In the last decade, research has been started due to accelerated growth in power demand has mainly concentrated on the large power production and quality of power. After the digital revolution, non-conventional energy sources, many state-of-art equipment, power electronics loads, reactive power compensating devices, sophisticated measuring devices, etc., entered the power industry. The reactive power compensating devices, connected electrical equipment, renewable energy sources can be anticipated/unanticipated action can cause considerable reactions may be failure issues to power grids. To deal with these challenges, the power sector crucially needs to design and implement new security systems to protect its systems. The Internet-of-Things (IoT) is treated as revolution technology after the invention of the digital machine and the internet. New developments in sensor devices with wireless technologies through embedded processors provide effective monitoring and different types of faults can be detected during electric power transmission. The wavelet (WT) is one of the mathematical tools to asses transient signals of different frequencies and provides crucial information in the form of detailed coefficients. Machine learning (ML) methods are recommended in the power systems community to simplify digital reform. ML and AI techniques can make effective and rapid decisions to improve the stability and safety of the power grid. This recommended approach can contribute critical information about symmetrical or asymmetrical faults through machine learning assessment of IoT supervised microgrid protection in the presence of SVC using the wavelet approach covers diversified types of faults combined with fault-inception-angles (FIA).
Citation: K.V. Dhana Lakshmi, P.K. Panigrahi, Ravi kumar Goli. Machine learning assessment of IoT managed microgrid protection in existence of SVC using wavelet methodology[J]. AIMS Electronics and Electrical Engineering, 2022, 6(4): 370-384. doi: 10.3934/electreng.2022022
[1] | Ishfaq Mallah, Idris Ahmed, Ali Akgul, Fahd Jarad, Subhash Alha . On ψ-Hilfer generalized proportional fractional operators. AIMS Mathematics, 2022, 7(1): 82-103. doi: 10.3934/math.2022005 |
[2] | M. J. Huntul . Inverse source problems for multi-parameter space-time fractional differential equations with bi-fractional Laplacian operators. AIMS Mathematics, 2024, 9(11): 32734-32756. doi: 10.3934/math.20241566 |
[3] | Kanagaraj Muthuselvan, Baskar Sundaravadivoo, Kottakkaran Sooppy Nisar, Suliman Alsaeed . Discussion on iterative process of nonlocal controllability exploration for Hilfer neutral impulsive fractional integro-differential equation. AIMS Mathematics, 2023, 8(7): 16846-16863. doi: 10.3934/math.2023861 |
[4] | Zihan Yue, Wei Jiang, Boying Wu, Biao Zhang . A meshless method based on the Laplace transform for multi-term time-space fractional diffusion equation. AIMS Mathematics, 2024, 9(3): 7040-7062. doi: 10.3934/math.2024343 |
[5] | Junseok Kim . A normalized Caputo–Fabrizio fractional diffusion equation. AIMS Mathematics, 2025, 10(3): 6195-6208. doi: 10.3934/math.2025282 |
[6] | Lahcene Rabhi, Mohammed Al Horani, Roshdi Khalil . Existence results of mild solutions for nonlocal fractional delay integro-differential evolution equations via Caputo conformable fractional derivative. AIMS Mathematics, 2022, 7(7): 11614-11634. doi: 10.3934/math.2022647 |
[7] | Choukri Derbazi, Hadda Hammouche . Caputo-Hadamard fractional differential equations with nonlocal fractional integro-differential boundary conditions via topological degree theory. AIMS Mathematics, 2020, 5(3): 2694-2709. doi: 10.3934/math.2020174 |
[8] | Amjid Ali, Teruya Minamoto, Rasool Shah, Kamsing Nonlaopon . A novel numerical method for solution of fractional partial differential equations involving the ψ-Caputo fractional derivative. AIMS Mathematics, 2023, 8(1): 2137-2153. doi: 10.3934/math.2023110 |
[9] | Jagdev Singh, Jitendra Kumar, Devendra Kumar, Dumitru Baleanu . A reliable numerical algorithm for fractional Lienard equation arising in oscillating circuits. AIMS Mathematics, 2024, 9(7): 19557-19568. doi: 10.3934/math.2024954 |
[10] | Apassara Suechoei, Parinya Sa Ngiamsunthorn . Extremal solutions of φ−Caputo fractional evolution equations involving integral kernels. AIMS Mathematics, 2021, 6(5): 4734-4757. doi: 10.3934/math.2021278 |
In the last decade, research has been started due to accelerated growth in power demand has mainly concentrated on the large power production and quality of power. After the digital revolution, non-conventional energy sources, many state-of-art equipment, power electronics loads, reactive power compensating devices, sophisticated measuring devices, etc., entered the power industry. The reactive power compensating devices, connected electrical equipment, renewable energy sources can be anticipated/unanticipated action can cause considerable reactions may be failure issues to power grids. To deal with these challenges, the power sector crucially needs to design and implement new security systems to protect its systems. The Internet-of-Things (IoT) is treated as revolution technology after the invention of the digital machine and the internet. New developments in sensor devices with wireless technologies through embedded processors provide effective monitoring and different types of faults can be detected during electric power transmission. The wavelet (WT) is one of the mathematical tools to asses transient signals of different frequencies and provides crucial information in the form of detailed coefficients. Machine learning (ML) methods are recommended in the power systems community to simplify digital reform. ML and AI techniques can make effective and rapid decisions to improve the stability and safety of the power grid. This recommended approach can contribute critical information about symmetrical or asymmetrical faults through machine learning assessment of IoT supervised microgrid protection in the presence of SVC using the wavelet approach covers diversified types of faults combined with fault-inception-angles (FIA).
Ostrowski proved the following interesting and useful integral inequality in 1938, see [18] and [15, page:468].
Theorem 1.1. Let f:I→R, where I⊆R is an interval, be a mapping differentiable in the interior I∘ of I and let a,b∈I∘ with a<b. If |f′(x)|≤M for all x∈[a,b], then the following inequality holds:
|f(x)−1b−a∫baf(t)dt|≤M(b−a)[14+(x−a+b2)2(b−a)2] | (1.1) |
for all x∈[a,b]. The constant 14 is the best possible in sense that it cannot be replaced by a smaller one.
This inequality gives an upper bound for the approximation of the integral average 1b−a∫baf(t)dt by the value of f(x) at point x∈[a,b]. In recent years, such inequalities were studied extensively by many researchers and numerous generalizations, extensions and variants of them appeared in a number of papers, see [1,2,10,11,19,20,21,22,23].
A function f:I⊆R→R is said to be convex (AA−convex) if the inequality
f(tx+(1−t)y)≤tf(x)+(1−t)f(y) |
holds for all x,y∈I and t∈[0,1].
In [4], Anderson et al. also defined generalized convexity as follows:
Definition 1.1. Let f:I→(0,∞) be continuous, where I is subinterval of (0,∞). Let M and N be any two Mean functions. We say f is MN-convex (concave) if
f(M(x,y))≤(≥)N(f(x),f(y)) |
for all x,y∈I.
Recall the definitions of AG−convex functions, GG−convex functions and GA− functions that are given in [16] by Niculescu:
The AG−convex functions (usually known as log−convex functions) are those functions f:I→(0,∞) for which
x,y∈I and λ∈[0,1]⟹f(λx+(1−λ)y)≤f(x)1−λf(y)λ, | (1.2) |
i.e., for which logf is convex.
The GG−convex functions (called in what follows multiplicatively convex functions) are those functions f:I→J (acting on subintervals of (0,∞)) such that
x,y∈I and λ∈[0,1]⟹f(x1−λyλ)≤f(x)1−λf(y)λ. | (1.3) |
The class of all GA−convex functions is constituted by all functions f:I→R (defined on subintervals of (0,∞)) for which
x,y∈I and λ∈[0,1]⟹f(x1−λyλ)≤(1−λ)f(x)+λf(y). | (1.4) |
The article organized three sections as follows: In the first section, some definitions an preliminaries for Riemann-Liouville and new fractional conformable integral operators are given. Also, some Ostrowski type results involving Riemann-Liouville fractional integrals are in this section. In the second section, an identity involving new fractional conformable integral operator is proved. Further, new Ostrowski type results involving fractional conformable integral operator are obtained by using some inequalities on established lemma and some well-known inequalities such that triangle inequality, Hölder inequality and power mean inequality. After the proof of theorems, it is pointed out that, in special cases, the results reduce the some results involving Riemann-Liouville fractional integrals given by Set in [27]. Finally, in the last chapter, some new results for AG-convex functions has obtained involving new fractional conformable integrals.
Let [a,b] (−∞<a<b<∞) be a finite interval on the real axis R. The Riemann-Liouville fractional integrals Jαa+f and Jαb−f of order α∈C (ℜ(α)>0) with a≥0 and b>0 are defined, respectively, by
Jαa+f(x):=1Γ(α)∫xa(x−t)α−1f(t)dt(x>a;ℜ(α)>0) | (1.5) |
and
Jαb−f(x):=1Γ(α)∫bx(t−x)α−1f(t)dt(x<b;ℜ(α)>0) | (1.6) |
where Γ(t)=∫∞0e−xxt−1dx is an Euler Gamma function.
We recall Beta function (see, e.g., [28, Section 1.1])
B(α,β)={∫10tα−1(1−t)β−1dt(ℜ(α)>0;ℜ(β)>0)Γ(α)Γ(β)Γ(α+β) (α,β∈C∖Z−0). | (1.7) |
and the incomplete gamma function, defined for real numbers a>0 and x≥0 by
Γ(a,x)=∫∞xe−tta−1dt. |
For more details and properties concerning the fractional integral operators (1.5) and (1.6), we refer the reader, for example, to the works [3,5,6,7,8,9,14,17] and the references therein. Also, several new and recent results of fractional derivatives can be found in the papers [29,30,31,32,33,34,35,36,37,38,39,40,41,42].
In [27], Set gave some Ostrowski type results involving Riemann-Liouville fractional integrals, as follows:
Lemma 1.1. Let f:[a,b]→R be a differentiable mapping on (a,b) with a<b. If f′∈L[a,b], then for all x∈[a,b] and α>0 we have:
(x−a)α+(b−x)αb−af(x)−Γ(α+1)b−a[Jαx−f(a)+Jαx+f(b)]=(x−a)α+1b−a∫10tαf′(tx+(1−t)a)dt−(b−x)α+1b−a∫10tαf′(tx+(1−t)b)dt |
where Γ(α) is Euler gamma function.
By using the above lemma, he obtained some new Ostrowski type results involving Riemann-Liouville fractional integral operators, which will generalized via new fractional integral operators in this paper.
Theorem 1.2. Let f:[a,b]⊂[0,∞)→R be a differentiable mapping on (a,b) with a<b such that f′∈L[a,b]. If |f′| is s−convex in the second sense on [a,b] for some fixed s∈(0,1] and |f′(x)|≤M, x∈[a,b], then the following inequality for fractional integrals with α>0 holds:
|(x−a)α+(b−x)αb−af(x)−Γ(α+1)b−a[Jαx−f(a)+Jαx+f(b)]|≤Mb−a(1+Γ(α+1)Γ(s+1)Γ(α+s+1))[(x−a)α+1+(b−x)α+1α+s+1] |
where Γ is Euler gamma function.
Theorem 1.3. Let f:[a,b]⊂[0,∞)→R be a differentiable mapping on (a,b) with a<b such that f′∈L[a,b]. If |f′|q is s−convex in the second sense on [a,b] for some fixed s∈(0,1],p,q>1 and |f′(x)|≤M, x∈[a,b], then the following inequality for fractional integrals holds:
|(x−a)α+(b−x)αb−af(x)−Γ(α+1)b−a[Jαx−f(a)+Jαx+f(b)]|≤M(1+pα)1p(2s+1)1q[(x−a)α+1+(b−x)α+1b−a] |
where 1p+1q=1, α>0 and Γ is Euler gamma function.
Theorem 1.4. Let f:[a,b]⊂[0,∞)→R be a differentiable mapping on (a,b) with a<b such that f′∈L[a,b]. If |f′|q is s−convex in the second sense on [a,b] for some fixed s∈(0,1],q≥1 and |f′(x)|≤M, x∈[a,b], then the following inequality for fractional integrals holds:
|(x−a)α+(b−x)αb−af(x)−Γ(α+1)b−a[Jαx−f(a)+Jαx+f(b)]|≤M(1+α)1−1q(1+Γ(α+1)Γ(s+1)Γ(α+s+1))1q[(x−a)α+1+(b−x)α+1b−a] |
where α>0 and Γ is Euler gamma function.
Theorem 1.5. Let f:[a,b]⊂[0,∞)→R be a differentiable mapping on (a,b) with a<b such that f′∈L[a,b]. If |f′|q is s−concave in the second sense on [a,b] for some fixed s∈(0,1],p,q>1, x∈[a,b], then the following inequality for fractional integrals holds:
|(x−a)α+(b−x)αb−af(x)−Γ(α+1)b−a[Jαx−f(a)+Jαx+f(b)]|≤2s−1q(1+pα)1p(b−a)[(x−a)α+1|f′(x+a2)|+(b−x)α+1|f′(b+x2)|] |
where 1p+1q=1, α>0 and Γ is Euler gamma function.
Some fractional integral operators generalize the some other fractional integrals, in special cases, as in the following integral operator. Jarad et. al. [13] has defined a new fractional integral operator. Also, they gave some properties and relations between the some other fractional integral operators, as Riemann-Liouville fractional integral, Hadamard fractional integrals, generalized fractional integral operators etc., with this operator.
Let β∈C,Re(β)>0, then the left and right sided fractional conformable integral operators has defined respectively, as follows;
βaJαf(x)=1Γ(β)∫xa((x−a)α−(t−a)αα)β−1f(t)(t−a)1−αdt; | (1.8) |
βJαbf(x)=1Γ(β)∫bx((b−x)α−(b−t)αα)β−1f(t)(b−t)1−αdt. | (1.9) |
The results presented here, being general, can be reduced to yield many relatively simple inequalities and identities for functions associated with certain fractional integral operators. For example, the case α=1 in the obtained results are found to yield the same results involving Riemann-Liouville fractional integrals, given before, in literatures. Further, getting more knowledge, see the paper given in [12]. Recently, some studies on this integral operators appeared in literature. Gözpınar [13] obtained Hermite-Hadamard type results for differentiable convex functions. Also, Set et. al. obtained some new results for quasi−convex, some different type convex functions and differentiable convex functions involving this new operator, see [24,25,26]. Motivating the new definition of fractional conformable integral operator and the studies given above, first aim of this study is obtaining new generalizations.
Lemma 2.1. Let f:[a,b]→R be a differentiable function on (a,b) with a<b and f′∈L[a,b]. Then the following equality for fractional conformable integrals holds:
(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]=(x−a)αβ+1b−a∫10(1−(1−t)αα)βf′(tx+(1−t)a)dt+(b−x)αβ+1b−a∫10(1−(1−t)αα)βf′(tx+(1−t)b)dt. |
where α,β>0 and Γ is Euler Gamma function.
Proof. Using the definition as in (1.8) and (1.9), integrating by parts and and changing variables with u=tx+(1−t)a and u=tx+(1−t)b in
I1=∫10(1−(1−t)αα)βf′(tx+(1−t)a)dt,I2=∫10(1−(1−t)αα)βf′(tx+(1−t)b)dt |
respectively, then we have
I1=∫10(1−(1−t)αα)βf′(tx+(1−t)a)dt=(1−(1−t)αα)βf(tx+(1−t)a)x−a|10−β∫10(1−(1−t)αα)β−1(1−t)α−1f(tx+(1−t)a)x−adt=f(x)αβ(x−a)−β∫xa(1−(x−ux−a)αα)β−1(x−ux−a)α−1f(u)x−adux−a=f(x)αβ(x−a)−β(x−a)αβ+1∫xa((x−a)α−(x−u)αα)β−1(x−u)α−1f(u)du=f(x)αβ(x−a)−Γ(β+1)(x−a)αβ+1βJαxf(a), |
similarly
I2=∫10(1−(1−t)αα)βf′(tx+(1−t)b)dt=−f(x)αβ(b−x)+Γ(β+1)(b−x)αβ+1βxJαf(b) |
By multiplying I1 with (x−a)αβ+1b−a and I2 with (b−x)αβ+1b−a we get desired result.
Remark 2.1. Taking α=1 in Lemma 2.1 is found to yield the same result as Lemma 1.1.
Theorem 2.1. Let f:[a,b]→R be a differentiable function on (a,b) with a<b and f′∈L[a,b]. If |f′| is convex on [a,b] and |f′(x)|≤M with x∈[a,b], then the following inequality for fractional conformable integrals holds:
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤Mαβ+1B(1α,β+1)[(x−a)αβ+1b−a+(b−x)αβ+1b−a] | (2.1) |
where α,β>0, B(x,y) and Γ are Euler beta and Euler gamma functions respectively.
Proof. From Lemma 2.1 we can write
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤(x−a)αβ+1b−a∫10(1−(1−t)αα)β|f′(tx+(1−t)a)|dt+(b−x)αβ+1b−a∫10(1−(1−t)αα)β|f′(tx+(1−t)b)|dt≤(x−a)αβ+1b−a[∫10(1−(1−t)αα)βt|f′(x)|dt+∫10(1−(1−t)αα)β(1−t)|f′(a)|dt]+(b−x)αβ+1b−a[∫10(1−(1−t)αα)βt|f′(x)|dt+∫10(1−(1−t)αα)β(1−t)|f′(b)|dt]. | (2.2) |
Notice that
∫10(1−(1−t)αα)βtdt=1αβ+1[B(1α,β+1)−B(2α,β+1)],∫10(1−(1−t)αα)β(1−t)dt=B(2α,β+1)αβ+1. | (2.3) |
Using the fact that, |f′(x)|≤M for x∈[a,b] and combining (2.3) with (2.2), we get desired result.
Remark 2.2. Taking α=1 in Theorem 3.1 and s=1 in Theorem 1.2 are found to yield the same results.
Theorem 2.2. Let f:[a,b]→R be a differentiable function on (a,b) with a<b and f′∈L[a,b]. If |f′|q is convex on [a,b], p,q>1 and |f′(x)|≤M with x∈[a,b], then the following inequality for fractional conformable integrals holds:
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤M[B(βp+1,1α)αβ+1]1p[(x−a)αβ+1b−a+(b−x)αβ+1b−a] | (2.4) |
where 1p+1q=1, α,β>0, B(x,y) and Γ are Euler beta and Euler gamma functions respectively.
Proof. By using Lemma 2.1, convexity of |f′|q and well-known Hölder's inequality, we have
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤(x−a)αβ+1b−a[(∫10(1−(1−t)αα)βp)1p(∫10|f′(tx+(1−t)a)|qdt)1q]+(b−x)αβ+1b−a[(∫10(1−(1−t)αα)βp)1p(∫10|f′(tx+(1−t)b)|qdt)1q]. | (2.5) |
Notice that, changing variables with x=1−(1−t)α, we get
∫10(1−(1−t)αα)βp=B(βp+1,1α)αβ+1. | (2.6) |
Since |f′|q is convex on [a,b] and |f′|q≤Mq, we can easily observe that,
∫10|f′(tx+(1−t)a)|qdt≤∫10t|f′(x)|qdt+∫10(1−t)|f′(a)|qdt≤Mq. | (2.7) |
As a consequence, combining the equality (2.6) and inequality (2.7) with the inequality (2.5), the desired result is obtained.
Remark 2.3. Taking α=1 in Theorem 3.2 and s=1 in Theorem 1.3 are found to yield the same results.
Theorem 2.3. Let f:[a,b]→R be a differentiable function on (a,b) with a<b and f′∈L[a,b]. If |f′|q is convex on [a,b], q≥1 and |f′(x)|≤M with x∈[a,b], then the following inequality for fractional conformable integrals holds:
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤Mαβ+1B(1α,β+1)[(x−a)αβ+1b−a+(b−x)αβ+1b−a] | (2.8) |
where α,β>0, B(x,y) and Γ are Euler Beta and Euler Gamma functions respectively.
Proof. By using Lemma 2.1, convexity of |f″|q and well-known power-mean inequality, we have
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤(x−a)αβ+1b−a(∫10(1−(1−t)αα)βdt)1−1q(∫10(1−(1−t)αα)β|f′(tx+(1−t)a)|qdt)1q+(b−x)αβ+1b−a(∫10(1−(1−t)αα)βdt)1−1q(∫10(1−(1−t)αα)β|f′(tx+(1−t)b)|qdt)1q. | (2.9) |
Since |f′|q is convex and |f′|q≤Mq, by using (2.3) we can easily observe that,
∫10(1−(1−t)αα)β|f′(tx+(1−t)a)|qdt≤∫10(1−(1−t)αα)β[t|f′(x)|q+(1−t)|f′(a)|q]dt≤Mqαβ+1B(1α,β+1). | (2.10) |
As a consequence,
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤(x−a)αβ+1b−a(1αβ+1B(1α,β+1))1−1q(Mqαβ+1B(1α,β+1))1q+(b−x)αβ+1b−a(1αβ+1B(1α,β+1))1−1q(Mqαβ+1B(1α,β+1))1q=Mαβ+1B(1α,β+1)[(x−a)αβ+1b−a+(b−x)αβ+1b−a]. | (2.11) |
This means that, the desired result is obtained.
Remark 2.4. Taking α=1 in Theorem 3.2 and s=1 in Theorem 1.4 are found to yield the same results.
Theorem 2.4. Let f:[a,b]→R be a differentiable function on (a,b) with a<b and f′∈L[a,b]. If |f′|q is concave on [a,b], p,q>1 and |f′(x)|≤M with x∈[a,b], then the following inequality for fractional conformable integrals holds:
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤[B(βp+1,1α)αβ+1]1p[(x−a)αβ+1b−a|f′(x+a2)|+(b−x)αβ+1b−a|f′(x+b2)|] | (2.12) |
where 1p+1q=1, α,β>0, B(x,y) and Γ are Euler Beta and Gamma functions respectively.
Proof. By using Lemma 2.1 and well-known Hölder's inequality, we have
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤(x−a)αβ+1b−a[(∫10(1−(1−t)αα)βp)1p(∫10|f′(tx+(1−t)a)|qdt)1q]+(b−x)αβ+1b−a[(∫10(1−(1−t)αα)βp)1p(∫10|f′(tx+(1−t)b)|qdt)1q]. | (2.13) |
Since |f″|q is concave, it can be easily observe that,
|f′(tx+(1−t)a)|qdt≤|f′(x+a2)|,|f′(tx+(1−t)b)|qdt≤|f′(b+x2)|. | (2.14) |
Notice that, changing variables with x=1−(1−t)α, as in (2.6), we get,
∫10(1−(1−t)αα)βp=B(βp+1,1α)αβ+1. | (2.15) |
As a consequence, substituting (2.14) and (2.15) in (2.13), the desired result is obtained.
Remark 2.5. Taking α=1 in Theorem 3.2 and s=1 in Theorem 1.5 are found to yield the same results.
Some new inequalities for AG-convex functions has obtained in this chapter. For the simplicity, we will denote |f′(x)||f′(a)|=ω and |f′(x)||f′(b)|=ψ.
Theorem 3.1. Let f:[a,b]→R be a differentiable function on (a,b) with a<b and f′∈L[a,b]. If |f′| is AG−convex on [a,b], then the following inequality for fractional conformable integrals holds:
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤|f′(a)|(x−a)αβ+1αβ(b−a)[ω−1lnω−(ωln−αβ−1(ω)(Γ(αβ+1)−Γ(αβ+1,lnω)))]+|f′(b)|(b−x)αβ+1αβ(b−a)[ψ−1lnψ−(ψln−αβ−1(ψ)(Γ(αβ+1)−Γ(αβ+1,lnψ)))] |
where α>0,β>1, Re(lnω)<0∧Re(lnψ)<0∧Re(αβ)>−1,B(x,y),Γ(x,y) and Γ are Euler Beta, Euler incomplete Gamma and Euler Gamma functions respectively.
Proof. From Lemma 2.1 and definition of AG−convexity, we have
(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]≤(x−a)αβ+1b−a∫10(1−(1−t)αα)β|f′(tx+(1−t)a)|dt+(b−x)αβ+1b−a∫10(1−(1−t)αα)β|f′(tx+(1−t)b)|dt≤(x−a)αβ+1b−a[∫10(1−(1−t)αα)β|f′(a)|(|f′(x)||f′(a)|)tdt]+(b−x)αβ+1b−a[∫10(1−(1−t)αα)β|f′(b)|(|f′(x)||f′(b)|)tdt]. | (3.1) |
By using the fact that |1−(1−t)α|β≤1−|1−t|αβ for α>0,β>1, we can write
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤(x−a)αβ+1αβ(b−a)[∫10(1−|1−t|αβ)|f′(a)|(|f′(x)||f′(a)|)tdt]+(b−x)αβ+1αβ(b−a)[∫10(1−|1−t|αβ)|f′(b)|(|f′(x)||f′(b)|)tdt]. |
By computing the above integrals, we get the desired result.
Theorem 3.2. Let f:[a,b]→R be a differentiable function on (a,b) with a<b and f′∈L[a,b]. If |f′|q is AG−convex on [a,b] and p,q>1, then the following inequality for fractional conformable integrals holds:
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤(B(βp+1,1α)αβ+1)1p[|f′(a)|(x−a)αβ+1b−a(ωq−1qlnω)1q+|f′(b)|(b−x)αβ+1b−a(ψq−1qlnψ)1q]. |
where 1p+1q=1, α,β>0, B(x,y) and Γ are Euler beta and Euler gamma functions respectively.
Proof. By using Lemma 2.1, AG−convexity of |f′|q and well-known Hölder's inequality, we can write
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤(x−a)αβ+1b−a[(∫10(1−(1−t)αα)βp)1p(|f′(a)|q∫10(|f′(x)||f′(a)|)qtdt)1q]+(b−x)αβ+1b−a[(∫10(1−(1−t)αα)βp)1p(|f′(b)|q∫10(|f′(x)||f′(b)|)qtdt)1q]. |
By a simple computation, one can obtain
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤(B(βp+1,1α)αβ+1)1p×[|f′(a)|(x−a)αβ+1b−a(ωq−1qlnω)1q+|f′(b)|(b−x)αβ+1b−a(ψq−1qlnψ)1q]. |
This completes the proof.
Corollary 3.1. In our results, some new Ostrowski type inequalities can be derived by choosing |f′|≤M. We omit the details.
The authors declare that no conflicts of interest in this paper.
[1] | IEEE Std C37.113-2015, IEEE Guide for Protective Relay Applications to Transmission Lines, 2015. |
[2] | Ebeed M, Aleem SHEM (2021) Overview of uncertainties in modern power systems: uncertainty models and methods. Uncertainties in Modern Power Systems, 1–34. https://doi.org/10.1016/B978-0-12-820491-7.00001-3 |
[3] |
Aprilia E, Meng K, Zeineldin HH, et al. (2020) Modeling of distributed generators and converters control for power flow analysis of networked islanded hybrid microgrids. Electr Pow Syst Res 184: 106343. https://doi.org/10.1016/j.epsr.2020.106343 doi: 10.1016/j.epsr.2020.106343
![]() |
[4] |
KOCAMAN B, Nurettin AB (2020) The role of static Var compensator at reactive power compensation. European Journal of Technique (EJT) 10: 143–152. https://doi.org/10.36222/ejt.659118 doi: 10.36222/ejt.659118
![]() |
[5] | Moore AT (2008) Distributed Generation (DG) Protection Overview. University of Western Ontario. |
[6] |
Hussain A, Bui VH, Kim HM (2019) Microgrids as a resilience resource and strategies used by microgrids for enhancing resilience. Appl Energ 240: 56–72. https://doi.org/10.1016/j.apenergy.2019.02.055 doi: 10.1016/j.apenergy.2019.02.055
![]() |
[7] |
González I, Calderón AJ, Portalo JM (2021) Innovative Multi-Layered Architecture for Heterogeneous Automation and Monitoring Systems: Application Case of a Photovoltaic Smart Microgrid. Sustainability 13: 2234. https://doi.org/10.3390/su13042234 doi: 10.3390/su13042234
![]() |
[8] |
Jiang A, Yuan H, Li D, et al. (2019) Key technologies of ubiquitous power Internet of Things-aided smart grid. J Renew Sustain Ener 11: 062702. https://doi.org/10.1063/1.5121856 doi: 10.1063/1.5121856
![]() |
[9] | Shaik M, Rao MN, Gafoor SA (2012) A wavelet based protection scheme for distribution networks with Distributed Generation. Proceedings - ICETEEEM 2012, International Conference on Emerging Trends in Electrical Engineering and Energy Management, 33–37. https://doi.org/10.1109/ICETEEEM.2012.6494440 |
[10] |
Anwar A, Mahmood A, Ray B, et al. (2020) Machine Learning to Ensure Data Integrity in Power System Topological Network Database. Electronics 9: 693. https://doi.org/10.3390/electronics9040693 doi: 10.3390/electronics9040693
![]() |
[11] |
Aminifar F, Abedini M, Amraee T, et al. (2021) A review of power system protection and asset management with machine learning techniques. Energy Systems, 1–38. https://doi.org/10.1007/s12667-021-00448-6 doi: 10.1007/s12667-021-00448-6
![]() |
[12] | Lasseter RH, Paigi P (2004) Microgrid: a conceptual solution. Proceedings of the IEEE 35th Annual Power Electronics Specialists Conference (PESC '04) 6: 4285–4290. |
[13] |
Pourbeik P, Sullivan DJ, Bostrom A, et al. (2012) Generic model structure for simulating static VAr systems in power system studies–-A WECC Task Force Effort. IEEE T Power Syst 27: 1618–1627. https://doi.org/10.1109/TPWRS.2011.2179322 doi: 10.1109/TPWRS.2011.2179322
![]() |
[14] |
Ghorbani A, Khederzadeh M, Mozafari B (2012) Impact of SVC on the protection of transmission lines. Int J Elec Power 42: 702–709. https://doi.org/10.1016/j.ijepes.2012.04.029 doi: 10.1016/j.ijepes.2012.04.029
![]() |
[15] |
Li Z, Shahidehpour M, Aminifar F, et al. (2017) Networked microgrids for enhancing the power system resilience. Proc IEEE 105: 1289–1310. https://doi.org/10.1109/JPROC.2017.2685558 doi: 10.1109/JPROC.2017.2685558
![]() |
[16] |
Yang H, Liu X, Zhang D, et al. (2021) Machine learning for power system protection and control. The Electricity Journal 34: 106881. https://doi.org/10.1016/j.tej.2020.106881 doi: 10.1016/j.tej.2020.106881
![]() |
[17] |
Aslan Y (2012) An alternative approach to fault location on power distribution feeders with embedded remote-end power generation using artificial neural networks. Electr Eng 94: 125–134. https://doi.org/10.1007/s00202-011-0218-2 doi: 10.1007/s00202-011-0218-2
![]() |
[18] |
Bedi G, Venayagamoorthy GK, Singh R, et al. (2018) Review of Internet of Things (IoT) in Electric Power and Energy Systems. IEEE Internet Things 5: 847–870. https://doi.org/10.1109/JIOT.2018.2802704 doi: 10.1109/JIOT.2018.2802704
![]() |
[19] |
Shekar SC, Kumar GR, Lalitha SVNL (2019) A transient current based micro-grid connected power system protection scheme using wavelet approach. International Journal of Electrical and Computer Engineering 9: 14. https://doi.org/10.11591/ijece.v9i1.pp14-22 doi: 10.11591/ijece.v9i1.pp14-22
![]() |
[20] |
Gururajapathy SS, Mokhlis H, Illias HA (2017) Fault location and detection techniques in power distribution systems with distributed generation: a review. Renew Sust Energy Rev 74: 949–958. https://doi.org/10.1016/j.rser.2017.03.021 doi: 10.1016/j.rser.2017.03.021
![]() |
1. | Anjali Upadhyay, Surendra Kumar, The exponential nature and solvability of stochastic multi-term fractional differential inclusions with Clarke’s subdifferential, 2023, 168, 09600779, 113202, 10.1016/j.chaos.2023.113202 | |
2. | Amadou Diop, Wei-Shih Du, Existence of Mild Solutions for Multi-Term Time-Fractional Random Integro-Differential Equations with Random Carathéodory Conditions, 2021, 10, 2075-1680, 252, 10.3390/axioms10040252 | |
3. | Yong-Kui Chang, Jianguo Zhao, Some new asymptotic properties on solutions to fractional evolution equations in Banach spaces, 2021, 0003-6811, 1, 10.1080/00036811.2021.1969016 | |
4. | Ahmad Al-Omari, Hanan Al-Saadi, António M. Lopes, Impulsive fractional order integrodifferential equation via fractional operators, 2023, 18, 1932-6203, e0282665, 10.1371/journal.pone.0282665 | |
5. | Hiba El Asraoui, Ali El Mfadel, M’hamed El Omari, Khalid Hilal, Existence of mild solutions for a multi-term fractional differential equation via ψ-(γ,σ)-resolvent operators, 2023, 16, 1793-5571, 10.1142/S1793557123502121 | |
6. | Zhiyuan Yuan, Luyao Wang, Wenchang He, Ning Cai, Jia Mu, Fractional Neutral Integro-Differential Equations with Nonlocal Initial Conditions, 2024, 12, 2227-7390, 1877, 10.3390/math12121877 | |
7. | Jia Mu, Zhiyuan Yuan, Yong Zhou, Mild Solutions of Fractional Integrodifferential Diffusion Equations with Nonlocal Initial Conditions via the Resolvent Family, 2023, 7, 2504-3110, 785, 10.3390/fractalfract7110785 |