This section is a discussion about other applications of various design algorithms which are not directly implemented in energy management systems, but are used in other feasible applications. They are considered important so that the mathematical applications of these algorithms can enhance further understanding of the various algorithms available. Two common terminologies used in this section are Convex Functions and Nonconvex Functions. Convex functions or convex optimization problems refer to problems with only one optimal solution which is globally optimal such as a quadratic function. On the other hand, nonconvex functions have more than one optimal solution and a typical example is a sine function.
In [81], the authors described an improved branch and bound algorithm for a ZERO-to-ONE MINLP optimisation with convex objective functions and constraints. The two novelties of the design included:
Deriving a method for obtaining lower bounds of a non-linear programming sub-problem without solving it to optimality.
Obtaining an early branching procedure thereby avoiding to solving sub- problems to optimality in some cases.
Computed results showed that these improvements effectively reduced processing time needed to solve MINLPs although for relatively small problems, no significant time change was recorded hence, the technique is best for computing increasing number of Sequential Quadratic Programming iterations. In [82], the authors proposed a generalised branch and cut (GBC) framework for solving MINLP optimisation problems which acts as a unifying framework for comparing branch and bound (BB) algorithms and decomposition algorithms. According to the authors, BB is the primary deterministic approach that can be used to successfully solve MINLP problems in which the participating functions are nonconvex. But with recent availability of decomposition algorithm in solving nonconvex MINLP problems, the authors proposed GBC as a means of comparing both methods for evaluation purposes. They came to a conclusion that BB and decomposition algorithms are the two classes of deterministic exponential time algorithms available to solve MINLPs in which the participating functions are non-convex, whereas deterministic polynomial algorithms were not known to solve MINLPs [83].
T. Yokota et.al in [66] were able to identify the shortcomings of the BB method which is the most widely used algorithm for solving Integer Programming (IP) problems, while proposing solving MINLP problems using GA and its applications. The major drawback as suggested by Taha is the inability of the BB in solving non-linear IP problem, mainly because the validity of the branching rules; as originally proposed in Land-Doig method, was based on an assumption of linearity [66]. There has been several modification of the original design with the aim of overcoming the deficiencies prevalent in it and one of
such was Dakin’s modification. This modification ensured that the branching rule is interdependent of the linearity condition. It was also observed that when the objective function and each constraint function with respect to decision variables are concave and convex respectively, a local optimum results to a global optimum. Further modification as proposed in the paper includes the use of GA for solving MINLP problems. It begins by selecting an initial set of random potential solutions and uses a process similar to biological evolution to improve upon them. It uses a special penalty multiplier of the evaluation function to modify infeasible solutions (chromosomes) in order to search the best solution more efficiently. Results showed that better solutions are available only within a constrained region, but due to the penalty multiplier involved, the population is forced to converge to the feasible region.
The authors in [84] described Simulated Annealing as a controlled- randomization process whereby the objective function to be minimized is gradually lowered by a series of improving moves to achieve optimal solution. This is analogous to annealing process involving gradual temperature reduction of a molten material in order to ensure better binding of the molecules. The authors aimed to develop optimal synthesis of a distillation column with intermediate heat exchangers using Simulated Annealing as the appropriate algorithm. Events in each column is scheduled and used as a learning strategy necessary for the development of artificial intelligence (AI) whereby probabilities were assigned to various decision rules at certain stages of the solution process. This causes the generation of various solutions to a particular problem and the best solution would be adopted depending on the quality of the results.
Furthermore in [85], the process of Simulated Annealing was used in the optimal synthesis of multi-component distillation systems aimed at minimizing cost for best investment results. Simulated Annealing was a preferred algorithmic process because the authors cited two reasons based on:
The difficulty of solving the non-convexity of the non-linear (MINLP) formulation.
The difficulty in solving problems with large size due to the combinatorial feature of such problems.
Branch and bound method could not be used on this instant because it is only very useful for solving small or moderate size problems [86]. Results showed the ability of the method in solving larger-scale MINLP optimisation problem without eliminating non-convexities and decompositions of the original problem into sub-problem, thereby improving efficiency.
Finally, the authors in [87] reviewed the future paths for integer programming (IP) whereby the contribution of BB approaches in the development of IP was considered more useful in practical applications due to the presence of integer variable constraints to satisfy the upper and lower bound conditions. The paper also discussed the futuristic impact of IP in solution strategies for diverse areas such as: Number Theory, Logics, Group Theory, Non-linear Functions, Convex Analysis, and Matroid Theory. Integer programming is also found to have links to Artificial Intelligence and as a result, the ability to solve a significantly increased number of IP problems effectively in the future was seen as a
possibility. In all cases, near-optimal solutions are readily obtainable with minimal computational iterations which is enhanced by the union of two disciplines which are: operations research and artificial intelligence. The paper also discussed four heuristic classifications capable of enhancing the development of AI and IP which includes: Controlled Randomization, Learning Strategies, Induced Decomposition and Tabu Search.
In summary, the lists or types of algorithms available for mathematical and engineering applications are numerous and in most cases, they are designed specifically to perform designated functions. In load scheduling applications the future lies on how intelligent these algorithms could be because there are a lot of dynamics and variables involved which could make it too tedious for humans to follow in an active and accurate manner. The next section is a review of the impacts of DR participation on various households and steps taken towards minimizing those impacts.