The generalized classical method (GCM) was developed on the basis of the differential Taylor transform method, which was first used by the Ukrainian scientist G.E. Pukhov. In this study, some applications of the GCM to the analysis of transient events in simple electrical circuits are examined. It is shown that, if the solution can be decomposed into a steady-state part and a transient part, the use of the GCM becomes more effective, and the transient regime of the circuit solution can be analyzed without a full solving process of differential equations. The efficiency of the proposed approximation method is illustrated by comparing it with prior arts on similar problems. The results reveal that the proposed method is simple and can be applied successfully for the analysis of both linear and nonlinear problems in mathematical physics by similar solving procedures.

Energy has been one of the most important concepts that we cannot remove from our lives with its many uses, from transportation, warming, and enlightenment to the tourism sector, being at the beginning of the most basic requirements for humanity for centuries. Renewable energy is a resource that does not decrease according to the increase of the population, can be obtained naturally on earth, can create an alternative solution to energy shortages and several problems. The concept of a smart grid, which provides an advantage for the use of renewable energy sources, ensures that the energy network can be controlled in the future under conditions of intermittent energy production and instantaneous demand. With the increase in distributed production in smart grids, production and consumption conditions have paved the way for energy pricing to be dynamic. In this study, price balancing was achieved by dynamic electricity pricing, which is one of the methods of achieving autonomous control in smart grids and obtaining energy safely. For this purpose, the coefficients of the Proportional-Integral-Derivative (PID) controller obtained using the pole placement method are modeled in the MATLAB/SIMULINK program with the energy supply-demand scenario. In order to examine the effect of change in energy demand on price, a temporary peak and fall are given to energy demand. According to changing demand conditions, the PID controller has been shown to provide price balance, regulate energy price online and maintain energy balance. Enerji, çağlar boyunca insanoğlu için en temel gereksinimlerinin başında gelip; ulaşım, ısınma ve aydınlanmadan turizm sektörüne kadar birçok kullanım alanıyla hayatımızdan çıkaramayacağımız en önemli kavramlardan biri olmuştur. Yenilenebilir enerji ise nüfusun artmasına göre azalmayan, yeryüzünde doğal olarak elde edilebilen, enerji yetersizliği ve birtakım problemlere alternatif çözüm oluşturabilecek kaynaklardır. Yenilenebilir enerji kaynaklarının kullanımı için avantaj sağlayan akıllı şebeke kavramı gelecekte enerjinin aralıklı üretim ve anlık olarak değişen talep koşulları altında enerji ağının kontrol edilebilir olmasını sağlamaktadır. Akıllı şebekelerde dağıtık üretimin de artmasıyla birlikte üretim ve tüketim koşulları enerji fiyatlandırmasının dinamik bir yapıda olmasının önünü açmıştır. Bu çalışmada akıllı şebekelerde otonom kontrolü sağlanabilen ve güvenli bir şekilde enerji elde etme yöntemlerinden biri olan, dinamik elektrik fiyatlandırması yapılarak fiyat dengelenmesi sağlanmıştır. Bu amaçla kutup yerleştirme metodu kullanılarak elde edilen Oransal-İntegral-Türev (PID) kontrolörün katsayıları enerji arz-talep senaryosu ile MATLAB/SIMULINK ortamında modellenmiştir. Enerji talebinin değişiminin fiyat üzerindeki etkisinin incelenmesi için enerji talebine anlık olarak zirve ve düşüş verilmiştir. Değişen talep koşullarına göre PID kontrolörün fiyat dengesini sağladığı, enerji fiyatını çevrimiçi olarak düzenlediği ve enerji dengesini koruduğu görülmüştür.

Renewable energy sources (RESs) enable sustainable and environmentally friendly electricity generation. However, their intermittent nature makes it difficult to maintain energy balance. Smart grids (SGs) address this challenge by enabling grid control under variable demand and fluctuating generation. With the growing share of distributed generation, dynamic energy pricing has become increasingly important for sustaining the supply–demand balance in SGs. This study aims to regulate the interaction between variable demand and distributed generation in SGs using control strategies. The dynamic pricing framework was analyzed using closed-loop Proportional–Integral–Derivative (PID), Fractional-Order PID (FOPID)- and Adaptive Neuro Fuzzy Inference System (ANFIS)-based FOPID controllers. PID and FOPID parameters were tuned by pole placement with reference model matching, while the FOPID parameters in the ANFIS-FOPID structure were adaptively optimized using ANFIS. Energy supply–demand models were developed in MATLAB/Simulink, and the effects of each controller on system dynamics and energy prices were comparatively examined. The results indicate that ANFIS-FOPID achieves lower overshoot, shorter settling time, and more stable balancing performance, owing to its fractional-order flexibility and optimized parameters. In the model established in the MATLAB/Simulink environment, the controllers were evaluated based on integral of squared error (ISE), time-weighted integral absolute error (ITAE), root mean square error (RMSE), average unit energy price, price volatility, and coefficient of variation. A virtual energy storage model was added to the system. Disturbance and load change scenarios were also examined. The results showed that the ANFIS-FOPID controller provided the most balanced performance in terms of error reduction, suppression of price fluctuations, and reduction of the average unit energy price.