10.1.1.1 Case Study

Schematic diagram of an isolated MG as the case study is illustrated in Fig. 10.1. It contains a DC voltage source, a DC–AC converter interfaced between the DC voltage source and distribution lines, a filter represented by and parameters to extract fundamental frequency of terminal voltage, a three-phase local load depicted by a parallel RLC, a three-phase transformer that transforms voltage from 600 V to 13.8 kV, and a local controller to maintain stability in both connected and disconnected modes. The main grid is described with Rs, Ls, and an AC voltage source. The MG is connected to the main grid via a circuit breaker (CB) at the point of common coupling (PCC) junction. If a disturbance occurs in the power system, such as a short circuit or unit outage, the MG may be unable to maintain its stability in the connected operation. Hence, the circuit breaker will be opened and the MG operating status will be transferred to the islanded mode.

The distributed generator (DG) unit and local loads must be in service in both connected and disconnected operations. In the connected mode, the main grid is responsible for maintaining system voltage and frequency in an acceptable range. In this mode, the voltage source converter (VSC) controls the active and reactive power exchange with the grid using direct-quadrature current control method. As mentioned in Chapter 9, this is known as the P–Q control method in which the DG units deliver constant active and reactive powers to the network. In disconnected mode, the MG should be able to control the system voltage and frequency as an independent power grid. This known as the VSI control method in which the inverter controls voltage and frequency in the isolated grid by changing absorbed active and reactive power from DG units.

In an islanded mode, the VSC can employ an internal oscillator with a constant frequency to generate the modulation signals. As shown in Fig. 10.1, the control unit uses rated frequency and three-phase of PCC voltage to control the voltage. For the sake of linear robust control design, the linearized model of the MG is needed. The state space model of the system under balanced condition in abc-frame is given in (10.1), where , , , and are three-phase terminal voltage, terminal currents, PCC voltages, and PCC currents, respectively.

(10.1)

Using α–β transformation and a rotating reference frame, which is presented in Ref. [20], the d- and q-axes of the state space variables system (Fig. 10.1) yield the following equations:

(10.2)

where

From (10.2), the transfer function of can be obtained as (10.3). The and are direct voltage component of the PCC and inverter terminal voltage, respectively.

(10.3)

where , and .

The N(s) and D(s) are numerator and denominator of the transfer function of the open-loop system, respectively. The , , , , , , and are numerator and denominator coefficients, which are expressed as follows:

As seen from (10.3), the system has two zeroes and four poles. By substituting the rated values of system parameters given in the Table 10.1, the nominal transfer function of the plant, is obtained as expressed in Equation (10.4). The plant is minimum phase and stable for the nominal values (Table 10.1). However, parametric uncertainties, unpredictable generation variation, load disturbance, and faults may lead the MG to an unstable condition. A robust control may guarantee acceptable performance of the system in this circumstance.

(10.4)

10.1.1.2 Controller Design

Bode diagrams of the open-loop system are shown in Fig. 10.2. The system gain margin and phase margin are 3.34 dB and 0.0241°, respectively; this indicates a relatively poor stability for the present case study. Based on Kharitonov's theorem [19], a polynomial such as K(s),

(10.5)

with real coefficients is Hurwitz if and only if the following four extreme polynomials are Hurwitz.

(10.6)

The and show the minimum and maximum bounds on the polynomial coefficients, respectively. In Kharitonov's theorem, the K(s) is a characteristic equation of the closed-loop system.

It is shown [19] that for polynomials with lower than fifth order, there are simpler criteria than (10.6). For a fifth-order plant, it is only sufficient to check the stability of , , and polynomials. For the problem at hand, the Kharitonov's theorem must be applied on the characteristic equation of the closed-loop system. Considering (10.3) with PI feedback control, the characteristic equation of the closed-loop system can be obtained as follows:

(10.7)

where

Here, and are coefficients of the open-loop transfer function system and, and are proportional and integral gains of the PI controller, respectively. According to (10.7), the order of the closed-loop system is 5. Therefore, it is just needed to test the Hurwitz criteria for the following three polynomials:

(10.8)

It is assumed that with ±10% change in rated values of the system parameters, the coefficients of the open-loop system (10.3) can be perturbed in the following ranges:

Following some manual algebraic operations to stabilize (10.8), a set of nine inequalities [4] is obtained. The resulting inequalities are satisfied for some and , which are presented in Fig. 10.3.

In addition to robust stability, to ensure a robust performance such as desirable time response, minimum overshoot/undershoot, and oscillation damping, it is important to maintain the roots of the characteristic equation in a specific region (D region). As seen from Fig. 10.3, there are numerous pairs of and . Using the D-stability concept, proper pairs of and , which remain the roots of the characteristic equation in a specific region are selected. Using this approach, the PI control parameters are tuned as and .

The basic geometry associated with the zero exclusion condition [19] for is fully demonstrated in Fig. 10.4. The designed PI controller is robust if and only if the rectangle plots do not include the origin. This issue is confirmed in Fig. 10.4. By substituting values of and in (10.7), the characteristic polynomial of the closed-loop system is determined as follows:

(10.9)

The closed-loop response of the MG system is evaluated against severe step load disturbances, changes in parameters, and operating mode. The simulation results are given in Ref. [4]. It is shown that the proposed control method can handle all test scenarios, effectively.

10.1.2.1 Neuro-Fuzzy Control Framework

The artificial neural network (ANN) is one of the intelligent algorithms that can be used in both identification and control. The ANNs have the ability to learn system behavior and are effectively used for the identification of nonlinear dynamic systems. On the other hand, fuzzy logic (FL) is a powerful tool in control engineering, which can be used to control variable structure systems in the real-time world. The ANNs can be trained by the training data, but the FL has no ability for training. Combining FL and ANNs leads to a useful and valuable result.

Adding the training ability of ANN to FL creates a new hybrid technique, known as ANFIS. In this method, the correction of fuzzy rules is possible when the system is being trained, and by setting the ANN appropriately, previous knowledge about the membership function (MF) and rules is not required, and the optimum MFs are sufficient for obtaining the input/output (I/O) data. The configuration of MFs depends on their parameters. The ANFIS selects these parameters automatically, and does not need a human to obtain these parameters. The parameters of the membership functions are set by means of a back-propagation (BP) algorithm and the least-squares error (LSE) method.

The overall structure of an ANFIS unit is shown in Fig. 10.5 [3], which includes five layers. Weights of the MFs are analyzed in the first layer, which is known as the MF layer. In this layer, the input variables are applied to the MFs. The output of the second layer is the multiplication of the input signals, which are equivalent to IF rules. In the third layer, which is known as the rule base layer, the activity level of even rule is calculated. The number of layers is equal to the number of fuzzy rules. The output of this layer is a normalized form of the previous layer. The fourth layer obtains the output values of the resulting rules. This layer is known as the defuzzification layer. Finally, the output value of the system is obtained from the output layer.

Here, a generalized droop control (GDC) concept [3] is modeled by means of ANFIS, and the validity of the model is examined. Design steps can be summarized as follows:

1. By applying and testing the GDC on the system shown in Fig. 10.6, and then saving the controller inputs/outputs, the training data for the ANFIS controller synthesis are collected. To obtain an accurate model, the training data under violent changes of active and reactive loads are considered.
2. After obtaining the training data set, the ANFIS structure is completed. The MFs of input and output are considered in the form of linear and Gaussian functions.
3. After creating the controller structure, using the optimal hybrid method (combination of the LSE and BP), the ANFIS is trained for 5 epochs (iterations) with a small error tolerance (i.e., 0.00001 ms).

10.1.2.2 Synthesis Procedure and Performance Evaluation

A simple voltage source inverter (VSI)-based MG, shown in Fig. 10.6, is used for designing/training of the ANFIS controller. The ANFIS model is created by using the GDC. The GDC, which is introduced in Ref. [3], consists of two inputs (active and reactive power) and two outputs (voltage and frequency). Since one output is allowed for the construction of ANFIS in the related toolbox in MATLAB software, two ANFIS blocks are used as shown in Fig. 10.7.

Therefore, the considered functions for each block consist of two inputs and one output. After generating the I/O data set and applying them to the ANFIS toolbox, the model is trained. Here, to reconstruct the system behavior effectively, the switching frequency of the inverter is fixed at 4000 Hz, and the simulation sampling time is considered as 100,000 samples per second. To evaluate the performance of the designed ANFIS model, two sets of data are selected from the real data and training data (as the test data). These two sets are compared and the results are shown in Fig. 10.8. Comparing the real and network data shows whether the training process is done accurately. Figure 10.9 shows the closed-loop control system for a VSI-based inverter.

Two trained ANFIS controllers are replaced with the GDC [3]. After that, under the violent changes of active and reactive loads, the voltage and frequency of the closed-loop system are examined and compared with the generalized droop control structure. The applied scenario for active and reactive power changes is shown in Fig. 10.10a. Figure 10.10b shows the voltage and frequency profiles for both ANFIS and GDC-based methodologies. This figure shows the validity of the ANFIS controller with a high level of accuracy.

To prove the reliability of the closed-loop system with the designed ANFIS controller, it is also tested on several MG systems. The results are given in Ref. [3].

10.2.1.1 Case Study

An isolated AC MG system is considered as the case study, which is shown in Fig. 10.11. The MG system contains conventional diesel engine generator (DEG), photovoltaic (PV) panel, wind turbine generator (WTG), fuel cell (FC) system, battery energy storage system (BESS), and flywheel energy storage system (FESS). As shown in Fig. 10.11, the DGs are connected to the MG by power electronic interfaces, which are used for synchronization in the AC sources like DEG and WTG and to reverse voltage in the DC sources like PV panel, FC, and energy storage devices.

Each microsource has a circuit breaker to disconnect from the network to avoid the impacts of severe disturbances through the MG or for maintaining purposes. To easily understand the MG frequency response, a simplified frequency response model is given in Fig. 10.12. The FC contains three fuel blocks, an inverter for converting DC to AC voltage, and an interconnection device. Although the FC and some other DGs have high-order characteristic models, for the present study the introduced model (Fig. 10.12) is sufficient.

This model can be useful to analyze/demonstrate the frequency behavior of the case study. The system details including PV and WTG models and the case study parameters are given in Ref. [2]. Since most of the energy sources have an intermittent nature with considerable uncertainty and fluctuation in the power system, efficient control methods must be employed to decrease the undesirable dynamic impacts.

10.2.1.2 Conventional and Fuzzy PI-Based Frequency Control

In traditional power systems, the secondary frequency control is mostly done by using conventional PI controllers that are usually tuned based on the specified operating points. In case of any change in the operating condition, the PI controllers cannot provide the assigned desirable performance. On the other hand, if the PI controller is able to track the changes that occur in the power system, the optimum performance will be always achieved. Fuzzy logic can be used as an intelligent method for online tuning of PI controller parameters.

In this section, the traditional PI controller for secondary frequency control is tuned by the well-known Ziegler–Nichols method. Then, a pure fuzzy PI controller is also designed. The result will be compared with the online PSO-fuzzy-based PI design methodology. A comprehensive study on classical PI/PID tuning methods like Ziegler–Nichols has been presented in Ref. [21]. Using the Ziegler–Nichols method, the PI parameters are obtained as given in Table 10.2.

Controller Parameter	Value
	4.095
	21.84

As described before, to achieve better performance, fuzzy logic is used as an intelligent method. Fuzzy logic is able to respond to the inability of the classic control theory for covering complex systems with their uncertainties and inaccuracies. The control framework for the application of the fuzzy logic system as an intelligent unit in order to fine-tune the traditional PI controller is shown in Fig. 10.13. The fuzzy PI controller has two levels: the first one is a traditional PI controller and the second one is a fuzzy system. As shown, the intelligent fuzzy system unit uses frequency deviation and load perturbation inputs to adjust the PI control parameters. In order to apply the fuzzy logic to the isolated MG system for tuning the PI control parameters, a set of fuzzy rules consisting of 18 rules is used to map input variables, (frequency deviation) and (load perturbation), to output variables, (proportional gain) and (integral gain).

The set of fuzzy rules are given in the Table 10.3 where membership functions corresponding to the input and output variables are arranged as negative large (NL), negative medium (NM), negative small (NS), positive small (PS), positive medium (PM), and positive large (PL). They have been arranged using the triangular membership function, which is the most traditional one. The antecedent parts of each rule are composed by using AND function (with interpretation of minimum). Here, the Mamdani fuzzy inference system is also used.

	Δf
	NL	NM	NS	PS	PM	PL
S	NL	NM	NS	PS	PS	PM
M	NL	NL	NM	PS	PM	PM
L	NL	NL	NL	PM	PM	PM

It is shown that the fuzzy-based PI performance highly depends on the membership functions [3]. Without precise information about the system, the membership functions cannot be carefully selected, and the designed fuzzy PI controller does not provide optimal performance in a wide range of operating conditions. Therefore, a complementary algorithm is used for online regulation of membership functions.

10.2.1.3 PSO-Fuzzy PI Frequency Control

Here, for online tuning of membership functions employed in the fuzzy PI controller, the PSO is used. The PSO is an optimization algorithm, based on the probability laws, which is inspired by the natural models. This algorithm belongs to a class of direct search methods and is used to find an optimal solution for the optimization problems in a given search space. The applied PSO algorithm is extensively explained in Ref. [2].

Up to now, many search algorithms have been proposed in order to solve the optimization problems, including genetic algorithm, ant colony, and bee colony. However, simplicity is an important advantage of the PSO in comparison with other methods, especially genetic algorithm. Several modifications have been proposed to improve the performance of the PSO algorithm. According to the description provided, what this research investigates is designing an online adaptive controller, using fuzzy logic and PSO, for the purpose of frequency regulation in an AC MG system. The overall control framework for online adjusting of membership functions for the fuzzy rules based on the PSO technique is shown in Fig. 10.14.

Considering the purpose of the algorithm, which is to find the extremum point of the cost function, if the cost function is not properly selected, the algorithm will be stopped in the local extremum points. Initialization of the algorithm parameters is also very important because if they are not carefully selected, the algorithm will never be convergent to the extremum point. The computational flow chart for the proposed online PSO-based optimal design approach is shown in Fig. 10.15.

10.2.1.4 Simulation Results

To compare the classic, fuzzy PI and the PSO-fuzzy PI controllers, several simulation tests are carried out and the performances of the proposed control methods are evaluated. To illustrate the dynamic response of the MG system, the closed-loop system is examined in the face of a multiple step load disturbance, which is plotted in Fig. 10.16a. The MG frequency response using the conventional fuzzy PI and PSO-fuzzy PI (optimal PI) controllers is also shown in Fig. 10.16b.

and are MG frequency deviation and load disturbance, respectively, having values in “pu.” As shown, the proposed optimal PSO fuzzy PI controller regulating the system frequency following a disturbance is quite better than the pure fuzzy PI and classical PI controllers. For the sake of a clear comparison between the performance of the PSO-fuzzy PI and fuzzy PI controllers, system frequency following a severe step load disturbance of 0.1 pu is shown in the Fig. 10.17. In this case also, the proposed optimal control method provides a much better performance.

Power system parameters are constantly changing and this may degrade the closed-loop system performance seriously. As indicated in the previous sections, one of the main advantages of the intelligent control methods is robustness against environmental and dynamical changes. To show the adaptive property of the PSO-fuzzy PI controller, the main power system parameters, in the frequency response model (Fig. 10.12), that is, D (damping coefficient), H (inertia constant), R (droop constant), (turbine time constant), (generator time constant), (FESS time constant), and (BESS time constant) are significantly changed according to Tables 10.4 and 10.5.

Parameter	Variation Range	Parameter	Variation Range
R	+30%		+50%
D	−40%		−45%
H	+50%		+55%
	−50%

Parameter	Variation Range	Parameter	Variation Range
R	−60%		−62%
D	−55%		−35%
H	+48%		−50%
	−53%

The closed-loop frequency response after applying these changes to the MG system parameters are shown in Figs. 10.18 and 10.19, respectively. It can be seen the conventional controller cannot handle the applied parameters perturbation. Figure 10.19 shows that the difference between the proposed optimal PSO-fuzzy PI controller with the other two controllers is more significant for a higher range of parameter variation.

Finally, two scenarios are examined for the secondary frequency control issue. First, only the DEG is considered as the responsible unit for frequency control in which the results are shown in the previous figures. The impact of the FC contribution in the secondary frequency control is considered as the second scenario. The output of the PSO-fuzzy PI controller is divided between the DEG and the FC units according to their participation factors. The result of this cooperation framework using the proposed intelligent technique is shown in Fig. 10.20.

10.2.2.1 ANN in Intelligent Control

Neural networks are formed by neurons. An ANN is a crude approximation to parts of the real brain. It is just a parallel computational system consisting of many simple processing elements connected together in a specific way in order to perform a particular task. The ANNs provide a very important tool in optimization tasks because they are extremely powerful computational devices with the capability of parallel processing, learning, generalization, and fault/noise tolerating. Based on configuration and connecting elements, there are several main applications in ANNs such as brain modeling, financial modeling, time series prediction, control systems, and optimization.

Indeed, an ANN consists of a finite number of interconnected neurons (as described earlier) and acts as a massively parallel distributed processor, inspired from biological neural networks, which can store experimental knowledge and makes it available for use. To use neural networks in optimization tasks, it is needed to have a mathematical model of neural networks. A simplified mathematical model of a neuron is given in (10.10). It consists of three basic components that include weights , threshold (or bias) , and a single activation function .

The values , ,…, are weight factors associated with each node to determine the strength of input row vector . Each input is multiplied by the associated weight of the neuron connection. Depending upon the activation function, if the weight is positive, the resulting signal commonly excites the output node; whereas, for negative weights, it tends to inhibit the output node. The node's internal threshold is the magnitude offset that affects the activation of the output node y as follows [22]:

(10.10)

The neurons could be combined together and they form a layer. Layers are constituted together and make a network. Updating the weights and training of neural networks is based on two basic feed-forward and feedback process. There are three methods for training the weights in feedback process: supervised, unsupervised, and reinforcement learning [22].

In supervised learning, the output is compared with the desired reference vector, and then error vectors applied for updating the weights. There is no desired reference vector for the reinforcement method and a revolutionary process is usually used for updating the weights. In the unsupervised learning method, updating is only based on the input data. In control structures, the supervised learning is usually used. There are several methods for supervised learning such as perceptron, Widrow–Hoff, correlation, and back-propagation learning methods. The most employed method is the back-propagation learning.

The main objective of intelligent control is to implement an autonomous system that could operate with increasing independence from human actions in an uncertain environment. The most common ANN-based intelligent control structures are well explained in Ref. [22]. In all existing structures, the control objectives could be achieved by learning from the environment through a feedback mechanism. The ANN has the capability to implement this kind of learning.

10.2.2.2 Proposed Control Scheme

The schematic diagram of the proposed control scheme for the ANN-based self-tuning frequency controller is shown in Fig. 10.21, where the ANN unit acts as an intelligent unit for optimal tuning of classic PI controller parameters, by getting input and output data based on certain rules. In the proposed intelligent control scheme, the ANN collects information about the plant (MG) response, adjusts weights via a learning algorithm, and recommends an appropriate control signal. In Fig. 10.21, the ANN performs an online automatic optimal tuner for the existing PI controller. The main components of the ANN as a fine-tuner for the PI controllers include a response recognition unit to monitor the controlled response and extract knowledge about the performance of the current controller gain setting, and an embedded unit to suggest suitable changes to be made to the controller gains.

The employed neural network structure for tuning the parameters of the PI controller used for frequency control (of the system given in Fig. 10.11) is shown in Fig. 10.22. In the ANN structure of Fig. 10.22, 20 linear neurons are considered for network input layer; 10 and 2 nonlinear neurons are also considered for hidden layer and output layer, respectively. The number of output layer's neurons is equal to the number of control parameters that must be adjusted. In Fig. 10.22, X is the input vector and W₁ and W₂ are connecting weight vectors between the layers.

Selection of initial conditions in an ANN-based control system is also known as an important issue. In multiobjective control problems, some initial values may not guarantee the achievement of objectives with a satisfactory value of the optimization function. The initial conditions are usually selected according to the a priori information about distributions at the already-known structure of the open-loop ANN and selected control strategy. Here, the initial quantities in the applied ANN scheme (Fig. 10.22) are considered as follows:

As mentioned, linear functions are considered for the first layer, and for the second and third layers, the sigmoid functions are chosen. In the output layer, different coefficients for sigmoid function are considered for tuning of the controller parameters. The main advantage of using these nonlinear functions is in performing a smooth updating of weights.

The learning process of the applied ANN for the MG test system is to minimize the performance function given by (10.11), where y_d represents reference signal and y represents the output of output layer.

(10.11)

The implemented algorithm for updating weights is based on back-propagation learning, which is described in the flowchart of Fig. 10.23. In the feed-forward process, by using the input vector (X), the values of hidden layer output (H) and output layer result (O) are provided, and then error value (E) obtained from the process is employed to update the weights as follows:

(10.12)

where is a learning rate given by a small positive constant, and

(10.13)

(10.14)

The learning process continues to reach the desired minimum error. This method is presented in detail in Ref. [22]. For testing of the proposed control methodology for tuning the PI controller parameters, the controller is applied to the case study (Fig. 10.11), and the results are compared with the response of a conventional PI controller.

To examine the proposed control strategy, system frequency response in the presence of the ANN-based self-tuning PI controller and conventional PI controller is tested. Here, a random step load disturbance, WTG mechanical output power variations, and sunlight flux variations are simultaneously considered in the MG test system to better evaluate the proposed closed-loop control performance. In this scenario, it is assumed that the BESS and FESS systems do not participate in the secondary frequency control issue. The results for the first scenario are plotted in Fig. 10.24a–e.

The considered power fluctuations, the WTG and PV output power variations, the DEG frequency response, and system frequency are shown in Fig. 10.24a–e, respectively. As shown, only the DEG participates in the secondary frequency control process and it injects more compensating power by using the proposed intelligent control method. Therefore, when the ANN adjusts the controller parameters, system frequency fluctuations are much less.

10.3.1.1 Distribution Network with Connected MGs

A simplified architecture of a distribution network organized by a distribution company (Disco1) is shown in Fig. 10.25. This network consists of N connected MGs in an LA, and NAs that may belong to another company (Discoi). The microsources and storage devices use power electronic circuits to connect to the MG. The MG can be connected to the network by a PCC via a static switch. This switch is capable of islanding the MG for maintenance purposes or when faults or a contingency occurs.

The DNO deals with some overall responsibilities for the distribution network (Disco) and the connected MGs, such as interchange power between the main grid and the MGs. This unit, which is located in the application layer of the distribution management system is acting in an economically based energy management between the main grid and the neighboring MGs.

As shown in Fig. 10.25, the DNO interfaces the main grid (Disco1) with the connected MGs (in the LA) as well as other neighbor grids (in NAs that may be covered by another Disco). The DNO also supervises the power flow control and market operation. This operator controls power flow from the main grid to the MGs to keep it close to the scheduled values. In the mentioned network, identifying the optimal consumption/generation schedule to minimize production costs and to balance the demand and supply, as well as online assessment of security and reliability, are the responsibilities of the DNO unit. As mentioned in Chapter 9, the DNO together with the MGCCs supervise the MG's market activities such as buying and selling active and reactive power to the grid and possible network congestions for transferring energy from a distribution network to the MGs in a local area and other neighboring areas.

10.3.1.2 Proposed Methodology and Results

Pricing of electricity can be used as a mechanism to encourage customers to follow a specified load scheduling. Various pricing schemes have been proposed by economists and regulatory agencies such as flat pricing, critical-peak pricing, time-of-use pricing, and real-time pricing. Among them, real-time pricing is motivated to be used in the next-generation power systems concerning its environmental and economic gains. Accordingly, in the present work, an energy scheduling approach based on real-time generation cost is proposed, which can be used to establish a real-time pricing scheme.

In the mentioned network, the objective of DNO to implement ECS could be either to minimize power generation cost or to minimize PAR. A solution is summarized as an adaptive cost-aware ECS (ACA-ECS) algorithm [7]. In order to minimize PAR of the total instantaneous power delivered to LA and NAs during the time period T, a minmax formulation is proposed with the objective of minimizing the peak of this power. The mathematical formulations are given in Ref. [7].

For investigation of the proposed methods, a distribution network is considered in connection with an LA consisting of N = 10 MGs and an NA. The ECS located in the DNO schedules energy consumption of MGs in LA during a time horizon of length 6 h.

The impact of the proposed ACA–ECS algorithm on the time domain curvature of the total grid demand in the presence of an unknown NA demand is evaluated and the result is compared with an optimal technique. A typical realization of T = 360 samples of NA demand with standard deviation of σ = 20 kWh and the corresponding optimal total demand are shown in Fig. 10.26a. As observed, the optimal solution schedules LA demand such that the system-wide total demand becomes smooth suitable for cost minimization. In fact, scheduling the LA demand provides a diversity for the ECS to mitigate the stochastic nature of NA demand. Total demand using the ACA–ECS scheme and the corresponding Lagrange multipliers [7] are also shown in Fig. 10.26b. Intuitively, after some initial time slots, the behavior of the total demand curve approximately converges to that of the optimal solution in Fig. 10.26a.

Performance measures of the ACA–ECS and the optimal ECS schemes, such as generation cost per kWh and PAR, versus the randomness of the NA demand could be interesting in the following. As another scheduling scheme, the results of uniform ECS scheme are also included. In this scheme, the demand of each MGn in LA is uniformly distributed over the whole time horizon, independent of the NA demand. This can also be considered as a deterministic solution. Cost and PAR performances versus the standard deviation are shown in Fig. 10.26c and d, respectively. For each instance, similar to the time-domain performance, first a data set with T = 360 samples are generated. This set is used to obtain the optimal ECS solution once in the beginning of the time horizon as well as to provide the ACA–ECS scheme with instantaneous realized NA demand. As shown in the first part of Fig. 10.26a, there is a typical generated data set with σ = 20 kWh. As a common observation in Fig. 10.26c and d, performance measures get worse as σ increases. In the case of cost measure, this is due to the fact that the considered squared cost function results in higher cost per kWh for high demand values in comparison with low demand values. The results in PAR are based on the fact that the averages of both LA and NA demands are made constant when σ increases. Considering PAR as a fractional term of the peak demand over the average demand, it is reasonable to conclude that PAR increases as σ increases. Moreover in Fig. 10.26c, with increase in σ, the performance gap between the compared ECS schemes and the optimal one increases. In the case of ACA–ECS, this is due to the fact that the stochastic estimator (given in Ref. [7]) would be far from optimality with the increase in the randomness of the uncertain part of power generation. In the case of uniform ECS, the degradation effect of high randomness would be more severe since this scheme does not take care of NA demand in the scheduling decisions.

Furthermore, in Fig. 10.26c and d, the generation cost and PAR performances of the ACA–ECS scheme outperform those of uniform ECS. This is reasonably expected as ACA–ECS takes advantage of the diversity in NA demand to smooth the total demand and therefore achieves a better performance. In the comparison between ACA–ECS and the optimal solution, it is observed that the optimal solution achieves lower cost. This is due to the fact that this solution fully takes into account the knowledge of NA demand at the beginning of the time horizon for the scheduling of LA demand. However, ACA–ECS makes a scheduling decision adaptively per a time unit, when the demand of NA is available in that unit. Remarkably, the PAR of ACA–ECS is comparable to that of the optimal solution. This implies that the optimality of generation cost does not necessarily imply the optimality of PAR too. This observation motivates the performance evaluation of PAR formulation in the following.

In order to evaluate the efficiency of PAR formulation, the generation cost per kWh and PAR performances of this formulation are illustrated in Fig. 10.26e and f, respectively. Similar to the cost formulation [7], the results of optimal solution in PAR formulation (optimal PAR) and uniform scheduling (uniform PAR) scheme are also included. Since the scheduling of the uniform strategy is independent of the objective function, the achieved results are the same in both cost and PAR formulations. We take advantage of this equality and take uniform strategy curves as references for comparison between these formulations.

Comparing Fig. 10.26c and e, it is observed that uniform scheduling was the worst in the former, whereas it is the best in the latter. Considering the results of uniform scheduling as reference in both figures, we conclude that cost minimization formulation is more cost efficient in comparison with PAR formulation. In terms of PAR, the optimal solution in the PAR formulation achieves the lowest PAR. This is reasonably expected as this solution takes NA demand into account a priori. In comparison with the uniform strategy, the PAR of PAR–ECS scheme is high. More importantly, this implies that PAR performance of the proposed adaptive approach in cost minimization formulation even outperforms its equivalent in the PAR minimization formulation.

This observation along with the lower generation cost in cost formulation demonstrates that our proposed adaptive approach achieves more efficient results with this formulation compared with the PAR one. Also, the proposed adaptive approach is a trade-off between the optimal (full NAs demand) and uniform (no NAs demand) schemes in terms of generation cost and PAR minimization.

10.3.2.1 Methodology and Results

As an example, consider a power grid consisting of small MGs starting from MG1 and ending with MG8 as shown in Ref. [8]. The distance between any two neighbor MGs is the same and is noted as one hop. The transmission price per unit of power between any two MGs is assumed to be the number of hops between them. Moreover, the power generation function is assumed to be a square function.

The simulation is run for 200 time slots; each slot with realization of a new demand and new maximum permitted supply values per MG. Produced powers at individual MGs by the proposed statistical cooperative power dispatching (SCPD) algorithm are shown in Fig. 10.27a. The s_n in this figure represents the produced power of MG_n. Despite the diverse power demands, the produced powers are close to each other. In comparison with their demands, low demand MGs produce higher power and high demand ones produce lower power. This is due to the fact that low demand MGs tend to sell power, whereas high demand ones tend to purchase power. Figure 10.27b illustrates Lagrange multipliers λ_i by which MGs interact within the grid. As previously mentioned, each λ_i in the SCPD algorithm can be interpreted as the price that announces at time slot to pay for a unit of power from any other MG in the system. As shown, even though all prices are initialized with the same value, they statistically converge to different levels during the simulation. The higher is the demand, the higher is the announced price for power purchasing.

The curves in this figure are interpreted relative to each other, that is, power flow direction within the grid is from MGs with low prices to ones with high prices. In other words, an MG with low purchase price in comparison with others is an indication of power selling and vice-versa.

The average demand, produced, sold, and purchased powers of individual MGs in the SCPD algorithm are shown in Fig. 10.27c. While demand increases linearly with MG indexes in accordance with assumed values, produced powers are approximately the same for all MGs, in compliance with Fig. 10.27a. Remarkably, the sum of produced and purchased powers at each MG is equal to the sum of demand plus sold power. This reveals the energy balance within the grid. Purchased and sold power curves vary in opposite directions versus the MG index, that is, purchased power increases with demand, whereas sold power decreases. Exceptionally, the decrease of purchased power is possibly due to its location in the network topology, which burdens high power transmission cost. In a logical statement, low demand MGs sell power to high demand ones.

This statement is investigated by comparing the average operational costs at individual MGs in these two modes, shown in Fig. 10.27d. To this end, the cost in autonomous mode is obtained from the square power production function. Furthermore, the cost in grid-connected mode implemented by SCPD algorithm is the production cost plus purchased cost minus the revenue from selling power to the other MGs. As shown, the grid-connected mode achieves lower cost for low demand and high demand MGs. The decrease in the cost of low demand MGs is the result of selling power to high demand ones. In particular, the cost of MG₁ and MG₂ even get negative as a result of high revenue from selling power that compensates their production cost. This outcome also decreases the cost of high demand MGs as they purchase a portion of their demand from the low demand ones. This is accomplished by means of high announced purchase prices in Fig. 10.27b. Furthermore, the purchased and sold power of moderate demand MGs are mostly the same. In summary, as a numerical indicator, the proposed SCPD in the grid connected mode achieves 20% cost reduction in comparison with stand-alone operation. Overall, this power sharing scheme transforms the parabolic cost curve to a linear one as shown in Fig. 10.27d.