Combined Heat and Power Dynamic Economic Emissions Dispatch with Valve Point Effects and Incentive Based Demand Response Programs

In this paper, the Combined Heat and Power Dynamic Economic Emissions Dispatch (CHPDEED) problem formulation is considered. This problem is a complicated nonlinear mathematical formulation with multiple, conflicting objective functions. The aim of this mathematical problem is to obtain the optimal quantities of heat and power output for the committed generating units which includes power and heat only units. Heat and load demand are expected to be satisfied throughout the total dispatch interval. In this paper, Valve Point effects are considered in the fuel cost function of the units which lead to a non-convex cost function. Furthermore, an Incentive Based Demand Response Program formulation is also simultaneously considered with the CHPDEED problem further complicating the mathematical problem. The decision variables are thus the optimal power and heat output of the generating units and the optimal power curbed and monetary incentive for the participating demand response consumers. The resulting mathematical formulations are tested on four practical scenarios depicting different system operating conditions and obtained results show the efficacy of the developed mathematical optimization model. Obtained results indicate that, when the Incentive-Based Demand Response (IBDR) program’s operational hours is unrestricted with a residential load profile, the energy curtailed is highest (2680 MWh), the energy produced by the generators is lowest (38,008.53 MWh), power losses are lowest (840.5291 MW) and both fuel costs and emissions are lowest.


Introduction
Combined Heat and Power (CHP) generating units also known as co-generation units produce electric power and heat simultaneously. The production of these two outputs as opposed to only power production significantly increases the efficiency of CHP units. Therefore, whilst conventional generation units have an efficiency of around 60% [1], CHP units have efficiency of around 90% [1]. An added advantage of CHP units over conventional units is that CHP units also yield lower emissions by about 13-18% [1]. The Combined Heat and Power Dynamic Economic Dispatch (CHPDED) problem seeks to minimize the fuel costs of committed units by determining their optimal power and heat output whilst also ensuring that both heat and power demand are satisfied during the whole scheduling interval [2]. This mathematical problem is constrained by practical mathematical constraints like ramp rate constraints, power balance constraint, generating units constraints, etc. Without the addition of ramp rate constraints, the CHPDED problem is simply referred to as the Combined Heat and Power Economic Dispatch (CHPED) [2]. The dynamic addition is due to the consideration of generator ramp rates which essentially sets a limit on heat and output power over consecutive time intervals in order to maintain the generator's useful life. When factoring emissions into the CHPDED problem, there are three approaches to the resultant problem.
The first approach is to minimize both fuel costs and emissions whilst ensuring that power and heat demands are met. This approach is termed Combined Heat and Power Dynamic Economic Emission Dispatch (CHPDEED). The second approach involves the minimization of only fuel costs and defining a constraint which limits the amount of allowable emissions. This approach is termed the Combined Heat and Power Emission Constrained Dynamic Economic Dispatch (CHPECDED). The third approach is termed Combined Heat and Power Pure Dynamic Emission Dispatch (CHPPDED) concerned with the minimization of only emissions whilst ensuring that power and heat demands are met. In this work, the focus is on the CHPDEED problem which is a multi-objective optimization problem with two conflicting objective functions (minimization of fuel costs and harmful emissions). The multi objective function is converted to a single objective function by assigning weights to both objectives which allows for the determination of a trade-off between fuel costs and emissions.
In essence, the CHPDEED problem determines the committed units power and heat outputs whilst minimizing the fuel costs and emissions and respecting system constraints [1,2]. There are two main research focus areas in the CHPED/CHPDED/CHPECDED and CHPDEED research fields. The first is concerned with the development and application of novel solution methodologies and approaches. These solution methodologies cover both classical optimization algorithms and heuristic algorithms. Examples include [3] where the real coded genetic algorithm with improved Muhlenbein mutation is deployed to solve the CHPED problem. The solution algorithm is tested on sample case studies and returns feasible solutions. In Reference [2], differential evolution and sequential quadratic programming is deployed in solving the CHPDED problem and in [1] the same solution methodology is used to solve the CHPDED, CHPDEED, and CHPPDED. Another example is [4] where the utilization of integrated civilized swarm optimization and Powell's pattern search method is used for the CHPED. Other examples include [5] and [6] where a "whale optimization method" is deployed to solve the CPHED problem. Other algorithms (solution methodologies) include the squirrel search algorithm [7], Kho-Kho optimization Algorithm [8], indicator and crowding distance-based evolutionary algorithm [9], cuckoo search algorithm [10,11], effective cuckoo search algorithm [12], exchange market algorithm [13], gravitational search algorithm [14], group search optimization algorithm [15], and modified group search optimizer [16]. A comprehensive review article on research works utilizing heuristic methods in solving the CHPDEED mathematical works is given in [17]. These heuristic solution algorithms are widely reported in literature and it is difficult to objectively report on the superiority of one algorithm over the other. In the final analysis, the choice of the heuristic solution algorithm is due to the bias of the researcher.
The second research focus area in the CHPED/CHPDED/CHPECDED and CHPDEED field deals with the incorporation of related power system sources or tasks when solving the mathematical optimization problem. In Reference [18], unit commitment which involves the determination of the ON/OFF status of generator units is performed together with economic dispatch for CHP units. In another work [19], dispatch is performed for a power system in China consisting of CHP units and wind turbines. Reference [20] performs dispatch for a CHP system including wind turbines and storage systems for both thermal and electrical energy. In [21], economic dispatch was performed for a micogrid consisting of CHP units, wind turbines, photovoltaic (PV) cells, battery storage, and gas fired boilers. Another work is [22] where a stochastic CHPED is performed for a system consisting of CHP units, wind turbines, and PV units. Chance constrained programming is utilized in solving the resultant model. In [23], the CHPDED problem is solved this time incorporating spinning reserve requirements and the resultant model is solved using an enhanced firefly algorithm.
The CHPDED problem has in recent times also been solved whilst incorporating demand response programs. Demand response programs are motivated primarily by the drive to curtail energy consumption on the demand side as opposed to increasing power generation on the supply side with its resultant financial implications [24][25][26]. Combining CHP and demand response programs is viewed as a cost effective way of maintaining today's power system as CHP's on the supply side have higher efficiency and lower emissions whilst demand response programs on the demand side introduce optimality and curtail consumer energy consumption. Hitherto, demand response programs have been incorporated into the economic dispatch of thermal units [25][26][27] and renewable energy systems [28]. However, only a few works focus on the joint optimization of CHP's and demand response programs with CHP's at the supply spectrum of the grid whilst demand response programs are at the demand spectrum of the grid. In Reference [29], a demand response scheme is integrated into a micro CHP system. A Model Predictive Control (MPC) solution methodology is utilized as the control algorithm and results indicate that the incorporation of demand response reduces cost about 1-14%. Reference [30] details a CHP system with demand response programs at the consumer side under a simulated energy hub. A comparison of various demand response schemes is provided and obtained results indicate that the incorporation of demand response programs reduce operation cots in the energy hub. In [31], the hourly scheduling of CHP units with demand response and storage facilities (energy and heat) is detailed. The case studies investigated from the paper shows that the implementation of demand response programs leads to cost reduction and improvement of grid reliability.
Recent works include [32] where a price based demand response program was formulated for combined heat and power consumers and [33] where a robust optimization framework was deployed for price based demand response programs integrated within a combined heat and power setup. An analysis of these prior works show that there is no integration of incentive based DR (IBDR) and CHP systems. Although it has been shown that incentive based demand response programs and CHP systems have the potential to be beneficial to utilities [34], practical investigations of this integration are lacking. This work therefore proposes a practical scheme for the optimal economic dispatch of CHP units with an incentive based demand response (IBDR) program. The resultant CHPDEED-IBDR problem has a non-smooth and non-convex objective function and provides an economic incentive for load curtailment in times of power system stress. The provided incentive is structured in such a way that it is greater than the curtailment cost and also factors in budgetary constraints amongst other practical constraints. Incorporating customer curtailment cost and ensuring that the incentive given is commensurate with the customer participation level and amount of curtailed power is referred to as "incentive compatibility".
Most CHP units have an intertwined relationship between heat and power generation, thereby adding a significant degree of complexity to the problem. Thus, we ensure that the curtailment of electrical power doesn't compromise the satisfaction of heat demand. Moreover, valve point effects and power losses are factored into the model. The developed multi-objective model with three objective models is converted into a single objective function with the use of a weighting method and the accuracy of the developed model is shown on four case studies with a high degree of success. The remainder of this article is thus given: The mathematical models for the CHPDEED-IBDR problem formulation is detailed in Section 2. Section 3 presents the methodology utilized for numerical simulations with Section 4 detailing results obtained via the simulations. Section 5 concludes the paper.

Combined Heat and Power Dynamic Economic Emission Dispatch Model
The CHPDEED mathematical model is made up of three distinct types of generators. They include: conventional thermal units (TU), CHP units, and heat-only units (H). Conventional thermal units and CHP units produce electric power whilst heat only units and CHP units produce heat. The CHPDEED mathematical problem has its objective as the minimization of the fuel costs and emissions of all units whilst satisfying the power and heat demand over the scheduling horizon under practical system constraints.
The individual fuel cost and emissions objective functions of all three types of generating units (thermal, CHP and heat) are detailed.

Thermal Units
The most common fuel function for thermal units is the quadratic representation [25][26][27]. A more accurate representation is one that incorporates valve point effects [1,2] given as: where a i , b i and c i are the positive fuel cost coefficients of generator i respectively; e i and f i are the fuel cost coefficients representing valve point effects of generator i, respectively; P TU i,t represents the power generated from thermal unit i at time t; P TU i,min represents the minimum capacity of thermal unit i; C i (P TU i,t ) represents the fuel cost of producing P TU i,t .
The emissions of thermal units are given by: This emission mathematical function is a combined quadratic and exponential representation of the thermal units power output. α i , β i , γ i , η i and δ i are the emission function coefficients of generator i and E i (P TU i,t ) represents the total emissions to produce P TU i,t .

CHP Units
The CHP unit produces both power and heat. Thus, the fuel cost is a product of both outputs. This is usually represented as a convex cost function given as: where a k , b k , c k , e k and f k are the fuel cost coefficients of CHP generator l respectively; C CHP k (P CHP k,t , H CHP k,t ) is the fuel cost for CHP generator l to produce heat and power (P CHP k,t , H CHP k,t ).
The total CHP units emissions is solely a function of the power generated and is given as: where α k and β k are emission function coefficients.

Heat Units
These units produce only heat and the fuel cost function is depicted by: where a l , b l and c l are the positive fuel cost coefficients of generator l, respectively; The emission function similarly is given by: where α l and β l are the emissions coefficients of heat units l.

Objective Functions
The total fuel cost (for thermal, CHP and heat units) is given by: where T, I, K, and L are the total scheduling interval, total number of thermal units, total number of CHP units, and total number of heat units, respectively. In a similar manner, the total emission function (for thermal, CHP and heat units) is given by: where T, I, K and L are the total scheduling interval, total number of thermal units, total number of CHP units, and total number of heat units, respectively.

Constraints
The constraints for the CHPDEED problem's objective function (Equations (7) and (8)) are given below: where P TU i,t is the power generated from thermal generator i at time t; P CHP k,t is the power generated from CHP generator k at time t; H CHP k,t is the heat produced from CHP generator k at time t; H H l,t is the heat produced from heat generator l at time t; D t is the total system power demand at time t; HD t is the total system heat demand at time t; loss t is the total system losses at time t; P TU i,min and P TU i,max are the minimum and maximum power capacity of thermal generator i respectively; H H l,min and H H l,max are the minimum and maximum heat capacities of generator l respectively; P CHP k,min (H CHP k,t ) and P CHP k,max (H CHP k,t ) are the minimum and maximum power capacities of CHP generator k, respectively. Both parameters are functions of the heat produced (H CHP k,t ). H CHP k,min (P CHP k,t ) and H CHP k,max (P CHP k,t ) are the minimum and maximum heat capacities of CHP generator k, respectively. Both parameters are functions of the power produced (P CHP k,t ). Equations (9)-(16) represent the constraints of the mathematical model and their interpretation is given as: • Constraint (9) is termed the "power balance constraint". Its role is to compel the total output power from both thermal and CHP units at each scheduling interval to satisfy the load demand and transmission line losses. Transmission line losses are determined by the B-coefficient method [1,2] and is represented mathematically in (17). B i,z is the izth element of the loss coefficient square matrix B of size I + L. This method has been used in [25][26][27]. • Constraint (10) is termed the "heat balance constraint" and its role is to compel the heat output from both CHP and heat-only units to match heat demand.

•
The third constraint is the thermal generation limits constraint (11). It compels the output power from thermal generators to not exceed allowed limits • The fourth constraint (12) limits power produced from CHP units within allowable units.

•
The fifth constraint (13) limits heat produced from CHP units within allowable limits. • Constraint (14) ensures that the heat produced from heat only units are within allowable limits. • Constraint (15) is the "generator ramp rate limits constraint" for thermal generators and compels the thermal generators output power for consecutive scheduling intervals to be within allowable ramp rate limits. • Constraint (16) is termed "generator ramp rate limits constraint" for CHP generators. Similar to constraint (15), it compels the output power for CHP units for consecutive scheduling intervals to be within allowable ramp rate limits.
The two objective functions (Equations (7) and (8)) can be concatenated into a single objective function via a weighting factor w. The resultant single objective function is still constrained by (9)- (17): where w and (1 − w) are weighting factors. The condition to be satisfied is [27,35]: Both weighting factors are non-negative and can be controlled by the modeler based on the preference given to objective functions. When the modeler seeks to minimize fuel costs alone, then w = 1 (CHPDED). However, when the modeler wants to minimize emissions alone, then w = 0 (CHPPDED). It is assumed for the purpose of this article that equal weights are given to both objective functions. Thus, w = (1 − w) = 0.5.

Incentive Based Demand Response Model
If we assume that an electric consumer/customer of type θ is willing to curb x MW of power. The customer benefit function can therefore be represented, thus: where y is the incentive (monetary value) the customer is given. Customer participation is only guaranteed if V 1 ≥ 0. In the same vein, the electric utility's benefit function is given as: λ is the Locational Marginal Price (LMP) or "value of power interruptibility" [26,27] and is calculated from Optimal Power Flow (OPF) routines. We can thus define the utility's benefit function (benefit maximization) as: max where x is the amount of power curbed by electric consumer/customer.
λ is the "value of power interruptibility" or LMP.
Customer Cost Function c(θ, x) is the cost to electric consumer/customer of type θ who curbs x MW of electric power. A quadratic customer cost function is assumed represented thus: where K 1 and K 2 are customer cost co-efficients. θ is the customer type [26,27] and classifies electric consumers/customers based on the amount of power they are curbing. θ is normalized in the interval 0 ≤ θ ≤ 1, thus θ = 1 is the keenest consumer/customer and θ = 0 is the least keen. The customer cost function satisfies the following conditions: Marginal cost decreases with an increase in θ: Customer (θ = 1), who is the keenest customer, will therefore have the lowest marginal cost and the largest marginal benefit. Customer (θ = 0), who is the least keen customer, will have the largest marginal cost and lowest marginal benefit: The marginal cost function is an increasing convex cost function. • When no power is curtailed, then the customer cost should be zero (c(θ, 0) = 0).
If y j is the incentive for customer j, customer benefit [26,27] is: The utility benefit is given as: The incentive Based DR program seeks to therefore maximize the utility benefit: • Constraint (27) is the "individual rationality constraint" and compels the customer benefit to be greater or at least zero. • Constraint (28) is the "incentive compatibility constraint" and compels customers to be compensated commensurate to the load they curtail.
There are two variables: customer power curtailed (x MW) and the customer incentive ($ y). Furthermore, we expand the model over more than one scheduling interval and incorporate other practical considerations into the model. The resulting model is detailed thus: where UB is the utility's total budget and CM j is the maximum amount of power customer j is willing to curb in a day; • The first constraint (30) makes sure that each customer's daily incentive is greater than their interruption cost.

•
The second constraint (31) makes sure that each customer's benefit is commensurate with their power curtailment.

•
The third constraint (32) compels the total monetary value of incentives paid by the electric utility to be within its budgeted amount.

•
The fourth constraint (33) compels the total daily power curbed by each customer to be within its allowable daily limits.

Combined Heat and Power Dynamic Economic Emissions Dispatch with Incentive Based Demand Response Model
The final combined mathematical model can be represented as follows: subject to

Numerical Simulations
In order to investigate the efficacy of our proposed mathematical formulations (CHPDEED-IBDR), four different cases are used. The cases differ based on their load profile and are detailed as: • CHPDEED-IBDR with residential load. • CHPDEED-IBDR with residential load with restrictions on DR operating hours. • CHPDEED-IBDR with commercial load.
• CHPDEED-IBDR with commercial load with restrictions on DR operating hours.
The load profile of Cases 1-4 are given in Figures 1-4, respectively. Figures 2 and 4 correspond to Case 2 and Case 4 with the colored areas depicting the allowed IBDR operating hours.    For all case studies, the eleven unit system consisting of (eight conventional units, two CHP units, and one heat-only unit) is utilized. The data for the conventional, CHP, and heat units are given in Tables 1-3, respectively, and is obtained from [1,2]. Feasible operating regions for the CHP units are given in Figures 5 and 6, respectively. The power and heat demand are given in Table 4 and the customer data are detailed in Table 5. The transmission loss formula coefficients for the thermal only units and the CHP units are given by Equations (48) and (49), respectively. The customer data (values of power interruptibility (LMP) and customer parameters: K 1,j and K 2,j ) are obtained from [26,27]. The daily limit of interruptible energy (CM j ) is utilized by the ISO to determine customer keenness θ j . The ISO's daily budget (UB) is given as $ 100,000. A key assumption made is that the heat demand is always satisfied and it is only the power demand that is curtailed via the demand response program. Cases 2 and 4 are cases when the DR programs have restricted operating hours. For Case 2, the IBDR program can only operate between 0900-1500 h and 2000-2200 h. Case 4 IBDR operation hours lies between 0900-1700 h. The Advanced Interactive Multidimensional Modeling System (AIMMS) via the CONOPT solver [36] is used to model and solve the developed optimization models.     Table 4. Heat demand and power demand. 1  390  1036  963  2  400  1110  1110  3  410  1258  1258  4  420  1406  1406  5  440  1480  1480  6  450  1628  1628  7  450  1702  1702  8  455  1776  1776  9  460  1924  1924  10  460  2022  2072  11  470  2106  2146  12  480  2150  2220  13  470  2072  2072  14  460  1924  2050  15  450  1776  2000  16  450  1554  1850  17  420  1480  1805  18  435  1628  1792  19  445  1776  1776  20  450  1972  1705  21  445  1924  1650  22  435  1628  1628  23  400  1332  1332  24  400  1184  1184  Table 5. Customer cost function coefficients, customer type, and daily customer energy limit.

Results and Discussion
The multi-objective optimization problem has three objective functions and we assume that equal objectives were given to all three objectives. Thus, w 1 = w 2 = w 3 = 0.333. Figures 7-10 give the initial load profiles and final load profiles after the IBDR program for cases 1, 2, 3, and 4, respectively. From the figures, it is obvious that the incorporation of the IBDR program leads to a reduction in the demand across all cases (commercial and industrial load profiles). The full results for all cases is given in Table 6. It shows the fuel cost ($), emissions (lb), total energy generated (MWh), total heat (MWth), total losses (MW), total incentive ($), and total energy saved/curtailed (MWh) for all cases over 24 h. In order to benchmark the CHPDEED-IBDR results, results from conventional CHPDEED using the data of Case 1 are also provided in the second column of Table 6.
For Case 1 and Case 3, when there is no restriction in IBDR operating hours (utilities might loathe requiring customers to restrict their demand for 24 h), the total energy generated by both the thermal units and CHP units over 24 h is 38,008.53 MWh and 38,732.62 MWh, respectively. Again, for both cases, the total energy saved or curtailed by the IBDR program is 2680 MWh. Both cases also have low power losses of 840.53 MW and 883.62 MW, respectively. Cases 2 and 4 are cases when the operational hours of the IBDR program are restricted to 0900-1500 h and 2000-2200 h (Case 2) and 0900-1700 (Case 4). To provide a fair comparison, the only difference between Cases 1 and 2 and Cases 3 and 4 is the IBDR operational hours. The utility budget is assumed constant in all cases at $100 000.
From the results obtained, Case 2 generates more energy than Case 1 (38,712.67 MWh to 38,008.53 MWh). In addition, Case 4 generates more energy than Case 3 (39,439.46 MWh to 38,732.62 MWh). This is expected and is due to the fact that the energy curtailed/energy saved when the IBDR program's operational hours is reduced is less than the case when the IBDR programs operate for 24 h. Correspondingly, the power losses for Case 2 and Case 4 are higher than for cases when there is no restriction on operating hours. The total fuel costs and emissions for all cases are closely related to the amount of energy generated. Thus, when the energy generated is high (Cases 2 and 4) because the IBDR program is restricted in its operational hours, then the fuel cost and emissions are correspondingly greater than cases without IBDR hours restrictions. Again, when the fuel cost is lowest with a value of $ 2,266,792 (Case 1), the emissions correspondingly gives the lowest value (458,955.4 lb). Case 4 returns the highest fuel cost and emissions ($ 2,376,601 and 494,630.5 lb).
As stated earlier, the heat demand is required to be always satisfied (heat balance constraint) and even though the power output and heat output for the CHP units are inter-related, a curtailment in power doesn't invalidate the heat balance constraint. The complete power flow and demand response results for all four cases are given in Tables A1-A8 in the Appendix A.
In order to provide additional analysis of the results, the daily operational Cost of Energy (CoE) [37] is calculated. It is the ratio of the daily total cost of generation to daily total energy generated in $/MWh. The results are shown in the last column of Table 6. These results show that, when the IBDR hours are restricted (Case 2 and Case 4), the CoE increases marginally when compared to unrestricted IBDR operational hours (Case 1 and Case 3). The difference is only marginal and therefore the case can be made that CHPDEED can perhaps be deployed with IBDR for restricted intervals specifically at times of severe power system constraints.

Conclusions
This work presented the incorporation of an Incentive Based Demand Response Program (IBDR) based on game theory with the Combined Heat and Power Dynamic Economic Emissions Dispatch (CHPDEED) problem. The CHPDEED problem incorporates valve point effects which leads to non-smooth and non-convex cost functions. The IBDR program has two important constraints: the individuality rationality constraint and the incentive compatibility constraint which are game theory (mechanism design) formulations and ensure that customers' incentives exceed their cost of curtailment and are commensurate with the quantity of power curtailed. Taken together, the CHPDEED-IBDR is a complicated and difficult formulation which ensures that there is optimality at both the supply side and demand side of the power grid. Cases were investigated with various load profiles that depicted both commercial and residential load profiles with restrictions in IBDR operating hours. Obtained results indicate that, when the IBDR program's operational hours is unrestricted with a residential load profile, the energy curtailed is highest (2680 MWh), the energy produced by the generators is lowest (38008.53 MWh), power losses are lowest (840.5291 MW), and both fuel costs and emissions are lowest. Case 3 (commercial load profile with unrestricted IBDR operational hours) returns the second best results. It is observed that restricting the IBDR program operational hours (Cases 2 and 4) leads to less power curtailed than Cases 1 and 3 without a commensurate reduction in incentives. Future work will consider the incorporation of heat energy storage devices to store the heat produced from the CHP and heat units.

Conflicts of Interest:
The author declares no conflict of interest.