, wind speed, and energy tariff are considered as uncertain parameters in the objective function. System dynamics and MCS are integrated into a time frame technology value approach for RES in the work of Jeon et al.35. The framework of these complicated interactions can be effectively modelled using system dynamics. Numerous researchers have applied the MCS approach for RES uncertainty such as probabilistic power flow analysis36, optimal sizing of power grids37, short-term RES production uncertainty38, and RES power production risk analysis39. Alternatively, probabilistic analytical strategies can model the unknown RES component for accurate and reliable results. The work of Zubo et al.40 has devised a convolution approach to model the hybrid power system for optimal sizing and placement of distributed RES generator uncertainties in power networks. Meng et al.41 suggested an optimization method, featuring the time-based Taylor series, for modeling the uncertain parameters in structural designs. Several other studies such as point estimation for hybrid power networks42, bi-stage approach43 and decentralized cooperative techniques44 have been used to model the RES uncertain behavior in energy grids.

To bolster the differential evolutionary algorithm’s adaptive capability, the author in45 proposed a self-adaptive variable search approach as well as a local search operator, for solving the EELD problem in high dimensions, containing highly nonlinear terms, non-smoothness, and non-convexity. However, the work ignores the fact that such algorithms are prone to local values and pushed to their limits, limiting their utility in solving the CPEED problem. In the light of the growing share of modern power generation systems, it is necessary to investigate the impact of photo-voltaic and wind power production on the energy and environmental benefits of power systems. The conventional methods like46,47,48,49,50 have several research gaps as these methods have not accounted a comprehensive, generic, and highly conflicting optimization problem of CPEED by incorporating wind and solar generation uncertainties, load demand limits, generating capacity limitations, and ramp-rate impacts of generation machines. In addition, these methods can also be improved for better cost value and for better convergence time.

To bridge the above-mentioned research gaps in the literature, we redesign the EELD objective model by taking into account both wind power generation and photo voltaic power generation as RES, in addition to the conflicting relationship between economic and environmental benefits over time. Integration of RES into conventional energy hubs will be beneficial in numerous ways. First, it contributes to the reduction of greenhouse gases emission in power hubs in order to maintain the environment. Consideration of international emissions control agendas, such as the United Nations sustainable development goals, will be facilitated by the scheme. In addition, it will help states to deal with geopolitical concerns by lowering their dependence on imported fuels. There are two main purposes of
the present study: First, the CPEED
must be solved, and a competitive solution must be
found to reduce power generation costs and pollutant emissions to the lowest possible levels. Second, effective solutions of CPEED must meet the non-convex, high dimensional constraints of EELD such as FOR, VLE, and power output balancing limitations with a part of renewable energy probabilistic model, featuring the combined influencing uncertainty parameter of wind and solar profiles. Our work’s primary philanthropy is as follows:

  1. 1.

    A practical and more generic multi-objective model of CPEED is considered herein for energy hubs, integrated with RES. The presented model considers most of the real-world complex system-level constraints such as capacity constraint, ramp-rate limitation, load balancing limitation, VLE limitation of thermal plants and RES uncertainties for both solar and wind as compared to46,47,48,49.

  2. 2.

    The existing ABWO approach is applied to solve the difficult probabilistic multi-objective CPEED model with multi-dimensional restrictions associated with traditional thermal plants and integrated RES. Owing its feature of elegance and adaptability, ABWO results in effective fuel cost and pollutant emissions reductions for the considered five test systems, ranging from small-to-large scales. It is clear from the results that ABWO outperforms other approaches as compared to51,52,53,54,55,56.

  3. 3.

    The proposed techniques provide better results in terms of early convergence in the presence of non-convex, non-linear, and high-dimensional constraints due to the cannibalism mechanism in ABWO, and they offer higher optimization performance and lower computational cost. Compared to current state-of-the-art algorithms, the proposed algorithm has better convergence and execution time. The mutation step validates the equilibrium between the exploitation and exploration phases. Therefore, the proposed approach is more effective at avoiding the local solutions within a given search space as compared to51,52,53,54,55,56,57.

Problem formulation and CPEED modeling

A framework for CPEED can reduce the cost of generation as well as the pollutant air emissions by scheduling the optimum generation from generating resources. There is no universal solution to obtain a single feasible optimized solution set for both models, simultaneously. These two conflicting issues yield various solution sets of Pareto optimal allocation, and choice goes to the decision-maker to attain a solution according to the priorities. The objective function of CPEED can be expressed as in Eq. (1)58.

$$\beginaligned Min\left( F \right) = \left\ F_TC,E_TE \right\ . \endaligned$$

(1)

where F denotes the multi-objective problem to be minimized and \(F_TC\) and \(E_TE\) are the total fuel cost and total emissions, respectively.

Fuel cost modeling

The model for economic generation cost can be expressed by quadratic function and is shown in Eq. (2)59.

$$\beginaligned f_EGC =\sum \limits _i = 1^N f_i(p_i) = \sum \limits _i = 1^N u_ip_i^2 + v_ip_i + w_i, \endaligned$$

(2)

where \(u_i,v_i\), and \(w_i\) are the fuel cost coefficients of particular thermal power plant. \(f_i\) denotes the thermal generation cost, and \(p_i\) is the generated power of N committed units.

Fuel cost modeling with VLE

VLE introduces the rippling effect on the cost curve of thermal power plants and makes the function as highly non-convex. The modelling of fuel cost with VLE can be shown in Eq. (3)60.

$$\beginaligned f_EGC= \sum \limits _i = 1^N f_i(p_i) = \sum \limits _i = 1^N u_ip_i^2 + v_ip_i + w + \left| e_i\sin \ f_i \times (p_i^\min – p_i)\ \right| , \endaligned$$

(3)

where \(\left| e_i\sin \ f_i \times (p_i^\min – p_i)\ \right|\) is the VLE on the cost curve of \(i^th\) unit.

Cost modeling of solar generation

The mathematical model for solar generation cost can be expressed as in Eq. (4)51.

$$\beginaligned f_SGC = \sum \limits _k = 1^N_s S_p,k \times b_ig_k, \endaligned$$

(4)

where \(f_SGC\) represents the solar generation cost. \(N_s\) and \(S_p,k\) are the number of solar panels and power in megawatts, respectively.

Cost modeling of wind generation

The mathematical model for wind generation cost has the form (5)51.

$$\beginaligned f_WGC = \sum \limits _j = 1^N_z W_p,j \times C_aj, \endaligned$$

(5)

where \(f_WGC\) represents the wind generation cost. \(N_z\) and \(W_p\) are the number of wind plants and power in megawatts, respectively.

Total cost model with VLE

The mathematical model for total cost with VLE can be expressed as in Eq. (6)61.

$$\beginaligned F_TC=\sum \limits _i = 1^N u_ip_i^2 + v_ip_i + w + \left| e_i\sin \ f_i \times (p_i^\min – p_i)\ \right| +\sum \limits _k = 1^N_s S_p,k \times b_ig_k+\sum \limits _j = 1^N_z W_p,j \times C_aj, \endaligned$$

(6)

where \(F_TC\) is the total cost as shown in Eq. (1) and the model consists of sum of all associated costs of load dispatch with VLE.

Total greenhouse gasses emission model

In this study, a comprehensive model of pollutant emissions is used to simulate the relationship between pollutant emissions and thermal power unit output, which is a quadratic function along with an exponential function. Usually, thermal power plants burn different fossil fuels to generate power, while the wind and solar energy systems are considered as clean sources because they do not emit pollutant. The emissions objective function does not need to consider the emissions characteristics of wind and photo voltaic power generation, and it can be written as mentioned in Eq. (7)5.

$$\beginaligned E_TE=\sum \limits _i = 1^N (X_i p_i^2 + Y_ip_i + Z_i + \theta _i\exp (\sigma _ip_i)), \endaligned$$

(7)

where \(X_i\), \(Y_i\), \(Z_i\), \(\theta _i\) and \(\sigma _i\) are the emissions coefficients of all committed generator. Scientists have focused on two aspects of solving multi-objective problems commonly known as the priori and posterior approaches. The priori approaches reduce a multi-objective problem to a single problem by applying appropriate weights to achieve a feasible compromised solution that meets the requirements. The mathematical modelling is shown in Eq. (8)61.

$$\beginaligned min \left( F \right) = h \times F_TC + E_TE\left( 1 – h \right) . \endaligned$$

(8)

CPEED is a complex engineering problem, and it consists of different nonlinear system limitations. To obtain the optimal set of solutions, the proposed ABWO strategy will satisfy all CPEED related constraints. The combined power generated from all sources, that is, thermal wind and solar must meet the load demand and transmission loss, as depicted in Eq. (9)51.

$$\beginaligned \sum \limits _i = 1^N^thermal {\sum \limits _i = 1^N^solar \sum \limits _i = 1^N^wind \left( p_i \right) = } p_i^Tx.Loss + p_i^load, \endaligned$$

(9)

where \(N^thermal\), \(N^wind\), and \(N^solar\) denote the number of thermal, wind and solar committed units. \(p_i\) is the generated power, \(p_i^Tx.Loss\) and \(p_i^load\) are the transmission losses and load power demand, respectively. Furthermore, the transmission loss of utility network can be modeled as in Eq. (10).

$$\beginaligned p_i^Tx.Loss = \sum \limits _i = 1^N {\sum \limits _j = 1^N p_i\beta _ijp_j + \sum \limits _i = 1^N \beta _oip_i + \beta _oo }, \endaligned$$

(10)

where \(\beta _ij\), \(\beta _oi\) and \(\beta _oo\) are loss coefficients matrix62.

Machine limitation of generating units

The machines limitation constraints that can be modeled as in Eq. (11)63.

$$\beginaligned p_i^\min \le p_i \le p_i^\max , \endaligned$$

(11)

where \(p_i^\min \) and \(p_i^\max \) are the minimum and maximum generated powers, machine can deliver.

Ramp-rate limitation

The ramp-rate limits can be modelled for generated \(p_i\) by a generator as given in Eq. (12)59.

$$\beginaligned \max \left\ p_i^\min ,p_i^o – DR_i \right\ \le p_i \le \min \left\ p_i^\max ,p_i^o + UR_i \right\ , \endaligned$$

(12)

where \(UR_i\) and \(DR_i\) are the upper and down ramp-rate limitations. Since some generating units have physical limitations, their actions are limited to specific operating regions, likely to result in input-output characteristics that are discontinuous in nature. The modeling for such type of regions in input-output characteristics is given in Eq. (16)64.

$$\beginaligned \p_i^\min \le p_i \le p_i,1^L\,\p_i,H – 1^U \le p_i \le p_i,H^L\,\p_i,nz^U \le p_i \le p_i^\max \, where H = 2, \ldots , nz. \endaligned$$

(13)

Wind probability density function

Wind energy generation is heavily reliant on wind speed variability. Numerous techniques have been used to portray the randomness of wind speed characteristics. The Weibull probability density function (WPDF) is used to model the wind speed attributes, and it is used to define the stochastic feature of wind speed profiles. The mathematical expression of WPDF is as shown in Eq. (14)65.

$$\beginaligned WPDF\left( V_CWD \right) = \left\{ {\frac\Pi \mathrmT\left( \fracV\mathrmT \right) ^\Pi -1 \cdot \exp \left( { – {\left( \fracV\mathrmT \right) ^\Pi }} \right) V > 0,} \right. \endaligned$$

(14)

where T and \(\Pi\) represent the scale and shape parameter, respectively. \(V_CWD\) shows the current speed of wind in (m/s). The power output of wind is computed using the speed-power curve via Eq. (15).

$$\beginaligned W_p = \left\{ \beginarrayl 0, \quad \left( V < V_inORV \ge V_out \right) \\ \frac{w_r \quad \left( V – V_in \right) }V_r – V_in\left( V_in \le V \le V_r \right) \\ w_r \quad \left( V_r \le V \le V_out \right) \endarray \right. \endaligned$$

(15)

where \(w_r\) represents the rated wind turbine power and \(W_p\) is the generated output power. \(V_r\), \(V_in\) and V show the cut in/off and rated speed respectively.

Solar probability density function

Unlike the wind energy, the electricity production of the photo-voltaic module is heavily reliant on solar radiations, environmental temperature and module performance characteristics. In this work, we apply beta distribution (BPDF) to represent percentages or proportions of outcomes. The mathematical modelling of solar energy using BPDF is depicted in Eq. (16)66.

$$\beginaligned f_\left( BPDF \right) \left( \Phi \right) = \left\{ \beginarrayl \frac\Gamma \left( R + TE \right) \Gamma \left( R \right) \Gamma \left( TE \right) \times \Phi ^R – 1\left( 1 – \Phi \right) ^TE – 1\\ for \quad 0 \le \Phi \le 1, \quad R \ge 0,TE \ge 0\\ 0,otherwise \endarray \right. \endaligned$$

(16)

Here R and TE represent the \(f_BPDF\) parameters, \(\Phi\) is the electromagnetic radiation power per area, and \(\Gamma\) is gamma function of BPDF. By using mean \((\mu )\) and standard deviation \((\varepsilon )\), the function can be written as follows:

$$\beginaligned R = \mu \left( {\frac\left( \mu \left( \mu + 1 \right) \right) \varepsilon ^2 – 1} \right) , \endaligned$$

(17)

$$\beginaligned TE = \left( 1 – \mu \right) \left( {\frac\mu \left( \mu + 1 \right) \varepsilon ^2 – 1} \right) . \endaligned$$

(18)

As stated earlier, the dependency for power generation in photo-voltaic modules relies on the solar irradiance and the temperature of cells. It can be formulated via Eq. (19).

$$\beginaligned S_p(t) = N_series \times N_Parallel[ S_p(stcon) \times \fracS(t)_radS_rad.stcon \times [ 1- \chi \times (U_cell – U_cell.stcon)]]. \endaligned$$

(19)

The temperature of cells can be determined from Eq. (20), where \(S_p(t)\) denotes the solar radiation incident on the photo-voltaic module at time interval t. \(S_p(stcon)\) and \(S_rad.stcon\) represent the solar power and radiance in the standard test conditions. \(\chi\) is the coefficient of temperature while \(U_cell\) and \(U_cell.stcon\) are the cell temperature and temperature at standard conditions. \(U_ambient\) and \(U_normal.temp\) are the environmental and normal temperature, respectively.

$$\beginaligned U_cell = U_ambient + \fracS(t)_radS_rad.stcon \times (U_normal.temp – 20). \endaligned$$

(20)

Traditional thermal power plants can provide electricity regardless of the weather, but RES rely on the climatic strength to generate power, making it imperative to have an accurate and reliable model to deal with the unpredictability of climate circumstances. In this study, we used beta-distribution function (16) and Weibull probability density function (14) to model the uncertainty in RES. The wind PDF is computed using (14), where T is the scaling factor and \(\Pi\) is the wind turbine blade profile shape factor. The values of T and V for the Kings park site are 3.23 and 5.8 metres per second, respectively. Once the unpredictability of the wind has been classified as a stochastic process, the output power of the wind generator can be measured as a random variable by transforming wind speed into output power. Equation (15) can be used to determine the output power of the wind based on the wind velocity \(W_P\), and can used in wind cost function Eq. (5) to compute the cost. To account for the uncertainty in the solar cost function, solar irradiance is classified as a random variable using the beta distribution. \(f_\left( BPDF \right) \left( \Phi \right)\) is the distribution function and is characterized as a random variable of solar irradiance (kw/\(m^2\)). Now, in order to acquire the power that the solar panel produces, we have used \(S_P,k=S_P0\times f_\left( BPDF \right) \left( \Phi \right)\), which further is used in the solar cost function equation (4) to compute the cost. It is pertinent to mention here that we have formulated a complex and more practical model of the economic emission dispatch which includes practical contagious system-level limitations for example, VPLE, ramp-rate limitations and system losses as compared to the similar ABWO framework50. Additionally, we have also integrated the renewable energy resources in our work in the form of wind and solar sources in comparison to the mentioned research study.